fof(t64_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v1_funct_7(A) )
     => k1_relat_1(k4_funct_7(k6_funct_7(A),A)) = k6_funct_7(A) ) ),
    inference(mizar_proof,[status(thm)],[t41_funct_7,t71_relat_1,t63_funct_7]),
    [file(funct_7,t64_funct_7),interesting(1.00)]).

fof(t63_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B)
        & v1_funct_7(B) )
     => ( B != k1_xboole_0
       => k1_relat_1(k4_funct_7(A,B)) = k3_xboole_0(k6_funct_7(B),A) ) ) ),
    inference(mizar_proof,[status(thm)],[t43_funct_7,t47_finseq_1,t47_funct_7,t60_funct_7,t37_funct_1,t63_finseq_1,t5_rewrite1,t45_finseq_1,t58_finseq_1,d6_funct_7,d8_funct_7,t57_finseq_1,t35_finseq_1,t29_nat_1,t27_finseq_3,d7_finseq_1,t57_finseq_1,t35_finseq_1,t59_finseq_1,d2_funct_2,d2_funct_2,t60_funct_7,t61_funct_7,t62_funct_7,t1_xboole_1,t62_funct_7,t46_relat_1,s5_funct_7]),
    [file(funct_7,t63_funct_7),interesting(0.94)]).

fof(t67_funct_7,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C)
        & v1_finseq_1(C)
        & v1_funct_7(C) )
     => ( ( r2_hidden(A,k6_funct_7(C))
          & r2_hidden(A,B) )
       => k1_funct_1(k5_funct_7(C,A),k1_nat_1(k3_finseq_1(C),1)) = k1_funct_1(k4_funct_7(B,C),A) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_funct_7,t44_funct_7,t43_funct_7,t62_funct_7,t47_funct_7,t49_funct_7,t57_finseq_1,t34_funct_1,t23_funct_1,t47_finseq_1,t61_finseq_1,d6_funct_7,t25_finseq_1,t19_nat_1,t38_nat_1,t27_finseq_3,d7_finseq_1,t63_funct_7,d3_xboole_0,d5_funct_7,t35_finseq_1,t62_funct_7,t59_finseq_1,d7_finseq_1,t23_funct_1,s5_funct_7]),
    [file(funct_7,t67_funct_7),interesting(0.92)]).

fof(t87_funct_7,theorem,(
    ! [A,B] : 
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ! [C] : 
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ! [D] : 
              ( ( v1_funct_1(D)
                & v1_funct_2(D,A,A)
                & m2_relset_1(D,A,A) )
             => k9_funct_7(k9_funct_7(D,B),C) = k9_funct_7(D,k2_nat_1(B,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[l80_funct_7,t81_funct_7]),
    [file(funct_7,t87_funct_7),interesting(0.90)]).

fof(t85_funct_7,theorem,(
    ! [A,B] : 
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ! [C] : 
          ( ( v1_funct_1(C)
            & v1_funct_2(C,A,A)
            & m2_relset_1(C,A,A) )
         => ( v1_funct_1(k9_funct_7(C,B))
            & v1_funct_2(k9_funct_7(C,B),A,A)
            & m2_relset_1(k9_funct_7(C,B),A,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_2,t76_funct_7,t11_relset_1,t76_funct_7,d1_funct_2]),
    [file(funct_7,t85_funct_7),interesting(0.88)]).

fof(t49_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => k5_funct_7(k9_finseq_1(B),A) = k10_finseq_1(A,k1_funct_1(B,A)) ) ),
    inference(mizar_proof,[status(thm)],[t42_funct_7,t25_finseq_1,t57_finseq_1,t47_finseq_1,t44_funct_7,d9_finseq_1]),
    [file(funct_7,t49_funct_7),interesting(0.87)]).

fof(t13_funct_7,theorem,(
    ! [A,B,C] : 
      ( ~ r1_xboole_0(A,k2_relat_1(k1_funct_4(k6_partfun1(B),k2_funcop_1(A,C))))
     => r2_hidden(C,A) ) ),
    inference(mizar_proof,[status(thm)],[t3_xboole_0,d5_funct_1,d1_funct_4,t19_funcop_1,t13_funcop_1,t14_funct_4,d2_xboole_0,t12_funct_4,t35_funct_1,t19_funcop_1]),
    [file(funct_7,t13_funct_7),interesting(0.86)]).

fof(t47_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => k4_funct_7(A,k9_finseq_1(B)) = k5_relat_1(k6_partfun1(A),B) ) ),
    inference(mizar_proof,[status(thm)],[t47_finseq_1,t43_funct_7,t41_funct_7]),
    [file(funct_7,t47_funct_7),interesting(0.86)]).

fof(t62_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( v1_funct_7(k7_finseq_1(A,B))
           => ( v1_funct_7(A)
              & v1_funct_7(B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_funct_7,t35_finseq_1,t19_finseq_1,t29_nat_1,t8_xreal_1,t21_finseq_1,t24_funct_7,d7_finseq_1,t24_finseq_3,t70_funct_1,d8_funct_7,t3_trees_1,d8_funct_7,t29_nat_1,t2_xreal_1,t21_finseq_1,t35_finseq_1,t24_funct_7,t41_finseq_1,d7_finseq_1,d7_finseq_1]),
    [file(funct_7,t62_funct_7),interesting(0.85)]).

fof(t89_funct_7,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ! [D] : 
          ( m2_subset_1(D,k1_numbers,k5_numbers)
         => ( ( r2_hidden(A,B)
              & C = k2_funcop_1(B,A) )
           => ( D = 0
              | k9_funct_7(C,D) = C ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t72_funct_7,t14_funcop_1,t19_funcop_1,t37_zfmisc_1,t76_funct_7,t13_funcop_1,t73_funct_7,t23_funct_1,t13_funcop_1,t9_funct_1,s1_nat_1,t22_nat_1]),
    [file(funct_7,t89_funct_7),interesting(0.85)]).

fof(t11_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => k5_relat_1(A,k1_funct_4(B,C)) = k1_funct_4(k5_relat_1(A,B),k5_relat_1(A,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,t21_funct_1,t21_funct_1,d1_funct_4,d2_xboole_0,t21_funct_1,d2_xboole_0,d10_xboole_0,d3_tarski,d2_xboole_0,t21_funct_1,t21_funct_1,d2_xboole_0,d1_funct_4,t21_funct_1,d2_xboole_0,t21_funct_1,t21_funct_1,t23_funct_1,t14_funct_4,t22_funct_1,t21_funct_1,t23_funct_1,t12_funct_4,t23_funct_1,d1_funct_4]),
    [file(funct_7,t11_funct_7),interesting(0.83)]).

fof(t32_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : k1_relat_1(k2_funct_7(A,C,B)) = k1_relat_1(A) ) ),
    inference(mizar_proof,[status(thm)],[t37_zfmisc_1,d3_funct_7,d1_funct_4,t5_cqc_lang,t12_xboole_1,d3_funct_7]),
    [file(funct_7,t32_funct_7),interesting(0.83)]).

fof(t53_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => k4_funct_7(A,k10_finseq_1(B,C)) = k5_relat_1(k6_partfun1(A),k5_relat_1(B,C)) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_finseq_1,t43_funct_7,t47_funct_7,t55_relat_1]),
    [file(funct_7,t53_funct_7),interesting(0.83)]).

fof(t72_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k9_funct_7(A,1) = A ) ),
    inference(mizar_proof,[status(thm)],[t71_funct_7,t70_funct_7,l83_funct_7]),
    [file(funct_7,t72_funct_7),interesting(0.83)]).

fof(t88_funct_7,theorem,(
    ! [A,B] : 
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ! [C] : 
          ( ( v1_funct_1(C)
            & m2_relset_1(C,A,A) )
         => ( v1_funct_1(k9_funct_7(C,B))
            & m2_relset_1(k9_funct_7(C,B),A,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t74_funct_7,t1_xboole_1,t11_relset_1]),
    [file(funct_7,t88_funct_7),interesting(0.83)]).

fof(t20_funct_7,theorem,(
    ! [A,B] : 
      ( r2_hidden(k9_finseq_1(A),k3_finseq_2(B))
    <=> r2_hidden(A,B) ) ),
    inference(mizar_proof,[status(thm)],[t55_finseq_1,d4_finseq_1,d11_finseq_1,t37_zfmisc_1]),
    [file(funct_7,t20_funct_7),interesting(0.82)]).

fof(t37_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : k2_funct_7(A,B,k1_funct_1(A,B)) = A ) ),
    inference(mizar_proof,[status(thm)],[t6_funct_7,d3_funct_7,t80_funct_4,d3_funct_7]),
    [file(funct_7,t37_funct_7),interesting(0.82)]).

fof(t41_funct_7,theorem,(
    ! [A] : k4_funct_7(A,k1_xboole_0) = k6_partfun1(A) ),
    inference(mizar_proof,[status(thm)],[t25_finseq_1,d4_funct_7]),
    [file(funct_7,t41_funct_7),interesting(0.81)]).

fof(t42_funct_7,theorem,(
    ! [A] : k5_funct_7(k1_xboole_0,A) = k9_finseq_1(A) ),
    inference(mizar_proof,[status(thm)],[t25_finseq_1,d5_funct_7,t57_finseq_1]),
    [file(funct_7,t42_funct_7),interesting(0.81)]).

fof(t61_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B)
        & v1_funcop_1(B) )
     => ( B != k1_xboole_0
       => r1_tarski(k2_relat_1(k4_funct_7(A,B)),k7_funct_7(B)) ) ) ),
    inference(mizar_proof,[status(thm)],[t43_funct_7,t45_relat_1,t60_funct_7,s5_funct_7]),
    [file(funct_7,t61_funct_7),interesting(0.81)]).

fof(t14_funct_7,theorem,(
    ! [A,B,C] : ~ ( A != B
      & r2_hidden(A,k2_relat_1(k1_funct_4(k6_partfun1(C),k3_cqc_lang(A,B)))) ) ),
    inference(mizar_proof,[status(thm)],[d1_tarski,t13_funct_7,t54_zfmisc_1]),
    [file(funct_7,t14_funct_7),interesting(0.80)]).

fof(t34_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C,D] : 
          ( C != D
         => k1_funct_1(k2_funct_7(A,C,B),D) = k1_funct_1(A,D) ) ) ),
    inference(mizar_proof,[status(thm)],[t5_cqc_lang,d1_tarski,d3_funct_7,t12_funct_4,d3_funct_7]),
    [file(funct_7,t34_funct_7),interesting(0.80)]).

fof(t35_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C,D,E] : 
          ( D != E
         => k2_funct_7(k2_funct_7(A,D,B),E,C) = k2_funct_7(k2_funct_7(A,E,C),D,B) ) ) ),
    inference(mizar_proof,[status(thm)],[t5_cqc_lang,t17_zfmisc_1,t32_funct_7,t32_funct_7,d3_funct_7,d3_funct_7,t15_funct_4,t36_funct_4,t15_funct_4,d3_funct_7,d3_funct_7,t32_funct_7,d3_funct_7,d3_funct_7,t32_funct_7,d3_funct_7,d3_funct_7,t32_funct_7,t32_funct_7,d3_funct_7,d3_funct_7,d3_funct_7,d3_funct_7]),
    [file(funct_7,t35_funct_7),interesting(0.79)]).

fof(t36_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C,D] : k2_funct_7(k2_funct_7(A,D,B),D,C) = k2_funct_7(A,D,C) ) ),
    inference(mizar_proof,[status(thm)],[t5_cqc_lang,t5_cqc_lang,t32_funct_7,d3_funct_7,d3_funct_7,t15_funct_4,t20_funct_4,d3_funct_7,d3_funct_7]),
    [file(funct_7,t36_funct_7),interesting(0.79)]).

fof(t33_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : 
          ( r2_hidden(C,k1_relat_1(A))
         => k1_funct_1(k2_funct_7(A,C,B),C) = B ) ) ),
    inference(mizar_proof,[status(thm)],[t5_cqc_lang,d1_tarski,d3_funct_7,t14_funct_4,t6_cqc_lang]),
    [file(funct_7,t33_funct_7),interesting(0.79)]).

fof(t48_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r1_tarski(k1_relat_1(B),A)
       => k4_funct_7(A,k9_finseq_1(B)) = B ) ) ),
    inference(mizar_proof,[status(thm)],[t47_funct_7,t77_relat_1]),
    [file(funct_7,t48_funct_7),interesting(0.79)]).

fof(t77_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k5_relat_1(k6_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),k9_funct_7(A,B)) = k9_funct_7(A,B) ) ) ),
    inference(mizar_proof,[status(thm)],[t74_funct_7,t77_relat_1]),
    [file(funct_7,t77_funct_7),interesting(0.77)]).

fof(t21_funct_7,theorem,(
    ! [A,B] : 
      ( r1_tarski(k3_finseq_2(A),k3_finseq_2(B))
     => r1_tarski(A,B) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,t20_funct_7,t20_funct_7]),
    [file(funct_7,t21_funct_7),interesting(0.76)]).

fof(t83_funct_7,theorem,(
    ! [A,B] : 
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => k9_funct_7(k6_partfun1(A),B) = k6_partfun1(A) ) ),
    inference(mizar_proof,[status(thm)],[t71_relat_1,t70_funct_7,t71_funct_7,t74_funct_2,s1_nat_1]),
    [file(funct_7,t83_funct_7),interesting(0.76)]).

fof(l80_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,A)
        & m2_relset_1(B,A,A) )
     => r1_tarski(k5_relset_1(A,A,B),k4_relset_1(A,A,B)) ) ),
    inference(mizar_proof,[status(thm)],[t67_funct_2]),
    [file(funct_7,l80_funct_7),interesting(0.76)]).

fof(t7_funct_7,theorem,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( r2_hidden(C,A)
         => k3_cqc_lang(C,k1_funct_1(B,C)) = k7_relat_1(B,k1_tarski(C)) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_pboole,t6_funct_7]),
    [file(funct_7,t7_funct_7),interesting(0.76)]).

fof(t19_funct_7,theorem,(
    k3_finseq_2(k1_xboole_0) = k1_tarski(k1_xboole_0) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d11_finseq_1,t64_relat_1,d1_xboole_0,d5_funct_1,t37_zfmisc_1,d10_xboole_0,d3_tarski,d1_tarski,d4_finseq_1,d11_finseq_1]),
    [file(funct_7,t19_funct_7),interesting(0.75)]).

fof(t27_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : r1_funct_7(A,A,B) ) ),
    inference(mizar_proof,[status(thm)],[d2_funct_7]),
    [file(funct_7,t27_funct_7),interesting(0.75)]).

fof(t69_funct_7,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_relat_1(A)
        & v2_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( m2_funct_7(B,A)
         => ( k1_relat_1(k4_funct_7(k1_funct_1(A,1),B)) = k1_funct_1(A,1)
            & r1_tarski(k2_relat_1(k4_funct_7(k1_funct_1(A,1),B)),k1_funct_1(A,k3_finseq_1(A))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d10_funct_7,t41_funct_7,t25_finseq_1,t71_relat_1,t61_funct_7,t68_funct_7,t64_funct_7,t1_xboole_1]),
    [file(funct_7,t69_funct_7),interesting(0.75)]).

fof(t82_funct_7,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k9_funct_7(k1_xboole_0,A) = k1_xboole_0 ) ),
    inference(mizar_proof,[status(thm)],[t70_funct_7,t81_relat_1,t71_funct_7,s1_nat_1]),
    [file(funct_7,t82_funct_7),interesting(0.75)]).

fof(t16_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C,D] : k1_funct_4(k1_funct_4(A,k2_funcop_1(B,C)),k2_funcop_1(B,D)) = k1_funct_4(A,k2_funcop_1(B,D)) ) ),
    inference(mizar_proof,[status(thm)],[t19_funcop_1,d1_funct_4,d1_funct_4,t4_xboole_1,t14_funct_4,t12_funct_4,t12_funct_4,t9_funct_1]),
    [file(funct_7,t16_funct_7),interesting(0.74)]).

fof(t18_funct_7,theorem,(
    k4_numbers != k3_finseq_2(k4_numbers) ),
    inference(mizar_proof,[status(thm)],[d11_finseq_1,t1_card_5,d1_tarski,d1_relat_1,d2_int_1]),
    [file(funct_7,t18_funct_7),interesting(0.74)]).

fof(t43_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B)
        & v1_funcop_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => k4_funct_7(A,k7_finseq_1(B,k9_finseq_1(C))) = k5_relat_1(k4_funct_7(A,B),C) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_funct_7,t39_finseq_1,d7_finseq_1,d4_funct_7,t4_finseq_1,t55_finseq_1,t57_finseq_1,d1_tarski,t35_finseq_1,t41_finseq_1,t59_finseq_1]),
    [file(funct_7,t43_funct_7),interesting(0.74)]).

fof(t46_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B)
        & v1_funcop_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => k5_funct_7(k7_finseq_1(k9_finseq_1(C),B),A) = k7_finseq_1(k9_finseq_1(A),k5_funct_7(B,k1_funct_1(C,A))) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_finseq_1,t47_finseq_1,t44_funct_7,t42_funct_7,t42_funct_7,t57_finseq_1,t42_funct_7,t44_funct_7,t57_finseq_1,d5_funct_7,t35_finseq_1,t29_nat_1,t27_finseq_3,d7_finseq_1,t45_finseq_1,t45_finseq_1,t44_funct_7,s5_funct_7]),
    [file(funct_7,t46_funct_7),interesting(0.74)]).

fof(t79_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k5_relat_1(k9_funct_7(A,C),k9_funct_7(A,B)) = k9_funct_7(A,k1_nat_1(B,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t70_funct_7,t77_funct_7,t71_funct_7,t55_relat_1,t71_funct_7,s1_nat_1]),
    [file(funct_7,t79_funct_7),interesting(0.74)]).

fof(t26_funct_7,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(A,k3_xboole_0(B,C))
     => k1_funct_7(A,B) = k1_funct_7(A,C) ) ),
    inference(mizar_proof,[status(thm)],[d3_xboole_0,d3_xboole_0,d1_funct_7,d1_funct_7]),
    [file(funct_7,t26_funct_7),interesting(0.73)]).

fof(t28_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : 
              ( r1_funct_7(A,B,C)
             => r1_funct_7(B,A,C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_funct_7,d2_funct_7]),
    [file(funct_7,t28_funct_7),interesting(0.73)]).

fof(t55_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => k5_funct_7(k10_finseq_1(B,C),A) = k11_finseq_1(A,k1_funct_1(B,A),k1_funct_1(C,k1_funct_1(B,A))) ) ) ),
    inference(mizar_proof,[status(thm)],[t49_funct_7,t57_finseq_1,d9_finseq_1,t44_funct_7,t61_finseq_1,t60_finseq_1]),
    [file(funct_7,t55_funct_7),interesting(0.73)]).

fof(t70_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k9_funct_7(A,0) = k6_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A))) ) ),
    inference(mizar_proof,[status(thm)],[d11_funct_7]),
    [file(funct_7,t70_funct_7),interesting(0.73)]).

fof(t73_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k9_funct_7(A,k1_nat_1(B,1)) = k5_relat_1(k9_funct_7(A,B),A) ) ) ),
    inference(mizar_proof,[status(thm)],[t72_funct_7,l83_funct_7,t70_funct_7,t71_funct_7,t55_relat_1,t71_funct_7,s1_nat_1]),
    [file(funct_7,t73_funct_7),interesting(0.73)]).

fof(t86_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,A)
        & m2_relset_1(B,A,A) )
     => k9_funct_7(B,0) = k6_partfun1(A) ) ),
    inference(mizar_proof,[status(thm)],[l80_funct_7,t70_funct_7,t67_funct_2,t12_xboole_1]),
    [file(funct_7,t86_funct_7),interesting(0.73)]).

fof(t90_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k9_funct_7(A,B) = k4_funct_7(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_finseq_2(B,A)) ) ) ),
    inference(mizar_proof,[status(thm)],[t70_funct_7,t41_funct_7,t72_finseq_2,t73_funct_7,t43_funct_7,t74_finseq_2,s1_nat_1]),
    [file(funct_7,t90_funct_7),interesting(0.73)]).

fof(t12_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( r1_xboole_0(k2_relat_1(A),k1_relat_1(B))
               => k5_relat_1(A,k1_funct_4(C,B)) = k5_relat_1(A,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t11_funct_7,t67_relat_1,t22_funct_4]),
    [file(funct_7,t12_funct_7),interesting(0.71)]).

fof(t29_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ! [D] : 
                  ( ( r1_funct_7(A,B,D)
                    & r1_funct_7(B,C,D) )
                 => r1_funct_7(A,C,D) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_funct_7,d2_funct_7]),
    [file(funct_7,t29_funct_7),interesting(0.71)]).

fof(t31_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : 
              ( r1_tarski(k1_relat_1(B),C)
             => r1_funct_7(A,k1_funct_4(A,B),C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_4,t42_xboole_1,t37_xboole_1,t13_prob_1,t63_xboole_1,t76_funct_4,d2_funct_7]),
    [file(funct_7,t31_funct_7),interesting(0.71)]).

fof(t54_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ( r1_tarski(k1_relat_1(B),A)
              | r1_tarski(k1_relat_1(k5_relat_1(B,C)),A) )
           => k4_funct_7(A,k10_finseq_1(B,C)) = k5_relat_1(B,C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t53_funct_7,t55_relat_1,t77_relat_1]),
    [file(funct_7,t54_funct_7),interesting(0.71)]).

fof(t56_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ! [D] : 
              ( ( v1_relat_1(D)
                & v1_funct_1(D) )
             => k4_funct_7(A,k11_finseq_1(B,C,D)) = k5_relat_1(k6_partfun1(A),k5_relat_1(B,k5_relat_1(C,D))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t60_finseq_1,t43_funct_7,t53_funct_7,t55_relat_1,t55_relat_1]),
    [file(funct_7,t56_funct_7),interesting(0.71)]).

fof(t6_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( r2_hidden(B,k1_relat_1(A))
         => k7_relat_1(A,k1_tarski(B)) = k3_cqc_lang(B,k1_funct_1(A,B)) ) ) ),
    inference(mizar_proof,[status(thm)],[t89_grfunc_1,t35_zfmisc_1]),
    [file(funct_7,t6_funct_7),interesting(0.70)]).

fof(t25_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( v1_funcop_1(k7_finseq_1(A,B))
           => ( v1_funcop_1(A)
              & v1_funcop_1(B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_funcop_1,d7_finseq_1,t24_finseq_3,d6_funcop_1,d6_funcop_1,t41_finseq_1,d7_finseq_1]),
    [file(funct_7,t25_funct_7),interesting(0.70)]).

fof(l83_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( k5_relat_1(A,k6_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)))) = A
        & k5_relat_1(k6_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),A) = A ) ) ),
    inference(mizar_proof,[status(thm)],[t7_xboole_1,t77_relat_1,t79_relat_1]),
    [file(funct_7,l83_funct_7),interesting(0.70)]).

fof(t84_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( r1_xboole_0(k2_relat_1(A),k1_relat_1(A))
       => k9_funct_7(A,2) = k1_xboole_0 ) ) ),
    inference(mizar_proof,[status(thm)],[t71_funct_7,t72_funct_7,t67_relat_1]),
    [file(funct_7,t84_funct_7),interesting(0.70)]).

fof(t59_funct_7,theorem,
    ( k6_funct_7(k1_xboole_0) = k1_xboole_0
    & k7_funct_7(k1_xboole_0) = k1_xboole_0 ),
    inference(mizar_proof,[status(thm)],[d6_funct_7,d7_funct_7]),
    [file(funct_7,t59_funct_7),interesting(0.69)]).

fof(t71_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k9_funct_7(A,k1_nat_1(B,1)) = k5_relat_1(A,k9_funct_7(A,B)) ) ) ),
    inference(mizar_proof,[status(thm)],[d11_funct_7,d11_funct_7]),
    [file(funct_7,t71_funct_7),interesting(0.69)]).

fof(t45_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B)
        & v1_funcop_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => k4_funct_7(A,k7_finseq_1(k9_finseq_1(C),B)) = k5_relat_1(k7_relat_1(C,A),k4_funct_7(k9_relat_1(C,A),B)) ) ) ),
    inference(mizar_proof,[status(thm)],[t47_finseq_1,t43_funct_7,t41_funct_7,t94_relat_1,t80_relat_1,t148_relat_1,t41_funct_7,t45_finseq_1,t43_funct_7,t55_relat_1,t43_funct_7,s5_funct_7]),
    [file(funct_7,t45_funct_7),interesting(0.68)]).

fof(t78_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k5_relat_1(k9_funct_7(A,B),k6_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)))) = k9_funct_7(A,B) ) ) ),
    inference(mizar_proof,[status(thm)],[t74_funct_7,t79_relat_1]),
    [file(funct_7,t78_funct_7),interesting(0.68)]).

fof(t23_funct_7,theorem,(
    ! [A] : 
      ( v1_finset_1(A)
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ? [C] : 
              ( m2_subset_1(C,k1_zfmisc_1(A),B)
              & ! [D] : 
                  ( m2_subset_1(D,k1_zfmisc_1(A),B)
                 => ( r1_tarski(D,C)
                   => D = C ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t46_setfam_1,t18_finset_1,d8_setfam_1,d8_setfam_1,t35_subset_1]),
    [file(funct_7,t23_funct_7),interesting(0.67)]).

fof(t30_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : 
              ( r1_funct_7(A,B,C)
             => k4_xboole_0(k1_relat_1(A),C) = k4_xboole_0(k1_relat_1(B),C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_funct_7,t36_xboole_1,t36_xboole_1,t28_xboole_1,t90_relat_1,t90_relat_1,t28_xboole_1]),
    [file(funct_7,t30_funct_7),interesting(0.67)]).

fof(t52_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B)
        & v1_funcop_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C)
            & v1_finseq_1(C)
            & v1_funcop_1(C) )
         => k5_funct_7(k7_finseq_1(B,C),A) = k1_rewrite1(k5_funct_7(B,A),k5_funct_7(C,k1_funct_1(k5_funct_7(B,A),k1_nat_1(k3_finseq_1(B),1)))) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_funct_7,t25_finseq_1,t63_finseq_1,t57_finseq_1,t35_finseq_1,t59_finseq_1,t47_finseq_1,t2_rewrite1,t42_funct_7,d5_funct_7,t25_finseq_1,d5_funct_7,t25_finseq_1,t51_funct_7,t45_finseq_1,t44_funct_7,t2_rewrite1,t45_finseq_1,t44_funct_7,t2_rewrite1,s5_funct_7]),
    [file(funct_7,t52_funct_7),interesting(0.67)]).

fof(t3_funct_7,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,A,B)
                & m2_relset_1(C,A,B) )
             => ( v2_funct_1(C)
               => ! [D] : 
                    ( m1_subset_1(D,A)
                   => ! [E] : 
                        ( m1_subset_1(E,k1_zfmisc_1(A))
                       => ( r2_hidden(k8_funct_2(A,B,C,D),k2_funct_2(A,B,C,E))
                         => r2_hidden(D,E) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t116_funct_2,t25_funct_2]),
    [file(funct_7,t3_funct_7),interesting(0.67)]).

fof(s2_funct_7,theorem,
    ( ( v1_finset_1(f1_s2_funct_7)
      & ! [A] : 
          ( m1_subset_1(A,f2_s2_funct_7)
         => ! [B] : 
              ( m1_subset_1(B,f2_s2_funct_7)
             => ( f4_s2_funct_7(A) = f4_s2_funct_7(B)
               => A = B ) ) ) )
   => v1_finset_1(a_0_0_funct_7) ),
    inference(mizar_proof,[status(thm)],[s7_domain_1,t13_finset_1,d6_tarski,t33_zfmisc_1,t33_zfmisc_1,t68_card_1,s21_fraenkel,d3_tarski,d10_xboole_0,d3_tarski]),
    [file(funct_7,s2_funct_7),interesting(0.65)]).

fof(t1_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( r1_tarski(k2_relat_1(A),B)
         => k5_relat_1(A,k6_partfun1(B)) = A ) ) ),
    inference(mizar_proof,[status(thm)],[t4_funct_2,t23_funct_2]),
    [file(funct_7,t1_funct_7),interesting(0.65)]).

fof(l85_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( r1_tarski(k2_relat_1(A),k1_relat_1(A))
       => k9_funct_7(A,0) = k6_partfun1(k1_relat_1(A)) ) ) ),
    inference(mizar_proof,[status(thm)],[t12_xboole_1,t70_funct_7]),
    [file(funct_7,l85_funct_7),interesting(0.65)]).

fof(s4_funct_7,theorem,
    ( ( r1_tarski(f1_s4_funct_7,f2_s4_funct_7)
      & ! [A] : 
          ( m1_subset_1(A,f2_s4_funct_7)
         => ! [B] : 
              ( m1_subset_1(B,f2_s4_funct_7)
             => ( f3_s4_funct_7(A) = f3_s4_funct_7(B)
               => A = B ) ) ) )
   => r2_tarski(f1_s4_funct_7,a_0_2_funct_7) ),
    inference(mizar_proof,[status(thm)],[t46_card_1,d6_tarski,t33_zfmisc_1,t33_zfmisc_1]),
    [file(funct_7,s4_funct_7),interesting(0.63)]).

fof(t44_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B)
        & v1_funcop_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => k5_funct_7(k7_finseq_1(B,k9_finseq_1(C)),A) = k7_finseq_1(k5_funct_7(B,A),k9_finseq_1(k1_funct_1(C,k1_funct_1(k5_funct_7(B,A),k1_nat_1(k3_finseq_1(B),1))))) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_funct_7,d5_funct_7,t57_finseq_1,t35_finseq_1,t57_finseq_1,d3_finseq_1,t59_finseq_1,t29_nat_1,t27_finseq_3,t27_finseq_3,t27_finseq_3,t38_nat_1,t2_xreal_1,t8_xreal_1,d5_funct_7,t38_nat_1,t27_finseq_3,t39_finseq_1,d7_finseq_1,d5_funct_7,t27_finseq_3,t19_nat_1,s1_nat_1,t55_finseq_1,t4_finseq_1,d1_tarski,d5_funct_7,t57_finseq_1,d7_finseq_1]),
    [file(funct_7,t44_funct_7),interesting(0.63)]).

fof(t8_funct_7,theorem,(
    ! [A,B,C] : 
      ( m1_pboole(C,k2_zfmisc_1(A,B))
     => ! [D,E] : 
          ( ( r2_hidden(D,A)
            & r2_hidden(E,B) )
         => k9_funct_2(D,E,k1_binop_1(C,D,E)) = k7_relat_1(C,k2_zfmisc_1(k1_tarski(D),k1_tarski(E))) ) ) ),
    inference(mizar_proof,[status(thm)],[t106_zfmisc_1,t42_cqc_lang,t7_funct_7,t35_zfmisc_1]),
    [file(funct_7,t8_funct_7),interesting(0.63)]).

fof(t22_funct_7,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_zfmisc_1(k5_numbers))
     => ( ! [B] : 
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r2_hidden(B,A)
                 => ( r1_xreal_0(B,C)
                    | r2_hidden(C,A) ) ) ) )
       => v1_card_1(A) ) ) ),
    inference(mizar_proof,[status(thm)],[t47_card_1,t10_subset_1,s6_nat_1,d3_tarski,t38_nat_1,d10_xboole_0,d3_tarski,t38_nat_1,t1_xreal_1,t30_axioms,d3_tarski,t83_card_1,d10_xboole_0]),
    [file(funct_7,t22_funct_7),interesting(0.62)]).

fof(t38_funct_7,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m2_finseq_1(B,A)
         => ! [C] : 
              ( m1_subset_1(C,A)
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( r2_hidden(D,k4_finseq_1(B))
                   => k4_finseq_4(k5_numbers,A,k3_funct_7(A,B,D,C),D) = C ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t32_funct_7,d4_finseq_4,t33_funct_7]),
    [file(funct_7,t38_funct_7),interesting(0.62)]).

fof(t40_funct_7,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m2_finseq_1(B,A)
         => ! [C] : 
              ( m1_subset_1(C,A)
             => ! [D] : 
                  ( m1_subset_1(D,A)
                 => ! [E] : 
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => k3_funct_7(A,B,E,k4_finseq_4(k5_numbers,A,B,E)) = B ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_finseq_4,t37_funct_7,d3_funct_7]),
    [file(funct_7,t40_funct_7),interesting(0.62)]).

fof(t4_funct_7,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,A,B)
                & m2_relset_1(C,A,B) )
             => ( v2_funct_1(C)
               => ! [D] : 
                    ( m1_subset_1(D,A)
                   => ! [E] : 
                        ( m1_subset_1(E,k1_zfmisc_1(A))
                       => ! [F] : 
                            ( m1_subset_1(F,k1_zfmisc_1(B))
                           => ( r2_hidden(k8_funct_2(A,B,C,D),k6_subset_1(B,k2_funct_2(A,B,C,E),F))
                             => r2_hidden(D,k6_subset_1(A,E,k3_funct_2(A,B,C,F))) ) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_xboole_0,t3_funct_7,d13_funct_1,d4_xboole_0,d4_xboole_0]),
    [file(funct_7,t4_funct_7),interesting(0.61)]).

fof(t15_funct_7,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( k1_relat_1(C) = k2_xboole_0(A,k1_tarski(B))
       => C = k1_funct_4(k7_relat_1(C,A),k3_cqc_lang(B,k1_funct_1(C,B))) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_tarski,d2_xboole_0,t74_funct_4,t6_funct_7]),
    [file(funct_7,t15_funct_7),interesting(0.58)]).

fof(t68_funct_7,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_relat_1(A)
        & v2_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( m2_funct_7(B,A)
         => ( B != k1_xboole_0
           => ( k6_funct_7(B) = k1_funct_1(A,1)
              & r1_tarski(k7_funct_7(B),k1_funct_1(A,k3_finseq_1(A))) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_finseq_1,t19_nat_1,t38_nat_1,t27_finseq_3,d10_funct_7,d10_funct_7,d2_funct_2,d10_funct_7,d2_funct_2,d6_funct_7,t21_funct_5,t21_funct_5,d7_funct_7]),
    [file(funct_7,t68_funct_7),interesting(0.58)]).

fof(t39_funct_7,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m2_finseq_1(B,A)
         => ! [C] : 
              ( m1_subset_1(C,A)
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] : 
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( r2_hidden(E,k4_finseq_1(B))
                       => ( D = E
                          | k4_finseq_4(k5_numbers,A,k3_funct_7(A,B,D,C),E) = k4_finseq_4(k5_numbers,A,B,E) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t32_funct_7,d4_finseq_4,t34_funct_7,d4_finseq_4]),
    [file(funct_7,t39_funct_7),interesting(0.56)]).

fof(t24_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( k3_finseq_1(A) = k1_nat_1(k3_finseq_1(B),1)
           => ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r2_hidden(C,k4_finseq_1(B))
                <=> ( r2_hidden(C,k4_finseq_1(A))
                    & r2_hidden(k1_nat_1(C,1),k4_finseq_1(A)) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t27_finseq_3,t29_nat_1,t2_xreal_1,t8_xreal_1,t29_nat_1,t27_finseq_3,t27_finseq_3,t8_xreal_1,t27_finseq_3]),
    [file(funct_7,t24_funct_7),interesting(0.56)]).

fof(t2_funct_7,theorem,(
    ! [A,B] : 
      ( ~ v1_xboole_0(B)
     => ! [C] : 
          ( ( v1_funct_1(C)
            & v1_funct_2(C,A,B)
            & m2_relset_1(C,A,B) )
         => ( v2_funct_1(C)
           => ! [D] : 
                ( m1_subset_1(D,k1_zfmisc_1(A))
               => ! [E] : 
                    ( m1_subset_1(E,k1_zfmisc_1(B))
                   => ( r1_tarski(E,k2_funct_2(A,B,C,D))
                     => r1_tarski(k3_funct_2(A,B,C,E),D) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t152_funct_1,t178_relat_1,t1_xboole_1]),
    [file(funct_7,t2_funct_7),interesting(0.55)]).

fof(t5_funct_7,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,A,B)
                & m2_relset_1(C,A,B) )
             => ( v2_funct_1(C)
               => ! [D] : 
                    ( m1_subset_1(D,B)
                   => ! [E] : 
                        ( m1_subset_1(E,k1_zfmisc_1(A))
                       => ! [F] : 
                            ( m1_subset_1(F,k1_zfmisc_1(B))
                           => ( r2_hidden(D,k6_subset_1(B,k2_funct_2(A,B,C,E),F))
                             => r2_hidden(k1_funct_1(k2_funct_1(C),D),k6_subset_1(A,E,k3_funct_2(A,B,C,F))) ) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_xboole_0,t144_relat_1,t54_funct_1,t57_funct_1,t4_funct_7]),
    [file(funct_7,t5_funct_7),interesting(0.48)]).

fof(s1_funct_7,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s1_funct_7)
       => ! [B] : 
            ( m1_subset_1(B,f2_s1_funct_7)
           => r2_hidden(f4_s1_funct_7(A,B),f3_s1_funct_7) ) )
   => ? [A] : 
        ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(f1_s1_funct_7,f2_s1_funct_7),f3_s1_funct_7)
        & m2_relset_1(A,k2_zfmisc_1(f1_s1_funct_7,f2_s1_funct_7),f3_s1_funct_7)
        & ! [B] : 
            ( m1_subset_1(B,f1_s1_funct_7)
           => ! [C] : 
                ( m1_subset_1(C,f2_s1_funct_7)
               => k8_funct_2(k2_zfmisc_1(f1_s1_funct_7,f2_s1_funct_7),f3_s1_funct_7,A,k1_domain_1(f1_s1_funct_7,f2_s1_funct_7,B,C)) = f4_s1_funct_7(B,C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s12_funct_2,t7_mcart_1]),
    [file(funct_7,s1_funct_7),interesting(0.42)]).

fof(t119_partfun1,theorem,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & m2_relset_1(C,A,B) )
     => r2_hidden(C,k4_partfun1(A,B)) ) ),
    file(partfun1,t119_partfun1),
    [interesting(0.00)]).

fof(s10_recdef_1,theorem,
    ( ( k8_funct_2(k5_numbers,f1_s10_recdef_1,f3_s10_recdef_1,0) = f2_s10_recdef_1
      & ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => p1_s10_recdef_1(A,k8_funct_2(k5_numbers,f1_s10_recdef_1,f3_s10_recdef_1,A),k8_funct_2(k5_numbers,f1_s10_recdef_1,f3_s10_recdef_1,k1_nat_1(A,1))) )
      & k8_funct_2(k5_numbers,f1_s10_recdef_1,f4_s10_recdef_1,0) = f2_s10_recdef_1
      & ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => p1_s10_recdef_1(A,k8_funct_2(k5_numbers,f1_s10_recdef_1,f4_s10_recdef_1,A),k8_funct_2(k5_numbers,f1_s10_recdef_1,f4_s10_recdef_1,k1_nat_1(A,1))) )
      & ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B] : 
              ( m1_subset_1(B,f1_s10_recdef_1)
             => ! [C] : 
                  ( m1_subset_1(C,f1_s10_recdef_1)
                 => ! [D] : 
                      ( m1_subset_1(D,f1_s10_recdef_1)
                     => ( ( p1_s10_recdef_1(A,B,C)
                          & p1_s10_recdef_1(A,B,D) )
                       => C = D ) ) ) ) ) )
   => f3_s10_recdef_1 = f4_s10_recdef_1 ),
    file(recdef_1,s10_recdef_1),
    [interesting(0.00)]).

fof(l81_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))))
                & m2_relset_1(C,k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A)))) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))))
                    & m2_relset_1(D,k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A)))) )
                 => ( ( k8_funct_2(k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),C,0) = k6_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)))
                      & ! [E] : 
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ? [F] : 
                              ( v1_relat_1(F)
                              & v1_funct_1(F)
                              & F = k8_funct_2(k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),C,E)
                              & k8_funct_2(k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),C,k1_nat_1(E,1)) = k5_relat_1(A,F) ) )
                      & k8_funct_2(k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),D,0) = k6_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)))
                      & ! [E] : 
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ? [F] : 
                              ( v1_relat_1(F)
                              & v1_funct_1(F)
                              & F = k8_funct_2(k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),D,E)
                              & k8_funct_2(k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),D,k1_nat_1(E,1)) = k5_relat_1(A,F) ) ) )
                   => C = D ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t119_partfun1,s10_recdef_1]),
    [file(funct_7,l81_funct_7),interesting(0.00)]).

fof(s12_funct_2,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s12_funct_2)
       => r2_hidden(f3_s12_funct_2(A),f2_s12_funct_2) )
   => ? [A] : 
        ( v1_funct_1(A)
        & v1_funct_2(A,f1_s12_funct_2,f2_s12_funct_2)
        & m2_relset_1(A,f1_s12_funct_2,f2_s12_funct_2)
        & ! [B] : 
            ( m1_subset_1(B,f1_s12_funct_2)
           => k8_funct_2(f1_s12_funct_2,f2_s12_funct_2,A,B) = f3_s12_funct_2(B) ) ) ),
    file(funct_2,s12_funct_2),
    [interesting(0.00)]).

fof(t7_mcart_1,theorem,(
    ! [A,B] : 
      ( k1_mcart_1(k4_tarski(A,B)) = A
      & k2_mcart_1(k4_tarski(A,B)) = B ) ),
    file(mcart_1,t7_mcart_1),
    [interesting(0.00)]).

fof(s7_domain_1,theorem,(
    m1_subset_1(a_0_0_domain_1,k1_zfmisc_1(f1_s7_domain_1)) ),
    file(domain_1,s7_domain_1),
    [interesting(0.00)]).

fof(t13_finset_1,theorem,(
    ! [A,B] : 
      ( ( r1_tarski(A,B)
        & v1_finset_1(B) )
     => v1_finset_1(A) ) ),
    file(finset_1,t13_finset_1),
    [interesting(0.00)]).

fof(d6_tarski,definition,(
    ! [A,B] : 
      ( r2_tarski(A,B)
    <=> ? [C] : 
          ( ! [D] : ~ ( r2_hidden(D,A)
              & ! [E] : ~ ( r2_hidden(E,B)
                  & r2_hidden(k4_tarski(D,E),C) ) )
          & ! [D] : ~ ( r2_hidden(D,B)
              & ! [E] : ~ ( r2_hidden(E,A)
                  & r2_hidden(k4_tarski(E,D),C) ) )
          & ! [D,E,F,G] : 
              ( ( r2_hidden(k4_tarski(D,E),C)
                & r2_hidden(k4_tarski(F,G),C) )
             => ( D = F
              <=> E = G ) ) ) ) ),
    file(tarski,d6_tarski),
    [interesting(0.00)]).

fof(t33_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( k4_tarski(A,B) = k4_tarski(C,D)
     => ( A = C
        & B = D ) ) ),
    file(zfmisc_1,t33_zfmisc_1),
    [interesting(0.00)]).

fof(t68_card_1,theorem,(
    ! [A,B] : 
      ( ( r2_wellord2(A,B)
        & v1_finset_1(A) )
     => v1_finset_1(B) ) ),
    file(card_1,t68_card_1),
    [interesting(0.00)]).

fof(s21_fraenkel,theorem,
    ( v1_finset_1(f2_s21_fraenkel)
   => v1_finset_1(a_0_38_fraenkel) ),
    file(fraenkel,s21_fraenkel),
    [interesting(0.00)]).

fof(d3_tarski,definition,(
    ! [A,B] : 
      ( r1_tarski(A,B)
    <=> ! [C] : 
          ( r2_hidden(C,A)
         => r2_hidden(C,B) ) ) ),
    file(tarski,d3_tarski),
    [interesting(0.00)]).

fof(d10_xboole_0,definition,(
    ! [A,B] : 
      ( A = B
    <=> ( r1_tarski(A,B)
        & r1_tarski(B,A) ) ) ),
    file(xboole_0,d10_xboole_0),
    [interesting(0.00)]).

fof(t46_card_1,theorem,(
    ! [A] : 
      ( r2_wellord2(A,k1_xboole_0)
    <=> A = k1_xboole_0 ) ),
    file(card_1,t46_card_1),
    [interesting(0.00)]).

fof(s3_funct_7,theorem,
    ( ( ! [A] : ~ ( r2_hidden(A,f1_s3_funct_7)
          & ! [B] : 
              ( m1_subset_1(B,f2_s3_funct_7)
             => A != f3_s3_funct_7(B) ) )
      & ! [A] : 
          ( m1_subset_1(A,f2_s3_funct_7)
         => ! [B] : 
              ( m1_subset_1(B,f2_s3_funct_7)
             => ( f3_s3_funct_7(A) = f3_s3_funct_7(B)
               => A = B ) ) ) )
   => r2_tarski(f1_s3_funct_7,a_0_1_funct_7) ),
    inference(mizar_proof,[status(thm)],[t46_card_1,d6_tarski,t33_zfmisc_1,t33_zfmisc_1]),
    [file(funct_7,s3_funct_7),interesting(0.00)]).

fof(t73_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ! [D] : 
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D) )
                 => ( ( r1_tarski(k2_relat_1(C),k1_relat_1(A))
                      & r1_tarski(k2_relat_1(D),k1_relat_1(B))
                      & r1_partfun1(A,B) )
                   => k5_relat_1(k1_funct_4(C,D),k1_funct_4(A,B)) = k1_funct_4(k5_relat_1(C,A),k5_relat_1(D,B)) ) ) ) ) ) ),
    file(funct_4,t73_funct_4),
    [interesting(0.00)]).

fof(t10_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( ( r1_tarski(k2_relat_1(B),k1_relat_1(A))
                  & r1_tarski(k2_relat_1(C),k1_relat_1(A)) )
               => k5_relat_1(k1_funct_4(B,C),A) = k1_funct_4(k5_relat_1(B,A),k5_relat_1(C,A)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t73_funct_4]),
    [file(funct_7,t10_funct_7),interesting(0.00)]).

fof(t21_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( r2_hidden(A,k1_relat_1(k5_relat_1(C,B)))
          <=> ( r2_hidden(A,k1_relat_1(C))
              & r2_hidden(k1_funct_1(C,A),k1_relat_1(B)) ) ) ) ) ),
    file(funct_1,t21_funct_1),
    [interesting(0.00)]).

fof(d1_funct_4,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( C = k1_funct_4(A,B)
              <=> ( k1_relat_1(C) = k2_xboole_0(k1_relat_1(A),k1_relat_1(B))
                  & ! [D] : 
                      ( r2_hidden(D,k2_xboole_0(k1_relat_1(A),k1_relat_1(B)))
                     => ( ( r2_hidden(D,k1_relat_1(B))
                         => k1_funct_1(C,D) = k1_funct_1(B,D) )
                        & ( ~ r2_hidden(D,k1_relat_1(B))
                         => k1_funct_1(C,D) = k1_funct_1(A,D) ) ) ) ) ) ) ) ) ),
    file(funct_4,d1_funct_4),
    [interesting(0.00)]).

fof(d2_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k2_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            | r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d2_xboole_0),
    [interesting(0.00)]).

fof(t23_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( r2_hidden(A,k1_relat_1(B))
           => k1_funct_1(k5_relat_1(B,C),A) = k1_funct_1(C,k1_funct_1(B,A)) ) ) ) ),
    file(funct_1,t23_funct_1),
    [interesting(0.00)]).

fof(t14_funct_4,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( r2_hidden(A,k1_relat_1(B))
           => k1_funct_1(k1_funct_4(C,B),A) = k1_funct_1(B,A) ) ) ) ),
    file(funct_4,t14_funct_4),
    [interesting(0.00)]).

fof(t22_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( r2_hidden(A,k1_relat_1(k5_relat_1(C,B)))
           => k1_funct_1(k5_relat_1(C,B),A) = k1_funct_1(B,k1_funct_1(C,A)) ) ) ) ),
    file(funct_1,t22_funct_1),
    [interesting(0.00)]).

fof(t12_funct_4,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ~ r2_hidden(A,k1_relat_1(B))
           => k1_funct_1(k1_funct_4(C,B),A) = k1_funct_1(C,A) ) ) ) ),
    file(funct_4,t12_funct_4),
    [interesting(0.00)]).

fof(t67_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => ( r1_xboole_0(k2_relat_1(A),k1_relat_1(B))
           => k5_relat_1(A,B) = k1_xboole_0 ) ) ) ),
    file(relat_1,t67_relat_1),
    [interesting(0.00)]).

fof(t22_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k1_funct_4(A,k1_xboole_0) = A ) ),
    file(funct_4,t22_funct_4),
    [interesting(0.00)]).

fof(d1_tarski,definition,(
    ! [A,B] : 
      ( B = k1_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> C = A ) ) ),
    file(tarski,d1_tarski),
    [interesting(0.00)]).

fof(t3_xboole_0,theorem,(
    ! [A,B] : 
      ( ~ ( ~ r1_xboole_0(A,B)
          & ! [C] : ~ ( r2_hidden(C,A)
              & r2_hidden(C,B) ) )
      & ~ ( ? [C] : 
              ( r2_hidden(C,A)
              & r2_hidden(C,B) )
          & r1_xboole_0(A,B) ) ) ),
    file(xboole_0,t3_xboole_0),
    [interesting(0.00)]).

fof(d5_funct_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( B = k2_relat_1(A)
        <=> ! [C] : 
              ( r2_hidden(C,B)
            <=> ? [D] : 
                  ( r2_hidden(D,k1_relat_1(A))
                  & C = k1_funct_1(A,D) ) ) ) ) ),
    file(funct_1,d5_funct_1),
    [interesting(0.00)]).

fof(t19_funcop_1,theorem,(
    ! [A,B] : 
      ( k1_relat_1(k2_funcop_1(A,B)) = A
      & r1_tarski(k2_relat_1(k2_funcop_1(A,B)),k1_tarski(B)) ) ),
    file(funcop_1,t19_funcop_1),
    [interesting(0.00)]).

fof(t13_funcop_1,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(B,A)
     => k1_funct_1(k2_funcop_1(A,C),B) = C ) ),
    file(funcop_1,t13_funcop_1),
    [interesting(0.00)]).

fof(t35_funct_1,theorem,(
    ! [A,B] : 
      ( r2_hidden(B,A)
     => k1_funct_1(k6_relat_1(A),B) = B ) ),
    file(funct_1,t35_funct_1),
    [interesting(0.00)]).

fof(t54_zfmisc_1,theorem,(
    ! [A,B] : ~ ( r1_xboole_0(k1_tarski(A),B)
      & r2_hidden(A,B) ) ),
    file(zfmisc_1,t54_zfmisc_1),
    [interesting(0.00)]).

fof(t74_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : 
          ( r1_tarski(k1_relat_1(A),k2_xboole_0(B,C))
         => k1_funct_4(k7_relat_1(A,B),k7_relat_1(A,C)) = A ) ) ),
    file(funct_4,t74_funct_4),
    [interesting(0.00)]).

fof(t89_grfunc_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( r2_hidden(B,k1_relat_1(A))
         => k7_relat_1(A,k1_tarski(B)) = k1_tarski(k4_tarski(B,k1_funct_1(A,B))) ) ) ),
    file(grfunc_1,t89_grfunc_1),
    [interesting(0.00)]).

fof(t35_zfmisc_1,theorem,(
    ! [A,B] : k2_zfmisc_1(k1_tarski(A),k1_tarski(B)) = k1_tarski(k4_tarski(A,B)) ),
    file(zfmisc_1,t35_zfmisc_1),
    [interesting(0.00)]).

fof(t4_xboole_1,theorem,(
    ! [A,B,C] : k2_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(A,k2_xboole_0(B,C)) ),
    file(xboole_1,t4_xboole_1),
    [interesting(0.00)]).

fof(t9_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( k1_relat_1(A) = k1_relat_1(B)
              & ! [C] : 
                  ( r2_hidden(C,k1_relat_1(A))
                 => k1_funct_1(A,C) = k1_funct_1(B,C) ) )
           => A = B ) ) ) ),
    file(funct_1,t9_funct_1),
    [interesting(0.00)]).

fof(t17_funct_7,theorem,(
    $true ),
    file(funct_7,t17_funct_7),
    [interesting(0.00)]).

fof(d11_finseq_1,definition,(
    ! [A,B] : 
      ( B = k13_finseq_1(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> m2_finseq_1(C,A) ) ) ),
    file(finseq_1,d11_finseq_1),
    [interesting(0.00)]).

fof(t1_card_5,theorem,
    ( 1 = k1_tarski(0)
    & 2 = k2_tarski(0,1) ),
    file(card_5,t1_card_5),
    [interesting(0.00)]).

fof(d1_relat_1,definition,(
    ! [A] : 
      ( v1_relat_1(A)
    <=> ! [B] : ~ ( r2_hidden(B,A)
          & ! [C,D] : B != k4_tarski(C,D) ) ) ),
    file(relat_1,d1_relat_1),
    [interesting(0.00)]).

fof(d2_int_1,definition,(
    ! [A] : 
      ( v1_int_1(A)
    <=> r2_hidden(A,k4_numbers) ) ),
    file(int_1,d2_int_1),
    [interesting(0.00)]).

fof(t64_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ( ( k1_relat_1(A) = k1_xboole_0
          | k2_relat_1(A) = k1_xboole_0 )
       => A = k1_xboole_0 ) ) ),
    file(relat_1,t64_relat_1),
    [interesting(0.00)]).

fof(d1_xboole_0,definition,(
    ! [A] : 
      ( A = k1_xboole_0
    <=> ! [B] : ~ r2_hidden(B,A) ) ),
    file(xboole_0,d1_xboole_0),
    [interesting(0.00)]).

fof(t37_zfmisc_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(k1_tarski(A),B)
    <=> r2_hidden(A,B) ) ),
    file(zfmisc_1,t37_zfmisc_1),
    [interesting(0.00)]).

fof(d4_finseq_1,definition,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => ( m1_finseq_1(B,A)
      <=> r1_tarski(k2_relat_1(B),A) ) ) ),
    file(finseq_1,d4_finseq_1),
    [interesting(0.00)]).

fof(t4_funct_2,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r1_tarski(k2_relat_1(B),A)
       => ( v1_funct_1(B)
          & v1_funct_2(B,k1_relat_1(B),A)
          & m2_relset_1(B,k1_relat_1(B),A) ) ) ) ),
    file(funct_2,t4_funct_2),
    [interesting(0.00)]).

fof(t23_funct_2,theorem,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => ( k7_relset_1(A,A,A,B,k6_partfun1(A),C) = C
        & k7_relset_1(A,B,B,B,C,k6_partfun1(B)) = C ) ) ),
    file(funct_2,t23_funct_2),
    [interesting(0.00)]).

fof(t55_finseq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => ( B = k9_finseq_1(A)
      <=> ( k4_finseq_1(B) = k2_finseq_1(1)
          & k2_relat_1(B) = k1_tarski(A) ) ) ) ),
    file(finseq_1,t55_finseq_1),
    [interesting(0.00)]).

fof(t47_card_1,theorem,(
    k1_card_1(k1_xboole_0) = k1_xboole_0 ),
    file(card_1,t47_card_1),
    [interesting(0.00)]).

fof(t10_subset_1,theorem,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ~ ( B != k1_xboole_0
          & ! [C] : 
              ( m1_subset_1(C,A)
             => ~ r2_hidden(C,B) ) ) ) ),
    file(subset_1,t10_subset_1),
    [interesting(0.00)]).

fof(s6_nat_1,theorem,
    ( ( ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( p1_s6_nat_1(A)
           => r1_xreal_0(A,f1_s6_nat_1) ) )
      & ? [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
          & p1_s6_nat_1(A) ) )
   => ? [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & p1_s6_nat_1(A)
        & ! [B] : 
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( p1_s6_nat_1(B)
             => r1_xreal_0(B,A) ) ) ) ),
    file(nat_1,s6_nat_1),
    [interesting(0.00)]).

fof(t38_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( ~ r1_xreal_0(k2_xcmplx_0(B,1),A)
          <=> r1_xreal_0(A,B) ) ) ) ),
    file(nat_1,t38_nat_1),
    [interesting(0.00)]).

fof(t1_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(A,B)
              & r1_xreal_0(B,A) )
           => A = B ) ) ) ),
    file(xreal_1,t1_xreal_1),
    [interesting(0.00)]).

fof(t30_axioms,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => A = a_1_0_axioms(A) ) ),
    file(axioms,t30_axioms),
    [interesting(0.00)]).

fof(t83_card_1,theorem,(
    k3_card_1(0) = k5_ordinal2 ),
    file(card_1,t83_card_1),
    [interesting(0.00)]).

fof(t46_setfam_1,theorem,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ~ ( B != k1_xboole_0
          & k7_setfam_1(A,B) = k1_xboole_0 ) ) ),
    file(setfam_1,t46_setfam_1),
    [interesting(0.00)]).

fof(t18_finset_1,theorem,(
    ! [A] : 
      ( v1_finset_1(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ~ ( B != k1_xboole_0
              & ! [C] : ~ ( r2_hidden(C,B)
                  & ! [D] : 
                      ( ( r2_hidden(D,B)
                        & r1_tarski(C,D) )
                     => D = C ) ) ) ) ) ),
    file(finset_1,t18_finset_1),
    [interesting(0.00)]).

fof(d8_setfam_1,definition,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] : 
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( C = k7_setfam_1(A,B)
          <=> ! [D] : 
                ( m1_subset_1(D,k1_zfmisc_1(A))
               => ( r2_hidden(D,C)
                <=> r2_hidden(k3_subset_1(A,D),B) ) ) ) ) ) ),
    file(setfam_1,d8_setfam_1),
    [interesting(0.00)]).

fof(t35_subset_1,theorem,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] : 
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( r1_tarski(B,k3_subset_1(A,C))
           => r1_tarski(C,k3_subset_1(A,B)) ) ) ) ),
    file(subset_1,t35_subset_1),
    [interesting(0.00)]).

fof(d3_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k3_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d3_xboole_0),
    [interesting(0.00)]).

fof(d1_funct_7,definition,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => k1_funct_7(A,B) = A ) ),
    file(funct_7,d1_funct_7),
    [interesting(0.00)]).

fof(d2_funct_7,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : 
              ( r1_funct_7(A,B,C)
            <=> k7_relat_1(A,k4_xboole_0(k1_relat_1(A),C)) = k7_relat_1(B,k4_xboole_0(k1_relat_1(B),C)) ) ) ) ),
    file(funct_7,d2_funct_7),
    [interesting(0.00)]).

fof(t152_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v2_funct_1(B)
       => r1_tarski(k10_relat_1(B,k9_relat_1(B,A)),A) ) ) ),
    file(funct_1,t152_funct_1),
    [interesting(0.00)]).

fof(t178_relat_1,theorem,(
    ! [A,B,C] : 
      ( v1_relat_1(C)
     => ( r1_tarski(A,B)
       => r1_tarski(k10_relat_1(C,A),k10_relat_1(C,B)) ) ) ),
    file(relat_1,t178_relat_1),
    [interesting(0.00)]).

fof(t1_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(B,C) )
     => r1_tarski(A,C) ) ),
    file(xboole_1,t1_xboole_1),
    [interesting(0.00)]).

fof(t36_xboole_1,theorem,(
    ! [A,B] : r1_tarski(k4_xboole_0(A,B),A) ),
    file(xboole_1,t36_xboole_1),
    [interesting(0.00)]).

fof(t28_xboole_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => k3_xboole_0(A,B) = A ) ),
    file(xboole_1,t28_xboole_1),
    [interesting(0.00)]).

fof(t90_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => k1_relat_1(k7_relat_1(B,A)) = k3_xboole_0(k1_relat_1(B),A) ) ),
    file(relat_1,t90_relat_1),
    [interesting(0.00)]).

fof(t42_xboole_1,theorem,(
    ! [A,B,C] : k4_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(k4_xboole_0(A,C),k4_xboole_0(B,C)) ),
    file(xboole_1,t42_xboole_1),
    [interesting(0.00)]).

fof(t37_xboole_1,theorem,(
    ! [A,B] : 
      ( k4_xboole_0(A,B) = k1_xboole_0
    <=> r1_tarski(A,B) ) ),
    file(xboole_1,t37_xboole_1),
    [interesting(0.00)]).

fof(t13_prob_1,theorem,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(B)
        & v2_finsub_1(B)
        & v1_prob_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => ! [C,D] : 
          ( r1_xboole_0(k4_xboole_0(C,D),D)
          & ( ( r2_hidden(C,B)
              & r2_hidden(D,B) )
           => r2_hidden(k2_xboole_0(k4_xboole_0(C,D),D),B) ) ) ) ),
    file(prob_1,t13_prob_1),
    [interesting(0.00)]).

fof(t63_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_xboole_0(B,C) )
     => r1_xboole_0(A,C) ) ),
    file(xboole_1,t63_xboole_1),
    [interesting(0.00)]).

fof(t76_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : 
              ( r1_xboole_0(C,k1_relat_1(B))
             => k7_relat_1(k1_funct_4(A,B),C) = k7_relat_1(A,C) ) ) ) ),
    file(funct_4,t76_funct_4),
    [interesting(0.00)]).

fof(t5_cqc_lang,theorem,(
    ! [A,B] : 
      ( k1_relat_1(k3_cqc_lang(A,B)) = k1_tarski(A)
      & k2_relat_1(k3_cqc_lang(A,B)) = k1_tarski(B) ) ),
    file(cqc_lang,t5_cqc_lang),
    [interesting(0.00)]).

fof(t17_zfmisc_1,theorem,(
    ! [A,B] : 
      ( A != B
     => r1_xboole_0(k1_tarski(A),k1_tarski(B)) ) ),
    file(zfmisc_1,t17_zfmisc_1),
    [interesting(0.00)]).

fof(d3_funct_7,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : 
          ( ( r2_hidden(B,k1_relat_1(A))
           => k2_funct_7(A,B,C) = k1_funct_4(A,k3_cqc_lang(B,C)) )
          & ( ~ r2_hidden(B,k1_relat_1(A))
           => k2_funct_7(A,B,C) = A ) ) ) ),
    file(funct_7,d3_funct_7),
    [interesting(0.00)]).

fof(t12_xboole_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => k2_xboole_0(A,B) = B ) ),
    file(xboole_1,t12_xboole_1),
    [interesting(0.00)]).

fof(t15_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => k1_funct_4(k1_funct_4(A,B),C) = k1_funct_4(A,k1_funct_4(B,C)) ) ) ) ),
    file(funct_4,t15_funct_4),
    [interesting(0.00)]).

fof(t36_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
           => k1_funct_4(A,B) = k1_funct_4(B,A) ) ) ) ),
    file(funct_4,t36_funct_4),
    [interesting(0.00)]).

fof(t20_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_tarski(k1_relat_1(A),k1_relat_1(B))
           => k1_funct_4(A,B) = B ) ) ) ),
    file(funct_4,t20_funct_4),
    [interesting(0.00)]).

fof(d4_finseq_4,definition,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & m2_relset_1(C,A,B) )
     => ! [D] : 
          ( r2_hidden(D,k1_relat_1(C))
         => k4_finseq_4(A,B,C,D) = k1_funct_1(C,D) ) ) ),
    file(finseq_4,d4_finseq_4),
    [interesting(0.00)]).

fof(t6_cqc_lang,theorem,(
    ! [A,B] : k1_funct_1(k3_cqc_lang(A,B),A) = B ),
    file(cqc_lang,t6_cqc_lang),
    [interesting(0.00)]).

fof(t80_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : k1_funct_4(A,k7_relat_1(A,B)) = A ) ),
    file(funct_4,t80_funct_4),
    [interesting(0.00)]).

fof(t47_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( k7_finseq_1(A,k1_xboole_0) = A
        & k7_finseq_1(k1_xboole_0,A) = A ) ) ),
    file(finseq_1,t47_finseq_1),
    [interesting(0.00)]).

fof(d4_funct_7,definition,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B)
        & v1_funcop_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( C = k4_funct_7(A,B)
          <=> ? [D] : 
                ( v1_funcop_1(D)
                & m1_pboole(D,k5_numbers)
                & C = k1_funct_1(D,k3_finseq_1(B))
                & k1_funct_1(D,0) = k6_partfun1(A)
                & ! [E] : 
                    ( m2_subset_1(E,k1_numbers,k5_numbers)
                   => ( r2_hidden(k1_nat_1(E,1),k4_finseq_1(B))
                     => ! [F] : 
                          ( ( v1_relat_1(F)
                            & v1_funct_1(F) )
                         => ! [G] : 
                              ( ( v1_relat_1(G)
                                & v1_funct_1(G) )
                             => ( ( F = k1_funct_1(D,E)
                                  & G = k1_funct_1(B,k1_nat_1(E,1)) )
                               => k1_funct_1(D,k1_nat_1(E,1)) = k5_relat_1(F,G) ) ) ) ) ) ) ) ) ) ),
    file(funct_7,d4_funct_7),
    [interesting(0.00)]).

fof(t39_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => r1_tarski(k4_finseq_1(A),k4_finseq_1(k7_finseq_1(A,B))) ) ) ),
    file(finseq_1,t39_finseq_1),
    [interesting(0.00)]).

fof(d7_finseq_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( C = k7_finseq_1(A,B)
              <=> ( k4_finseq_1(C) = k2_finseq_1(k1_nat_1(k3_finseq_1(A),k3_finseq_1(B)))
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(A))
                       => k1_funct_1(C,D) = k1_funct_1(A,D) ) )
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(B))
                       => k1_funct_1(C,k1_nat_1(k3_finseq_1(A),D)) = k1_funct_1(B,D) ) ) ) ) ) ) ) ),
    file(finseq_1,d7_finseq_1),
    [interesting(0.00)]).

fof(t4_finseq_1,theorem,
    ( k2_finseq_1(0) = k1_xboole_0
    & k2_finseq_1(1) = k1_tarski(1)
    & k2_finseq_1(2) = k2_tarski(1,2) ),
    file(finseq_1,t4_finseq_1),
    [interesting(0.00)]).

fof(t57_finseq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => ( B = k9_finseq_1(A)
      <=> ( k3_finseq_1(B) = 1
          & k1_funct_1(B,1) = A ) ) ) ),
    file(finseq_1,t57_finseq_1),
    [interesting(0.00)]).

fof(t35_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => k3_finseq_1(k7_finseq_1(A,B)) = k1_nat_1(k3_finseq_1(A),k3_finseq_1(B)) ) ) ),
    file(finseq_1,t35_finseq_1),
    [interesting(0.00)]).

fof(t41_finseq_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( r2_hidden(A,k4_finseq_1(B))
               => r2_hidden(k1_nat_1(k3_finseq_1(C),A),k4_finseq_1(k7_finseq_1(C,B))) ) ) ) ) ),
    file(finseq_1,t41_finseq_1),
    [interesting(0.00)]).

fof(t59_finseq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => k1_funct_1(k7_finseq_1(B,k9_finseq_1(A)),k1_nat_1(k3_finseq_1(B),1)) = A ) ),
    file(finseq_1,t59_finseq_1),
    [interesting(0.00)]).

fof(t25_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( k3_finseq_1(A) = 0
      <=> A = k1_xboole_0 ) ) ),
    file(finseq_1,t25_finseq_1),
    [interesting(0.00)]).

fof(t94_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => k7_relat_1(B,A) = k5_relat_1(k6_relat_1(A),B) ) ),
    file(relat_1,t94_relat_1),
    [interesting(0.00)]).

fof(t80_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => k5_relat_1(A,k6_relat_1(k2_relat_1(A))) = A ) ),
    file(relat_1,t80_relat_1),
    [interesting(0.00)]).

fof(t148_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => k2_relat_1(k7_relat_1(B,A)) = k9_relat_1(B,A) ) ),
    file(relat_1,t148_relat_1),
    [interesting(0.00)]).

fof(t45_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => k7_finseq_1(k7_finseq_1(A,B),C) = k7_finseq_1(A,k7_finseq_1(B,C)) ) ) ) ),
    file(finseq_1,t45_finseq_1),
    [interesting(0.00)]).

fof(t55_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => ! [C] : 
              ( v1_relat_1(C)
             => k5_relat_1(k5_relat_1(A,B),C) = k5_relat_1(A,k5_relat_1(B,C)) ) ) ) ),
    file(relat_1,t55_relat_1),
    [interesting(0.00)]).

fof(d6_funcop_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_funcop_1(A)
      <=> ! [B] : 
            ( r2_hidden(B,k1_relat_1(A))
           => ( v1_relat_1(k1_funct_1(A,B))
              & v1_funct_1(k1_funct_1(A,B)) ) ) ) ) ),
    file(funcop_1,d6_funcop_1),
    [interesting(0.00)]).

fof(t24_finseq_3,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] : 
              ( r2_hidden(C,k4_finseq_1(A))
             => r2_hidden(C,k4_finseq_1(k7_finseq_1(A,B))) ) ) ) ),
    file(finseq_3,t24_finseq_3),
    [interesting(0.00)]).

fof(s3_finseq_1,theorem,
    ( ( p1_s3_finseq_1(k1_xboole_0)
      & ! [A] : 
          ( ( v1_relat_1(A)
            & v1_funct_1(A)
            & v1_finseq_1(A) )
         => ! [B] : 
              ( p1_s3_finseq_1(A)
             => p1_s3_finseq_1(k7_finseq_1(A,k9_finseq_1(B))) ) ) )
   => ! [A] : 
        ( ( v1_relat_1(A)
          & v1_funct_1(A)
          & v1_finseq_1(A) )
       => p1_s3_finseq_1(A) ) ),
    file(finseq_1,s3_finseq_1),
    [interesting(0.00)]).

fof(s5_funct_7,theorem,
    ( ( p1_s5_funct_7(k1_xboole_0)
      & ! [A] : 
          ( ( v1_relat_1(A)
            & v1_funct_1(A)
            & v1_finseq_1(A)
            & v1_funcop_1(A) )
         => ( p1_s5_funct_7(A)
           => ! [B] : 
                ( ( v1_relat_1(B)
                  & v1_funct_1(B) )
               => p1_s5_funct_7(k7_finseq_1(A,k9_finseq_1(B))) ) ) ) )
   => ! [A] : 
        ( ( v1_relat_1(A)
          & v1_funct_1(A)
          & v1_finseq_1(A)
          & v1_funcop_1(A) )
       => p1_s5_funct_7(A) ) ),
    inference(mizar_proof,[status(thm)],[t4_finseq_1,t55_finseq_1,t57_finseq_1,t25_funct_7,d1_tarski,d6_funcop_1,s3_finseq_1]),
    [file(funct_7,s5_funct_7),interesting(0.00)]).

fof(d5_funct_7,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v1_funcop_1(A) )
     => ! [B,C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C)
            & v1_finseq_1(C) )
         => ( C = k5_funct_7(A,B)
          <=> ( k3_finseq_1(C) = k1_nat_1(k3_finseq_1(A),1)
              & k1_funct_1(C,1) = B
              & ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] : 
                      ( ( v1_relat_1(E)
                        & v1_funct_1(E) )
                     => ( ( r2_hidden(D,k4_finseq_1(A))
                          & E = k1_funct_1(A,D) )
                       => k1_funct_1(C,k1_nat_1(D,1)) = k1_funct_1(E,k1_funct_1(C,D)) ) ) ) ) ) ) ) ),
    file(funct_7,d5_funct_7),
    [interesting(0.00)]).

fof(d3_finseq_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k3_finseq_1(A)
          <=> k2_finseq_1(B) = k1_relat_1(A) ) ) ) ),
    file(finseq_1,d3_finseq_1),
    [interesting(0.00)]).

fof(t29_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => r1_xreal_0(A,k2_xcmplx_0(A,B)) ) ) ),
    file(nat_1,t29_nat_1),
    [interesting(0.00)]).

fof(t27_finseq_3,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( r2_hidden(B,k4_finseq_1(A))
          <=> ( r1_xreal_0(1,B)
              & r1_xreal_0(B,k3_finseq_1(A)) ) ) ) ) ),
    file(finseq_3,t27_finseq_3),
    [interesting(0.00)]).

fof(t2_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( ( r1_xreal_0(A,B)
                  & r1_xreal_0(B,C) )
               => r1_xreal_0(A,C) ) ) ) ) ),
    file(xreal_1,t2_xreal_1),
    [interesting(0.00)]).

fof(t8_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( r1_xreal_0(A,B)
              <=> r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,C)) ) ) ) ) ),
    file(xreal_1,t8_xreal_1),
    [interesting(0.00)]).

fof(t19_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ~ ( 0 != A
          & r1_xreal_0(A,0) ) ) ),
    file(nat_1,t19_nat_1),
    [interesting(0.00)]).

fof(s1_nat_1,theorem,
    ( ( p1_s1_nat_1(0)
      & ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( p1_s1_nat_1(A)
           => p1_s1_nat_1(k1_nat_1(A,1)) ) ) )
   => ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => p1_s1_nat_1(A) ) ),
    file(nat_1,s1_nat_1),
    [interesting(0.00)]).

fof(t77_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => ( r1_tarski(k1_relat_1(B),A)
       => k5_relat_1(k6_relat_1(A),B) = B ) ) ),
    file(relat_1,t77_relat_1),
    [interesting(0.00)]).

fof(t79_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => ( r1_tarski(k2_relat_1(B),A)
       => k5_relat_1(B,k6_relat_1(A)) = B ) ) ),
    file(relat_1,t79_relat_1),
    [interesting(0.00)]).

fof(t50_funct_7,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C)
        & v1_finseq_1(C)
        & v1_funcop_1(C) )
     => ! [D] : 
          ( ( v1_relat_1(D)
            & v1_funct_1(D)
            & v1_finseq_1(D)
            & v1_funcop_1(D) )
         => ( r1_tarski(k2_relat_1(k4_funct_7(A,C)),B)
           => k4_funct_7(A,k7_finseq_1(C,D)) = k5_relat_1(k4_funct_7(A,C),k4_funct_7(B,D)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t47_finseq_1,t79_relat_1,t41_funct_7,t45_finseq_1,t43_funct_7,t55_relat_1,t43_funct_7,s5_funct_7]),
    [file(funct_7,t50_funct_7),interesting(0.00)]).

fof(t63_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ~ ( A != k1_xboole_0
          & ! [B] : 
              ( ( v1_relat_1(B)
                & v1_funct_1(B)
                & v1_finseq_1(B) )
             => ! [C] : A != k7_finseq_1(B,k9_finseq_1(C)) ) ) ) ),
    file(finseq_1,t63_finseq_1),
    [interesting(0.00)]).

fof(t2_rewrite1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] : 
              ( A != k1_xboole_0
             => k1_rewrite1(k7_finseq_1(B,k9_finseq_1(C)),A) = k7_finseq_1(B,A) ) ) ) ),
    file(rewrite1,t2_rewrite1),
    [interesting(0.00)]).

fof(t51_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B)
        & v1_funcop_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C)
            & v1_finseq_1(C)
            & v1_funcop_1(C) )
         => k1_funct_1(k5_funct_7(k7_finseq_1(B,C),A),k1_nat_1(k3_finseq_1(k7_finseq_1(B,C)),1)) = k1_funct_1(k5_funct_7(C,k1_funct_1(k5_funct_7(B,A),k1_nat_1(k3_finseq_1(B),1))),k1_nat_1(k3_finseq_1(C),1)) ) ) ),
    inference(mizar_proof,[status(thm)],[t42_funct_7,t25_finseq_1,t47_finseq_1,t57_finseq_1,t45_finseq_1,t44_funct_7,t57_finseq_1,d5_funct_7,t35_finseq_1,t44_funct_7,d5_funct_7,t35_finseq_1,t59_finseq_1,t59_finseq_1,s5_funct_7]),
    [file(funct_7,t51_funct_7),interesting(0.00)]).

fof(d9_finseq_1,definition,(
    ! [A,B] : k10_finseq_1(A,B) = k7_finseq_1(k9_finseq_1(A),k9_finseq_1(B)) ),
    file(finseq_1,d9_finseq_1),
    [interesting(0.00)]).

fof(t60_finseq_1,theorem,(
    ! [A,B,C] : 
      ( k11_finseq_1(A,B,C) = k7_finseq_1(k9_finseq_1(A),k10_finseq_1(B,C))
      & k11_finseq_1(A,B,C) = k7_finseq_1(k10_finseq_1(A,B),k9_finseq_1(C)) ) ),
    file(finseq_1,t60_finseq_1),
    [interesting(0.00)]).

fof(t57_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ! [D] : 
              ( ( v1_relat_1(D)
                & v1_funct_1(D) )
             => ( ~ ( ~ r1_tarski(k1_relat_1(B),A)
                    & ~ r1_tarski(k1_relat_1(k5_relat_1(B,C)),A)
                    & ~ r1_tarski(k1_relat_1(k5_relat_1(B,k5_relat_1(C,D))),A) )
               => k4_funct_7(A,k11_finseq_1(B,C,D)) = k5_relat_1(B,k5_relat_1(C,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t56_funct_7,t55_relat_1,t77_relat_1]),
    [file(funct_7,t57_funct_7),interesting(0.00)]).

fof(t61_finseq_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C)
        & v1_finseq_1(C) )
     => ( C = k10_finseq_1(A,B)
      <=> ( k3_finseq_1(C) = 2
          & k1_funct_1(C,1) = A
          & k1_funct_1(C,2) = B ) ) ) ),
    file(finseq_1,t61_finseq_1),
    [interesting(0.00)]).

fof(t62_finseq_1,theorem,(
    ! [A,B,C,D] : 
      ( ( v1_relat_1(D)
        & v1_funct_1(D)
        & v1_finseq_1(D) )
     => ( D = k11_finseq_1(A,B,C)
      <=> ( k3_finseq_1(D) = 3
          & k1_funct_1(D,1) = A
          & k1_funct_1(D,2) = B
          & k1_funct_1(D,3) = C ) ) ) ),
    file(finseq_1,t62_finseq_1),
    [interesting(0.00)]).

fof(t58_funct_7,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ! [D] : 
              ( ( v1_relat_1(D)
                & v1_funct_1(D) )
             => k5_funct_7(k11_finseq_1(B,C,D),A) = k7_finseq_1(k9_finseq_1(A),k11_finseq_1(k1_funct_1(B,A),k1_funct_1(C,k1_funct_1(B,A)),k1_funct_1(D,k1_funct_1(C,k1_funct_1(B,A))))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t55_funct_7,t61_finseq_1,t60_finseq_1,t44_funct_7,t62_finseq_1,t60_finseq_1,t45_finseq_1,t60_finseq_1]),
    [file(funct_7,t58_funct_7),interesting(0.00)]).

fof(d4_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k4_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & ~ r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d4_xboole_0),
    [interesting(0.00)]).

fof(t144_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => r1_tarski(k9_relat_1(B,A),k2_relat_1(B)) ) ),
    file(relat_1,t144_relat_1),
    [interesting(0.00)]).

fof(t54_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => ! [B] : 
            ( ( v1_relat_1(B)
              & v1_funct_1(B) )
           => ( B = k2_funct_1(A)
            <=> ( k1_relat_1(B) = k2_relat_1(A)
                & ! [C,D] : 
                    ( ( ( r2_hidden(C,k2_relat_1(A))
                        & D = k1_funct_1(B,C) )
                     => ( r2_hidden(D,k1_relat_1(A))
                        & C = k1_funct_1(A,D) ) )
                    & ( ( r2_hidden(D,k1_relat_1(A))
                        & C = k1_funct_1(A,D) )
                     => ( r2_hidden(C,k2_relat_1(A))
                        & D = k1_funct_1(B,C) ) ) ) ) ) ) ) ) ),
    file(funct_1,t54_funct_1),
    [interesting(0.00)]).

fof(t57_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( ( v2_funct_1(B)
          & r2_hidden(A,k2_relat_1(B)) )
       => ( A = k1_funct_1(B,k1_funct_1(k2_funct_1(B),A))
          & A = k1_funct_1(k5_relat_1(k2_funct_1(B),B),A) ) ) ) ),
    file(funct_1,t57_funct_1),
    [interesting(0.00)]).

fof(t116_funct_2,theorem,(
    ! [A,B,C,D] : 
      ( ( v1_funct_1(D)
        & v1_funct_2(D,A,B)
        & m2_relset_1(D,A,B) )
     => ! [E] : ~ ( r2_hidden(E,k2_funct_2(A,B,D,C))
          & ! [F] : 
              ( m1_subset_1(F,A)
             => ~ ( r2_hidden(F,C)
                  & E = k1_funct_1(D,F) ) ) ) ) ),
    file(funct_2,t116_funct_2),
    [interesting(0.00)]).

fof(t25_funct_2,theorem,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B) )
     => ( ( B = k1_xboole_0
         => A = k1_xboole_0 )
       => ( v2_funct_1(C)
        <=> ! [D,E] : 
              ( ( r2_hidden(D,A)
                & r2_hidden(E,A)
                & k1_funct_1(C,D) = k1_funct_1(C,E) )
             => D = E ) ) ) ) ),
    file(funct_2,t25_funct_2),
    [interesting(0.00)]).

fof(d13_funct_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : 
          ( C = k10_relat_1(A,B)
        <=> ! [D] : 
              ( r2_hidden(D,C)
            <=> ( r2_hidden(D,k1_relat_1(A))
                & r2_hidden(k1_funct_1(A,D),B) ) ) ) ) ),
    file(funct_1,d13_funct_1),
    [interesting(0.00)]).

fof(d8_funct_7,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( v1_funct_7(A)
      <=> ? [B] : 
            ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B)
            & k3_finseq_1(B) = k1_nat_1(k3_finseq_1(A),1)
            & ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r2_hidden(C,k4_finseq_1(A))
                 => r2_hidden(k1_funct_1(A,C),k1_funct_2(k1_funct_1(B,C),k1_funct_1(B,k1_nat_1(C,1)))) ) ) ) ) ) ),
    file(funct_7,d8_funct_7),
    [interesting(0.00)]).

fof(d2_funct_2,definition,(
    ! [A,B,C] : 
      ( C = k1_funct_2(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ? [E] : 
              ( v1_relat_1(E)
              & v1_funct_1(E)
              & D = E
              & k1_relat_1(E) = A
              & r1_tarski(k2_relat_1(E),B) ) ) ) ),
    file(funct_2,d2_funct_2),
    [interesting(0.00)]).

fof(d6_funct_7,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_xboole_0(A)
           => ( B = k6_funct_7(A)
            <=> v1_xboole_0(B) ) )
          & ( ~ v1_xboole_0(A)
           => ( B = k6_funct_7(A)
            <=> B = k1_funct_5(k1_funct_1(A,1)) ) ) ) ) ),
    file(funct_7,d6_funct_7),
    [interesting(0.00)]).

fof(t21_funct_5,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ( k1_funct_5(A) = k1_relat_1(A)
        & k2_funct_5(A) = k2_relat_1(A) ) ) ),
    file(funct_5,t21_funct_5),
    [interesting(0.00)]).

fof(t2_xboole_1,theorem,(
    ! [A] : r1_tarski(k1_xboole_0,A) ),
    file(xboole_1,t2_xboole_1),
    [interesting(0.00)]).

fof(d7_funct_7,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_xboole_0(A)
           => ( B = k7_funct_7(A)
            <=> v1_xboole_0(B) ) )
          & ( ~ v1_xboole_0(A)
           => ( B = k7_funct_7(A)
            <=> B = k2_funct_5(k1_funct_1(A,k3_finseq_1(A))) ) ) ) ) ),
    file(funct_7,d7_funct_7),
    [interesting(0.00)]).

fof(t58_finseq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => k1_funct_1(k7_finseq_1(k9_finseq_1(A),B),1) = A ) ),
    file(finseq_1,t58_finseq_1),
    [interesting(0.00)]).

fof(d5_real_1,definition,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
          <=> ~ ( r1_xreal_0(B,A)
                & B != A ) ) ) ) ),
    file(real_1,d5_real_1),
    [interesting(0.00)]).

fof(t28_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ~ ( r1_xreal_0(A,B)
              & ! [C] : 
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => B != k2_xcmplx_0(A,C) ) ) ) ) ),
    file(nat_1,t28_nat_1),
    [interesting(0.00)]).

fof(t65_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v1_funct_7(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_tarski(k2_relat_1(B),k6_funct_7(A))
           => ( v1_relat_1(k7_finseq_1(k9_finseq_1(B),A))
              & v1_funct_1(k7_finseq_1(k9_finseq_1(B),A))
              & v1_finseq_1(k7_finseq_1(k9_finseq_1(B),A))
              & v1_funct_7(k7_finseq_1(k9_finseq_1(B),A)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_funct_7,t57_finseq_1,t35_finseq_1,t25_finseq_1,t19_nat_1,t38_nat_1,t27_finseq_3,d2_funct_2,d6_funct_7,t21_funct_5,d8_funct_7,t27_finseq_3,t29_nat_1,t27_finseq_3,t2_xboole_1,t59_funct_7,t1_xboole_1,t58_finseq_1,d7_finseq_1,d2_funct_2,d5_real_1,t38_nat_1,t28_nat_1,t29_nat_1,t8_xreal_1,t27_finseq_3,t24_funct_7,d7_finseq_1,d7_finseq_1]),
    [file(funct_7,t65_funct_7),interesting(0.00)]).

fof(t29_finseq_3,theorem,(
    ! [A,B] : k4_finseq_1(k10_finseq_1(A,B)) = k2_finseq_1(2) ),
    file(finseq_3,t29_finseq_3),
    [interesting(0.00)]).

fof(d2_tarski,definition,(
    ! [A,B,C] : 
      ( C = k2_tarski(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( D = A
            | D = B ) ) ) ),
    file(tarski,d2_tarski),
    [interesting(0.00)]).

fof(t26_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( A = k1_xboole_0
      <=> k4_finseq_1(A) = k1_xboole_0 ) ) ),
    file(finseq_1,t26_finseq_1),
    [interesting(0.00)]).

fof(t26_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ~ ( r1_xreal_0(A,k2_xcmplx_0(B,1))
              & ~ r1_xreal_0(A,B)
              & A != k2_xcmplx_0(B,1) ) ) ) ),
    file(nat_1,t26_nat_1),
    [interesting(0.00)]).

fof(t66_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v1_funct_7(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_tarski(k7_funct_7(A),k1_relat_1(B))
           => ( v1_relat_1(k7_finseq_1(A,k9_finseq_1(B)))
              & v1_funct_1(k7_finseq_1(A,k9_finseq_1(B)))
              & v1_finseq_1(k7_finseq_1(A,k9_finseq_1(B)))
              & v1_funct_7(k7_finseq_1(A,k9_finseq_1(B))) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_funct_7,t25_finseq_1,t63_finseq_1,t57_finseq_1,t61_finseq_1,t35_finseq_1,t29_finseq_3,t4_finseq_1,t61_finseq_1,d2_tarski,d2_funct_2,t26_finseq_1,d7_funct_7,t21_funct_5,d2_funct_2,d8_funct_7,t27_finseq_3,t26_nat_1,t59_finseq_1,d7_finseq_1,d2_funct_2,t27_finseq_3,d7_finseq_1,t29_nat_1,t38_nat_1,t27_finseq_3,d7_finseq_1,d5_real_1]),
    [file(funct_7,t66_funct_7),interesting(0.00)]).

fof(t19_finseq_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( v1_relat_1(k7_relat_1(B,k2_finseq_1(A)))
            & v1_funct_1(k7_relat_1(B,k2_finseq_1(A)))
            & v1_finseq_1(k7_relat_1(B,k2_finseq_1(A))) ) ) ) ),
    file(finseq_1,t19_finseq_1),
    [interesting(0.00)]).

fof(t21_finseq_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( ( r1_xreal_0(A,k3_finseq_1(B))
                  & C = k7_relat_1(B,k2_finseq_1(A)) )
               => ( k3_finseq_1(C) = A
                  & k4_finseq_1(C) = k2_finseq_1(A) ) ) ) ) ) ),
    file(finseq_1,t21_finseq_1),
    [interesting(0.00)]).

fof(t70_funct_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(B,k1_relat_1(k7_relat_1(C,A)))
       => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ),
    file(funct_1,t70_funct_1),
    [interesting(0.00)]).

fof(t3_trees_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ~ ( C = k7_relat_1(B,k2_finseq_1(A))
                  & ! [D] : 
                      ( ( v1_relat_1(D)
                        & v1_funct_1(D)
                        & v1_finseq_1(D) )
                     => B != k7_finseq_1(C,D) ) ) ) ) ) ),
    file(trees_1,t3_trees_1),
    [interesting(0.00)]).

fof(t34_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( B = k6_relat_1(A)
      <=> ( k1_relat_1(B) = A
          & ! [C] : 
              ( r2_hidden(C,A)
             => k1_funct_1(B,C) = C ) ) ) ) ),
    file(funct_1,t34_funct_1),
    [interesting(0.00)]).

fof(t60_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( k6_funct_7(k7_finseq_1(k9_finseq_1(A),B)) = k1_relat_1(A)
            & k7_funct_7(k7_finseq_1(B,k9_finseq_1(A))) = k2_relat_1(A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_funct_7,t58_finseq_1,t21_funct_5,t57_finseq_1,t35_finseq_1,d7_funct_7,t59_finseq_1,t21_funct_5]),
    [file(funct_7,t60_funct_7),interesting(0.00)]).

fof(t37_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => k1_relat_1(k5_relat_1(k6_relat_1(A),B)) = k3_xboole_0(k1_relat_1(B),A) ) ),
    file(funct_1,t37_funct_1),
    [interesting(0.00)]).

fof(t5_rewrite1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ~ ( A != k1_xboole_0
          & ! [B,C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ~ ( A = k7_finseq_1(k9_finseq_1(B),C)
                  & k3_finseq_1(A) = k1_nat_1(k3_finseq_1(C),1) ) ) ) ) ),
    file(rewrite1,t5_rewrite1),
    [interesting(0.00)]).

fof(t45_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => r1_tarski(k2_relat_1(k5_relat_1(A,B)),k2_relat_1(B)) ) ) ),
    file(relat_1,t45_relat_1),
    [interesting(0.00)]).

fof(t46_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => ( r1_tarski(k2_relat_1(A),k1_relat_1(B))
           => k1_relat_1(k5_relat_1(A,B)) = k1_relat_1(A) ) ) ) ),
    file(relat_1,t46_relat_1),
    [interesting(0.00)]).

fof(d10_funct_7,definition,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_relat_1(A)
        & v2_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( m2_funct_7(B,A)
          <=> ( k1_nat_1(k3_finseq_1(B),1) = k3_finseq_1(A)
              & ! [C] : 
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k4_finseq_1(B))
                   => r2_hidden(k1_funct_1(B,C),k1_funct_2(k1_funct_1(A,C),k1_funct_1(A,k1_nat_1(C,1)))) ) ) ) ) ) ) ),
    file(funct_7,d10_funct_7),
    [interesting(0.00)]).

fof(t71_relat_1,theorem,(
    ! [A] : 
      ( k1_relat_1(k6_relat_1(A)) = A
      & k2_relat_1(k6_relat_1(A)) = A ) ),
    file(relat_1,t71_relat_1),
    [interesting(0.00)]).

fof(d11_funct_7,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( C = k9_funct_7(A,B)
              <=> ? [D] : 
                    ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))))
                    & m2_relset_1(D,k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))))
                    & C = k8_funct_2(k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),D,B)
                    & k8_funct_2(k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),D,0) = k6_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)))
                    & ! [E] : 
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ? [F] : 
                            ( v1_relat_1(F)
                            & v1_funct_1(F)
                            & F = k8_funct_2(k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),D,E)
                            & k8_funct_2(k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),D,k1_nat_1(E,1)) = k5_relat_1(A,F) ) ) ) ) ) ) ) ),
    file(funct_7,d11_funct_7),
    [interesting(0.00)]).

fof(t7_xboole_1,theorem,(
    ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ),
    file(xboole_1,t7_xboole_1),
    [interesting(0.00)]).

fof(t44_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => r1_tarski(k1_relat_1(k5_relat_1(A,B)),k1_relat_1(A)) ) ) ),
    file(relat_1,t44_relat_1),
    [interesting(0.00)]).

fof(t22_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ~ ( A != 0
          & ! [B] : 
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => A != k1_nat_1(B,1) ) ) ) ),
    file(nat_1,t22_nat_1),
    [interesting(0.00)]).

fof(t75_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B != 0
           => ( r1_tarski(k1_relat_1(k9_funct_7(A,B)),k1_relat_1(A))
              & r1_tarski(k2_relat_1(k9_funct_7(A,B)),k2_relat_1(A)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t72_funct_7,t71_funct_7,t73_funct_7,t44_relat_1,t45_relat_1,s1_nat_1,t22_nat_1]),
    [file(funct_7,t75_funct_7),interesting(0.00)]).

fof(t74_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_tarski(k1_relat_1(k9_funct_7(A,B)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A)))
            & r1_tarski(k2_relat_1(k9_funct_7(A,B)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t70_funct_7,t71_relat_1,t71_funct_7,t73_funct_7,t44_relat_1,t45_relat_1,t1_xboole_1,s1_nat_1]),
    [file(funct_7,t74_funct_7),interesting(0.00)]).

fof(l95_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( k9_funct_7(k9_funct_7(A,B),C) = k9_funct_7(A,k2_nat_1(B,C))
               => k9_funct_7(k9_funct_7(A,B),k1_nat_1(C,1)) = k9_funct_7(A,k2_nat_1(B,k1_nat_1(C,1))) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t71_funct_7,t79_funct_7]),
    [file(funct_7,l95_funct_7),interesting(0.00)]).

fof(t80_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( B != 0
               => k9_funct_7(k9_funct_7(A,C),B) = k9_funct_7(A,k2_nat_1(C,B)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t72_funct_7,l95_funct_7,s1_nat_1,t22_nat_1]),
    [file(funct_7,t80_funct_7),interesting(0.00)]).

fof(t81_relat_1,theorem,(
    k6_relat_1(k1_xboole_0) = k1_xboole_0 ),
    file(relat_1,t81_relat_1),
    [interesting(0.00)]).

fof(t74_funct_2,theorem,(
    ! [A,B] : 
      ( m2_relset_1(B,A,A)
     => ( k7_relset_1(A,A,A,A,B,k6_partfun1(A)) = B
        & k7_relset_1(A,A,A,A,k6_partfun1(A),B) = B ) ) ),
    file(funct_2,t74_funct_2),
    [interesting(0.00)]).

fof(d1_funct_2,definition,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => ( ( ( B = k1_xboole_0
           => A = k1_xboole_0 )
         => ( v1_funct_2(C,A,B)
          <=> A = k4_relset_1(A,B,C) ) )
        & ( B = k1_xboole_0
         => ( A = k1_xboole_0
            | ( v1_funct_2(C,A,B)
            <=> C = k1_xboole_0 ) ) ) ) ) ),
    file(funct_2,d1_funct_2),
    [interesting(0.00)]).

fof(t76_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_tarski(k2_relat_1(A),k1_relat_1(A))
           => ( k1_relat_1(k9_funct_7(A,B)) = k1_relat_1(A)
              & r1_tarski(k2_relat_1(k9_funct_7(A,B)),k1_relat_1(A)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[l85_funct_7,t71_relat_1,t71_funct_7,t73_funct_7,t45_relat_1,t46_relat_1,t1_xboole_1,s1_nat_1]),
    [file(funct_7,t76_funct_7),interesting(0.00)]).

fof(t11_relset_1,theorem,(
    ! [A,B,C] : 
      ( v1_relat_1(C)
     => ( ( r1_tarski(k1_relat_1(C),A)
          & r1_tarski(k2_relat_1(C),B) )
       => m2_relset_1(C,A,B) ) ) ),
    file(relset_1,t11_relset_1),
    [interesting(0.00)]).

fof(t67_funct_2,theorem,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,A)
        & m2_relset_1(B,A,A) )
     => ( k4_relset_1(A,A,B) = A
        & r1_tarski(k2_relat_1(B),A) ) ) ),
    file(funct_2,t67_funct_2),
    [interesting(0.00)]).

fof(t81_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_tarski(k2_relat_1(A),k1_relat_1(A))
               => k9_funct_7(k9_funct_7(A,B),C) = k9_funct_7(A,k2_nat_1(B,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t76_funct_7,t12_xboole_1,t70_funct_7,t12_xboole_1,t70_funct_7,l95_funct_7,s1_nat_1]),
    [file(funct_7,t81_funct_7),interesting(0.00)]).

fof(t14_funcop_1,theorem,(
    ! [A,B] : 
      ( A != k1_xboole_0
     => k2_relat_1(k2_funcop_1(A,B)) = k1_tarski(B) ) ),
    file(funcop_1,t14_funcop_1),
    [interesting(0.00)]).

fof(t106_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D))
    <=> ( r2_hidden(A,C)
        & r2_hidden(B,D) ) ) ),
    file(zfmisc_1,t106_zfmisc_1),
    [interesting(0.00)]).

fof(t42_cqc_lang,theorem,(
    ! [A,B,C] : k9_funct_2(A,B,C) = k3_cqc_lang(k4_tarski(A,B),C) ),
    file(cqc_lang,t42_cqc_lang),
    [interesting(0.00)]).

fof(d3_pboole,definition,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( m1_pboole(B,A)
      <=> k1_relat_1(B) = A ) ) ),
    file(pboole,d3_pboole),
    [interesting(0.00)]).

fof(t72_finseq_2,theorem,(
    ! [A] : k2_finseq_2(0,A) = k1_xboole_0 ),
    file(finseq_2,t72_finseq_2),
    [interesting(0.00)]).

fof(t74_finseq_2,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : k2_finseq_2(k1_nat_1(A,1),B) = k7_finseq_1(k2_finseq_2(A,B),k9_finseq_1(B)) ) ),
    file(finseq_2,t74_finseq_2),
    [interesting(0.00)]).

fof(t9_funct_7,theorem,(
    $true ),
    file(funct_7,t9_funct_7),
    [interesting(0.00)]).