fof(t71_funct_4,theorem,(
    ! [A,B,C,D] : 
      ( A != C
     => k4_funct_4(A,C,B,D) = k2_tarski(k4_tarski(A,B),k4_tarski(C,D)) ) ),
    inference(mizar_proof,[status(thm)],[t70_funct_4,t17_zfmisc_1,t19_funcop_1,t32_funct_4,t41_enumset1]),
    [file(funct_4,t71_funct_4),interesting(1.00)]).

fof(t67_funct_4,theorem,(
    ! [A,B,C,D] : 
      ( A != B
     => k2_relat_1(k4_funct_4(A,B,C,D)) = k2_tarski(C,D) ) ),
    inference(mizar_proof,[status(thm)],[t65_funct_4,d10_xboole_0,d3_tarski,t65_funct_4,d2_tarski,t66_funct_4,d2_tarski,d5_funct_1]),
    [file(funct_4,t67_funct_4),interesting(0.94)]).

fof(t68_funct_4,theorem,(
    ! [A,B,C] : k4_funct_4(A,B,C,C) = k2_funcop_1(k2_tarski(A,B),C) ),
    inference(mizar_proof,[status(thm)],[t65_funct_4,t19_funcop_1,t12_funct_4,t19_funcop_1,t13_funcop_1,t14_funct_4,t19_funcop_1,t13_funcop_1,t13_funct_4,t9_funct_1]),
    [file(funct_4,t68_funct_4),interesting(0.91)]).

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 ) ),
    inference(mizar_proof,[status(thm)],[t88_relat_1,t79_funct_4]),
    [file(funct_4,t80_funct_4),interesting(0.89)]).

fof(t31_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_partfun1(A,B)
          <=> k2_xboole_0(A,B) = k1_funct_4(A,B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t26_funct_4,t29_funct_4,t30_funct_4,t8_xboole_1,d10_xboole_0,t130_partfun1]),
    [file(funct_4,t31_funct_4),interesting(0.88)]).

fof(t72_funct_4,theorem,(
    ! [A,B,C,D,E,F] : 
      ( k4_funct_4(A,B,C,D) = k4_funct_4(A,B,E,F)
     => ( A = B
        | ( C = E
          & D = F ) ) ) ),
    inference(mizar_proof,[status(thm)],[t66_funct_4,t66_funct_4,t66_funct_4,t66_funct_4]),
    [file(funct_4,t72_funct_4),interesting(0.88)]).

fof(t33_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))
           => r1_tarski(A,k1_funct_4(A,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t32_funct_4,t7_xboole_1]),
    [file(funct_4,t33_funct_4),interesting(0.80)]).

fof(t32_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))
           => k2_xboole_0(A,B) = k1_funct_4(A,B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t138_partfun1,t31_funct_4]),
    [file(funct_4,t32_funct_4),interesting(0.78)]).

fof(t29_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_partfun1(A,B)
          <=> r1_tarski(A,k1_funct_4(A,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_4,t16_funct_4,t7_xboole_1,t8_grfunc_1,t135_partfun1,t28_funct_4]),
    [file(funct_4,t29_funct_4),interesting(0.77)]).

fof(t79_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_tarski(A,B)
           => ( k1_funct_4(A,B) = B
              & k1_funct_4(B,A) = B ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t8_grfunc_1,t20_funct_4,t135_partfun1,t35_funct_4]),
    [file(funct_4,t79_funct_4),interesting(0.77)]).

fof(t22_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k1_funct_4(A,k1_xboole_0) = A ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_4,t12_funct_4,t9_funct_1]),
    [file(funct_4,t22_funct_4),interesting(0.76)]).

fof(t52_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => r1_tarski(k2_funct_4(k2_funct_4(A)),A) ) ),
    inference(mizar_proof,[status(thm)],[d2_funct_4,t43_funct_4,d3_tarski,d2_funct_4,d2_funct_4,t8_grfunc_1]),
    [file(funct_4,t52_funct_4),interesting(0.75)]).

fof(t6_funct_4,theorem,(
    ! [A,B,C,D] : 
      ( r1_tarski(k2_funcop_1(A,B),k2_funcop_1(C,D))
     => r1_tarski(A,C) ) ),
    inference(mizar_proof,[status(thm)],[t19_funcop_1,t25_relat_1]),
    [file(funct_4,t6_funct_4),interesting(0.75)]).

fof(t35_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_partfun1(A,B)
          <=> k1_funct_4(A,B) = k1_funct_4(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_4,d1_funct_4,d1_funct_4,t14_funct_4,d3_xboole_0,d6_partfun1,t12_funct_4,t14_funct_4,t12_funct_4,t14_funct_4,d2_xboole_0,t9_funct_1,d6_partfun1,d3_xboole_0,t14_funct_4]),
    [file(funct_4,t35_funct_4),interesting(0.74)]).

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)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_4,d1_funct_4,t4_xboole_1,d1_funct_4,t12_funct_4,t14_funct_4,t12_funct_4,t14_funct_4,t14_funct_4,t13_funct_4,t13_funct_4,t12_funct_4,t12_funct_4,d1_funct_4]),
    [file(funct_4,t15_funct_4),interesting(0.74)]).

fof(t26_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => r1_tarski(A,k1_funct_4(B,A)) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_4,t14_funct_4,t7_xboole_1,t8_grfunc_1]),
    [file(funct_4,t26_funct_4),interesting(0.72)]).

fof(t66_funct_4,theorem,(
    ! [A,B,C,D] : 
      ( A != B
     => ( k1_funct_1(k4_funct_4(A,B,C,D),A) = C
        & k1_funct_1(k4_funct_4(A,B,C,D),B) = D ) ) ),
    inference(mizar_proof,[status(thm)],[d1_tarski,t19_funcop_1,d1_tarski,t12_funct_4,t13_funcop_1,t14_funct_4,t13_funcop_1]),
    [file(funct_4,t66_funct_4),interesting(0.72)]).

fof(t21_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k1_funct_4(k1_xboole_0,A) = A ) ),
    inference(mizar_proof,[status(thm)],[t2_xboole_1,t20_funct_4]),
    [file(funct_4,t21_funct_4),interesting(0.71)]).

fof(t30_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => r1_tarski(k1_funct_4(A,B),k2_xboole_0(A,B)) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d1_relat_1,t8_funct_1,t13_funct_4,t12_funct_4,t14_funct_4,t8_funct_1,t8_funct_1,d2_xboole_0]),
    [file(funct_4,t30_funct_4),interesting(0.69)]).

fof(t54_funct_4,theorem,(
    ! [A,B,C,D] : 
      ( ( v1_funct_1(D)
        & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
     => k2_funct_4(k2_funct_4(D)) = D ) ),
    inference(mizar_proof,[status(thm)],[t53_funct_4]),
    [file(funct_4,t54_funct_4),interesting(0.69)]).

fof(t28_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) )
             => ( r1_partfun1(A,k1_funct_4(B,C))
               => r1_partfun1(A,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_partfun1,d3_xboole_0,t13_funct_4,t14_funct_4,d3_xboole_0,d6_partfun1]),
    [file(funct_4,t28_funct_4),interesting(0.65)]).

fof(l1_funct_4,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(k4_tarski(A,B),C)
     => ( r2_hidden(A,k3_tarski(k3_tarski(C)))
        & r2_hidden(B,k3_tarski(k3_tarski(C))) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_tarski,d4_tarski,d1_tarski,d2_tarski,d4_tarski]),
    [file(funct_4,l1_funct_4),interesting(0.65)]).

fof(t4_funct_4,theorem,(
    ! [A,B] : 
      ( r1_tarski(k6_partfun1(A),k6_partfun1(B))
    <=> r1_tarski(A,B) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d10_relat_1,d10_relat_1,d3_tarski,t103_zfmisc_1,d10_relat_1,d10_relat_1]),
    [file(funct_4,t4_funct_4),interesting(0.65)]).

fof(t5_funct_4,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,B)
     => r1_tarski(k2_funcop_1(A,C),k2_funcop_1(B,C)) ) ),
    inference(mizar_proof,[status(thm)],[t19_funcop_1,t13_funcop_1,t8_grfunc_1]),
    [file(funct_4,t5_funct_4),interesting(0.65)]).

fof(t23_funct_4,theorem,(
    ! [A,B] : k1_funct_4(k6_partfun1(A),k6_partfun1(B)) = k6_partfun1(k2_xboole_0(A,B)) ),
    inference(mizar_proof,[status(thm)],[d1_funct_4,t71_relat_1,t71_relat_1,t71_relat_1,t71_relat_1,t14_funct_4,t35_funct_1,t35_funct_1,d2_xboole_0,t71_relat_1,t12_funct_4,t35_funct_1,t35_funct_1,t9_funct_1]),
    [file(funct_4,t23_funct_4),interesting(0.64)]).

fof(t41_funct_4,theorem,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & m2_relset_1(C,A,B) )
     => ! [D] : 
          ( ( v1_funct_1(D)
            & m2_relset_1(D,A,B) )
         => ( v1_funct_1(k1_funct_4(C,D))
            & m2_relset_1(k1_funct_4(C,D),A,B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_4,t18_funct_4,t1_xboole_1,t11_relset_1]),
    [file(funct_4,t41_funct_4),interesting(0.64)]).

fof(t70_funct_4,theorem,(
    ! [A,B] : k2_funcop_1(k1_tarski(A),B) = k1_tarski(k4_tarski(A,B)) ),
    inference(mizar_proof,[status(thm)],[d2_funcop_1,t35_zfmisc_1]),
    [file(funct_4,t70_funct_4),interesting(0.64)]).

fof(t1_funct_4,theorem,(
    ! [A] : ~ ( ! [B] : ~ ( r2_hidden(B,A)
          & ! [C,D] : B != k4_tarski(C,D) )
      & ! [B,C] : ~ r1_tarski(A,k2_zfmisc_1(B,C)) ) ),
    inference(mizar_proof,[status(thm)],[s1_xboole_0,s1_xboole_0,d3_tarski,l1_funct_4,t106_zfmisc_1]),
    [file(funct_4,t1_funct_4),interesting(0.63)]).

fof(t75_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : k7_relat_1(k1_funct_4(A,B),C) = k1_funct_4(k7_relat_1(A,C),k7_relat_1(B,C)) ) ) ),
    inference(mizar_proof,[status(thm)],[t90_relat_1,d1_funct_4,t23_xboole_1,t90_relat_1,t90_relat_1,d2_xboole_0,t90_relat_1,d3_xboole_0,t90_relat_1,d3_xboole_0,t70_funct_1,t14_funct_4,t70_funct_1,t90_relat_1,d3_xboole_0,d2_xboole_0,t90_relat_1,d3_xboole_0,t13_funct_4,d3_xboole_0,t90_relat_1,t70_funct_1,t12_funct_4,t70_funct_1,d1_funct_4]),
    [file(funct_4,t75_funct_4),interesting(0.62)]).

fof(t53_funct_4,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r1_tarski(k1_relat_1(C),k2_zfmisc_1(A,B))
       => k2_funct_4(k2_funct_4(C)) = C ) ) ),
    inference(mizar_proof,[status(thm)],[t52_funct_4,t25_relat_1,d10_xboole_0,d3_tarski,t103_zfmisc_1,t43_funct_4,t43_funct_4,t9_grfunc_1]),
    [file(funct_4,t53_funct_4),interesting(0.61)]).

fof(t65_funct_4,theorem,(
    ! [A,B,C,D] : 
      ( k1_relat_1(k4_funct_4(A,B,C,D)) = k2_tarski(A,B)
      & r1_tarski(k2_relat_1(k4_funct_4(A,B,C,D)),k2_tarski(C,D)) ) ),
    inference(mizar_proof,[status(thm)],[t19_funcop_1,t41_enumset1,d1_funct_4,t19_funcop_1,t13_xboole_1,t18_funct_4,t41_enumset1,t1_xboole_1]),
    [file(funct_4,t65_funct_4),interesting(0.61)]).

fof(t42_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => r1_tarski(k2_relat_1(k2_funct_4(A)),k2_relat_1(A)) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d5_funct_1,d2_funct_4,d2_funct_4,d5_funct_1]),
    [file(funct_4,t42_funct_4),interesting(0.60)]).

fof(t78_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_relat_1(B) = k1_relat_1(C)
               => k1_funct_4(k1_funct_4(A,B),C) = k1_funct_4(A,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t15_funct_4,t20_funct_4]),
    [file(funct_4,t78_funct_4),interesting(0.60)]).

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) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d7_xboole_0,t90_relat_1,t55_funct_2,t57_partfun1,t75_funct_4,t22_funct_4]),
    [file(funct_4,t76_funct_4),interesting(0.59)]).

fof(t49_funct_4,theorem,(
    ! [A,B,C,D] : 
      ( ( v1_funct_1(D)
        & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
     => ( v1_funct_1(k2_funct_4(D))
        & m2_relset_1(k2_funct_4(D),k2_zfmisc_1(B,A),C) ) ) ),
    inference(mizar_proof,[status(thm)],[t46_funct_4,t48_funct_4,t11_relset_1]),
    [file(funct_4,t49_funct_4),interesting(0.56)]).

fof(t7_funct_4,theorem,(
    ! [A,B,C,D] : 
      ( r1_tarski(k2_funcop_1(A,B),k2_funcop_1(C,D))
     => ( A = k1_xboole_0
        | B = D ) ) ),
    inference(mizar_proof,[status(thm)],[t6_funct_4,t19_funcop_1,d3_tarski,t13_funcop_1,t8_grfunc_1]),
    [file(funct_4,t7_funct_4),interesting(0.55)]).

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) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t138_partfun1,t35_funct_4]),
    [file(funct_4,t36_funct_4),interesting(0.53)]).

fof(t18_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => r1_tarski(k2_relat_1(k1_funct_4(A,B)),k2_xboole_0(k2_relat_1(A),k2_relat_1(B))) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d5_funct_1,t13_funct_4,t12_funct_4,t14_funct_4,d5_funct_1,d2_xboole_0]),
    [file(funct_4,t18_funct_4),interesting(0.53)]).

fof(t13_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(k1_funct_4(B,C)))
          <=> ( r2_hidden(A,k1_relat_1(B))
              | r2_hidden(A,k1_relat_1(C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_4,d2_xboole_0]),
    [file(funct_4,t13_funct_4),interesting(0.52)]).

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 ) ) ) ),
    inference(mizar_proof,[status(thm)],[t12_xboole_1,d1_funct_4,t9_funct_1]),
    [file(funct_4,t20_funct_4),interesting(0.52)]).

fof(t55_funct_4,theorem,(
    ! [A,B,C,D] : 
      ( ( v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,B),C)
        & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
     => ( C != k1_xboole_0
       => k2_funct_4(k2_funct_4(D)) = D ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_2,t53_funct_4]),
    [file(funct_4,t55_funct_4),interesting(0.52)]).

fof(t47_funct_4,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( k1_relat_1(C) = k2_zfmisc_1(A,B)
       => k1_relat_1(k2_funct_4(C)) = k2_zfmisc_1(B,A) ) ) ),
    inference(mizar_proof,[status(thm)],[t46_funct_4,d10_xboole_0,d3_tarski,d2_zfmisc_1,d2_zfmisc_1,d2_funct_4]),
    [file(funct_4,t47_funct_4),interesting(0.51)]).

fof(l2_funct_4,theorem,(
    ! [A,B,C,D,E,F,G,H] : 
      ( k4_tarski(k4_tarski(A,B),k4_tarski(C,D)) = k4_tarski(k4_tarski(E,F),k4_tarski(G,H))
     => ( A = E
        & C = G
        & B = F
        & D = H ) ) ),
    inference(mizar_proof,[status(thm)],[t33_zfmisc_1,t33_zfmisc_1]),
    [file(funct_4,l2_funct_4),interesting(0.51)]).

fof(t19_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => r1_tarski(k2_relat_1(A),k2_relat_1(k1_funct_4(B,A))) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d5_funct_1,t13_funct_4,t14_funct_4,d5_funct_1]),
    [file(funct_4,t19_funct_4),interesting(0.48)]).

fof(t24_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => k7_relat_1(k1_funct_4(A,B),k1_relat_1(B)) = B ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_4,t7_xboole_1,t91_relat_1,t14_funct_4,t70_funct_1,t9_funct_1]),
    [file(funct_4,t24_funct_4),interesting(0.48)]).

fof(t25_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => r1_tarski(k7_relat_1(k1_funct_4(A,B),k4_xboole_0(k1_relat_1(A),k1_relat_1(B))),A) ) ) ),
    inference(mizar_proof,[status(thm)],[t87_relat_1,t36_xboole_1,t1_xboole_1,t87_relat_1,d4_xboole_0,t12_funct_4,t70_funct_1,t8_grfunc_1]),
    [file(funct_4,t25_funct_4),interesting(0.48)]).

fof(t39_funct_4,theorem,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] : 
          ( ( v1_funct_1(C)
            & v1_funct_2(C,A,A)
            & m2_relset_1(C,A,A) )
         => k1_funct_4(B,C) = C ) ) ),
    inference(mizar_proof,[status(thm)],[t38_funct_4]),
    [file(funct_4,t39_funct_4),interesting(0.48)]).

fof(t43_funct_4,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(k4_tarski(A,B),k1_relat_1(C))
      <=> r2_hidden(k4_tarski(B,A),k1_relat_1(k2_funct_4(C))) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_funct_4,d2_funct_4,t33_zfmisc_1]),
    [file(funct_4,t43_funct_4),interesting(0.48)]).

fof(t59_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => r1_tarski(k2_relat_1(k3_funct_4(A,B)),k2_zfmisc_1(k2_relat_1(A),k2_relat_1(B))) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d5_funct_1,d3_funct_4,d5_funct_1,t106_zfmisc_1,d3_funct_4]),
    [file(funct_4,t59_funct_4),interesting(0.48)]).

fof(t37_funct_4,theorem,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & m2_relset_1(C,A,B) )
     => ! [D] : 
          ( ( v1_funct_1(D)
            & m2_relset_1(D,A,B) )
         => ( v1_partfun1(D,A,B)
           => k1_funct_4(C,D) = D ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_partfun1,t20_funct_4]),
    [file(funct_4,t37_funct_4),interesting(0.47)]).

fof(t46_funct_4,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r1_tarski(k1_relat_1(C),k2_zfmisc_1(A,B))
       => r1_tarski(k1_relat_1(k2_funct_4(C)),k2_zfmisc_1(B,A)) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_funct_4,t107_zfmisc_1]),
    [file(funct_4,t46_funct_4),interesting(0.47)]).

fof(t83_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : k7_relat_1(A,k2_xboole_0(B,C)) = k1_funct_4(k7_relat_1(A,B),k7_relat_1(A,C)) ) ),
    inference(mizar_proof,[status(thm)],[t100_relat_1,t21_xboole_1,t100_relat_1,t21_xboole_1,t87_relat_1,t74_funct_4]),
    [file(funct_4,t83_funct_4),interesting(0.47)]).

fof(t2_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => k5_relat_1(B,A) = k5_relat_1(B,k7_relat_1(A,k2_relat_1(B))) ) ) ),
    inference(mizar_proof,[status(thm)],[t90_relat_1,t21_funct_1,d5_funct_1,d3_xboole_0,t21_funct_1,t21_funct_1,d3_xboole_0,t21_funct_1,t2_tarski,t21_funct_1,d5_funct_1,t22_funct_1,t72_funct_1,t23_funct_1,t9_funct_1]),
    [file(funct_4,t2_funct_4),interesting(0.46)]).

fof(t9_funct_4,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => r1_tarski(k7_relat_1(k8_relat_1(A,C),B),C) ) ),
    inference(mizar_proof,[status(thm)],[t88_relat_1,t117_relat_1,t1_xboole_1]),
    [file(funct_4,t9_funct_4),interesting(0.46)]).

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) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_xboole_0,d1_funct_4,d2_xboole_0,d1_funct_4,d4_funct_1,d4_funct_1]),
    [file(funct_4,t12_funct_4),interesting(0.45)]).

fof(t38_funct_4,theorem,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B) )
     => ! [D] : 
          ( ( v1_funct_1(D)
            & v1_funct_2(D,A,B)
            & m2_relset_1(D,A,B) )
         => ( ( B = k1_xboole_0
             => A = k1_xboole_0 )
           => k1_funct_4(C,D) = D ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_2,t20_funct_4]),
    [file(funct_4,t38_funct_4),interesting(0.45)]).

fof(t77_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(k1_relat_1(A),C)
             => k7_relat_1(k1_funct_4(A,B),C) = k7_relat_1(B,C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d7_xboole_0,t90_relat_1,t55_funct_2,t57_partfun1,t75_funct_4,t21_funct_4]),
    [file(funct_4,t77_funct_4),interesting(0.45)]).

fof(t45_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ? [B,C] : r1_tarski(k1_relat_1(k2_funct_4(A)),k2_zfmisc_1(B,C)) ) ),
    inference(mizar_proof,[status(thm)],[d2_funct_4,t1_funct_4]),
    [file(funct_4,t45_funct_4),interesting(0.44)]).

fof(t56_funct_4,theorem,(
    ! [A,B,C] : 
      ( ~ v1_xboole_0(C)
     => ! [D] : 
          ( ( v1_funct_1(D)
            & v1_funct_2(D,k2_zfmisc_1(A,B),C)
            & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
         => k2_funct_4(k2_funct_4(D)) = D ) ) ),
    inference(mizar_proof,[status(thm)],[t55_funct_4]),
    [file(funct_4,t56_funct_4),interesting(0.44)]).

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) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_xboole_0,d1_funct_4]),
    [file(funct_4,t14_funct_4),interesting(0.43)]).

fof(t50_funct_4,theorem,(
    ! [A,B,C,D] : 
      ( ( v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,B),C)
        & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
     => ( C != k1_xboole_0
       => ( v1_funct_1(k2_funct_4(D))
          & v1_funct_2(k2_funct_4(D),k2_zfmisc_1(B,A),C)
          & m2_relset_1(k2_funct_4(D),k2_zfmisc_1(B,A),C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_2,t47_funct_4,t48_funct_4,t11_relset_1,d1_funct_2,t47_funct_4,d1_funct_2]),
    [file(funct_4,t50_funct_4),interesting(0.42)]).

fof(t62_funct_4,theorem,(
    ! [A,B,C,D,E,F,G] : 
      ( ( v1_funct_1(G)
        & m2_relset_1(G,k2_zfmisc_1(A,B),C) )
     => ! [H] : 
          ( ( v1_funct_1(H)
            & m2_relset_1(H,k2_zfmisc_1(D,E),F) )
         => ( v1_funct_1(k3_funct_4(G,H))
            & m2_relset_1(k3_funct_4(G,H),k2_zfmisc_1(k2_zfmisc_1(A,D),k2_zfmisc_1(B,E)),k2_zfmisc_1(C,F)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t60_funct_4,t59_funct_4,t119_zfmisc_1,t1_xboole_1,t11_relset_1]),
    [file(funct_4,t62_funct_4),interesting(0.42)]).

fof(t16_funct_4,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ( r1_partfun1(B,C)
              & r2_hidden(A,k1_relat_1(B)) )
           => k1_funct_1(k1_funct_4(B,C),A) = k1_funct_1(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t14_funct_4,d3_xboole_0,d6_partfun1,t12_funct_4]),
    [file(funct_4,t16_funct_4),interesting(0.40)]).

fof(t40_funct_4,theorem,(
    ! [A,B] : 
      ( ~ v1_xboole_0(B)
     => ! [C] : 
          ( ( v1_funct_1(C)
            & v1_funct_2(C,A,B)
            & m2_relset_1(C,A,B) )
         => ! [D] : 
              ( ( v1_funct_1(D)
                & v1_funct_2(D,A,B)
                & m2_relset_1(D,A,B) )
             => k1_funct_4(C,D) = D ) ) ) ),
    inference(mizar_proof,[status(thm)],[t38_funct_4]),
    [file(funct_4,t40_funct_4),interesting(0.39)]).

fof(t51_funct_4,theorem,(
    ! [A,B,C] : 
      ( ~ v1_xboole_0(C)
     => ! [D] : 
          ( ( v1_funct_1(D)
            & v1_funct_2(D,k2_zfmisc_1(A,B),C)
            & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
         => ( v1_funct_1(k2_funct_4(D))
            & v1_funct_2(k2_funct_4(D),k2_zfmisc_1(B,A),C)
            & m2_relset_1(k2_funct_4(D),k2_zfmisc_1(B,A),C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t50_funct_4]),
    [file(funct_4,t51_funct_4),interesting(0.39)]).

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 ) ) ),
    inference(mizar_proof,[status(thm)],[t90_relat_1,t28_xboole_1,t23_xboole_1,t70_funct_1,d2_xboole_0,t70_funct_1,d1_funct_4]),
    [file(funct_4,t74_funct_4),interesting(0.39)]).

fof(t10_funct_4,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ! [D] : 
          ( ( v1_relat_1(D)
            & v1_funct_1(D) )
         => ( r1_tarski(C,D)
           => r1_tarski(k7_relat_1(k8_relat_1(A,C),B),k7_relat_1(k8_relat_1(A,D),B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t132_relat_1,t105_relat_1]),
    [file(funct_4,t10_funct_4),interesting(0.36)]).

fof(t48_funct_4,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r1_tarski(k1_relat_1(C),k2_zfmisc_1(A,B))
       => k2_relat_1(k2_funct_4(C)) = k2_relat_1(C) ) ) ),
    inference(mizar_proof,[status(thm)],[t42_funct_4,d10_xboole_0,d3_tarski,d5_funct_1,t103_zfmisc_1,d2_funct_4,d5_funct_1]),
    [file(funct_4,t48_funct_4),interesting(0.36)]).

fof(t81_funct_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C,D] : 
              ( ( r1_tarski(k1_relat_1(A),C)
                & r1_tarski(k1_relat_1(B),D)
                & r1_xboole_0(C,D) )
             => ( k7_relat_1(k1_funct_4(A,B),C) = A
                & k7_relat_1(k1_funct_4(A,B),D) = B ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t63_xboole_1,d7_xboole_0,t63_xboole_1,d7_xboole_0,t90_relat_1,t55_funct_2,t57_partfun1,t90_relat_1,t55_funct_2,t57_partfun1,t75_funct_4,t22_funct_4,t97_relat_1,t75_funct_4,t21_funct_4,t97_relat_1]),
    [file(funct_4,t81_funct_4),interesting(0.35)]).

fof(t82_funct_4,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(A),C)
                & r1_xboole_0(k1_relat_1(B),C) )
             => k7_relat_1(k1_funct_4(A,B),C) = A ) ) ) ),
    inference(mizar_proof,[status(thm)],[t76_funct_4,t97_relat_1]),
    [file(funct_4,t82_funct_4),interesting(0.35)]).

fof(t11_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(k1_funct_4(A,B)))
            & r1_tarski(k1_relat_1(B),k1_relat_1(k1_funct_4(A,B))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_4,t7_xboole_1]),
    [file(funct_4,t11_funct_4),interesting(0.33)]).

fof(t34_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))
           => k7_relat_1(k1_funct_4(A,B),k1_relat_1(A)) = A ) ) ) ),
    inference(mizar_proof,[status(thm)],[t83_xboole_1,t90_relat_1,d1_funct_4,t21_xboole_1,t25_funct_4,t9_grfunc_1]),
    [file(funct_4,t34_funct_4),interesting(0.33)]).

fof(t27_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) )
             => ( r1_partfun1(A,k1_funct_4(B,C))
               => r1_partfun1(k7_relat_1(A,k4_xboole_0(k1_relat_1(A),k1_relat_1(C))),B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_partfun1,d3_xboole_0,t87_relat_1,d4_xboole_0,t13_funct_4,d4_xboole_0,t12_funct_4,d3_xboole_0,d6_partfun1,t72_funct_1]),
    [file(funct_4,t27_funct_4),interesting(0.22)]).

fof(t69_funct_4,theorem,(
    ! [A,B,C,D,E] : 
      ( ( v1_relat_1(E)
        & v1_funct_1(E) )
     => ( ( k1_relat_1(E) = k2_tarski(A,B)
          & k1_funct_1(E,A) = C
          & k1_funct_1(E,B) = D )
       => E = k4_funct_4(A,B,C,D) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_tarski,t19_funcop_1,t41_enumset1,d1_tarski,t13_funcop_1,d2_xboole_0,d1_tarski,t13_funcop_1,d1_funct_4]),
    [file(funct_4,t69_funct_4),interesting(0.21)]).

fof(t8_funct_4,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r2_hidden(A,k1_relat_1(B))
       => r1_tarski(k2_funcop_1(k1_tarski(A),k1_funct_1(B,A)),B) ) ) ),
    inference(mizar_proof,[status(thm)],[t19_funcop_1,d3_tarski,d1_tarski,t13_funcop_1,d1_tarski,t8_grfunc_1]),
    [file(funct_4,t8_funct_4),interesting(0.17)]).

fof(t17_funct_4,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ( r1_xboole_0(k1_relat_1(B),k1_relat_1(C))
              & r2_hidden(A,k1_relat_1(B)) )
           => k1_funct_1(k1_funct_4(B,C),A) = k1_funct_1(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d7_xboole_0,d3_xboole_0,t12_funct_4]),
    [file(funct_4,t17_funct_4),interesting(0.12)]).

fof(t44_funct_4,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(k4_tarski(A,B),k1_relat_1(k2_funct_4(C)))
       => k1_funct_1(k2_funct_4(C),k4_tarski(A,B)) = k1_funct_1(C,k4_tarski(B,A)) ) ) ),
    inference(mizar_proof,[status(thm)],[t43_funct_4,d2_funct_4]),
    [file(funct_4,t44_funct_4),interesting(0.11)]).

fof(t63_funct_4,theorem,(
    ! [A,B,C,D,E,F,G] : 
      ( ( v1_funct_1(G)
        & v1_funct_2(G,k2_zfmisc_1(A,B),C)
        & m2_relset_1(G,k2_zfmisc_1(A,B),C) )
     => ! [H] : 
          ( ( v1_funct_1(H)
            & v1_funct_2(H,k2_zfmisc_1(D,E),F)
            & m2_relset_1(H,k2_zfmisc_1(D,E),F) )
         => ~ ( C != k1_xboole_0
              & F != k1_xboole_0
              & ~ ( v1_funct_1(k3_funct_4(G,H))
                  & v1_funct_2(k3_funct_4(G,H),k2_zfmisc_1(k2_zfmisc_1(A,D),k2_zfmisc_1(B,E)),k2_zfmisc_1(C,F))
                  & m2_relset_1(k3_funct_4(G,H),k2_zfmisc_1(k2_zfmisc_1(A,D),k2_zfmisc_1(B,E)),k2_zfmisc_1(C,F)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_2,t61_funct_4,t59_funct_4,t119_zfmisc_1,t1_xboole_1,t11_relset_1,d1_funct_2,t61_funct_4,t113_zfmisc_1,d1_funct_2]),
    [file(funct_4,t63_funct_4),interesting(0.08)]).

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)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t18_funct_4,d1_funct_4,t13_xboole_1,t1_xboole_1,t46_relat_1,d1_funct_4,t46_relat_1,d1_funct_4,t22_funct_1,d2_xboole_0,t22_funct_1,d5_funct_1,d1_funct_4,t14_funct_4,t16_funct_4,t9_funct_1]),
    [file(funct_4,t73_funct_4),interesting(0.01)]).

fof(t132_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => ! [C] : 
          ( v1_relat_1(C)
         => ( r1_tarski(B,C)
           => r1_tarski(k8_relat_1(A,B),k8_relat_1(A,C)) ) ) ) ),
    file(relat_1,t132_relat_1),
    [interesting(0.00)]).

fof(t105_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => ! [C] : 
          ( v1_relat_1(C)
         => ( r1_tarski(B,C)
           => r1_tarski(k7_relat_1(B,A),k7_relat_1(C,A)) ) ) ) ),
    file(relat_1,t105_relat_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(t7_xboole_1,theorem,(
    ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ),
    file(xboole_1,t7_xboole_1),
    [interesting(0.00)]).

fof(d7_xboole_0,definition,(
    ! [A,B] : 
      ( r1_xboole_0(A,B)
    <=> k3_xboole_0(A,B) = k1_xboole_0 ) ),
    file(xboole_0,d7_xboole_0),
    [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(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(d4_funct_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : 
          ( ( r2_hidden(B,k1_relat_1(A))
           => ( C = k1_funct_1(A,B)
            <=> r2_hidden(k4_tarski(B,C),A) ) )
          & ( ~ r2_hidden(B,k1_relat_1(A))
           => ( C = k1_funct_1(A,B)
            <=> C = k1_xboole_0 ) ) ) ) ),
    file(funct_1,d4_funct_1),
    [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(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(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(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(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(t91_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => ( r1_tarski(A,k1_relat_1(B))
       => k1_relat_1(k7_relat_1(B,A)) = A ) ) ),
    file(relat_1,t91_relat_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(d6_partfun1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_partfun1(A,B)
          <=> ! [C] : 
                ( r2_hidden(C,k3_xboole_0(k1_relat_1(A),k1_relat_1(B)))
               => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) ) ) ),
    file(partfun1,d6_partfun1),
    [interesting(0.00)]).

fof(t87_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => r1_tarski(k1_relat_1(k7_relat_1(B,A)),A) ) ),
    file(relat_1,t87_relat_1),
    [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(t72_funct_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(B,A)
       => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ),
    file(funct_1,t72_funct_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(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(t2_tarski,theorem,(
    ! [A,B] : 
      ( ! [C] : 
          ( r2_hidden(C,A)
        <=> r2_hidden(C,B) )
     => A = B ) ),
    file(tarski,t2_tarski),
    [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(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(t138_partfun1,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))
           => r1_partfun1(A,B) ) ) ) ),
    file(partfun1,t138_partfun1),
    [interesting(0.00)]).

fof(t8_grfunc_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_tarski(A,B)
          <=> ( r1_tarski(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) ) ) ) ) ) ),
    file(grfunc_1,t8_grfunc_1),
    [interesting(0.00)]).

fof(t135_partfun1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_tarski(A,B)
           => r1_partfun1(A,B) ) ) ) ),
    file(partfun1,t135_partfun1),
    [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(t8_funct_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(k4_tarski(A,B),C)
      <=> ( r2_hidden(A,k1_relat_1(C))
          & B = k1_funct_1(C,A) ) ) ) ),
    file(funct_1,t8_funct_1),
    [interesting(0.00)]).

fof(t8_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,B) )
     => r1_tarski(k2_xboole_0(A,C),B) ) ),
    file(xboole_1,t8_xboole_1),
    [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(t130_partfun1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_partfun1(A,B)
          <=> ? [C] : 
                ( v1_relat_1(C)
                & v1_funct_1(C)
                & k2_xboole_0(A,B) = C ) ) ) ) ),
    file(partfun1,t130_partfun1),
    [interesting(0.00)]).

fof(t83_xboole_1,theorem,(
    ! [A,B] : 
      ( r1_xboole_0(A,B)
    <=> k4_xboole_0(A,B) = A ) ),
    file(xboole_1,t83_xboole_1),
    [interesting(0.00)]).

fof(t21_xboole_1,theorem,(
    ! [A,B] : k3_xboole_0(A,k2_xboole_0(A,B)) = A ),
    file(xboole_1,t21_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(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(t9_grfunc_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)
              & r1_tarski(A,B) )
           => A = B ) ) ) ),
    file(grfunc_1,t9_grfunc_1),
    [interesting(0.00)]).

fof(d4_partfun1,definition,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => ( v1_partfun1(C,A,B)
      <=> k4_relset_1(A,B,C) = A ) ) ),
    file(partfun1,d4_partfun1),
    [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(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(t3_funct_4,theorem,(
    $true ),
    file(funct_4,t3_funct_4),
    [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(d2_funct_4,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( B = k2_funct_4(A)
          <=> ( ! [C] : 
                  ( r2_hidden(C,k1_relat_1(B))
                <=> ? [D,E] : 
                      ( C = k4_tarski(E,D)
                      & r2_hidden(k4_tarski(D,E),k1_relat_1(A)) ) )
              & ! [C,D] : 
                  ( r2_hidden(k4_tarski(C,D),k1_relat_1(A))
                 => k1_funct_1(B,k4_tarski(D,C)) = k1_funct_1(A,k4_tarski(C,D)) ) ) ) ) ) ),
    file(funct_4,d2_funct_4),
    [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(s1_xboole_0,theorem,(
    ? [A] : 
    ! [B] : 
      ( r2_hidden(B,A)
    <=> ( r2_hidden(B,f1_s1_xboole_0)
        & p1_s1_xboole_0(B) ) ) ),
    file(xboole_0,s1_xboole_0),
    [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(d4_tarski,definition,(
    ! [A,B] : 
      ( B = k3_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> ? [D] : 
              ( r2_hidden(C,D)
              & r2_hidden(D,A) ) ) ) ),
    file(tarski,d4_tarski),
    [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(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(t107_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D))
     => r2_hidden(k4_tarski(B,A),k2_zfmisc_1(D,C)) ) ),
    file(zfmisc_1,t107_zfmisc_1),
    [interesting(0.00)]).

fof(t103_zfmisc_1,theorem,(
    ! [A,B,C,D] : ~ ( r1_tarski(A,k2_zfmisc_1(B,C))
      & r2_hidden(D,A)
      & ! [E,F] : ~ ( r2_hidden(E,B)
          & r2_hidden(F,C)
          & D = k4_tarski(E,F) ) ) ),
    file(zfmisc_1,t103_zfmisc_1),
    [interesting(0.00)]).

fof(d10_relat_1,definition,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => ( B = k6_relat_1(A)
      <=> ! [C,D] : 
            ( r2_hidden(k4_tarski(C,D),B)
          <=> ( r2_hidden(C,A)
              & C = D ) ) ) ) ),
    file(relat_1,d10_relat_1),
    [interesting(0.00)]).

fof(d2_zfmisc_1,definition,(
    ! [A,B,C] : 
      ( C = k2_zfmisc_1(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ? [E,F] : 
              ( r2_hidden(E,A)
              & r2_hidden(F,B)
              & D = k4_tarski(E,F) ) ) ) ),
    file(zfmisc_1,d2_zfmisc_1),
    [interesting(0.00)]).

fof(t25_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => ( r1_tarski(A,B)
           => ( r1_tarski(k1_relat_1(A),k1_relat_1(B))
              & r1_tarski(k2_relat_1(A),k2_relat_1(B)) ) ) ) ) ),
    file(relat_1,t25_relat_1),
    [interesting(0.00)]).

fof(d3_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 = k3_funct_4(A,B)
              <=> ( ! [D] : 
                      ( r2_hidden(D,k1_relat_1(C))
                    <=> ? [E,F,G,H] : 
                          ( D = k4_tarski(k4_tarski(E,G),k4_tarski(F,H))
                          & r2_hidden(k4_tarski(E,F),k1_relat_1(A))
                          & r2_hidden(k4_tarski(G,H),k1_relat_1(B)) ) )
                  & ! [D,E,F,G] : 
                      ( ( r2_hidden(k4_tarski(D,E),k1_relat_1(A))
                        & r2_hidden(k4_tarski(F,G),k1_relat_1(B)) )
                     => k1_funct_1(C,k4_tarski(k4_tarski(D,F),k4_tarski(E,G))) = k4_tarski(k1_funct_1(A,k4_tarski(D,E)),k1_funct_1(B,k4_tarski(F,G))) ) ) ) ) ) ) ),
    file(funct_4,d3_funct_4),
    [interesting(0.00)]).

fof(t57_funct_4,theorem,(
    ! [A,B,C,D,E] : 
      ( ( v1_relat_1(E)
        & v1_funct_1(E) )
     => ! [F] : 
          ( ( v1_relat_1(F)
            & v1_funct_1(F) )
         => ( r2_hidden(k4_tarski(k4_tarski(A,B),k4_tarski(C,D)),k1_relat_1(k3_funct_4(E,F)))
          <=> ( r2_hidden(k4_tarski(A,C),k1_relat_1(E))
              & r2_hidden(k4_tarski(B,D),k1_relat_1(F)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_funct_4,l2_funct_4,d3_funct_4]),
    [file(funct_4,t57_funct_4),interesting(0.00)]).

fof(t58_funct_4,theorem,(
    ! [A,B,C,D,E] : 
      ( ( v1_relat_1(E)
        & v1_funct_1(E) )
     => ! [F] : 
          ( ( v1_relat_1(F)
            & v1_funct_1(F) )
         => ( r2_hidden(k4_tarski(k4_tarski(A,B),k4_tarski(C,D)),k1_relat_1(k3_funct_4(E,F)))
           => k1_funct_1(k3_funct_4(E,F),k4_tarski(k4_tarski(A,B),k4_tarski(C,D))) = k4_tarski(k1_funct_1(E,k4_tarski(A,C)),k1_funct_1(F,k4_tarski(B,D))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t57_funct_4,d3_funct_4]),
    [file(funct_4,t58_funct_4),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(t60_funct_4,theorem,(
    ! [A,B,C,D,E] : 
      ( ( v1_relat_1(E)
        & v1_funct_1(E) )
     => ! [F] : 
          ( ( v1_relat_1(F)
            & v1_funct_1(F) )
         => ( ( r1_tarski(k1_relat_1(E),k2_zfmisc_1(A,B))
              & r1_tarski(k1_relat_1(F),k2_zfmisc_1(C,D)) )
           => r1_tarski(k1_relat_1(k3_funct_4(E,F)),k2_zfmisc_1(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d3_funct_4,t106_zfmisc_1,t106_zfmisc_1,t106_zfmisc_1]),
    [file(funct_4,t60_funct_4),interesting(0.00)]).

fof(t119_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D) )
     => r1_tarski(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) ) ),
    file(zfmisc_1,t119_zfmisc_1),
    [interesting(0.00)]).

fof(t61_funct_4,theorem,(
    ! [A,B,C,D,E] : 
      ( ( v1_relat_1(E)
        & v1_funct_1(E) )
     => ! [F] : 
          ( ( v1_relat_1(F)
            & v1_funct_1(F) )
         => ( ( k1_relat_1(E) = k2_zfmisc_1(A,B)
              & k1_relat_1(F) = k2_zfmisc_1(C,D) )
           => k1_relat_1(k3_funct_4(E,F)) = k2_zfmisc_1(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t60_funct_4,d10_xboole_0,d3_tarski,d2_zfmisc_1,d2_zfmisc_1,d2_zfmisc_1,t106_zfmisc_1,d3_funct_4]),
    [file(funct_4,t61_funct_4),interesting(0.00)]).

fof(t113_zfmisc_1,theorem,(
    ! [A,B] : 
      ( k2_zfmisc_1(A,B) = k1_xboole_0
    <=> ( A = k1_xboole_0
        | B = k1_xboole_0 ) ) ),
    file(zfmisc_1,t113_zfmisc_1),
    [interesting(0.00)]).

fof(t64_funct_4,theorem,(
    ! [A,B,C,D,E] : 
      ( ~ v1_xboole_0(E)
     => ! [F] : 
          ( ~ v1_xboole_0(F)
         => ! [G] : 
              ( ( v1_funct_1(G)
                & v1_funct_2(G,k2_zfmisc_1(A,B),E)
                & m2_relset_1(G,k2_zfmisc_1(A,B),E) )
             => ! [H] : 
                  ( ( v1_funct_1(H)
                    & v1_funct_2(H,k2_zfmisc_1(C,D),F)
                    & m2_relset_1(H,k2_zfmisc_1(C,D),F) )
                 => ( v1_funct_1(k3_funct_4(G,H))
                    & v1_funct_2(k3_funct_4(G,H),k2_zfmisc_1(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)),k2_zfmisc_1(E,F))
                    & m2_relset_1(k3_funct_4(G,H),k2_zfmisc_1(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)),k2_zfmisc_1(E,F)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t63_funct_4]),
    [file(funct_4,t64_funct_4),interesting(0.00)]).

fof(t41_enumset1,theorem,(
    ! [A,B] : k2_tarski(A,B) = k2_xboole_0(k1_tarski(A),k1_tarski(B)) ),
    file(enumset1,t41_enumset1),
    [interesting(0.00)]).

fof(t13_xboole_1,theorem,(
    ! [A,B,C,D] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D) )
     => r1_tarski(k2_xboole_0(A,C),k2_xboole_0(B,D)) ) ),
    file(xboole_1,t13_xboole_1),
    [interesting(0.00)]).

fof(d2_funcop_1,definition,(
    ! [A,B] : k2_funcop_1(A,B) = k2_zfmisc_1(A,k1_tarski(B)) ),
    file(funcop_1,d2_funcop_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(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(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(t55_funct_2,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( k1_relat_1(B) = k1_xboole_0
       => ( v1_funct_1(B)
          & v1_funct_2(B,k1_xboole_0,A)
          & m2_relset_1(B,k1_xboole_0,A) ) ) ) ),
    file(funct_2,t55_funct_2),
    [interesting(0.00)]).

fof(t57_partfun1,theorem,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_xboole_0,A) )
     => B = k1_xboole_0 ) ),
    file(partfun1,t57_partfun1),
    [interesting(0.00)]).

fof(t23_xboole_1,theorem,(
    ! [A,B,C] : k3_xboole_0(A,k2_xboole_0(B,C)) = k2_xboole_0(k3_xboole_0(A,B),k3_xboole_0(A,C)) ),
    file(xboole_1,t23_xboole_1),
    [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(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(t88_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => r1_tarski(k7_relat_1(B,A),B) ) ),
    file(relat_1,t88_relat_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(t97_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => ( r1_tarski(k1_relat_1(B),A)
       => k7_relat_1(B,A) = B ) ) ),
    file(relat_1,t97_relat_1),
    [interesting(0.00)]).

fof(t100_relat_1,theorem,(
    ! [A,B,C] : 
      ( v1_relat_1(C)
     => k7_relat_1(k7_relat_1(C,A),B) = k7_relat_1(C,k3_xboole_0(A,B)) ) ),
    file(relat_1,t100_relat_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(t117_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => r1_tarski(k8_relat_1(A,B),B) ) ),
    file(relat_1,t117_relat_1),
    [interesting(0.00)]).