fof(l43_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( ( v4_group_1(A)
          & v13_monoid_0(A) )
       => v3_group_1(A) ) ) ),
    inference(mizar_proof,[status(thm)],[t17_monoid_0,t6_monoid_0,d3_group_1,d16_monoid_0,d5_monoid_0,l29_monoid_0]),
    [file(monoid_0,l43_monoid_0),interesting(1.00)]).

fof(t17_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v4_group_1(A)
       => ( v13_monoid_0(A)
        <=> ( v2_group_1(A)
            & v1_finseqop(u1_group_1(A),u1_struct_0(A)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t12_monoid_0,t12_monoid_0,l29_monoid_0,l29_monoid_0,t6_monoid_0,t12_monoid_0,l29_monoid_0,s3_funct_2,d2_finseqop,d1_finseqop,t11_binop_1,d8_binop_1,d10_monoid_0,d2_finseqop,d5_monoid_0,d16_monoid_0,l29_monoid_0,d1_finseqop,t23_setwiseo,l29_monoid_0,d1_finseqop,t23_setwiseo]),
    [file(monoid_0,t17_monoid_0),interesting(0.87)]).

fof(t39_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v1_vectsp_2(A)
        & v17_monoid_0(A)
        & l1_vectsp_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m3_monoid_0(B,A) )
         => v17_monoid_0(B) ) ) ),
    inference(mizar_proof,[status(thm)],[d25_monoid_0,t19_monoid_0,t23_monoid_0,t38_monoid_0]),
    [file(monoid_0,t39_monoid_0),interesting(0.85)]).

fof(t88_monoid_0,theorem,(
    ! [A,B] : 
      ( m1_subset_1(B,u1_struct_0(k20_monoid_0(A)))
     => k3_group_1(k20_monoid_0(A),B) = k2_funct_1(B) ) ),
    inference(mizar_proof,[status(thm)],[t86_monoid_0,t86_monoid_0,t79_monoid_0,t87_monoid_0,t88_funct_2,d6_group_1]),
    [file(monoid_0,t88_monoid_0),interesting(0.85)]).

fof(t52_monoid_0,theorem,(
    k8_monoid_0 = g1_vectsp_1(k5_numbers,k47_binop_2,0) ),
    inference(mizar_proof,[status(thm)],[d22_monoid_0,t19_monoid_0,t44_monoid_0,t49_monoid_0]),
    [file(monoid_0,t52_monoid_0),interesting(0.83)]).

fof(t84_monoid_0,theorem,(
    ! [A] : k3_binop_1(u1_struct_0(k18_monoid_0(A)),u1_group_1(k18_monoid_0(A))) = k6_partfun1(A) ),
    inference(mizar_proof,[status(thm)],[d40_monoid_0,t12_funct_2,t80_monoid_0]),
    [file(monoid_0,t84_monoid_0),interesting(0.83)]).

fof(t65_monoid_0,theorem,(
    k11_monoid_0 = g1_vectsp_1(k5_numbers,k48_binop_2,1) ),
    inference(mizar_proof,[status(thm)],[d22_monoid_0,t19_monoid_0,d31_monoid_0,d33_monoid_0,t63_monoid_0]),
    [file(monoid_0,t65_monoid_0),interesting(0.80)]).

fof(t54_monoid_0,theorem,
    ( r3_binop_1(k5_numbers,0,k47_binop_2)
    & v9_monoid_0(k47_binop_2,k5_numbers) ),
    inference(mizar_proof,[status(thm)],[t44_monoid_0,t49_monoid_0,d2_setwiseo,d8_binop_1,d20_monoid_0,t49_monoid_0]),
    [file(monoid_0,t54_monoid_0),interesting(0.79)]).

fof(t63_monoid_0,theorem,(
    k3_binop_1(u1_struct_0(k10_monoid_0),u1_group_1(k10_monoid_0)) = 1 ),
    inference(mizar_proof,[status(thm)],[d31_monoid_0,t32_monoid_0,t7_binop_2]),
    [file(monoid_0,t63_monoid_0),interesting(0.79)]).

fof(t67_monoid_0,theorem,
    ( r3_binop_1(k5_numbers,1,k48_binop_2)
    & v9_monoid_0(k48_binop_2,k5_numbers) ),
    inference(mizar_proof,[status(thm)],[d2_setwiseo,t61_monoid_0,t63_monoid_0,d8_binop_1,d20_monoid_0,t61_monoid_0]),
    [file(monoid_0,t67_monoid_0),interesting(0.78)]).

fof(t35_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ( v4_group_1(A)
           => v4_group_1(B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_monoid_0,d3_tarski,t27_monoid_0,t27_monoid_0,l29_monoid_0,t27_monoid_0,t27_monoid_0,l29_monoid_0]),
    [file(monoid_0,t35_monoid_0),interesting(0.75)]).

fof(t23_monoid_0,theorem,(
    ! [A] : 
      ( l1_group_1(A)
     => ! [B] : 
          ( m3_monoid_0(B,A)
         => m2_monoid_0(B,A) ) ) ),
    inference(mizar_proof,[status(thm)],[d24_monoid_0,d23_monoid_0]),
    [file(monoid_0,t23_monoid_0),interesting(0.74)]).

fof(t38_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ( ( r2_hidden(k3_binop_1(u1_struct_0(A),u1_group_1(A)),u1_struct_0(B))
              & v17_monoid_0(A) )
           => v17_monoid_0(B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_monoid_0,d20_monoid_0,d10_monoid_0,t32_monoid_0,d10_monoid_0,d9_monoid_0,d20_monoid_0,t25_monoid_0,d3_tarski,t27_monoid_0]),
    [file(monoid_0,t38_monoid_0),interesting(0.74)]).

fof(t44_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & m2_monoid_0(A,k5_monoid_0) )
     => k3_binop_1(u1_struct_0(A),u1_group_1(A)) = 0 ) ),
    inference(mizar_proof,[status(thm)],[t6_monoid_0,t25_monoid_0,d3_tarski,t43_monoid_0,d1_group_1,t11_binop_1,d8_binop_1]),
    [file(monoid_0,t44_monoid_0),interesting(0.74)]).

fof(t37_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ( v16_monoid_0(A)
           => v16_monoid_0(B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t15_monoid_0,t25_monoid_0,d3_tarski,t27_monoid_0,t15_monoid_0,t15_monoid_0]),
    [file(monoid_0,t37_monoid_0),interesting(0.72)]).

fof(l42_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v16_monoid_0(A)
      <=> ( v14_monoid_0(A)
          & v15_monoid_0(A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d19_monoid_0,t2_monoid_0,d17_monoid_0,d18_monoid_0,d17_monoid_0,d18_monoid_0,t2_monoid_0,d19_monoid_0]),
    [file(monoid_0,l42_monoid_0),interesting(0.71)]).

fof(t34_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ( v7_group_1(A)
           => v7_group_1(B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_monoid_0,d3_tarski,t27_monoid_0,l28_monoid_0,t27_monoid_0,l28_monoid_0]),
    [file(monoid_0,t34_monoid_0),interesting(0.71)]).

fof(t36_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ( v10_monoid_0(A)
           => v10_monoid_0(B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t9_monoid_0,t25_monoid_0,d3_tarski,t27_monoid_0,t9_monoid_0,t9_monoid_0]),
    [file(monoid_0,t36_monoid_0),interesting(0.71)]).

fof(l41_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v13_monoid_0(A)
      <=> ( v11_monoid_0(A)
          & v12_monoid_0(A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d16_monoid_0,t1_monoid_0,d14_monoid_0,d15_monoid_0,d14_monoid_0,d15_monoid_0,t1_monoid_0,d16_monoid_0]),
    [file(monoid_0,l41_monoid_0),interesting(0.70)]).

fof(t33_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v1_vectsp_2(A)
        & l1_vectsp_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m3_monoid_0(B,A) )
         => v1_vectsp_2(B) ) ) ),
    inference(mizar_proof,[status(thm)],[t23_monoid_0,d21_monoid_0,d24_monoid_0,t25_monoid_0,d3_tarski,t27_monoid_0,t11_binop_1,t27_monoid_0,t11_binop_1,t11_binop_1,d21_monoid_0]),
    [file(monoid_0,t33_monoid_0),interesting(0.70)]).

fof(l128_monoid_0,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,k1_funct_2(B,B))
     => ( v1_funct_1(A)
        & v1_funct_2(A,B,B)
        & m2_relset_1(A,B,B) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_funct_2,t4_funct_2]),
    [file(monoid_0,l128_monoid_0),interesting(0.70)]).

fof(t87_monoid_0,theorem,(
    ! [A] : 
      ( k3_binop_1(u1_struct_0(k20_monoid_0(A)),u1_group_1(k20_monoid_0(A))) = k6_partfun1(A)
      & k2_group_1(k20_monoid_0(A)) = k6_partfun1(A) ) ),
    inference(mizar_proof,[status(thm)],[t86_monoid_0,t86_monoid_0,t79_monoid_0,t74_funct_2,t11_binop_1,d8_binop_1,t33_group_1]),
    [file(monoid_0,t87_monoid_0),interesting(0.70)]).

fof(t49_monoid_0,theorem,(
    k7_monoid_0 = g1_group_1(k5_numbers,k47_binop_2) ),
    inference(mizar_proof,[status(thm)],[d27_monoid_0,d29_monoid_0]),
    [file(monoid_0,t49_monoid_0),interesting(0.68)]).

fof(t61_monoid_0,theorem,(
    k10_monoid_0 = g1_group_1(k5_numbers,k48_binop_2) ),
    inference(mizar_proof,[status(thm)],[d31_monoid_0,d33_monoid_0]),
    [file(monoid_0,t61_monoid_0),interesting(0.68)]).

fof(t81_monoid_0,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => m2_monoid_0(k15_monoid_0(A),k15_monoid_0(B)) ) ),
    inference(mizar_proof,[status(thm)],[t128_partfun1,t119_zfmisc_1,d37_monoid_0,d1_funct_2,t10_mcart_1,d37_monoid_0,d37_monoid_0,t23_mcart_1,d37_monoid_0,d37_monoid_0,t23_mcart_1,t8_grfunc_1,d23_monoid_0]),
    [file(monoid_0,t81_monoid_0),interesting(0.68)]).

fof(t86_monoid_0,theorem,(
    ! [A,B] : 
      ( m1_subset_1(A,u1_struct_0(k20_monoid_0(B)))
    <=> ( v1_funct_1(A)
        & v1_funct_2(A,B,B)
        & v3_funct_2(A,B,B)
        & m2_relset_1(A,B,B) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_monoid_0,d40_monoid_0,t12_funct_2,d3_tarski,d42_monoid_0]),
    [file(monoid_0,t86_monoid_0),interesting(0.68)]).

fof(t78_monoid_0,theorem,(
    ! [A] : k3_binop_1(u1_struct_0(k15_monoid_0(A)),u1_group_1(k15_monoid_0(A))) = k6_partfun1(A) ),
    inference(mizar_proof,[status(thm)],[d37_monoid_0,t119_partfun1,t120_partfun1,d37_monoid_0,t37_partfun1,d37_monoid_0,t36_partfun1,t11_binop_1,d8_binop_1]),
    [file(monoid_0,t78_monoid_0),interesting(0.67)]).

fof(t83_monoid_0,theorem,(
    ! [A] : u1_group_1(k18_monoid_0(A)) = k1_realset1(k17_monoid_0(A),k1_funct_2(A,A)) ),
    inference(mizar_proof,[status(thm)],[d23_monoid_0,d40_monoid_0,d1_funct_2,t64_grfunc_1]),
    [file(monoid_0,t83_monoid_0),interesting(0.67)]).

fof(t2_monoid_0,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v8_monoid_0(B,A)
          <=> ( v6_monoid_0(B,A)
              & v7_monoid_0(B,A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_monoid_0,d6_monoid_0,d7_monoid_0,d6_monoid_0,d7_monoid_0,d8_monoid_0]),
    [file(monoid_0,t2_monoid_0),interesting(0.65)]).

fof(t66_monoid_0,theorem,(
    k48_binop_2 = k1_realset1(k35_binop_2,k5_numbers) ),
    inference(mizar_proof,[status(thm)],[t26_monoid_0,t61_monoid_0]),
    [file(monoid_0,t66_monoid_0),interesting(0.65)]).

fof(t68_monoid_0,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => k12_monoid_0(A) = g1_group_1(k3_finseq_2(A),k14_monoid_0(A)) ) ),
    inference(mizar_proof,[status(thm)],[d34_monoid_0]),
    [file(monoid_0,t68_monoid_0),interesting(0.65)]).

fof(t80_monoid_0,theorem,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(B)
        & m2_monoid_0(B,k15_monoid_0(A)) )
     => ( m1_subset_1(k6_partfun1(A),u1_struct_0(B))
       => ( v2_group_1(B)
          & k3_binop_1(u1_struct_0(B),u1_group_1(B)) = k6_partfun1(A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t78_monoid_0,t32_monoid_0]),
    [file(monoid_0,t80_monoid_0),interesting(0.65)]).

fof(t85_monoid_0,theorem,(
    ! [A] : 
      ( u1_struct_0(k19_monoid_0(A)) = k1_funct_2(A,A)
      & u1_group_1(k19_monoid_0(A)) = k1_realset1(k17_monoid_0(A),k1_funct_2(A,A))
      & u1_vectsp_1(k19_monoid_0(A)) = k6_partfun1(A) ) ),
    inference(mizar_proof,[status(thm)],[d22_monoid_0,t84_monoid_0,d40_monoid_0,t19_monoid_0,t83_monoid_0]),
    [file(monoid_0,t85_monoid_0),interesting(0.65)]).

fof(t89_monoid_0,theorem,(
    ! [A] : 
      ( l1_struct_0(A)
     => ( v1_fraenkel(u1_struct_0(A))
       => v1_monoid_0(A) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_monoid_0,d2_subset_1,d1_fraenkel]),
    [file(monoid_0,t89_monoid_0),interesting(0.65)]).

fof(t30_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ! [C] : 
              ( ( ~ v3_struct_0(C)
                & m2_monoid_0(C,A) )
             => ( r1_tarski(u1_struct_0(B),u1_struct_0(C))
               => m5_monoid_0(B,A,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d23_monoid_0,d1_funct_2,t119_zfmisc_1,t82_funct_1,t64_grfunc_1,t88_relat_1,d23_monoid_0]),
    [file(monoid_0,t30_monoid_0),interesting(0.64)]).

fof(t1_monoid_0,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v5_monoid_0(B,A)
          <=> ( v3_monoid_0(B,A)
              & v4_monoid_0(B,A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_monoid_0,d3_monoid_0,d4_monoid_0,d3_monoid_0,d4_monoid_0,d5_monoid_0]),
    [file(monoid_0,t1_monoid_0),interesting(0.63)]).

fof(l46_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ( v1_vectsp_2(A)
       => v2_group_1(A) ) ) ),
    inference(mizar_proof,[status(thm)],[d21_monoid_0,d2_setwiseo,d10_monoid_0]),
    [file(monoid_0,l46_monoid_0),interesting(0.63)]).

fof(t6_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v2_group_1(A)
      <=> ? [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
            & ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ( k1_group_1(A,B,C) = C
                  & k1_group_1(A,C,B) = C ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_setwiseo,d10_monoid_0,d7_binop_1,d5_binop_1,d6_binop_1,d2_setwiseo,d10_monoid_0,d16_binop_1,d7_binop_1,d17_binop_1]),
    [file(monoid_0,t6_monoid_0),interesting(0.62)]).

fof(t47_monoid_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,u1_struct_0(k6_monoid_0))
    <=> v1_int_1(A) ) ),
    inference(mizar_proof,[status(thm)],[d4_gr_cy_1,d2_int_1]),
    [file(monoid_0,t47_monoid_0),interesting(0.62)]).

fof(t26_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => u1_group_1(B) = k1_realset1(u1_group_1(A),u1_struct_0(B)) ) ) ),
    inference(mizar_proof,[status(thm)],[d23_monoid_0,d1_funct_2,t64_grfunc_1]),
    [file(monoid_0,t26_monoid_0),interesting(0.62)]).

fof(t75_monoid_0,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ( r1_tarski(A,B)
           => m2_monoid_0(k12_monoid_0(A),k12_monoid_0(B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t83_finseq_1,t119_zfmisc_1,d34_monoid_0,d1_funct_2,t10_mcart_1,d34_monoid_0,d34_monoid_0,t23_mcart_1,d34_monoid_0,d34_monoid_0,t23_mcart_1,t8_grfunc_1,d23_monoid_0]),
    [file(monoid_0,t75_monoid_0),interesting(0.62)]).

fof(t9_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v10_monoid_0(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => k1_group_1(A,B,B) = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[d15_binop_1,d13_monoid_0,d15_binop_1,d13_monoid_0]),
    [file(monoid_0,t9_monoid_0),interesting(0.61)]).

fof(t69_monoid_0,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => k3_binop_1(u1_struct_0(k12_monoid_0(A)),u1_group_1(k12_monoid_0(A))) = k1_xboole_0 ) ),
    inference(mizar_proof,[status(thm)],[d34_monoid_0,d11_finseq_1,d34_monoid_0,t47_finseq_1,d34_monoid_0,t47_finseq_1,t11_binop_1,d8_binop_1]),
    [file(monoid_0,t69_monoid_0),interesting(0.61)]).

fof(t77_monoid_0,theorem,(
    ! [A,B] : 
      ( m1_subset_1(A,u1_struct_0(k15_monoid_0(B)))
    <=> ( v1_funct_1(A)
        & m2_relset_1(A,B,B) ) ) ),
    inference(mizar_proof,[status(thm)],[d37_monoid_0,t119_partfun1,t120_partfun1]),
    [file(monoid_0,t77_monoid_0),interesting(0.61)]).

fof(t31_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m3_monoid_0(B,A) )
         => ! [C] : 
              ( ( ~ v3_struct_0(C)
                & m3_monoid_0(C,A) )
             => ( r1_tarski(u1_struct_0(B),u1_struct_0(C))
               => m7_monoid_0(B,A,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t23_monoid_0,t30_monoid_0,d23_monoid_0,d25_monoid_0,d25_monoid_0]),
    [file(monoid_0,t31_monoid_0),interesting(0.60)]).

fof(t25_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ! [C] : 
              ( ( ~ v3_struct_0(C)
                & m3_monoid_0(C,A) )
             => ( r1_tarski(u1_struct_0(B),u1_struct_0(A))
                & r1_tarski(u1_struct_0(C),u1_struct_0(A)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d23_monoid_0,d24_monoid_0,d1_funct_2,t8_grfunc_1,d3_tarski,t106_zfmisc_1,t106_zfmisc_1,d3_tarski,t106_zfmisc_1,t106_zfmisc_1]),
    [file(monoid_0,t25_monoid_0),interesting(0.60)]).

fof(t4_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v2_group_1(A)
       => r3_binop_1(u1_struct_0(A),k3_binop_1(u1_struct_0(A),u1_group_1(A)),u1_group_1(A)) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_setwiseo,d10_monoid_0,d8_binop_1]),
    [file(monoid_0,t4_monoid_0),interesting(0.60)]).

fof(t82_monoid_0,theorem,(
    ! [A,B] : 
      ( m1_subset_1(A,u1_struct_0(k18_monoid_0(B)))
    <=> ( v1_funct_1(A)
        & v1_funct_2(A,B,B)
        & m2_relset_1(A,B,B) ) ) ),
    inference(mizar_proof,[status(thm)],[d40_monoid_0,l128_monoid_0,t12_funct_2]),
    [file(monoid_0,t82_monoid_0),interesting(0.60)]).

fof(t79_monoid_0,theorem,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(B)
        & m2_monoid_0(B,k15_monoid_0(A)) )
     => ! [C] : 
          ( m1_subset_1(C,u1_struct_0(B))
         => ! [D] : 
              ( m1_subset_1(D,u1_struct_0(B))
             => k4_monoid_0(B,C,D) = k5_relat_1(D,C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_monoid_0,d3_tarski,t27_monoid_0,d37_monoid_0]),
    [file(monoid_0,t79_monoid_0),interesting(0.60)]).

fof(t24_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & l1_vectsp_1(B) )
         => ( m2_monoid_0(A,A)
            & m3_monoid_0(B,B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d23_monoid_0,d25_monoid_0]),
    [file(monoid_0,t24_monoid_0),interesting(0.59)]).

fof(l28_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v7_group_1(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => k1_group_1(A,B,C) = k1_group_1(A,C,B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d13_binop_1,d11_monoid_0,d13_binop_1,d11_monoid_0]),
    [file(monoid_0,l28_monoid_0),interesting(0.58)]).

fof(t50_monoid_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,u1_struct_0(k8_monoid_0))
    <=> m2_subset_1(A,k1_numbers,k5_numbers) ) ),
    inference(mizar_proof,[status(thm)],[t20_monoid_0,d27_monoid_0]),
    [file(monoid_0,t50_monoid_0),interesting(0.58)]).

fof(t19_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ( v1_vectsp_2(A)
       => u1_vectsp_1(A) = k3_binop_1(u1_struct_0(A),u1_group_1(A)) ) ) ),
    inference(mizar_proof,[status(thm)],[d21_monoid_0,d8_binop_1]),
    [file(monoid_0,t19_monoid_0),interesting(0.57)]).

fof(t32_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ( ( v2_group_1(A)
              & r2_hidden(k3_binop_1(u1_struct_0(A),u1_group_1(A)),u1_struct_0(B)) )
           => ( v2_group_1(B)
              & k3_binop_1(u1_struct_0(A),u1_group_1(A)) = k3_binop_1(u1_struct_0(B),u1_group_1(B)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t4_monoid_0,t25_monoid_0,d3_tarski,t27_monoid_0,t11_binop_1,t27_monoid_0,t11_binop_1,t6_monoid_0,t11_binop_1,d8_binop_1]),
    [file(monoid_0,t32_monoid_0),interesting(0.57)]).

fof(t72_monoid_0,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,k12_monoid_0(A)) )
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(B))
                 => k4_monoid_0(B,C,D) = k7_finseq_1(C,D) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_monoid_0,d3_tarski,t27_monoid_0,d34_monoid_0]),
    [file(monoid_0,t72_monoid_0),interesting(0.56)]).

fof(l29_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v4_group_1(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] : 
                    ( m1_subset_1(D,u1_struct_0(A))
                   => k1_group_1(A,k1_group_1(A,B,C),D) = k1_group_1(A,B,k1_group_1(A,C,D)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d14_binop_1,d12_monoid_0,d14_binop_1,d12_monoid_0]),
    [file(monoid_0,l29_monoid_0),interesting(0.54)]).

fof(t53_monoid_0,theorem,
    ( k47_binop_2 = k1_realset1(k33_binop_2,k5_numbers)
    & k47_binop_2 = k1_realset1(k44_binop_2,k5_numbers) ),
    inference(mizar_proof,[status(thm)],[d27_monoid_0,d29_monoid_0,d4_gr_cy_1,t26_monoid_0]),
    [file(monoid_0,t53_monoid_0),interesting(0.53)]).

fof(t42_monoid_0,theorem,(
    k3_binop_1(u1_struct_0(k5_monoid_0),u1_group_1(k5_monoid_0)) = 0 ),
    inference(mizar_proof,[status(thm)],[t2_binop_2]),
    [file(monoid_0,t42_monoid_0),interesting(0.52)]).

fof(t57_monoid_0,theorem,(
    k3_binop_1(u1_struct_0(k9_monoid_0),u1_group_1(k9_monoid_0)) = 1 ),
    inference(mizar_proof,[status(thm)],[t7_binop_2]),
    [file(monoid_0,t57_monoid_0),interesting(0.52)]).

fof(t22_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( ( v1_vectsp_1(B)
            & v1_vectsp_2(B)
            & m1_monoid_0(B,A) )
         => ! [C] : 
              ( ( v1_vectsp_1(C)
                & v1_vectsp_2(C)
                & m1_monoid_0(C,A) )
             => B = C ) ) ) ),
    inference(mizar_proof,[status(thm)],[d22_monoid_0,t19_monoid_0]),
    [file(monoid_0,t22_monoid_0),interesting(0.49)]).

fof(t46_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & m2_monoid_0(A,k5_monoid_0) )
     => ( A = k6_monoid_0
      <=> u1_struct_0(A) = k4_numbers ) ) ),
    inference(mizar_proof,[status(thm)],[t28_monoid_0,d4_gr_cy_1]),
    [file(monoid_0,t46_monoid_0),interesting(0.49)]).

fof(t43_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & m2_monoid_0(A,k5_monoid_0) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] : 
                      ( m1_subset_1(E,k1_numbers)
                     => ( ( B = D
                          & C = E )
                       => k10_group_1(A,B,C) = k9_binop_2(D,E) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_monoid_0,d3_tarski,t27_monoid_0,t40_monoid_0]),
    [file(monoid_0,t43_monoid_0),interesting(0.49)]).

fof(t73_monoid_0,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v2_group_1(B)
            & m2_monoid_0(B,k12_monoid_0(A)) )
         => k3_binop_1(u1_struct_0(B),u1_group_1(B)) = k1_xboole_0 ) ) ),
    inference(mizar_proof,[status(thm)],[d10_monoid_0,t47_finseq_1,t23_setwiseo,t72_monoid_0,t46_finseq_1]),
    [file(monoid_0,t73_monoid_0),interesting(0.49)]).

fof(t74_monoid_0,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,k12_monoid_0(A)) )
         => ( m1_subset_1(k1_xboole_0,u1_struct_0(B))
           => ( v2_group_1(B)
              & k3_binop_1(u1_struct_0(B),u1_group_1(B)) = k1_xboole_0 ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t69_monoid_0,t32_monoid_0]),
    [file(monoid_0,t74_monoid_0),interesting(0.49)]).

fof(t71_monoid_0,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k13_monoid_0(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k13_monoid_0(A)))
             => k4_monoid_0(k13_monoid_0(A),B,C) = k7_finseq_1(B,C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d22_monoid_0,t20_monoid_0,d34_monoid_0]),
    [file(monoid_0,t71_monoid_0),interesting(0.48)]).

fof(t58_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & m2_monoid_0(A,k9_monoid_0) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] : 
                      ( m1_subset_1(E,k1_numbers)
                     => ( ( B = D
                          & C = E )
                       => k10_group_1(A,B,C) = k11_binop_2(D,E) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_monoid_0,d3_tarski,t27_monoid_0,t55_monoid_0]),
    [file(monoid_0,t58_monoid_0),interesting(0.47)]).

fof(t60_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & m2_monoid_0(A,k9_monoid_0) )
     => ( k3_binop_1(u1_struct_0(A),u1_group_1(A)) = 0
        | k3_binop_1(u1_struct_0(A),u1_group_1(A)) = 1 ) ) ),
    inference(mizar_proof,[status(thm)],[t25_monoid_0,d3_tarski,t4_monoid_0,t11_binop_1,t58_monoid_0,t5_xcmplx_1]),
    [file(monoid_0,t60_monoid_0),interesting(0.47)]).

fof(t76_monoid_0,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ( v1_setwiseo(k14_monoid_0(A),k3_finseq_2(A))
        & k3_binop_1(k3_finseq_2(A),k14_monoid_0(A)) = k1_xboole_0
        & v2_binop_1(k14_monoid_0(A),k3_finseq_2(A)) ) ) ),
    inference(mizar_proof,[status(thm)],[t68_monoid_0,d10_monoid_0,d12_monoid_0,t69_monoid_0]),
    [file(monoid_0,t76_monoid_0),interesting(0.46)]).

fof(t10_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v11_monoid_0(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] : 
                    ( m1_subset_1(D,u1_struct_0(A))
                    & k1_group_1(A,D,B) = C ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_monoid_0,d14_monoid_0,d3_monoid_0,d14_monoid_0]),
    [file(monoid_0,t10_monoid_0),interesting(0.45)]).

fof(t11_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v12_monoid_0(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] : 
                    ( m1_subset_1(D,u1_struct_0(A))
                    & k1_group_1(A,B,D) = C ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_monoid_0,d15_monoid_0,d4_monoid_0,d15_monoid_0]),
    [file(monoid_0,t11_monoid_0),interesting(0.45)]).

fof(t70_monoid_0,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ( u1_struct_0(k13_monoid_0(A)) = k3_finseq_2(A)
        & u1_group_1(k13_monoid_0(A)) = k14_monoid_0(A)
        & u1_vectsp_1(k13_monoid_0(A)) = k1_xboole_0 ) ) ),
    inference(mizar_proof,[status(thm)],[d22_monoid_0,t19_monoid_0,t69_monoid_0,d34_monoid_0]),
    [file(monoid_0,t70_monoid_0),interesting(0.44)]).

fof(t13_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v14_monoid_0(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] : 
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( k1_group_1(A,B,C) = k1_group_1(A,B,D)
                     => C = D ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_monoid_0,d17_monoid_0,d6_monoid_0,d17_monoid_0]),
    [file(monoid_0,t13_monoid_0),interesting(0.43)]).

fof(t14_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v15_monoid_0(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] : 
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( k1_group_1(A,C,B) = k1_group_1(A,D,B)
                     => C = D ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d7_monoid_0,d18_monoid_0,d7_monoid_0,d18_monoid_0]),
    [file(monoid_0,t14_monoid_0),interesting(0.43)]).

fof(s1_monoid_0,theorem,
    ( ! [A] : 
        ( m1_struct_0(A,f1_s1_monoid_0,f2_s1_monoid_0)
       => ! [B] : 
            ( m1_struct_0(B,f1_s1_monoid_0,f2_s1_monoid_0)
           => r2_hidden(k1_group_1(f1_s1_monoid_0,A,B),f2_s1_monoid_0) ) )
   => ? [A] : 
        ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & m2_monoid_0(A,f1_s1_monoid_0)
        & u1_struct_0(A) = f2_s1_monoid_0 ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_2,t91_relat_1,d3_tarski,d5_funct_1,t91_relat_1,t23_mcart_1,t70_funct_1,d1_funct_2,t11_relset_1,t88_relat_1,d23_monoid_0]),
    [file(monoid_0,s1_monoid_0),interesting(0.42)]).

fof(t18_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ( v1_vectsp_2(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ( k1_group_1(A,u1_vectsp_1(A),B) = B
              & k1_group_1(A,B,u1_vectsp_1(A)) = B ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d21_monoid_0,t11_binop_1,t11_binop_1]),
    [file(monoid_0,t18_monoid_0),interesting(0.42)]).

fof(t21_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( m1_monoid_0(B,A)
         => ( ( v2_group_1(A)
             => v2_group_1(B) )
            & ( v7_group_1(A)
             => v7_group_1(B) )
            & ( v4_group_1(A)
             => v4_group_1(B) )
            & ( v13_monoid_0(A)
             => v13_monoid_0(B) )
            & ( v17_monoid_0(A)
             => v17_monoid_0(B) )
            & ( v16_monoid_0(A)
             => v16_monoid_0(B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d22_monoid_0,d10_monoid_0,d10_monoid_0,d11_monoid_0,d11_monoid_0,d12_monoid_0,d12_monoid_0,d16_monoid_0,d16_monoid_0,d20_monoid_0,d20_monoid_0,d19_monoid_0,d19_monoid_0]),
    [file(monoid_0,t21_monoid_0),interesting(0.36)]).

fof(s3_monoid_0,theorem,
    ( ( ! [A] : 
          ( m1_struct_0(A,f1_s3_monoid_0,f2_s3_monoid_0)
         => ! [B] : 
              ( m1_struct_0(B,f1_s3_monoid_0,f2_s3_monoid_0)
             => r2_hidden(k1_group_1(f1_s3_monoid_0,A,B),f2_s3_monoid_0) ) )
      & r2_hidden(u1_vectsp_1(f1_s3_monoid_0),f2_s3_monoid_0) )
   => ? [A] : 
        ( ~ v3_struct_0(A)
        & v1_vectsp_1(A)
        & m3_monoid_0(A,f1_s3_monoid_0)
        & u1_struct_0(A) = f2_s3_monoid_0 ) ),
    inference(mizar_proof,[status(thm)],[s1_monoid_0,d23_monoid_0,d25_monoid_0]),
    [file(monoid_0,s3_monoid_0),interesting(0.29)]).

fof(t16_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v17_monoid_0(A)
      <=> ( v1_setwiseo(u1_group_1(A),u1_struct_0(A))
          & ! [B] : 
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] : 
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ( k1_group_1(A,B,C) = k3_binop_1(u1_struct_0(A),u1_group_1(A))
                   => ( B = C
                      & C = k3_binop_1(u1_struct_0(A),u1_group_1(A)) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_monoid_0,d20_monoid_0,d9_monoid_0,d20_monoid_0]),
    [file(monoid_0,t16_monoid_0),interesting(0.25)]).

fof(l3_monoid_0,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) )
           => ( v1_funct_1(k4_funct_2(A,C,B))
              & v1_funct_2(k4_funct_2(A,C,B),A,A)
              & m2_relset_1(k4_funct_2(A,C,B),A,A) ) ) )
    & ! [A,B] : 
        ( ( v1_funct_1(B)
          & v1_funct_2(B,A,A)
          & v3_funct_2(B,A,A)
          & m2_relset_1(B,A,A) )
       => ! [C] : 
            ( ( v1_funct_1(C)
              & v1_funct_2(C,A,A)
              & v3_funct_2(C,A,A)
              & m2_relset_1(C,A,A) )
           => ( v1_funct_1(k5_funct_2(A,C,B))
              & v1_funct_2(k5_funct_2(A,C,B),A,A)
              & v3_funct_2(k5_funct_2(A,C,B),A,A)
              & m2_relset_1(k5_funct_2(A,C,B),A,A) ) ) )
    & ! [A,B,C,D] : 
        ( ( v1_funct_1(D)
          & m2_relset_1(D,A,B) )
       => ! [E] : 
            ( ( v1_funct_1(E)
              & m2_relset_1(E,B,C) )
           => ( v1_funct_1(k1_partfun1(A,B,B,C,D,E))
              & m2_relset_1(k1_partfun1(A,B,B,C,D,E),A,C) ) ) )
    & ! [A,B] : 
        ( ~ v1_xboole_0(B)
       => ! [C] : 
            ( ~ v1_xboole_0(C)
           => ! [D] : 
                ( ( v1_funct_1(D)
                  & v1_funct_2(D,A,B)
                  & m2_relset_1(D,A,B) )
               => ! [E] : 
                    ( ( v1_funct_1(E)
                      & v1_funct_2(E,B,C)
                      & m2_relset_1(E,B,C) )
                   => ( v1_funct_1(k7_funct_2(A,B,C,D,E))
                      & v1_funct_2(k7_funct_2(A,B,C,D,E),A,C)
                      & m2_relset_1(k7_funct_2(A,B,C,D,E),A,C) ) ) ) ) ) ),
    file(monoid_0,l3_monoid_0),
    [interesting(0.00)]).

fof(d16_monoid_0,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v13_monoid_0(A)
      <=> v5_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) ),
    file(monoid_0,d16_monoid_0),
    [interesting(0.00)]).

fof(d5_monoid_0,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v5_monoid_0(B,A)
          <=> ! [C] : 
                ( m1_subset_1(C,A)
               => ! [D] : 
                    ( m1_subset_1(D,A)
                   => ? [E] : 
                        ( m1_subset_1(E,A)
                        & ? [F] : 
                            ( m1_subset_1(F,A)
                            & k2_binop_1(A,A,A,B,C,E) = D
                            & k2_binop_1(A,A,A,B,F,C) = D ) ) ) ) ) ) ) ),
    file(monoid_0,d5_monoid_0),
    [interesting(0.00)]).

fof(d3_monoid_0,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v3_monoid_0(B,A)
          <=> ! [C] : 
                ( m1_subset_1(C,A)
               => ! [D] : 
                    ( m1_subset_1(D,A)
                   => ? [E] : 
                        ( m1_subset_1(E,A)
                        & k2_binop_1(A,A,A,B,E,C) = D ) ) ) ) ) ) ),
    file(monoid_0,d3_monoid_0),
    [interesting(0.00)]).

fof(d4_monoid_0,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v4_monoid_0(B,A)
          <=> ! [C] : 
                ( m1_subset_1(C,A)
               => ! [D] : 
                    ( m1_subset_1(D,A)
                   => ? [E] : 
                        ( m1_subset_1(E,A)
                        & k2_binop_1(A,A,A,B,C,E) = D ) ) ) ) ) ) ),
    file(monoid_0,d4_monoid_0),
    [interesting(0.00)]).

fof(d14_monoid_0,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v11_monoid_0(A)
      <=> v3_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) ),
    file(monoid_0,d14_monoid_0),
    [interesting(0.00)]).

fof(d15_monoid_0,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v12_monoid_0(A)
      <=> v4_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) ),
    file(monoid_0,d15_monoid_0),
    [interesting(0.00)]).

fof(d19_monoid_0,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v16_monoid_0(A)
      <=> v8_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) ),
    file(monoid_0,d19_monoid_0),
    [interesting(0.00)]).

fof(d8_monoid_0,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v8_monoid_0(B,A)
          <=> ! [C] : 
                ( m1_subset_1(C,A)
               => ! [D] : 
                    ( m1_subset_1(D,A)
                   => ! [E] : 
                        ( m1_subset_1(E,A)
                       => ( ( k2_binop_1(A,A,A,B,C,D) = k2_binop_1(A,A,A,B,C,E)
                            | k2_binop_1(A,A,A,B,D,C) = k2_binop_1(A,A,A,B,E,C) )
                         => D = E ) ) ) ) ) ) ) ),
    file(monoid_0,d8_monoid_0),
    [interesting(0.00)]).

fof(d6_monoid_0,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v6_monoid_0(B,A)
          <=> ! [C] : 
                ( m1_subset_1(C,A)
               => ! [D] : 
                    ( m1_subset_1(D,A)
                   => ! [E] : 
                        ( m1_subset_1(E,A)
                       => ( k2_binop_1(A,A,A,B,C,D) = k2_binop_1(A,A,A,B,C,E)
                         => D = E ) ) ) ) ) ) ) ),
    file(monoid_0,d6_monoid_0),
    [interesting(0.00)]).

fof(d7_monoid_0,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v7_monoid_0(B,A)
          <=> ! [C] : 
                ( m1_subset_1(C,A)
               => ! [D] : 
                    ( m1_subset_1(D,A)
                   => ! [E] : 
                        ( m1_subset_1(E,A)
                       => ( k2_binop_1(A,A,A,B,D,C) = k2_binop_1(A,A,A,B,E,C)
                         => D = E ) ) ) ) ) ) ) ),
    file(monoid_0,d7_monoid_0),
    [interesting(0.00)]).

fof(d17_monoid_0,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v14_monoid_0(A)
      <=> v6_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) ),
    file(monoid_0,d17_monoid_0),
    [interesting(0.00)]).

fof(d18_monoid_0,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v15_monoid_0(A)
      <=> v7_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) ),
    file(monoid_0,d18_monoid_0),
    [interesting(0.00)]).

fof(t12_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v13_monoid_0(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] : 
                    ( m1_subset_1(D,u1_struct_0(A))
                    & ? [E] : 
                        ( m1_subset_1(E,u1_struct_0(A))
                        & k1_group_1(A,B,D) = C
                        & k1_group_1(A,E,B) = C ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_monoid_0,d16_monoid_0,d5_monoid_0,d16_monoid_0]),
    [file(monoid_0,t12_monoid_0),interesting(0.00)]).

fof(d14_binop_1,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v2_binop_1(B,A)
          <=> ! [C] : 
                ( m1_subset_1(C,A)
               => ! [D] : 
                    ( m1_subset_1(D,A)
                   => ! [E] : 
                        ( m1_subset_1(E,A)
                       => k2_binop_1(A,A,A,B,C,k2_binop_1(A,A,A,B,D,E)) = k2_binop_1(A,A,A,B,k2_binop_1(A,A,A,B,C,D),E) ) ) ) ) ) ) ),
    file(binop_1,d14_binop_1),
    [interesting(0.00)]).

fof(d12_monoid_0,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v4_group_1(A)
      <=> v2_binop_1(u1_group_1(A),u1_struct_0(A)) ) ) ),
    file(monoid_0,d12_monoid_0),
    [interesting(0.00)]).

fof(d2_setwiseo,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v1_setwiseo(B,A)
          <=> ? [C] : 
                ( m1_subset_1(C,A)
                & r3_binop_1(A,C,B) ) ) ) ) ),
    file(setwiseo,d2_setwiseo),
    [interesting(0.00)]).

fof(d10_monoid_0,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v2_group_1(A)
      <=> v1_setwiseo(u1_group_1(A),u1_struct_0(A)) ) ) ),
    file(monoid_0,d10_monoid_0),
    [interesting(0.00)]).

fof(d7_binop_1,definition,(
    ! [A,B] : 
      ( m1_subset_1(B,A)
     => ! [C] : 
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(A,A),A)
            & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
         => ( r3_binop_1(A,B,C)
          <=> ( r1_binop_1(A,B,C)
              & r2_binop_1(A,B,C) ) ) ) ) ),
    file(binop_1,d7_binop_1),
    [interesting(0.00)]).

fof(d5_binop_1,definition,(
    ! [A,B] : 
      ( m1_subset_1(B,A)
     => ! [C] : 
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(A,A),A)
            & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
         => ( r1_binop_1(A,B,C)
          <=> ! [D] : 
                ( m1_subset_1(D,A)
               => k1_binop_1(C,B,D) = D ) ) ) ) ),
    file(binop_1,d5_binop_1),
    [interesting(0.00)]).

fof(d6_binop_1,definition,(
    ! [A,B] : 
      ( m1_subset_1(B,A)
     => ! [C] : 
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(A,A),A)
            & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
         => ( r2_binop_1(A,B,C)
          <=> ! [D] : 
                ( m1_subset_1(D,A)
               => k1_binop_1(C,D,B) = D ) ) ) ) ),
    file(binop_1,d6_binop_1),
    [interesting(0.00)]).

fof(d16_binop_1,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( r1_binop_1(A,B,C)
              <=> ! [D] : 
                    ( m1_subset_1(D,A)
                   => k2_binop_1(A,A,A,C,B,D) = D ) ) ) ) ) ),
    file(binop_1,d16_binop_1),
    [interesting(0.00)]).

fof(d17_binop_1,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( r2_binop_1(A,B,C)
              <=> ! [D] : 
                    ( m1_subset_1(D,A)
                   => k2_binop_1(A,A,A,C,D,B) = D ) ) ) ) ) ),
    file(binop_1,d17_binop_1),
    [interesting(0.00)]).

fof(s3_funct_2,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s3_funct_2)
       => ? [B] : 
            ( m1_subset_1(B,f2_s3_funct_2)
            & p1_s3_funct_2(A,B) ) )
   => ? [A] : 
        ( v1_funct_1(A)
        & v1_funct_2(A,f1_s3_funct_2,f2_s3_funct_2)
        & m2_relset_1(A,f1_s3_funct_2,f2_s3_funct_2)
        & ! [B] : 
            ( m1_subset_1(B,f1_s3_funct_2)
           => p1_s3_funct_2(B,k8_funct_2(f1_s3_funct_2,f2_s3_funct_2,A,B)) ) ) ),
    file(funct_2,s3_funct_2),
    [interesting(0.00)]).

fof(d2_finseqop,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v1_finseqop(B,A)
          <=> ? [C] : 
                ( v1_funct_1(C)
                & v1_funct_2(C,A,A)
                & m2_relset_1(C,A,A)
                & r1_finseqop(A,C,B) ) ) ) ) ),
    file(finseqop,d2_finseqop),
    [interesting(0.00)]).

fof(d1_finseqop,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(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,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( r1_finseqop(A,B,C)
              <=> ! [D] : 
                    ( m1_subset_1(D,A)
                   => ( k2_binop_1(A,A,A,C,D,k8_funct_2(A,A,B,D)) = k3_binop_1(A,C)
                      & k2_binop_1(A,A,A,C,k8_funct_2(A,A,B,D),D) = k3_binop_1(A,C) ) ) ) ) ) ) ),
    file(finseqop,d1_finseqop),
    [interesting(0.00)]).

fof(t11_binop_1,theorem,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
     => ! [C] : 
          ( m1_subset_1(C,A)
         => ( r3_binop_1(A,C,B)
          <=> ! [D] : 
                ( m1_subset_1(D,A)
               => ( k1_binop_1(B,C,D) = D
                  & k1_binop_1(B,D,C) = D ) ) ) ) ) ),
    file(binop_1,t11_binop_1),
    [interesting(0.00)]).

fof(d8_binop_1,definition,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
     => ( ? [C] : 
            ( m1_subset_1(C,A)
            & r3_binop_1(A,C,B) )
       => ! [C] : 
            ( m1_subset_1(C,A)
           => ( C = k3_binop_1(A,B)
            <=> r3_binop_1(A,C,B) ) ) ) ) ),
    file(binop_1,d8_binop_1),
    [interesting(0.00)]).

fof(t23_setwiseo,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v1_setwiseo(B,A)
           => ! [C] : 
                ( m1_subset_1(C,A)
               => ( k2_binop_1(A,A,A,B,k3_binop_1(A,B),C) = C
                  & k2_binop_1(A,A,A,B,C,k3_binop_1(A,B)) = C ) ) ) ) ) ),
    file(setwiseo,t23_setwiseo),
    [interesting(0.00)]).

fof(d3_group_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v3_group_1(A)
      <=> ? [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
            & ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ( k1_group_1(A,C,B) = C
                  & k1_group_1(A,B,C) = C
                  & ? [D] : 
                      ( m1_subset_1(D,u1_struct_0(A))
                      & k1_group_1(A,C,D) = B
                      & k1_group_1(A,D,C) = B ) ) ) ) ) ) ),
    file(group_1,d3_group_1),
    [interesting(0.00)]).

fof(d21_monoid_0,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ( v1_vectsp_2(A)
      <=> r3_binop_1(u1_struct_0(A),u1_vectsp_1(A),u1_group_1(A)) ) ) ),
    file(monoid_0,d21_monoid_0),
    [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(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(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(t23_mcart_1,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(A,k2_zfmisc_1(B,C))
     => A = k4_tarski(k1_mcart_1(A),k2_mcart_1(A)) ) ),
    file(mcart_1,t23_mcart_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(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(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(d23_monoid_0,definition,(
    ! [A] : 
      ( l1_group_1(A)
     => ! [B] : 
          ( l1_group_1(B)
         => ( m2_monoid_0(B,A)
          <=> r1_tarski(u1_group_1(B),u1_group_1(A)) ) ) ) ),
    file(monoid_0,d23_monoid_0),
    [interesting(0.00)]).

fof(d25_monoid_0,definition,(
    ! [A] : 
      ( l1_vectsp_1(A)
     => ! [B] : 
          ( l1_vectsp_1(B)
         => ( m3_monoid_0(B,A)
          <=> ( r1_tarski(u1_group_1(B),u1_group_1(A))
              & u1_vectsp_1(B) = u1_vectsp_1(A) ) ) ) ) ),
    file(monoid_0,d25_monoid_0),
    [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(s2_monoid_0,theorem,
    ( ( ! [A] : 
          ( m1_subset_1(A,u1_struct_0(f1_s2_monoid_0))
         => ! [B] : 
              ( m1_subset_1(B,u1_struct_0(f1_s2_monoid_0))
             => ( ( p1_s2_monoid_0(A)
                  & p1_s2_monoid_0(B) )
               => p1_s2_monoid_0(k1_group_1(f1_s2_monoid_0,A,B)) ) ) )
      & ? [A] : 
          ( m1_subset_1(A,u1_struct_0(f1_s2_monoid_0))
          & p1_s2_monoid_0(A) ) )
   => ? [A] : 
        ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & m2_monoid_0(A,f1_s2_monoid_0)
        & ! [B] : 
            ( m1_subset_1(B,u1_struct_0(f1_s2_monoid_0))
           => ( r2_hidden(B,u1_struct_0(A))
            <=> p1_s2_monoid_0(B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s1_xboole_0,d3_tarski,s1_monoid_0]),
    [file(monoid_0,s2_monoid_0),interesting(0.00)]).

fof(s4_monoid_0,theorem,
    ( ( ! [A] : 
          ( m1_subset_1(A,u1_struct_0(f1_s4_monoid_0))
         => ! [B] : 
              ( m1_subset_1(B,u1_struct_0(f1_s4_monoid_0))
             => ( ( p1_s4_monoid_0(A)
                  & p1_s4_monoid_0(B) )
               => p1_s4_monoid_0(k1_group_1(f1_s4_monoid_0,A,B)) ) ) )
      & p1_s4_monoid_0(u1_vectsp_1(f1_s4_monoid_0)) )
   => ? [A] : 
        ( ~ v3_struct_0(A)
        & v1_vectsp_1(A)
        & m3_monoid_0(A,f1_s4_monoid_0)
        & ! [B] : 
            ( m1_subset_1(B,u1_struct_0(f1_s4_monoid_0))
           => ( r2_hidden(B,u1_struct_0(A))
            <=> p1_s4_monoid_0(B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s2_monoid_0,d23_monoid_0,d25_monoid_0]),
    [file(monoid_0,s4_monoid_0),interesting(0.00)]).

fof(d9_monoid_0,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v9_monoid_0(B,A)
          <=> ( v1_setwiseo(B,A)
              & ! [C] : 
                  ( m1_subset_1(C,A)
                 => ! [D] : 
                      ( m1_subset_1(D,A)
                     => ( k2_binop_1(A,A,A,B,C,D) = k3_binop_1(A,B)
                       => ( C = D
                          & D = k3_binop_1(A,B) ) ) ) ) ) ) ) ) ),
    file(monoid_0,d9_monoid_0),
    [interesting(0.00)]).

fof(d20_monoid_0,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v17_monoid_0(A)
      <=> v9_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) ),
    file(monoid_0,d20_monoid_0),
    [interesting(0.00)]).

fof(d22_monoid_0,definition,(
    ! [A] : 
      ( l1_group_1(A)
     => ! [B] : 
          ( l1_vectsp_1(B)
         => ( m1_monoid_0(B,A)
          <=> g1_group_1(u1_struct_0(B),u1_group_1(B)) = g1_group_1(u1_struct_0(A),u1_group_1(A)) ) ) ) ),
    file(monoid_0,d22_monoid_0),
    [interesting(0.00)]).

fof(d11_monoid_0,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v7_group_1(A)
      <=> v1_binop_1(u1_group_1(A),u1_struct_0(A)) ) ) ),
    file(monoid_0,d11_monoid_0),
    [interesting(0.00)]).

fof(d24_monoid_0,definition,(
    ! [A] : 
      ( l1_group_1(A)
     => ! [B] : 
          ( l1_vectsp_1(B)
         => ( m3_monoid_0(B,A)
          <=> ( r1_tarski(u1_group_1(B),u1_group_1(A))
              & ! [C] : 
                  ( l1_vectsp_1(C)
                 => ( A = C
                   => u1_vectsp_1(B) = u1_vectsp_1(C) ) ) ) ) ) ) ),
    file(monoid_0,d24_monoid_0),
    [interesting(0.00)]).

fof(t64_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)
           => k7_relat_1(B,k1_relat_1(A)) = A ) ) ) ),
    file(grfunc_1,t64_grfunc_1),
    [interesting(0.00)]).

fof(t28_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ! [C] : 
              ( ( ~ v3_struct_0(C)
                & m2_monoid_0(C,A) )
             => ( u1_struct_0(B) = u1_struct_0(C)
               => g1_group_1(u1_struct_0(B),u1_group_1(B)) = g1_group_1(u1_struct_0(C),u1_group_1(C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d23_monoid_0,d1_funct_2,t64_grfunc_1]),
    [file(monoid_0,t28_monoid_0),interesting(0.00)]).

fof(t29_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m3_monoid_0(B,A) )
         => ! [C] : 
              ( ( ~ v3_struct_0(C)
                & m3_monoid_0(C,A) )
             => ( u1_struct_0(B) = u1_struct_0(C)
               => g1_vectsp_1(u1_struct_0(B),u1_group_1(B),u1_vectsp_1(B)) = g1_vectsp_1(u1_struct_0(C),u1_group_1(C),u1_vectsp_1(C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t23_monoid_0,d25_monoid_0,t28_monoid_0]),
    [file(monoid_0,t29_monoid_0),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(t82_funct_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r1_tarski(A,B)
       => ( k7_relat_1(k7_relat_1(C,A),B) = k7_relat_1(C,A)
          & k7_relat_1(k7_relat_1(C,B),A) = k7_relat_1(C,A) ) ) ) ),
    file(funct_1,t82_funct_1),
    [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(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(t27_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] : 
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ! [F] : 
                          ( m1_subset_1(F,u1_struct_0(A))
                         => ( ( C = E
                              & D = F )
                           => k1_group_1(B,C,D) = k1_group_1(A,E,F) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d23_monoid_0,d1_funct_2,t8_grfunc_1]),
    [file(monoid_0,t27_monoid_0),interesting(0.00)]).

fof(d13_binop_1,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v1_binop_1(B,A)
          <=> ! [C] : 
                ( m1_subset_1(C,A)
               => ! [D] : 
                    ( m1_subset_1(D,A)
                   => k2_binop_1(A,A,A,B,C,D) = k2_binop_1(A,A,A,B,D,C) ) ) ) ) ) ),
    file(binop_1,d13_binop_1),
    [interesting(0.00)]).

fof(d15_binop_1,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v3_binop_1(B,A)
          <=> ! [C] : 
                ( m1_subset_1(C,A)
               => k2_binop_1(A,A,A,B,C,C) = C ) ) ) ) ),
    file(binop_1,d15_binop_1),
    [interesting(0.00)]).

fof(d13_monoid_0,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v10_monoid_0(A)
      <=> v3_binop_1(u1_group_1(A),u1_struct_0(A)) ) ) ),
    file(monoid_0,d13_monoid_0),
    [interesting(0.00)]).

fof(t15_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v16_monoid_0(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] : 
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( ( k1_group_1(A,B,C) = k1_group_1(A,B,D)
                        | k1_group_1(A,C,B) = k1_group_1(A,D,B) )
                     => C = D ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_monoid_0,d19_monoid_0,d8_monoid_0,d19_monoid_0]),
    [file(monoid_0,t15_monoid_0),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(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(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(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(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(t3_monoid_0,theorem,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(k1_tarski(A),k1_tarski(A)),k1_tarski(A))
        & m2_relset_1(B,k2_zfmisc_1(k1_tarski(A),k1_tarski(A)),k1_tarski(A)) )
     => ( B = k2_funcop_1(k1_tarski(k4_tarski(A,A)),A)
        & v1_setwiseo(B,k1_tarski(A))
        & v1_binop_1(B,k1_tarski(A))
        & v2_binop_1(B,k1_tarski(A))
        & v3_binop_1(B,k1_tarski(A))
        & v5_monoid_0(B,k1_tarski(A))
        & v8_monoid_0(B,k1_tarski(A))
        & v9_monoid_0(B,k1_tarski(A)) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_tarski,t19_funcop_1,d1_funct_2,t35_zfmisc_1,t35_zfmisc_1,d1_tarski,t13_funcop_1,t9_funct_1,d1_tarski,t11_binop_1,d2_setwiseo,d13_binop_1,d14_binop_1,d15_binop_1,d5_monoid_0,d8_monoid_0,d2_setwiseo,d9_monoid_0]),
    [file(monoid_0,t3_monoid_0),interesting(0.00)]).

fof(t41_monoid_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,u1_struct_0(k5_monoid_0))
    <=> m1_subset_1(A,k1_numbers) ) ),
    file(monoid_0,t41_monoid_0),
    [interesting(0.00)]).

fof(t2_binop_2,theorem,(
    k3_binop_1(k1_numbers,k33_binop_2) = 0 ),
    file(binop_2,t2_binop_2),
    [interesting(0.00)]).

fof(t45_monoid_0,theorem,(
    $true ),
    file(monoid_0,t45_monoid_0),
    [interesting(0.00)]).

fof(d4_gr_cy_1,definition,(
    k3_gr_cy_1 = g1_group_1(k4_numbers,k44_binop_2) ),
    file(gr_cy_1,d4_gr_cy_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(t48_monoid_0,theorem,(
    $true ),
    file(monoid_0,t48_monoid_0),
    [interesting(0.00)]).

fof(t20_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( m1_monoid_0(B,A)
         => ( u1_struct_0(B) = u1_struct_0(A)
            & u1_group_1(B) = u1_group_1(A)
            & ! [C] : 
                ( m1_subset_1(C,u1_struct_0(B))
               => ! [D] : 
                    ( m1_subset_1(D,u1_struct_0(B))
                   => ! [E] : 
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ! [F] : 
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ( ( C = E
                                & D = F )
                             => k1_group_1(B,C,D) = k1_group_1(A,E,F) ) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d22_monoid_0]),
    [file(monoid_0,t20_monoid_0),interesting(0.00)]).

fof(d27_monoid_0,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v2_group_1(A)
        & v17_monoid_0(A)
        & m5_monoid_0(A,k5_monoid_0,k6_monoid_0) )
     => ( A = k7_monoid_0
      <=> u1_struct_0(A) = k5_numbers ) ) ),
    file(monoid_0,d27_monoid_0),
    [interesting(0.00)]).

fof(d9_binop_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) )
     => ( A = k33_binop_2
      <=> ! [B] : 
            ( m1_subset_1(B,k1_numbers)
           => ! [C] : 
                ( m1_subset_1(C,k1_numbers)
               => k2_binop_1(k1_numbers,k1_numbers,k1_numbers,A,B,C) = k9_binop_2(B,C) ) ) ) ) ),
    file(binop_2,d9_binop_2),
    [interesting(0.00)]).

fof(t40_monoid_0,theorem,
    ( u1_struct_0(k5_monoid_0) = k1_numbers
    & u1_group_1(k5_monoid_0) = k33_binop_2
    & ! [A] : 
        ( m1_subset_1(A,u1_struct_0(k5_monoid_0))
       => ! [B] : 
            ( m1_subset_1(B,u1_struct_0(k5_monoid_0))
           => ! [C] : 
                ( m1_subset_1(C,k1_numbers)
               => ! [D] : 
                    ( m1_subset_1(D,k1_numbers)
                   => ( ( A = C
                        & B = D )
                     => k10_group_1(k5_monoid_0,A,B) = k9_binop_2(C,D) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_binop_2]),
    [file(monoid_0,t40_monoid_0),interesting(0.00)]).

fof(t51_monoid_0,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k8_monoid_0))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(k8_monoid_0))
                 => ( ( A = C
                      & B = D )
                   => k10_group_1(k8_monoid_0,C,D) = k23_binop_2(A,B) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t20_monoid_0,d23_monoid_0,t43_monoid_0]),
    [file(monoid_0,t51_monoid_0),interesting(0.00)]).

fof(d1_group_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => k1_group_1(A,B,C) = k2_binop_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_group_1(A),B,C) ) ) ) ),
    file(group_1,d1_group_1),
    [interesting(0.00)]).

fof(d29_monoid_0,definition,(
    k47_binop_2 = u1_group_1(k7_monoid_0) ),
    file(monoid_0,d29_monoid_0),
    [interesting(0.00)]).

fof(t56_monoid_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,u1_struct_0(k9_monoid_0))
    <=> m1_subset_1(A,k1_numbers) ) ),
    file(monoid_0,t56_monoid_0),
    [interesting(0.00)]).

fof(t7_binop_2,theorem,(
    k3_binop_1(k1_numbers,k35_binop_2) = 1 ),
    file(binop_2,t7_binop_2),
    [interesting(0.00)]).

fof(t59_monoid_0,theorem,(
    $true ),
    file(monoid_0,t59_monoid_0),
    [interesting(0.00)]).

fof(t5_monoid_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v2_group_1(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ( k1_group_1(A,k3_binop_1(u1_struct_0(A),u1_group_1(A)),B) = B
              & k1_group_1(A,B,k3_binop_1(u1_struct_0(A),u1_group_1(A))) = B ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_setwiseo,d10_monoid_0,d8_binop_1,t11_binop_1,d2_setwiseo,d10_monoid_0,d16_binop_1,d7_binop_1,d17_binop_1]),
    [file(monoid_0,t5_monoid_0),interesting(0.00)]).

fof(d11_binop_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) )
     => ( A = k35_binop_2
      <=> ! [B] : 
            ( m1_subset_1(B,k1_numbers)
           => ! [C] : 
                ( m1_subset_1(C,k1_numbers)
               => k2_binop_1(k1_numbers,k1_numbers,k1_numbers,A,B,C) = k11_binop_2(B,C) ) ) ) ) ),
    file(binop_2,d11_binop_2),
    [interesting(0.00)]).

fof(t55_monoid_0,theorem,
    ( u1_struct_0(k9_monoid_0) = k1_numbers
    & u1_group_1(k9_monoid_0) = k35_binop_2
    & ! [A] : 
        ( m1_subset_1(A,u1_struct_0(k9_monoid_0))
       => ! [B] : 
            ( m1_subset_1(B,u1_struct_0(k9_monoid_0))
           => ! [C] : 
                ( m1_subset_1(C,k1_numbers)
               => ! [D] : 
                    ( m1_subset_1(D,k1_numbers)
                   => ( ( A = C
                        & B = D )
                     => k10_group_1(k9_monoid_0,A,B) = k11_binop_2(C,D) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d11_binop_2]),
    [file(monoid_0,t55_monoid_0),interesting(0.00)]).

fof(t5_xcmplx_1,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => ! [B] : 
          ( v1_xcmplx_0(B)
         => ! [C] : 
              ( v1_xcmplx_0(C)
             => ( k3_xcmplx_0(B,A) = k3_xcmplx_0(C,A)
               => ( A = 0
                  | B = C ) ) ) ) ) ),
    file(xcmplx_1,t5_xcmplx_1),
    [interesting(0.00)]).

fof(t62_monoid_0,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k10_monoid_0))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(k10_monoid_0))
                 => ( ( A = C
                      & B = D )
                   => k10_group_1(k10_monoid_0,C,D) = k24_binop_2(A,B) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t58_monoid_0]),
    [file(monoid_0,t62_monoid_0),interesting(0.00)]).

fof(t64_monoid_0,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k11_monoid_0))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(k11_monoid_0))
                 => ( ( A = C
                      & B = D )
                   => k10_group_1(k11_monoid_0,C,D) = k24_binop_2(A,B) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d22_monoid_0,t20_monoid_0,t58_monoid_0]),
    [file(monoid_0,t64_monoid_0),interesting(0.00)]).

fof(d31_monoid_0,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v2_group_1(A)
        & v17_monoid_0(A)
        & m2_monoid_0(A,k9_monoid_0) )
     => ( A = k10_monoid_0
      <=> u1_struct_0(A) = k5_numbers ) ) ),
    file(monoid_0,d31_monoid_0),
    [interesting(0.00)]).

fof(d33_monoid_0,definition,(
    k48_binop_2 = u1_group_1(k10_monoid_0) ),
    file(monoid_0,d33_monoid_0),
    [interesting(0.00)]).

fof(d34_monoid_0,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v1_group_1(B)
            & v2_group_1(B)
            & v4_group_1(B)
            & v2_monoid_0(B)
            & v16_monoid_0(B)
            & v17_monoid_0(B)
            & l1_group_1(B) )
         => ( B = k12_monoid_0(A)
          <=> ( u1_struct_0(B) = k3_finseq_2(A)
              & ! [C] : 
                  ( m1_subset_1(C,u1_struct_0(B))
                 => ! [D] : 
                      ( m1_subset_1(D,u1_struct_0(B))
                     => k4_monoid_0(B,C,D) = k7_finseq_1(C,D) ) ) ) ) ) ) ),
    file(monoid_0,d34_monoid_0),
    [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(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(t46_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(A,B) = k7_finseq_1(C,B)
                  | k7_finseq_1(B,A) = k7_finseq_1(B,C) )
               => A = C ) ) ) ) ),
    file(finseq_1,t46_finseq_1),
    [interesting(0.00)]).

fof(t83_finseq_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => r1_tarski(k13_finseq_1(A),k13_finseq_1(B)) ) ),
    file(finseq_1,t83_finseq_1),
    [interesting(0.00)]).

fof(t10_mcart_1,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(A,k2_zfmisc_1(B,C))
     => ( r2_hidden(k1_mcart_1(A),B)
        & r2_hidden(k2_mcart_1(A),C) ) ) ),
    file(mcart_1,t10_mcart_1),
    [interesting(0.00)]).

fof(d37_monoid_0,definition,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(B)
        & v1_group_1(B)
        & v2_group_1(B)
        & v4_group_1(B)
        & v1_monoid_0(B)
        & l1_group_1(B) )
     => ( B = k15_monoid_0(A)
      <=> ( u1_struct_0(B) = k4_partfun1(A,A)
          & ! [C] : 
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(B))
                 => k4_monoid_0(B,C,D) = k5_relat_1(D,C) ) ) ) ) ) ),
    file(monoid_0,d37_monoid_0),
    [interesting(0.00)]).

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(t120_partfun1,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(C,k4_partfun1(A,B))
     => ( v1_funct_1(C)
        & m2_relset_1(C,A,B) ) ) ),
    file(partfun1,t120_partfun1),
    [interesting(0.00)]).

fof(t7_monoid_0,theorem,(
    $true ),
    file(monoid_0,t7_monoid_0),
    [interesting(0.00)]).

fof(t128_partfun1,theorem,(
    ! [A,B,C,D] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D) )
     => r1_tarski(k4_partfun1(A,C),k4_partfun1(B,D)) ) ),
    file(partfun1,t128_partfun1),
    [interesting(0.00)]).

fof(d40_monoid_0,definition,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(B)
        & v1_group_1(B)
        & v2_group_1(B)
        & m2_monoid_0(B,k15_monoid_0(A)) )
     => ( B = k18_monoid_0(A)
      <=> u1_struct_0(B) = k1_funct_2(A,A) ) ) ),
    file(monoid_0,d40_monoid_0),
    [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(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(t12_funct_2,theorem,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,A)
        & m2_relset_1(B,A,A) )
     => r2_hidden(B,k1_funct_2(A,A)) ) ),
    file(funct_2,t12_funct_2),
    [interesting(0.00)]).

fof(t37_partfun1,theorem,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => k5_relat_1(C,k6_relat_1(B)) = C ) ),
    file(partfun1,t37_partfun1),
    [interesting(0.00)]).

fof(t36_partfun1,theorem,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => k5_relat_1(k6_relat_1(A),C) = C ) ),
    file(partfun1,t36_partfun1),
    [interesting(0.00)]).

fof(d42_monoid_0,definition,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(B)
        & v1_group_1(B)
        & v2_group_1(B)
        & v13_monoid_0(B)
        & m5_monoid_0(B,k15_monoid_0(A),k18_monoid_0(A)) )
     => ( B = k20_monoid_0(A)
      <=> ! [C] : 
            ( m1_subset_1(C,u1_struct_0(k18_monoid_0(A)))
           => ( r2_hidden(C,u1_struct_0(B))
            <=> ( v1_funct_1(C)
                & v1_funct_2(C,A,A)
                & v3_funct_2(C,A,A)
                & m2_relset_1(C,A,A) ) ) ) ) ) ),
    file(monoid_0,d42_monoid_0),
    [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(t33_group_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => k3_binop_1(u1_struct_0(A),u1_group_1(A)) = k2_group_1(A) ) ),
    file(group_1,t33_group_1),
    [interesting(0.00)]).

fof(t88_funct_2,theorem,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,A)
        & v3_funct_2(B,A,A)
        & m2_relset_1(B,A,A) )
     => ( k5_funct_2(A,B,k6_funct_2(A,B)) = k6_partfun1(A)
        & k5_funct_2(A,k6_funct_2(A,B),B) = k6_partfun1(A) ) ) ),
    file(funct_2,t88_funct_2),
    [interesting(0.00)]).

fof(d6_group_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ( C = k3_group_1(A,B)
              <=> ( k1_group_1(A,B,C) = k2_group_1(A)
                  & k1_group_1(A,C,B) = k2_group_1(A) ) ) ) ) ) ),
    file(group_1,d6_group_1),
    [interesting(0.00)]).

fof(d1_monoid_0,definition,(
    ! [A] : 
      ( l1_struct_0(A)
     => ( v1_monoid_0(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ( v1_relat_1(B)
              & v1_funct_1(B) ) ) ) ) ),
    file(monoid_0,d1_monoid_0),
    [interesting(0.00)]).

fof(d2_subset_1,definition,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
       => ( m1_subset_1(B,A)
        <=> r2_hidden(B,A) ) )
      & ( v1_xboole_0(A)
       => ( m1_subset_1(B,A)
        <=> v1_xboole_0(B) ) ) ) ),
    file(subset_1,d2_subset_1),
    [interesting(0.00)]).

fof(d1_fraenkel,definition,(
    ! [A] : 
      ( v1_fraenkel(A)
    <=> ! [B] : 
          ( r2_hidden(B,A)
         => ( v1_relat_1(B)
            & v1_funct_1(B) ) ) ) ),
    file(fraenkel,d1_fraenkel),
    [interesting(0.00)]).

fof(t8_monoid_0,theorem,(
    $true ),
    file(monoid_0,t8_monoid_0),
    [interesting(0.00)]).