fof(t26_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v1_algstr_1(A)
        & v1_binom(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_binom(A,k4_rlvect_1(A,B,C),D) = k9_rlvect_1(A,k8_binom(A,B,C,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t8_binom,t3_binom,t23_binom,t9_binom,d11_vectsp_1,t5_binom,t5_binom,d3_finseq_1,d3_polynom1,d3_finseq_1,d3_finseq_1,d3_polynom1,d3_finseq_1,t35_finseq_1,t57_finseq_1,t8_finseq_1,d10_binom,t110_finseq_2,t35_finseq_1,t57_finseq_1,t8_finseq_1,d10_binom,t110_finseq_2,t109_finseq_2,t109_finseq_2,d3_finseq_1,d3_finseq_1,t58_rlvect_1,t3_binom,d7_rlvect_1,t58_rlvect_1,t3_binom,d5_algstr_1,t7_binom,d3_finseq_1,d4_binom,d3_finseq_1,t8_finseq_1,d10_binom,d3_finseq_1,t3_finseq_1,t3_finseq_1,t18_int_1,d3_finseq_1,d3_finseq_1,d3_finseq_1,t37_nat_1,d10_binom,d1_tarski,t18_nat_1,t8_xreal_1,t3_finseq_1,d10_binom,d3_finseq_1,d3_finseq_1,d4_finseq_4,d4_finseq_4,d7_finseq_1,d4_finseq_4,d3_polynom1,t24_binom,t9_binom,d4_finseq_4,t58_finseq_1,d4_finseq_4,d4_binom,d7_rlvect_1,t24_binom,d1_tarski,d3_finseq_1,t3_finseq_1,d1_tarski,d10_binom,t8_finseq_1,t6_finseq_1,d10_binom,d3_finseq_1,d3_finseq_1,d4_finseq_4,t25_binom,d4_finseq_4,t59_finseq_1,d4_finseq_4,t56_finseq_1,d7_finseq_1,d4_finseq_4,d3_polynom1,t9_binom,d4_finseq_4,d4_binom,d5_algstr_1,t25_binom,t38_nat_1,t3_finseq_1,d10_binom,d3_finseq_1,d3_finseq_1,t18_int_1,t11_xreal_1,t18_int_1,t38_nat_1,t8_xreal_1,t38_nat_1,t3_finseq_1,d10_binom,d3_finseq_1,d3_finseq_1,t18_int_1,t11_xreal_1,t18_int_1,d3_finseq_1,d4_finseq_4,d4_finseq_4,d7_finseq_1,d4_finseq_4,d3_polynom1,t57_finseq_1,d7_finseq_1,d4_finseq_4,d3_polynom1,d4_finseq_4,d4_binom,d10_binom,t18_binom,t22_binom,d4_group_1,t9_binom,d10_binom,d4_group_1,t9_binom,t21_binom,t21_binom,t18_binom,d18_vectsp_1,t18_binom,t16_binom,t32_newton,d10_binom,d4_finseq_4,d2_xboole_0,d2_xboole_0,t5_newton,d10_binom,t18_finseq_1,s1_nat_1]),
    [file(binom,t26_binom),interesting(1.00)]).

fof(t3_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v1_algstr_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => k9_rlvect_1(A,k12_finseq_1(u1_struct_0(A),B)) = B ) ) ),
    inference(mizar_proof,[status(thm)],[d12_rlvect_1,t56_finseq_1,t39_nat_1,t39_nat_1,t57_finseq_1,d5_algstr_1,t56_finseq_1]),
    [file(binom,t3_binom),interesting(0.94)]).

fof(t14_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => k5_binom(A,B,1) = B ) ) ),
    inference(mizar_proof,[status(thm)],[d6_binom,d6_binom,d7_rlvect_1]),
    [file(binom,t14_binom),interesting(0.94)]).

fof(t15_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v1_algstr_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => k6_binom(A,B,1) = B ) ) ),
    inference(mizar_proof,[status(thm)],[d7_binom,d7_binom,d5_algstr_1]),
    [file(binom,t15_binom),interesting(0.94)]).

fof(t13_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ( k5_binom(A,B,0) = k1_rlvect_1(A)
            & k6_binom(A,B,0) = k1_rlvect_1(A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_binom,d7_binom]),
    [file(binom,t13_binom),interesting(0.90)]).

fof(t1_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => k1_group_1(A,k1_rlvect_1(A),B) = k1_rlvect_1(A) ) ) ),
    inference(mizar_proof,[status(thm)],[d7_rlvect_1,d12_vectsp_1,d7_rlvect_1,d6_algstr_1]),
    [file(binom,t1_binom),interesting(0.87)]).

fof(t2_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v3_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_algstr_1,d11_vectsp_1,d5_algstr_1,d7_algstr_1]),
    [file(binom,t2_binom),interesting(0.87)]).

fof(t22_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_algstr_1(A)
        & v1_binom(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k1_group_1(A,k6_binom(A,B,D),C) = k1_group_1(A,B,k5_binom(A,C,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t18_binom,t20_binom,t18_binom,t21_binom]),
    [file(binom,t22_binom),interesting(0.85)]).

fof(t23_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v2_group_1(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => k8_binom(A,B,C,0) = k12_finseq_1(u1_struct_0(A),k2_group_1(A)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d10_binom,t4_finseq_1,d3_finseq_1,d1_tarski,d1_tarski,t18_int_1,d4_finseq_4,d10_binom,t8_binom,t27_newton,t8_binom,t14_binom,d5_group_1,t57_finseq_1]),
    [file(binom,t23_binom),interesting(0.84)]).

fof(t8_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ( k2_binom(A,B,0) = k2_group_1(A)
            & k2_binom(A,B,1) = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_group_1,d8_group_1,d8_group_1,d5_group_1]),
    [file(binom,t8_binom),interesting(0.83)]).

fof(t5_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m2_finseq_1(C,u1_struct_0(A))
             => k9_rlvect_1(A,k7_polynom1(A,C,B)) = k1_group_1(A,k9_rlvect_1(A,C),B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d12_rlvect_1,d12_rlvect_1,d3_finseq_1,d3_polynom1,d3_finseq_1,t1_binom,t8_finseq_1,t38_nat_1,t8_xreal_1,t3_finseq_1,d3_finseq_1,d4_finseq_4,t3_finseq_1,d3_finseq_1,d4_finseq_4,d3_polynom1,d12_vectsp_1,s4_polynom2,t18_nat_1,t8_finseq_1]),
    [file(binom,t5_binom),interesting(0.83)]).

fof(t19_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k5_binom(A,B,C) = k6_binom(A,B,C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t13_binom,t13_binom,l25_binom,l26_binom,s1_nat_1]),
    [file(binom,t19_binom),interesting(0.80)]).

fof(t24_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v2_group_1(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k1_funct_1(k8_binom(A,B,C,D),1) = k2_binom(A,B,D) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t27_newton,t18_nat_1,d10_binom,d3_finseq_1,t8_xreal_1,t3_finseq_1,d4_finseq_4,d10_binom,t29_newton,t14_binom,t8_binom,d5_group_1]),
    [file(binom,t24_binom),interesting(0.75)]).

fof(t16_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v1_algstr_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k5_binom(A,B,k1_nat_1(C,D)) = k2_rlvect_1(A,k5_binom(A,B,C),k5_binom(A,B,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_algstr_1,t13_binom,l25_binom,d6_rlvect_1,l25_binom,s1_nat_1]),
    [file(binom,t16_binom),interesting(0.75)]).

fof(l25_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k5_binom(A,B,k1_nat_1(C,1)) = k2_rlvect_1(A,B,k5_binom(A,B,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_binom]),
    [file(binom,l25_binom),interesting(0.74)]).

fof(l26_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k6_binom(A,B,k1_nat_1(C,1)) = k2_rlvect_1(A,k6_binom(A,B,C),B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d7_binom]),
    [file(binom,l26_binom),interesting(0.74)]).

fof(t20_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v1_algstr_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k1_group_1(A,k5_binom(A,B,D),C) = k5_binom(A,k1_group_1(A,B,C),D) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t13_binom,t1_binom,t13_binom,l25_binom,d12_vectsp_1,t14_binom,t16_binom,s1_nat_1]),
    [file(binom,t20_binom),interesting(0.74)]).

fof(t6_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v7_group_1(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m2_finseq_1(C,u1_struct_0(A))
             => k9_rlvect_1(A,k7_polynom1(A,C,B)) = k9_rlvect_1(A,k6_polynom1(A,C,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d12_rlvect_1,d12_rlvect_1,d3_finseq_1,d2_polynom1,d3_finseq_1,d3_finseq_1,d3_polynom1,d3_finseq_1,t8_finseq_1,t38_nat_1,t8_xreal_1,t3_finseq_1,d3_finseq_1,d4_finseq_4,t8_finseq_1,t3_finseq_1,d3_finseq_1,d2_polynom1,d4_finseq_4,d3_polynom1,d2_polynom1,s4_polynom2,t18_nat_1,t8_finseq_1,t8_finseq_1]),
    [file(binom,t6_binom),interesting(0.74)]).

fof(t21_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_algstr_1(A)
        & v3_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k1_group_1(A,C,k5_binom(A,B,D)) = k6_binom(A,k1_group_1(A,C,B),D) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t13_binom,t2_binom,t13_binom,l25_binom,d11_vectsp_1,t15_binom,t17_binom,s1_nat_1]),
    [file(binom,t21_binom),interesting(0.72)]).

fof(t9_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k2_binom(A,B,k1_nat_1(C,1)) = k1_group_1(A,k2_binom(A,B,C),B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_group_1]),
    [file(binom,t9_binom),interesting(0.70)]).

fof(t25_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v2_group_1(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k1_funct_1(k8_binom(A,B,C,D),k1_nat_1(D,1)) = k2_binom(A,C,D) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t18_int_1,d10_binom,d3_finseq_1,t6_finseq_1,d4_finseq_4,d10_binom,t31_newton,t8_binom,t14_binom,d5_group_1]),
    [file(binom,t25_binom),interesting(0.70)]).

fof(t4_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v3_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m2_finseq_1(C,u1_struct_0(A))
             => k9_rlvect_1(A,k6_polynom1(A,C,B)) = k1_group_1(A,B,k9_rlvect_1(A,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d12_rlvect_1,d12_rlvect_1,d3_finseq_1,d2_polynom1,d3_finseq_1,t2_binom,t8_finseq_1,t38_nat_1,t38_nat_1,t8_xreal_1,t3_finseq_1,d3_finseq_1,d4_finseq_4,t3_finseq_1,d3_finseq_1,d4_finseq_4,d2_polynom1,d11_vectsp_1,s4_polynom2,t18_nat_1,t8_finseq_1]),
    [file(binom,t4_binom),interesting(0.66)]).

fof(t7_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m2_finseq_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m2_finseq_1(C,u1_struct_0(A))
             => ( k4_relset_1(k5_numbers,u1_struct_0(A),B) = k4_relset_1(k5_numbers,u1_struct_0(A),C)
               => k9_rlvect_1(A,k1_binom(A,B,C)) = k4_rlvect_1(A,k9_rlvect_1(A,B),k9_rlvect_1(A,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_finseq_1,d3_finseq_1,t8_finseq_1,d3_finseq_1,d4_binom,d3_finseq_1,t8_finseq_1,d12_rlvect_1,d12_rlvect_1,d12_rlvect_1,d7_rlvect_1,t38_nat_1,t38_nat_1,t8_xreal_1,t3_finseq_1,d3_finseq_1,d4_finseq_4,t3_finseq_1,d3_finseq_1,d4_finseq_4,t3_finseq_1,d3_finseq_1,d4_finseq_4,d4_binom,d6_rlvect_1,d6_rlvect_1,d6_rlvect_1,s4_polynom2,t18_nat_1,t8_finseq_1]),
    [file(binom,t7_binom),interesting(0.65)]).

fof(t18_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v1_algstr_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k5_binom(A,B,C) = k6_binom(A,B,C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t13_binom,t13_binom,t16_binom,t14_binom,t15_binom,t17_binom,s1_nat_1]),
    [file(binom,t18_binom),interesting(0.55)]).

fof(t11_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_binom(A,B,k1_nat_1(C,D)) = k1_group_1(A,k2_binom(A,B,C),k2_binom(A,B,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_group_1,d8_group_1,d8_group_1,d4_group_1,d8_group_1,s1_nat_1]),
    [file(binom,t11_binom),interesting(0.50)]).

fof(t10_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_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))
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_binom(A,k10_group_1(A,B,C),D) = k10_group_1(A,k2_binom(A,B,D),k2_binom(A,C,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_group_1,d5_group_1,d8_group_1,d8_group_1,d8_group_1,d4_group_1,d4_group_1,d4_group_1,d8_group_1,d8_group_1,s1_nat_1]),
    [file(binom,t10_binom),interesting(0.49)]).

fof(t17_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k6_binom(A,B,k1_nat_1(C,D)) = k2_rlvect_1(A,k6_binom(A,B,C),k6_binom(A,B,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d7_rlvect_1,t13_binom,l26_binom,d6_rlvect_1,l26_binom,s1_nat_1]),
    [file(binom,t17_binom),interesting(0.37)]).

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

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

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

fof(s1_binom,theorem,
    ( ( p1_s1_binom(f1_s1_binom)
      & ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(f1_s1_binom,A)
              & p1_s1_binom(A) )
           => p1_s1_binom(k1_nat_1(A,1)) ) ) )
   => ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( r1_xreal_0(f1_s1_binom,A)
         => p1_s1_binom(A) ) ) ),
    inference(mizar_proof,[status(thm)],[t29_nat_1,s1_nat_1,t28_nat_1]),
    [file(binom,s1_binom),interesting(0.00)]).

fof(s2_recdef_1,theorem,
    ( ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ! [B] : 
            ( m1_subset_1(B,f1_s2_recdef_1)
           => ? [C] : 
                ( m1_subset_1(C,f1_s2_recdef_1)
                & p1_s2_recdef_1(A,B,C) ) ) )
   => ? [A] : 
        ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,f1_s2_recdef_1)
        & m2_relset_1(A,k5_numbers,f1_s2_recdef_1)
        & k8_funct_2(k5_numbers,f1_s2_recdef_1,A,0) = f2_s2_recdef_1
        & ! [B] : 
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => p1_s2_recdef_1(B,k8_funct_2(k5_numbers,f1_s2_recdef_1,A,B),k8_funct_2(k5_numbers,f1_s2_recdef_1,A,k1_nat_1(B,1))) ) ) ),
    file(recdef_1,s2_recdef_1),
    [interesting(0.00)]).

fof(d1_mcart_1,definition,(
    ! [A] : 
      ( ? [B,C] : A = k4_tarski(B,C)
     => ! [B] : 
          ( B = k1_mcart_1(A)
        <=> ! [C,D] : 
              ( A = k4_tarski(C,D)
             => B = C ) ) ) ),
    file(mcart_1,d1_mcart_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(d2_mcart_1,definition,(
    ! [A] : 
      ( ? [B,C] : A = k4_tarski(B,C)
     => ! [B] : 
          ( B = k2_mcart_1(A)
        <=> ! [C,D] : 
              ( A = k4_tarski(C,D)
             => B = D ) ) ) ),
    file(mcart_1,d2_mcart_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(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(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(d1_relset_1,definition,(
    ! [A,B,C] : 
      ( m1_relset_1(C,A,B)
    <=> r1_tarski(C,k2_zfmisc_1(A,B)) ) ),
    file(relset_1,d1_relset_1),
    [interesting(0.00)]).

fof(t11_funct_2,theorem,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B) )
     => ( ( B = k1_xboole_0
         => A = k1_xboole_0 )
       => r2_hidden(C,k1_funct_2(A,B)) ) ) ),
    file(funct_2,t11_funct_2),
    [interesting(0.00)]).

fof(d4_relat_1,definition,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( B = k1_relat_1(A)
        <=> ! [C] : 
              ( r2_hidden(C,B)
            <=> ? [D] : r2_hidden(k4_tarski(C,D),A) ) ) ) ),
    file(relat_1,d4_relat_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(d1_funct_1,definition,(
    ! [A] : 
      ( v1_funct_1(A)
    <=> ! [B,C,D] : 
          ( ( r2_hidden(k4_tarski(B,C),A)
            & r2_hidden(k4_tarski(B,D),A) )
         => C = D ) ) ),
    file(funct_1,d1_funct_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(s2_binom,theorem,(
    ? [A] : 
      ( v1_funct_1(A)
      & v1_funct_2(A,k2_zfmisc_1(k5_numbers,f1_s2_binom),f2_s2_binom)
      & m2_relset_1(A,k2_zfmisc_1(k5_numbers,f1_s2_binom),f2_s2_binom)
      & ! [B] : 
          ( m1_subset_1(B,f1_s2_binom)
         => ( k2_binop_1(k5_numbers,f1_s2_binom,f2_s2_binom,A,0,B) = f3_s2_binom
            & ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_binop_1(k5_numbers,f1_s2_binom,f2_s2_binom,A,k1_nat_1(C,1),B) = k2_binop_1(f1_s2_binom,f2_s2_binom,f2_s2_binom,f4_s2_binom,B,k2_binop_1(k5_numbers,f1_s2_binom,f2_s2_binom,A,C,B)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s2_recdef_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d1_funct_2,s1_nat_1,d2_mcart_1,d2_mcart_1,t9_funct_1,d2_mcart_1,d2_mcart_1,d2_zfmisc_1,d3_tarski,d1_relset_1,t11_funct_2,d4_relat_1,t2_tarski,d1_funct_1,d1_funct_2,d1_funct_2,t8_funct_1,d1_mcart_1,d1_mcart_1,d2_mcart_1,d2_mcart_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d2_mcart_1,d2_mcart_1,d2_mcart_1,d2_mcart_1,d2_zfmisc_1,d2_zfmisc_1,d3_tarski,d1_relset_1,d2_zfmisc_1,d4_relat_1,t2_tarski,d1_funct_1,d1_funct_2,d2_zfmisc_1,d1_funct_2,d4_relat_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d2_mcart_1,d2_mcart_1,d2_mcart_1,d2_mcart_1,t8_funct_1,d2_zfmisc_1,d1_funct_2,d4_relat_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d2_mcart_1,d2_mcart_1,d2_mcart_1,d2_mcart_1,d2_zfmisc_1,d1_funct_2,d4_relat_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d2_mcart_1,d2_mcart_1,d2_mcart_1,d2_mcart_1,t8_funct_1,t8_funct_1]),
    [file(binom,s2_binom),interesting(0.00)]).

fof(s3_binom,theorem,(
    ? [A] : 
      ( v1_funct_1(A)
      & v1_funct_2(A,k2_zfmisc_1(f1_s3_binom,k5_numbers),f2_s3_binom)
      & m2_relset_1(A,k2_zfmisc_1(f1_s3_binom,k5_numbers),f2_s3_binom)
      & ! [B] : 
          ( m1_subset_1(B,f1_s3_binom)
         => ( k2_binop_1(f1_s3_binom,k5_numbers,f2_s3_binom,A,B,0) = f3_s3_binom
            & ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_binop_1(f1_s3_binom,k5_numbers,f2_s3_binom,A,B,k1_nat_1(C,1)) = k2_binop_1(f2_s3_binom,f1_s3_binom,f2_s3_binom,f4_s3_binom,k2_binop_1(f1_s3_binom,k5_numbers,f2_s3_binom,A,B,C),B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s2_recdef_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d1_funct_2,s1_nat_1,d2_mcart_1,d2_mcart_1,t9_funct_1,d2_mcart_1,d2_mcart_1,d2_zfmisc_1,d3_tarski,d1_relset_1,t11_funct_2,d4_relat_1,t2_tarski,d1_funct_1,d1_funct_2,d1_funct_2,t8_funct_1,d1_mcart_1,d1_mcart_1,d2_mcart_1,d2_mcart_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d1_mcart_1,d2_mcart_1,d2_mcart_1,d2_mcart_1,d2_mcart_1,d2_zfmisc_1,d2_zfmisc_1,d3_tarski,d1_relset_1,d2_zfmisc_1,d4_relat_1,t2_tarski,d1_funct_1,d1_funct_2,d2_zfmisc_1,d1_funct_2,d4_relat_1,d1_mcart_1,d1_mcart_1,d2_mcart_1,d2_mcart_1,d2_mcart_1,d2_mcart_1,d1_mcart_1,d1_mcart_1,t8_funct_1,d2_zfmisc_1,d1_funct_2,d4_relat_1,d1_mcart_1,d1_mcart_1,d2_mcart_1,d2_mcart_1,d2_mcart_1,d2_mcart_1,d1_mcart_1,d1_mcart_1,d2_zfmisc_1,d1_funct_2,d4_relat_1,d1_mcart_1,d1_mcart_1,d2_mcart_1,d2_mcart_1,d2_mcart_1,d2_mcart_1,d1_mcart_1,d1_mcart_1,t8_funct_1,t8_funct_1]),
    [file(binom,s3_binom),interesting(0.00)]).

fof(d8_group_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),k5_numbers),u1_struct_0(A))
            & m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),k5_numbers),u1_struct_0(A)) )
         => ( B = k5_group_1(A)
          <=> ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ( k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),B,C,0) = k2_group_1(A)
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),B,C,k1_nat_1(D,1)) = k1_group_1(A,k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),B,C,D),C) ) ) ) ) ) ) ),
    file(group_1,d8_group_1),
    [interesting(0.00)]).

fof(d5_group_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v2_group_1(A)
       => ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ( B = k2_group_1(A)
            <=> ! [C] : 
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ( k1_group_1(A,C,B) = C
                    & k1_group_1(A,B,C) = C ) ) ) ) ) ) ),
    file(group_1,d5_group_1),
    [interesting(0.00)]).

fof(d4_group_1,definition,(
    ! [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)) ) ) ) ) ) ),
    file(group_1,d4_group_1),
    [interesting(0.00)]).

fof(t12_binom,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_binom(A,k2_binom(A,B,C),D) = k2_binom(A,B,k2_nat_1(C,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_group_1,d8_group_1,d8_group_1,t11_binom,s1_nat_1]),
    [file(binom,t12_binom),interesting(0.00)]).

fof(d6_binom,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k5_numbers,u1_struct_0(A)),u1_struct_0(A))
            & m2_relset_1(B,k2_zfmisc_1(k5_numbers,u1_struct_0(A)),u1_struct_0(A)) )
         => ( B = k3_binom(A)
          <=> ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ( k2_binop_1(k5_numbers,u1_struct_0(A),u1_struct_0(A),B,0,C) = k1_rlvect_1(A)
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_binop_1(k5_numbers,u1_struct_0(A),u1_struct_0(A),B,k1_nat_1(D,1),C) = k2_rlvect_1(A,C,k2_binop_1(k5_numbers,u1_struct_0(A),u1_struct_0(A),B,D,C)) ) ) ) ) ) ) ),
    file(binom,d6_binom),
    [interesting(0.00)]).

fof(d7_binom,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),k5_numbers),u1_struct_0(A))
            & m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),k5_numbers),u1_struct_0(A)) )
         => ( B = k4_binom(A)
          <=> ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ( k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),B,C,0) = k1_rlvect_1(A)
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),B,C,k1_nat_1(D,1)) = k2_rlvect_1(A,k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),B,C,D),C) ) ) ) ) ) ) ),
    file(binom,d7_binom),
    [interesting(0.00)]).

fof(d12_rlvect_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m2_finseq_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ( C = k9_rlvect_1(A,B)
              <=> ? [D] : 
                    ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A))
                    & C = k8_funct_2(k5_numbers,u1_struct_0(A),D,k3_finseq_1(B))
                    & k8_funct_2(k5_numbers,u1_struct_0(A),D,0) = k1_rlvect_1(A)
                    & ! [E] : 
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ! [F] : 
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ( F = k1_funct_1(B,k1_nat_1(E,1))
                             => ( r1_xreal_0(k3_finseq_1(B),E)
                                | k8_funct_2(k5_numbers,u1_struct_0(A),D,k1_nat_1(E,1)) = k2_rlvect_1(A,k8_funct_2(k5_numbers,u1_struct_0(A),D,E),F) ) ) ) ) ) ) ) ) ) ),
    file(rlvect_1,d12_rlvect_1),
    [interesting(0.00)]).

fof(t56_finseq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => ( B = k9_finseq_1(A)
      <=> ( k3_finseq_1(B) = 1
          & k2_relat_1(B) = k1_tarski(A) ) ) ) ),
    file(finseq_1,t56_finseq_1),
    [interesting(0.00)]).

fof(t39_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(1,A)
       => A = 0 ) ) ),
    file(nat_1,t39_nat_1),
    [interesting(0.00)]).

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

fof(d5_algstr_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v1_algstr_1(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => k2_rlvect_1(A,k1_rlvect_1(A),B) = B ) ) ) ),
    file(algstr_1,d5_algstr_1),
    [interesting(0.00)]).

fof(d10_binom,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] : 
                      ( m2_finseq_1(E,u1_struct_0(A))
                     => ( E = k8_binom(A,B,C,D)
                      <=> ( k3_finseq_1(E) = k1_nat_1(D,1)
                          & ! [F] : 
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ! [G] : 
                                  ( m2_subset_1(G,k1_numbers,k5_numbers)
                                 => ! [H] : 
                                      ( m2_subset_1(H,k1_numbers,k5_numbers)
                                     => ( ( r2_hidden(F,k4_relset_1(k5_numbers,u1_struct_0(A),E))
                                          & H = k6_xcmplx_0(F,1)
                                          & G = k6_xcmplx_0(D,H) )
                                       => k4_finseq_4(k5_numbers,u1_struct_0(A),E,F) = k1_group_1(A,k5_binom(A,k2_binom(A,B,G),k7_binom(H,D)),k2_binom(A,C,H)) ) ) ) ) ) ) ) ) ) ) ) ),
    file(binom,d10_binom),
    [interesting(0.00)]).

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

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

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

fof(t18_int_1,theorem,(
    ! [A] : 
      ( v1_int_1(A)
     => ! [B] : 
          ( v1_int_1(B)
         => ( r1_xreal_0(A,B)
           => r2_hidden(k6_xcmplx_0(B,A),k5_numbers) ) ) ) ),
    file(int_1,t18_int_1),
    [interesting(0.00)]).

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

fof(t27_newton,theorem,(
    k8_newton(0,0) = 1 ),
    file(newton,t27_newton),
    [interesting(0.00)]).

fof(d7_rlvect_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v5_rlvect_1(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => k2_rlvect_1(A,B,k1_rlvect_1(A)) = B ) ) ) ),
    file(rlvect_1,d7_rlvect_1),
    [interesting(0.00)]).

fof(d11_vectsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( v4_vectsp_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,B,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,D)) ) ) ) ) ) ),
    file(vectsp_1,d11_vectsp_1),
    [interesting(0.00)]).

fof(d3_polynom1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( m2_finseq_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m2_finseq_1(D,u1_struct_0(A))
                 => ( D = k7_polynom1(A,B,C)
                  <=> ( k4_finseq_1(D) = k4_finseq_1(B)
                      & ! [E] : 
                          ( r2_hidden(E,k4_finseq_1(B))
                         => k4_finseq_4(k5_numbers,u1_struct_0(A),D,E) = k1_group_1(A,k4_finseq_4(k5_numbers,u1_struct_0(A),B,E),C) ) ) ) ) ) ) ) ),
    file(polynom1,d3_polynom1),
    [interesting(0.00)]).

fof(d12_vectsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( v5_vectsp_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,k2_rlvect_1(A,C,D),B) = k2_rlvect_1(A,k1_group_1(A,C,B),k1_group_1(A,D,B)) ) ) ) ) ) ),
    file(vectsp_1,d12_vectsp_1),
    [interesting(0.00)]).

fof(d6_algstr_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v2_algstr_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))
                   => ( k2_rlvect_1(A,B,C) = k2_rlvect_1(A,B,D)
                     => C = D ) ) ) ) ) ) ),
    file(algstr_1,d6_algstr_1),
    [interesting(0.00)]).

fof(t8_finseq_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( k2_finseq_1(A) = k2_finseq_1(B)
           => A = B ) ) ) ),
    file(finseq_1,t8_finseq_1),
    [interesting(0.00)]).

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

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

fof(t3_finseq_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( r2_hidden(A,k2_finseq_1(B))
          <=> ( r1_xreal_0(1,A)
              & r1_xreal_0(A,B) ) ) ) ) ),
    file(finseq_1,t3_finseq_1),
    [interesting(0.00)]).

fof(s4_polynom2,theorem,
    ( ( p1_s4_polynom2(f1_s4_polynom2)
      & ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(f1_s4_polynom2,A)
              & p1_s4_polynom2(A) )
           => ( r1_xreal_0(f2_s4_polynom2,A)
              | p1_s4_polynom2(k1_nat_1(A,1)) ) ) ) )
   => ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( ( r1_xreal_0(f1_s4_polynom2,A)
            & r1_xreal_0(A,f2_s4_polynom2) )
         => p1_s4_polynom2(A) ) ) ),
    file(polynom2,s4_polynom2),
    [interesting(0.00)]).

fof(t18_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => r1_xreal_0(0,A) ) ),
    file(nat_1,t18_nat_1),
    [interesting(0.00)]).

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

fof(t110_finseq_2,theorem,(
    ! [A,B] : 
      ( m2_finseq_1(B,A)
     => m1_subset_1(B,k4_finseq_2(k3_finseq_1(B),A)) ) ),
    file(finseq_2,t110_finseq_2),
    [interesting(0.00)]).

fof(t109_finseq_2,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( m2_finseq_2(C,B,k4_finseq_2(A,B))
             => k3_finseq_1(C) = A ) ) ) ),
    file(finseq_2,t109_finseq_2),
    [interesting(0.00)]).

fof(t58_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m2_finseq_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m2_finseq_1(C,u1_struct_0(A))
             => k9_rlvect_1(A,k8_finseq_1(u1_struct_0(A),B,C)) = k2_rlvect_1(A,k9_rlvect_1(A,B),k9_rlvect_1(A,C)) ) ) ) ),
    file(rlvect_1,t58_rlvect_1),
    [interesting(0.00)]).

fof(d4_binom,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m2_finseq_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m2_finseq_1(C,u1_struct_0(A))
             => ( k4_relset_1(k5_numbers,u1_struct_0(A),B) = k4_relset_1(k5_numbers,u1_struct_0(A),C)
               => ! [D] : 
                    ( m2_finseq_1(D,u1_struct_0(A))
                   => ( D = k1_binom(A,B,C)
                    <=> ( k4_relset_1(k5_numbers,u1_struct_0(A),D) = k4_relset_1(k5_numbers,u1_struct_0(A),B)
                        & ! [E] : 
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ( ( r1_xreal_0(1,E)
                                & r1_xreal_0(E,k3_finseq_1(D)) )
                             => k4_finseq_4(k5_numbers,u1_struct_0(A),D,E) = k2_rlvect_1(A,k4_finseq_4(k5_numbers,u1_struct_0(A),B,E),k4_finseq_4(k5_numbers,u1_struct_0(A),C,E)) ) ) ) ) ) ) ) ) ) ),
    file(binom,d4_binom),
    [interesting(0.00)]).

fof(d6_rlvect_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v4_rlvect_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))
                   => k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) ) ) ) ) ) ),
    file(rlvect_1,d6_rlvect_1),
    [interesting(0.00)]).

fof(t37_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ! [C] : 
              ( v4_ordinal2(C)
             => ( r1_xreal_0(A,B)
               => r1_xreal_0(A,k2_xcmplx_0(B,C)) ) ) ) ) ),
    file(nat_1,t37_nat_1),
    [interesting(0.00)]).

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

fof(t29_newton,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => k8_newton(0,A) = 1 ) ),
    file(newton,t29_newton),
    [interesting(0.00)]).

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

fof(t6_finseq_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => r2_hidden(k2_xcmplx_0(A,1),k2_finseq_1(k2_xcmplx_0(A,1))) ) ),
    file(finseq_1,t6_finseq_1),
    [interesting(0.00)]).

fof(t31_newton,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => k8_newton(A,A) = 1 ) ),
    file(newton,t31_newton),
    [interesting(0.00)]).

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

fof(t11_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( r1_xreal_0(A,B)
              <=> r1_xreal_0(k6_xcmplx_0(A,C),k6_xcmplx_0(B,C)) ) ) ) ) ),
    file(xreal_1,t11_xreal_1),
    [interesting(0.00)]).

fof(d7_algstr_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v3_algstr_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))
                   => ( k2_rlvect_1(A,C,B) = k2_rlvect_1(A,D,B)
                     => C = D ) ) ) ) ) ) ),
    file(algstr_1,d7_algstr_1),
    [interesting(0.00)]).

fof(d18_vectsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( v7_vectsp_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,B,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,D))
                      & k1_group_1(A,k2_rlvect_1(A,C,D),B) = k2_rlvect_1(A,k1_group_1(A,C,B),k1_group_1(A,D,B)) ) ) ) ) ) ) ),
    file(vectsp_1,d18_vectsp_1),
    [interesting(0.00)]).

fof(t32_newton,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => k8_newton(k2_xcmplx_0(B,1),k2_xcmplx_0(A,1)) = k2_xcmplx_0(k8_newton(k2_xcmplx_0(B,1),A),k8_newton(B,A)) ) ) ),
    file(newton,t32_newton),
    [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(t5_newton,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(1,A)
       => k2_finseq_1(A) = k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(A)),k1_tarski(A)) ) ) ),
    file(newton,t5_newton),
    [interesting(0.00)]).

fof(t18_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( ( k3_finseq_1(A) = k3_finseq_1(B)
              & ! [C] : 
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( ( r1_xreal_0(1,C)
                      & r1_xreal_0(C,k3_finseq_1(A)) )
                   => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) )
           => A = B ) ) ) ),
    file(finseq_1,t18_finseq_1),
    [interesting(0.00)]).

fof(d2_polynom1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] : 
          ( m2_finseq_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m2_finseq_1(D,u1_struct_0(A))
                 => ( D = k6_polynom1(A,B,C)
                  <=> ( k4_finseq_1(D) = k4_finseq_1(B)
                      & ! [E] : 
                          ( r2_hidden(E,k4_finseq_1(B))
                         => k4_finseq_4(k5_numbers,u1_struct_0(A),D,E) = k1_group_1(A,C,k4_finseq_4(k5_numbers,u1_struct_0(A),B,E)) ) ) ) ) ) ) ) ),
    file(polynom1,d2_polynom1),
    [interesting(0.00)]).