fof(t41_afinsq_1,theorem,(
    ! [A,B,C] : 
      ( k8_afinsq_1(A,B,C) = k1_ordinal4(k6_afinsq_1(A),k7_afinsq_1(B,C))
      & k8_afinsq_1(A,B,C) = k1_ordinal4(k7_afinsq_1(A,B),k6_afinsq_1(C)) ) ),
    inference(mizar_proof,[status(thm)],[t30_afinsq_1]),
    [file(afinsq_1,t41_afinsq_1),interesting(1.00)]).

fof(t5_afinsq_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r1_tarski(k2_finseq_1(A),k1_nat_1(A,1)) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,t3_finseq_1,t38_nat_1,t1_euler_1]),
    [file(afinsq_1,t5_afinsq_1),interesting(0.97)]).

fof(t1_afinsq_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r2_hidden(A,k1_nat_1(A,1)) ) ),
    inference(mizar_proof,[status(thm)],[t38_nat_1,t30_axioms]),
    [file(afinsq_1,t1_afinsq_1),interesting(0.89)]).

fof(t27_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => r1_tarski(k2_relat_1(A),k2_relat_1(k1_ordinal4(A,B))) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_funct_1,t24_afinsq_1,d4_afinsq_1,t12_funct_1,d3_tarski]),
    [file(afinsq_1,t27_afinsq_1),interesting(0.88)]).

fof(t44_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ~ ( A != k1_xboole_0
          & ! [B] : 
              ( ( v1_relat_1(B)
                & v1_funct_1(B)
                & v5_ordinal1(B)
                & v1_finset_1(B) )
             => ! [C] : A != k1_ordinal4(B,k6_afinsq_1(C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t18_afinsq_1,t22_nat_1,t12_afinsq_1,t90_relat_1,d1_afinsq_1,t29_nat_1,t56_card_1,t28_xboole_1,d1_afinsq_1,t20_afinsq_1,t37_afinsq_1,d1_afinsq_1,d1_afinsq_1,d1_afinsq_1,t4_afinsq_1,t70_funct_1,d4_afinsq_1,t1_afinsq_1,d5_afinsq_1,d1_tarski,d1_afinsq_1,d4_afinsq_1,d5_afinsq_1,d1_tarski,d2_xboole_0,t9_funct_1]),
    [file(afinsq_1,t44_afinsq_1),interesting(0.87)]).

fof(t47_afinsq_1,theorem,(
    ! [A] : r2_hidden(k1_xboole_0,k10_afinsq_1(A)) ),
    inference(mizar_proof,[status(thm)],[d8_afinsq_1]),
    [file(afinsq_1,t47_afinsq_1),interesting(0.87)]).

fof(l38_afinsq_1,theorem,(
    ! [A,B,C,D] : 
      ( r2_hidden(k4_tarski(A,B),k1_tarski(k4_tarski(C,D)))
     => ( A = C
        & B = D ) ) ),
    inference(mizar_proof,[status(thm)],[d1_tarski,t33_zfmisc_1]),
    [file(afinsq_1,l38_afinsq_1),interesting(0.85)]).

fof(t19_afinsq_1,theorem,(
    ! [A] : 
      ( v1_finset_1(k1_xboole_0)
      & m1_ordinal1(k1_xboole_0,A) ) ),
    inference(mizar_proof,[status(thm)],[t2_xboole_1,d8_ordinal1]),
    [file(afinsq_1,t19_afinsq_1),interesting(0.84)]).

fof(t28_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => r1_tarski(k2_relat_1(A),k2_relat_1(k1_ordinal4(B,A))) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_funct_1,t26_afinsq_1,d4_afinsq_1,t12_funct_1,d3_tarski]),
    [file(afinsq_1,t28_afinsq_1),interesting(0.84)]).

fof(t46_afinsq_1,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,k10_afinsq_1(B))
    <=> ( v1_finset_1(A)
        & m1_ordinal1(A,B) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_afinsq_1]),
    [file(afinsq_1,t46_afinsq_1),interesting(0.82)]).

fof(t15_afinsq_1,theorem,(
    ! [A,B] : 
      ( ( v1_finset_1(B)
        & m1_ordinal1(B,A) )
     => ( v1_funct_1(B)
        & m2_relset_1(B,k5_numbers,A) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_ordinal1,t11_relset_1]),
    [file(afinsq_1,t15_afinsq_1),interesting(0.81)]).

fof(t24_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B) )
         => r1_ordinal1(k1_relat_1(A),k1_relat_1(k1_ordinal4(A,B))) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_ordinal4,t27_ordinal3]),
    [file(afinsq_1,t24_afinsq_1),interesting(0.81)]).

fof(t43_afinsq_1,theorem,(
    ! [A,B,C,D] : 
      ( ( v1_relat_1(D)
        & v1_funct_1(D)
        & v5_ordinal1(D)
        & v1_finset_1(D) )
     => ( D = k8_afinsq_1(A,B,C)
      <=> ( k1_afinsq_1(D) = 3
          & k1_funct_1(D,0) = A
          & k1_funct_1(D,1) = B
          & k1_funct_1(D,2) = C ) ) ) ),
    inference(mizar_proof,[status(thm)],[t20_afinsq_1,t42_afinsq_1,t37_afinsq_1,d1_tarski,d5_afinsq_1,t1_card_5,t41_afinsq_1,d4_afinsq_1,d5_afinsq_1,t1_afinsq_1,t42_afinsq_1,d1_afinsq_1,d4_afinsq_1,t42_afinsq_1,d5_afinsq_1,t1_card_5,t42_afinsq_1,d4_afinsq_1,d5_afinsq_1,d1_afinsq_1,t42_afinsq_1,t37_afinsq_1,t42_afinsq_1,d1_afinsq_1,t42_afinsq_1,t42_afinsq_1,t1_card_5,d2_tarski,d5_afinsq_1,t1_card_5,d1_tarski,t42_afinsq_1,d5_afinsq_1,d4_afinsq_1]),
    [file(afinsq_1,t43_afinsq_1),interesting(0.81)]).

fof(t32_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ( k1_ordinal4(A,k1_xboole_0) = A
        & k1_ordinal4(k1_xboole_0,A) = A ) ) ),
    inference(mizar_proof,[status(thm)],[d1_afinsq_1,t20_afinsq_1,t18_afinsq_1,d1_afinsq_1,d4_afinsq_1,t10_afinsq_1,d1_afinsq_1,t20_afinsq_1,t18_afinsq_1,d1_afinsq_1,t18_afinsq_1,d4_afinsq_1,t10_afinsq_1]),
    [file(afinsq_1,t32_afinsq_1),interesting(0.76)]).

fof(t16_afinsq_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B,C] : 
          ( ( v1_finset_1(C)
            & m1_ordinal1(C,B) )
         => ( v1_finset_1(k2_ordinal1(C,A))
            & m1_ordinal1(k2_ordinal1(C,A),B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_ordinal1,t1_xboole_1,t12_afinsq_1,d8_ordinal1]),
    [file(afinsq_1,t16_afinsq_1),interesting(0.75)]).

fof(t17_afinsq_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ? [C] : 
              ( v1_finset_1(C)
              & m1_ordinal1(C,B)
              & k1_afinsq_1(C) = A ) ) ) ),
    inference(mizar_proof,[status(thm)],[t13_funcop_1,t19_funcop_1,t7_afinsq_1,t19_funcop_1,t37_zfmisc_1,t1_xboole_1,d8_ordinal1,d1_afinsq_1]),
    [file(afinsq_1,t17_afinsq_1),interesting(0.75)]).

fof(t29_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => k2_relat_1(k1_ordinal4(A,B)) = k2_xboole_0(k2_relat_1(A),k2_relat_1(B)) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_funct_1,d4_afinsq_1,t1_euler_1,t21_afinsq_1,t28_nat_1,t92_real_1,t1_euler_1,d1_afinsq_1,t12_funct_1,t1_euler_1,d1_afinsq_1,d4_afinsq_1,t12_funct_1,d2_xboole_0,d3_tarski,t27_afinsq_1,t28_afinsq_1,t8_xboole_1,d10_xboole_0]),
    [file(afinsq_1,t29_afinsq_1),interesting(0.74)]).

fof(t4_afinsq_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_xboole_0(A,k1_tarski(A)) = k1_nat_1(A,1) ) ),
    inference(mizar_proof,[status(thm)],[t52_card_1,d1_ordinal1]),
    [file(afinsq_1,t4_afinsq_1),interesting(0.74)]).

fof(t2_afinsq_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(A,B)
           => A = k3_xboole_0(A,B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t56_card_1,t28_xboole_1]),
    [file(afinsq_1,t2_afinsq_1),interesting(0.72)]).

fof(s3_afinsq_1,theorem,
    ( ( p1_s3_afinsq_1(k1_xboole_0)
      & ! [A] : 
          ( ( v1_relat_1(A)
            & v1_funct_1(A)
            & v5_ordinal1(A)
            & v1_finset_1(A) )
         => ! [B] : 
              ( p1_s3_afinsq_1(A)
             => p1_s3_afinsq_1(k1_ordinal4(A,k6_afinsq_1(B))) ) ) )
   => ! [A] : 
        ( ( v1_relat_1(A)
          & v1_funct_1(A)
          & v5_ordinal1(A)
          & v1_finset_1(A) )
       => p1_s3_afinsq_1(A) ) ),
    inference(mizar_proof,[status(thm)],[s1_real_1,t18_afinsq_1,t18_afinsq_1,t44_afinsq_1,t20_afinsq_1,t37_afinsq_1,s1_nat_1]),
    [file(afinsq_1,s3_afinsq_1),interesting(0.70)]).

fof(t7_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ( v1_finset_1(A)
          & v5_ordinal1(A) )
      <=> ? [B] : 
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & k1_relat_1(A) = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[t29_finset_1,d7_ordinal1,t7_card_4,t29_finset_1,d7_ordinal1]),
    [file(afinsq_1,t7_afinsq_1),interesting(0.69)]).

fof(t18_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ( k1_afinsq_1(A) = 0
      <=> A = k1_xboole_0 ) ) ),
    inference(mizar_proof,[status(thm)],[d5_card_1,t46_card_1]),
    [file(afinsq_1,t18_afinsq_1),interesting(0.69)]).

fof(t39_afinsq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B)
        & v1_finset_1(B) )
     => k1_funct_1(k1_ordinal4(k6_afinsq_1(A),B),0) = A ) ),
    inference(mizar_proof,[status(thm)],[t1_card_5,d1_tarski,d5_afinsq_1,d4_afinsq_1,d5_afinsq_1]),
    [file(afinsq_1,t39_afinsq_1),interesting(0.69)]).

fof(t36_afinsq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B)
        & v1_finset_1(B) )
     => ( B = k6_afinsq_1(A)
      <=> ( k2_afinsq_1(B) = 1
          & k2_relat_1(B) = k1_tarski(A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_afinsq_1,d5_afinsq_1,t1_card_5,t14_funct_1,d5_afinsq_1,t1_afinsq_1,t12_funct_1,d1_tarski,d5_afinsq_1]),
    [file(afinsq_1,t36_afinsq_1),interesting(0.69)]).

fof(t6_afinsq_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_nat_1(A,1) = k2_xboole_0(k1_tarski(0),k2_finseq_1(A)) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,t30_axioms,t38_nat_1,t31_xreal_1,t38_nat_1,t3_finseq_1,d1_tarski,d2_xboole_0,d10_xboole_0,t29_nat_1,t56_card_1,t1_card_5,t5_afinsq_1,t8_xboole_1]),
    [file(afinsq_1,t6_afinsq_1),interesting(0.69)]).

fof(t20_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => k1_afinsq_1(k1_ordinal4(A,B)) = k1_nat_1(k1_afinsq_1(A),k1_afinsq_1(B)) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_afinsq_1,d1_afinsq_1]),
    [file(afinsq_1,t20_afinsq_1),interesting(0.68)]).

fof(t8_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ~ ( ? [B] : 
              ( m2_subset_1(B,k1_numbers,k5_numbers)
              & r1_tarski(k1_relat_1(A),B) )
          & ! [B] : 
              ( ( v1_relat_1(B)
                & v1_funct_1(B)
                & v5_ordinal1(B)
                & v1_finset_1(B) )
             => ~ r1_tarski(A,B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s3_funct_1,t7_afinsq_1,d3_tarski,d1_relat_1,d4_relat_1,d4_funct_1,t8_funct_1]),
    [file(afinsq_1,t8_afinsq_1),interesting(0.68)]).

fof(t40_afinsq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B)
        & v1_finset_1(B) )
     => k1_funct_1(k1_ordinal4(B,k6_afinsq_1(A)),k1_afinsq_1(B)) = A ) ),
    inference(mizar_proof,[status(thm)],[d5_afinsq_1,t1_afinsq_1,d4_afinsq_1,d5_afinsq_1]),
    [file(afinsq_1,t40_afinsq_1),interesting(0.66)]).

fof(t42_afinsq_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C)
        & v5_ordinal1(C)
        & v1_finset_1(C) )
     => ( C = k7_afinsq_1(A,B)
      <=> ( k1_afinsq_1(C) = 2
          & k1_funct_1(C,0) = A
          & k1_funct_1(C,1) = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[t20_afinsq_1,t37_afinsq_1,t37_afinsq_1,d1_tarski,d5_afinsq_1,t1_card_5,d4_afinsq_1,d5_afinsq_1,d5_afinsq_1,t1_card_5,t37_afinsq_1,d4_afinsq_1,d5_afinsq_1,d1_afinsq_1,t37_afinsq_1,t37_afinsq_1,d5_afinsq_1,t1_card_5,d1_tarski,d5_afinsq_1,d5_afinsq_1,t1_card_5,t37_afinsq_1,d1_tarski,d5_afinsq_1,d1_tarski,d4_afinsq_1]),
    [file(afinsq_1,t42_afinsq_1),interesting(0.66)]).

fof(t3_afinsq_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( A = k3_xboole_0(A,B)
           => r1_xreal_0(A,B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t2_afinsq_1]),
    [file(afinsq_1,t3_afinsq_1),interesting(0.65)]).

fof(t37_afinsq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B)
        & v1_finset_1(B) )
     => ( B = k6_afinsq_1(A)
      <=> ( k1_afinsq_1(B) = 1
          & k2_relat_1(B) = k1_tarski(A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_afinsq_1,t36_afinsq_1]),
    [file(afinsq_1,t37_afinsq_1),interesting(0.64)]).

fof(s4_afinsq_1,theorem,(
    ? [A] : 
    ! [B] : 
      ( r2_hidden(B,A)
    <=> ? [C] : 
          ( v1_relat_1(C)
          & v1_funct_1(C)
          & v5_ordinal1(C)
          & v1_finset_1(C)
          & r2_hidden(C,k10_afinsq_1(f1_s4_afinsq_1))
          & p1_s4_afinsq_1(C)
          & B = C ) ) ),
    inference(mizar_proof,[status(thm)],[s1_xboole_0]),
    [file(afinsq_1,s4_afinsq_1),interesting(0.51)]).

fof(t25_afinsq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B)
        & v1_finset_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C)
            & v5_ordinal1(C)
            & v1_finset_1(C) )
         => ~ ( r2_hidden(A,k2_afinsq_1(B))
              & ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ~ ( D = A
                      & r2_hidden(k1_nat_1(k1_afinsq_1(C),D),k2_afinsq_1(k1_ordinal4(C,B))) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_afinsq_1,t1_euler_1,t10_xreal_1,t1_euler_1,d4_afinsq_1]),
    [file(afinsq_1,t25_afinsq_1),interesting(0.43)]).

fof(t26_afinsq_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C)
                & v1_finset_1(C) )
             => ( r2_hidden(A,k2_afinsq_1(B))
               => r2_hidden(k1_nat_1(k1_afinsq_1(C),A),k2_afinsq_1(k1_ordinal4(C,B))) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_afinsq_1]),
    [file(afinsq_1,t26_afinsq_1),interesting(0.43)]).

fof(t38_afinsq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B)
        & v1_finset_1(B) )
     => ( B = k6_afinsq_1(A)
      <=> ( k1_afinsq_1(B) = 1
          & k1_funct_1(B,0) = A ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_afinsq_1,d5_afinsq_1]),
    [file(afinsq_1,t38_afinsq_1),interesting(0.39)]).

fof(t10_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => ( ( k2_afinsq_1(A) = k2_afinsq_1(B)
              & ! [C] : 
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k2_afinsq_1(A))
                   => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) )
           => A = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[t9_funct_1]),
    [file(afinsq_1,t10_afinsq_1),interesting(0.35)]).

fof(t21_afinsq_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C)
                & v1_finset_1(C) )
             => ( r1_xreal_0(k1_afinsq_1(B),A)
               => ( r1_xreal_0(k1_nat_1(k1_afinsq_1(B),k1_afinsq_1(C)),A)
                  | k1_funct_1(k1_ordinal4(B,C),A) = k1_funct_1(C,k5_real_1(A,k1_afinsq_1(B))) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t28_nat_1,t92_real_1,t1_euler_1,d1_afinsq_1,d4_afinsq_1]),
    [file(afinsq_1,t21_afinsq_1),interesting(0.31)]).

fof(t9_afinsq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B)
        & v1_finset_1(B) )
     => ~ ( r2_hidden(A,B)
          & ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ~ ( r2_hidden(C,k2_afinsq_1(B))
                  & A = k4_tarski(C,k1_funct_1(B,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_relat_1,t8_funct_1,t8_funct_1]),
    [file(afinsq_1,t9_afinsq_1),interesting(0.28)]).

fof(s2_afinsq_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v5_ordinal1(A)
      & v1_finset_1(A)
      & k1_afinsq_1(A) = f1_s2_afinsq_1
      & ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_hidden(B,f1_s2_afinsq_1)
           => k1_funct_1(A,B) = f2_s2_afinsq_1(B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s3_funct_1,t7_afinsq_1,d1_afinsq_1]),
    [file(afinsq_1,s2_afinsq_1),interesting(0.25)]).

fof(t23_afinsq_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C)
                & v1_finset_1(C) )
             => ~ ( r2_hidden(A,k2_afinsq_1(k1_ordinal4(B,C)))
                  & ~ r2_hidden(A,k2_afinsq_1(B))
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ~ ( r2_hidden(D,k2_afinsq_1(C))
                          & A = k1_nat_1(k1_afinsq_1(B),D) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_afinsq_1,t20_afinsq_1,t1_euler_1,t1_euler_1,d1_afinsq_1,t28_nat_1,t92_real_1,t1_euler_1,d1_afinsq_1]),
    [file(afinsq_1,t23_afinsq_1),interesting(0.22)]).

fof(t11_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => ( ( k1_afinsq_1(A) = k1_afinsq_1(B)
              & ! [C] : 
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( ~ r1_xreal_0(k1_afinsq_1(A),C)
                   => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) )
           => A = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_afinsq_1,t1_euler_1,t9_funct_1]),
    [file(afinsq_1,t11_afinsq_1),interesting(0.21)]).

fof(t22_afinsq_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C)
                & v1_finset_1(C) )
             => ( r1_xreal_0(k1_afinsq_1(B),A)
               => ( r1_xreal_0(k1_afinsq_1(k1_ordinal4(B,C)),A)
                  | k1_funct_1(k1_ordinal4(B,C),A) = k1_funct_1(C,k5_real_1(A,k1_afinsq_1(B))) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t20_afinsq_1,t21_afinsq_1]),
    [file(afinsq_1,t22_afinsq_1),interesting(0.16)]).

fof(t14_afinsq_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C)
                & v1_finset_1(C) )
             => ( C = k2_ordinal1(B,A)
               => ( r1_xreal_0(k1_afinsq_1(B),A)
                  | ( k1_afinsq_1(C) = A
                    & k2_afinsq_1(C) = A ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t56_card_1,d1_afinsq_1,t91_relat_1,d1_afinsq_1]),
    [file(afinsq_1,t14_afinsq_1),interesting(0.15)]).

fof(t48_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( k1_afinsq_1(k2_funct_7(A,B,C)) = k1_afinsq_1(A)
              & ( ~ r1_xreal_0(k1_afinsq_1(A),B)
               => k1_funct_1(k2_funct_7(A,B,C),B) = C )
              & ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( D != B
                   => k1_funct_1(k2_funct_7(A,B,C),D) = k1_funct_1(A,D) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_afinsq_1,t32_funct_7,t56_card_1,t26_ordinal1,t33_funct_7,t34_funct_7]),
    [file(afinsq_1,t48_afinsq_1),interesting(0.10)]).

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

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

fof(s2_funct_1,theorem,
    ( ( ! [A,B,C] : 
          ( ( r2_hidden(A,f1_s2_funct_1)
            & p1_s2_funct_1(A,B)
            & p1_s2_funct_1(A,C) )
         => B = C )
      & ! [A] : ~ ( r2_hidden(A,f1_s2_funct_1)
          & ! [B] : ~ p1_s2_funct_1(A,B) ) )
   => ? [A] : 
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & k1_relat_1(A) = f1_s2_funct_1
        & ! [B] : 
            ( r2_hidden(B,f1_s2_funct_1)
           => p1_s2_funct_1(B,k1_funct_1(A,B)) ) ) ),
    file(funct_1,s2_funct_1),
    [interesting(0.00)]).

fof(t29_finset_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_finset_1(k1_relat_1(A))
      <=> v1_finset_1(A) ) ) ),
    file(finset_1,t29_finset_1),
    [interesting(0.00)]).

fof(d7_ordinal1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v5_ordinal1(A)
      <=> v3_ordinal1(k1_relat_1(A)) ) ) ),
    file(ordinal1,d7_ordinal1),
    [interesting(0.00)]).

fof(t7_card_4,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( v1_finset_1(A)
      <=> r2_hidden(A,k5_ordinal2) ) ) ),
    file(card_4,t7_card_4),
    [interesting(0.00)]).

fof(s1_afinsq_1,theorem,
    ( ( ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B,C] : 
              ( ( r2_hidden(A,f1_s1_afinsq_1)
                & p1_s1_afinsq_1(A,B)
                & p1_s1_afinsq_1(A,C) )
             => B = C ) )
      & ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ~ ( r2_hidden(A,f1_s1_afinsq_1)
              & ! [B] : ~ p1_s1_afinsq_1(A,B) ) ) )
   => ? [A] : 
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A)
        & k2_afinsq_1(A) = f1_s1_afinsq_1
        & ! [B] : 
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r2_hidden(B,f1_s1_afinsq_1)
             => p1_s1_afinsq_1(B,k1_funct_1(A,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t30_axioms,t30_axioms,s2_funct_1,t7_afinsq_1]),
    [file(afinsq_1,s1_afinsq_1),interesting(0.00)]).

fof(s3_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & k1_relat_1(A) = f1_s3_funct_1
      & ! [B] : 
          ( r2_hidden(B,f1_s3_funct_1)
         => k1_funct_1(A,B) = f2_s3_funct_1(B) ) ) ),
    file(funct_1,s3_funct_1),
    [interesting(0.00)]).

fof(d1_afinsq_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k1_afinsq_1(A)
          <=> B = k1_relat_1(A) ) ) ) ),
    file(afinsq_1,d1_afinsq_1),
    [interesting(0.00)]).

fof(s1_xboole_0,theorem,(
    ? [A] : 
    ! [B] : 
      ( r2_hidden(B,A)
    <=> ( r2_hidden(B,f1_s1_xboole_0)
        & p1_s1_xboole_0(B) ) ) ),
    file(xboole_0,s1_xboole_0),
    [interesting(0.00)]).

fof(t1_euler_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( r2_hidden(A,B)
          <=> ~ r1_xreal_0(B,A) ) ) ) ),
    file(euler_1,t1_euler_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(t46_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => ( r1_tarski(k2_relat_1(A),k1_relat_1(B))
           => k1_relat_1(k5_relat_1(A,B)) = k1_relat_1(A) ) ) ) ),
    file(relat_1,t46_relat_1),
    [interesting(0.00)]).

fof(t13_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => ( r1_tarski(k2_relat_1(B),k1_relat_1(A))
           => ( v1_relat_1(k5_relat_1(B,A))
              & v1_funct_1(k5_relat_1(B,A))
              & v5_ordinal1(k5_relat_1(B,A))
              & v1_finset_1(k5_relat_1(B,A)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t46_relat_1,d1_afinsq_1,t7_afinsq_1]),
    [file(afinsq_1,t13_afinsq_1),interesting(0.00)]).

fof(t56_card_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(A,B)
          <=> r1_ordinal1(A,B) ) ) ) ),
    file(card_1,t56_card_1),
    [interesting(0.00)]).

fof(t91_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => ( r1_tarski(A,k1_relat_1(B))
       => k1_relat_1(k7_relat_1(B,A)) = A ) ) ),
    file(relat_1,t91_relat_1),
    [interesting(0.00)]).

fof(d8_ordinal1,definition,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B) )
     => ( m1_ordinal1(B,A)
      <=> r1_tarski(k2_relat_1(B),A) ) ) ),
    file(ordinal1,d8_ordinal1),
    [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(t1_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(B,C) )
     => r1_tarski(A,C) ) ),
    file(xboole_1,t1_xboole_1),
    [interesting(0.00)]).

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

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

fof(t12_afinsq_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => ( v1_relat_1(k2_ordinal1(B,A))
            & v1_funct_1(k2_ordinal1(B,A))
            & v5_ordinal1(k2_ordinal1(B,A))
            & v1_finset_1(k2_ordinal1(B,A)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t90_relat_1,d1_afinsq_1,t2_afinsq_1,t7_afinsq_1]),
    [file(afinsq_1,t12_afinsq_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(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(t37_zfmisc_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(k1_tarski(A),B)
    <=> r2_hidden(A,B) ) ),
    file(zfmisc_1,t37_zfmisc_1),
    [interesting(0.00)]).

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

fof(d4_afinsq_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C) )
             => ( C = k1_ordinal4(A,B)
              <=> ( k1_relat_1(C) = k1_nat_1(k1_afinsq_1(A),k1_afinsq_1(B))
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k2_afinsq_1(A))
                       => k1_funct_1(C,D) = k1_funct_1(A,D) ) )
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k2_afinsq_1(B))
                       => k1_funct_1(C,k1_nat_1(k1_afinsq_1(A),D)) = k1_funct_1(B,D) ) ) ) ) ) ) ) ),
    file(afinsq_1,d4_afinsq_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(t92_real_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ! [D] : 
                  ( v1_xreal_0(D)
                 => ( ( ( r1_xreal_0(A,B)
                        & r1_xreal_0(C,D) )
                     => r1_xreal_0(k6_xcmplx_0(A,D),k6_xcmplx_0(B,C)) )
                    & ~ ( ( ( ~ r1_xreal_0(B,A)
                            & r1_xreal_0(C,D) )
                          | ( r1_xreal_0(A,B)
                            & ~ r1_xreal_0(D,C) ) )
                        & r1_xreal_0(k6_xcmplx_0(B,C),k6_xcmplx_0(A,D)) ) ) ) ) ) ) ),
    file(real_1,t92_real_1),
    [interesting(0.00)]).

fof(t31_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C)
                & v1_finset_1(C) )
             => ( ( k1_ordinal4(A,B) = k1_ordinal4(C,B)
                  | k1_ordinal4(B,A) = k1_ordinal4(B,C) )
               => A = C ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t20_afinsq_1,t20_afinsq_1,d1_afinsq_1,d1_afinsq_1,d4_afinsq_1,d4_afinsq_1,t10_afinsq_1,t20_afinsq_1,t20_afinsq_1,d1_afinsq_1,d1_afinsq_1,d4_afinsq_1,d4_afinsq_1,t10_afinsq_1]),
    [file(afinsq_1,t31_afinsq_1),interesting(0.00)]).

fof(d5_card_1,definition,(
    ! [A,B] : 
      ( v1_card_1(B)
     => ( B = k1_card_1(A)
      <=> r2_wellord2(A,B) ) ) ),
    file(card_1,d5_card_1),
    [interesting(0.00)]).

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

fof(t23_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( k2_xcmplx_0(A,B) = 0
           => ( A = 0
              & B = 0 ) ) ) ) ),
    file(nat_1,t23_nat_1),
    [interesting(0.00)]).

fof(t33_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => ( k1_ordinal4(A,B) = k1_xboole_0
           => ( A = k1_xboole_0
              & B = k1_xboole_0 ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t18_afinsq_1,t20_afinsq_1,t23_nat_1,t18_afinsq_1]),
    [file(afinsq_1,t33_afinsq_1),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(t12_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r2_hidden(A,k1_relat_1(B))
       => r2_hidden(k1_funct_1(B,A),k2_relat_1(B)) ) ) ),
    file(funct_1,t12_funct_1),
    [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(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_ordinal4,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C) )
             => ( C = k1_ordinal4(A,B)
              <=> ( k1_relat_1(C) = k14_ordinal2(k1_relat_1(A),k1_relat_1(B))
                  & ! [D] : 
                      ( v3_ordinal1(D)
                     => ( r2_hidden(D,k1_relat_1(A))
                       => k1_funct_1(C,D) = k1_funct_1(A,D) ) )
                  & ! [D] : 
                      ( v3_ordinal1(D)
                     => ( r2_hidden(D,k1_relat_1(B))
                       => k1_funct_1(C,k14_ordinal2(k1_relat_1(A),D)) = k1_funct_1(B,D) ) ) ) ) ) ) ) ),
    file(ordinal4,d1_ordinal4),
    [interesting(0.00)]).

fof(t27_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r1_ordinal1(A,k14_ordinal2(A,B))
            & r1_ordinal1(B,k14_ordinal2(A,B)) ) ) ) ),
    file(ordinal3,t27_ordinal3),
    [interesting(0.00)]).

fof(t10_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ! [D] : 
                  ( v1_xreal_0(D)
                 => ~ ( ~ r1_xreal_0(B,A)
                      & r1_xreal_0(C,D)
                      & r1_xreal_0(k2_xcmplx_0(B,D),k2_xcmplx_0(A,C)) ) ) ) ) ) ),
    file(xreal_1,t10_xreal_1),
    [interesting(0.00)]).

fof(t8_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,B) )
     => r1_tarski(k2_xboole_0(A,C),B) ) ),
    file(xboole_1,t8_xboole_1),
    [interesting(0.00)]).

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

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

fof(t34_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => ! [C] : 
              ( ( v1_finset_1(k1_ordinal4(A,B))
                & m1_ordinal1(k1_ordinal4(A,B),C) )
             => ( v1_finset_1(A)
                & m1_ordinal1(A,C)
                & v1_finset_1(B)
                & m1_ordinal1(B,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_ordinal1,t29_afinsq_1,t7_xboole_1,t1_xboole_1,d8_ordinal1,t7_xboole_1,t1_xboole_1,d8_ordinal1]),
    [file(afinsq_1,t34_afinsq_1),interesting(0.00)]).

fof(t35_afinsq_1,theorem,(
    ! [A] : k6_afinsq_1(A) = k1_tarski(k4_tarski(0,A)) ),
    file(afinsq_1,t35_afinsq_1),
    [interesting(0.00)]).

fof(d5_afinsq_1,definition,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( B = k6_afinsq_1(A)
      <=> ( k1_relat_1(B) = 1
          & k1_funct_1(B,0) = A ) ) ) ),
    file(afinsq_1,d5_afinsq_1),
    [interesting(0.00)]).

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

fof(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(t14_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( k1_relat_1(B) = k1_tarski(A)
       => k2_relat_1(B) = k1_tarski(k1_funct_1(B,A)) ) ) ),
    file(funct_1,t14_funct_1),
    [interesting(0.00)]).

fof(t30_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C)
                & v1_finset_1(C) )
             => k1_ordinal4(k1_ordinal4(A,B),C) = k1_ordinal4(A,k1_ordinal4(B,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_afinsq_1,t20_afinsq_1,t20_afinsq_1,t24_afinsq_1,d4_afinsq_1,d4_afinsq_1,t26_afinsq_1,d4_afinsq_1,d4_afinsq_1,d4_afinsq_1,t23_afinsq_1,t20_afinsq_1,d4_afinsq_1,d4_afinsq_1,d4_afinsq_1]),
    [file(afinsq_1,t30_afinsq_1),interesting(0.00)]).

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

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

fof(s1_real_1,theorem,(
    ? [A] : 
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
      & ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ( r2_hidden(B,A)
          <=> p1_s1_real_1(B) ) ) ) ),
    file(real_1,s1_real_1),
    [interesting(0.00)]).

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

fof(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(t52_card_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_ordinal1(A) = k1_nat_1(A,1) ) ),
    file(card_1,t52_card_1),
    [interesting(0.00)]).

fof(d1_ordinal1,definition,(
    ! [A] : k1_ordinal1(A) = k2_xboole_0(A,k1_tarski(A)) ),
    file(ordinal1,d1_ordinal1),
    [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(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(t45_afinsq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_finset_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_finset_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C)
                & v1_finset_1(C) )
             => ! [D] : 
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v5_ordinal1(D)
                    & v1_finset_1(D) )
                 => ~ ( k1_ordinal4(A,B) = k1_ordinal4(C,D)
                      & r1_xreal_0(k1_afinsq_1(A),k1_afinsq_1(C))
                      & ! [E] : 
                          ( ( v1_relat_1(E)
                            & v1_funct_1(E)
                            & v5_ordinal1(E)
                            & v1_finset_1(E) )
                         => k1_ordinal4(A,E) != C ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t18_afinsq_1,t32_afinsq_1,t18_afinsq_1,d1_afinsq_1,d1_afinsq_1,d4_afinsq_1,d4_afinsq_1,t32_afinsq_1,t10_afinsq_1,t20_afinsq_1,t37_afinsq_1,t20_afinsq_1,t37_afinsq_1,t26_nat_1,t30_afinsq_1,t30_afinsq_1,s3_afinsq_1]),
    [file(afinsq_1,t45_afinsq_1),interesting(0.00)]).

fof(d8_afinsq_1,definition,(
    ! [A,B] : 
      ( B = k10_afinsq_1(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> ( v1_finset_1(C)
            & m1_ordinal1(C,A) ) ) ) ),
    file(afinsq_1,d8_afinsq_1),
    [interesting(0.00)]).

fof(t32_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : k1_relat_1(k2_funct_7(A,C,B)) = k1_relat_1(A) ) ),
    file(funct_7,t32_funct_7),
    [interesting(0.00)]).

fof(t26_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r1_ordinal1(A,B)
            | r2_hidden(B,A) ) ) ) ),
    file(ordinal1,t26_ordinal1),
    [interesting(0.00)]).

fof(t33_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : 
          ( r2_hidden(C,k1_relat_1(A))
         => k1_funct_1(k2_funct_7(A,C,B),C) = B ) ) ),
    file(funct_7,t33_funct_7),
    [interesting(0.00)]).

fof(t34_funct_7,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C,D] : 
          ( C != D
         => k1_funct_1(k2_funct_7(A,C,B),D) = k1_funct_1(A,D) ) ) ),
    file(funct_7,t34_funct_7),
    [interesting(0.00)]).

fof(t31_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ~ ( ~ r1_xreal_0(A,0)
              & r1_xreal_0(k2_xcmplx_0(B,A),B) ) ) ) ),
    file(xreal_1,t31_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(d1_relat_1,definition,(
    ! [A] : 
      ( v1_relat_1(A)
    <=> ! [B] : ~ ( r2_hidden(B,A)
          & ! [C,D] : B != k4_tarski(C,D) ) ) ),
    file(relat_1,d1_relat_1),
    [interesting(0.00)]).

fof(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(d4_funct_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : 
          ( ( r2_hidden(B,k1_relat_1(A))
           => ( C = k1_funct_1(A,B)
            <=> r2_hidden(k4_tarski(B,C),A) ) )
          & ( ~ r2_hidden(B,k1_relat_1(A))
           => ( C = k1_funct_1(A,B)
            <=> C = k1_xboole_0 ) ) ) ) ),
    file(funct_1,d4_funct_1),
    [interesting(0.00)]).

fof(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)]).