fof(t45_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v3_toprns_1(B,A)
           => v3_toprns_1(k3_toprns_1(A,B),A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t43_toprns_1,t12_toprns_1]),
    [file(toprns_1,t45_toprns_1),interesting(1.00)]).

fof(t24_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k5_toprns_1(A,k16_euclid(A)) = 0 ) ),
    inference(mizar_proof,[status(thm)],[d9_euclid,d6_toprns_1,t10_euclid]),
    [file(toprns_1,t24_toprns_1),interesting(0.98)]).

fof(t18_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => k3_toprns_1(A,k3_toprns_1(A,B)) = B ) ) ),
    inference(mizar_proof,[status(thm)],[t12_toprns_1,t12_toprns_1,t14_toprns_1,t17_toprns_1]),
    [file(toprns_1,t18_toprns_1),interesting(0.94)]).

fof(t27_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => k5_toprns_1(A,k19_euclid(A,B)) = k5_toprns_1(A,B) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_euclid,d12_euclid,d6_toprns_1,t13_euclid,d6_toprns_1]),
    [file(toprns_1,t27_toprns_1),interesting(0.92)]).

fof(t46_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v3_toprns_1(B,A)
           => k7_toprns_1(A,k3_toprns_1(A,B)) = k19_euclid(A,k7_toprns_1(A,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t44_toprns_1,t44_euclid,t33_euclid,t12_toprns_1]),
    [file(toprns_1,t46_toprns_1),interesting(0.91)]).

fof(t43_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( v3_toprns_1(C,A)
               => v3_toprns_1(k2_toprns_1(B,A,C),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_toprns_1,t33_euclid,t28_toprns_1,t45_euclid,t31_euclid,t27_toprns_1,d3_toprns_1,t33_euclid,t24_toprns_1,t133_complex1,t133_complex1,t73_real_1,d3_toprns_1,t53_euclid,t8_toprns_1,d9_xcmplx_0,d7_xcmplx_0,t70_xreal_1,d9_toprns_1]),
    [file(toprns_1,t43_toprns_1),interesting(0.88)]).

fof(t28_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => k5_toprns_1(A,k20_euclid(A,B,C)) = k5_toprns_1(A,k20_euclid(A,C,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t27_toprns_1,t48_euclid]),
    [file(toprns_1,t28_toprns_1),interesting(0.86)]).

fof(t23_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v1_toprns_1(B,A)
           => v1_toprns_1(k3_toprns_1(A,B),A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t22_toprns_1,t12_toprns_1]),
    [file(toprns_1,t23_toprns_1),interesting(0.85)]).

fof(t26_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => r1_xreal_0(0,k5_toprns_1(A,B)) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_euclid,t12_euclid,d6_toprns_1]),
    [file(toprns_1,t26_toprns_1),interesting(0.84)]).

fof(t25_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ( k5_toprns_1(A,B) = 0
           => B = k16_euclid(A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_euclid,d6_toprns_1,t11_euclid,d9_euclid]),
    [file(toprns_1,t25_toprns_1),interesting(0.82)]).

fof(t50_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v3_toprns_1(B,A)
           => v2_toprns_1(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_toprns_1,t26_toprns_1,t10_xreal_1,t33_toprns_1,t2_xreal_1,t8_xreal_1,t39_toprns_1,t10_xreal_1,t10_xreal_1,t10_xreal_1,t2_xreal_1,t2_xreal_1,d8_toprns_1]),
    [file(toprns_1,t50_toprns_1),interesting(0.81)]).

fof(l36_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ( k5_toprns_1(A,k20_euclid(A,B,C)) = 0
               => B = C ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_toprns_1,t47_euclid]),
    [file(toprns_1,l36_toprns_1),interesting(0.80)]).

fof(l37_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ( B = C
               => k5_toprns_1(A,k20_euclid(A,B,C)) = 0 ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t46_euclid,t24_toprns_1]),
    [file(toprns_1,l37_toprns_1),interesting(0.77)]).

fof(t44_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( v3_toprns_1(C,A)
               => k7_toprns_1(A,k2_toprns_1(B,A,C)) = k18_euclid(B,A,k7_toprns_1(A,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t43_toprns_1,t33_euclid,d3_toprns_1,t28_toprns_1,t45_euclid,t31_euclid,t27_toprns_1,t33_euclid,t24_toprns_1,t133_complex1,t133_complex1,t73_real_1,d10_toprns_1,d3_toprns_1,t53_euclid,t8_toprns_1,d9_xcmplx_0,d7_xcmplx_0,t70_xreal_1,d10_toprns_1]),
    [file(toprns_1,t44_toprns_1),interesting(0.77)]).

fof(t29_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ( k5_toprns_1(A,k20_euclid(A,B,C)) = 0
              <=> B = C ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[l36_toprns_1,l37_toprns_1]),
    [file(toprns_1,t29_toprns_1),interesting(0.74)]).

fof(t8_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => k4_real_1(k18_complex1(B),k5_toprns_1(A,C)) = k5_toprns_1(A,k18_euclid(B,A,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_euclid,d11_euclid,d6_toprns_1,t14_euclid]),
    [file(toprns_1,t8_toprns_1),interesting(0.73)]).

fof(t47_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( ( v3_toprns_1(B,A)
                  & v3_toprns_1(C,A) )
               => v3_toprns_1(k4_toprns_1(A,B,C),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t45_toprns_1,t41_toprns_1]),
    [file(toprns_1,t47_toprns_1),interesting(0.72)]).

fof(t48_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( ( v3_toprns_1(B,A)
                  & v3_toprns_1(C,A) )
               => k7_toprns_1(A,k4_toprns_1(A,B,C)) = k20_euclid(A,k7_toprns_1(A,B),k7_toprns_1(A,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t45_toprns_1,t42_toprns_1,t46_toprns_1,t45_euclid]),
    [file(toprns_1,t48_toprns_1),interesting(0.72)]).

fof(t22_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( v1_toprns_1(C,A)
               => ( B = 0
                  | v1_toprns_1(k2_toprns_1(B,A,C),A) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t4_toprns_1,d3_toprns_1,t4_toprns_1,t35_euclid]),
    [file(toprns_1,t22_toprns_1),interesting(0.67)]).

fof(t35_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => r1_xreal_0(k5_toprns_1(A,k20_euclid(A,B,C)),k3_real_1(k5_toprns_1(A,k20_euclid(A,B,D)),k5_toprns_1(A,k20_euclid(A,D,C)))) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t46_euclid,t45_euclid,t31_euclid,t30_euclid,t45_euclid,t49_euclid,t30_toprns_1]),
    [file(toprns_1,t35_toprns_1),interesting(0.63)]).

fof(t12_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => k3_toprns_1(A,B) = k2_toprns_1(k1_real_1(1),A,B) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_toprns_1,t43_euclid,d4_toprns_1,t6_toprns_1]),
    [file(toprns_1,t12_toprns_1),interesting(0.62)]).

fof(t34_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ~ ( B != C
                  & r1_xreal_0(k5_toprns_1(A,k20_euclid(A,B,C)),0) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_euclid,d13_euclid,t20_euclid,d6_toprns_1]),
    [file(toprns_1,t34_toprns_1),interesting(0.62)]).

fof(t4_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v1_toprns_1(B,A)
          <=> ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_normsp_1(k15_euclid(A),B,C) != k16_euclid(A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t3_toprns_1,t3_toprns_1]),
    [file(toprns_1,t4_toprns_1),interesting(0.61)]).

fof(t3_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v1_toprns_1(B,A)
          <=> ! [C] : ~ ( r2_hidden(C,k5_numbers)
                & k1_funct_1(B,C) = k16_euclid(A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_toprns_1,t2_toprns_1,d5_funct_1,d4_xboole_0,d1_tarski,d5_funct_1,t17_normsp_1,t17_normsp_1,d1_tarski,d4_xboole_0,d3_tarski,d1_toprns_1]),
    [file(toprns_1,t3_toprns_1),interesting(0.56)]).

fof(t42_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( ( v3_toprns_1(B,A)
                  & v3_toprns_1(C,A) )
               => k7_toprns_1(A,k1_toprns_1(A,B,C)) = k17_euclid(A,k7_toprns_1(A,B),k7_toprns_1(A,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t41_toprns_1,t3_seq_2,d10_toprns_1,d10_toprns_1,t37_nat_1,t2_xreal_1,t37_nat_1,t2_xreal_1,t10_xreal_1,t50_euclid,d2_toprns_1,t49_euclid,t45_euclid,t30_euclid,t45_euclid,t49_euclid,t30_toprns_1,t2_xreal_1,d10_toprns_1]),
    [file(toprns_1,t42_toprns_1),interesting(0.54)]).

fof(t15_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
                 => k2_toprns_1(B,A,k4_toprns_1(A,C,D)) = k4_toprns_1(A,k2_toprns_1(B,A,C),k2_toprns_1(B,A,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t13_toprns_1,t12_toprns_1,t14_toprns_1,t14_toprns_1,t12_toprns_1]),
    [file(toprns_1,t15_toprns_1),interesting(0.53)]).

fof(t19_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => k4_toprns_1(A,B,k3_toprns_1(A,C)) = k1_toprns_1(A,B,C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t18_toprns_1]),
    [file(toprns_1,t19_toprns_1),interesting(0.53)]).

fof(t39_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ? [D] : 
                  ( m1_subset_1(D,k1_numbers)
                  & ~ r1_xreal_0(D,0)
                  & ! [E] : 
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ~ ( r1_xreal_0(E,C)
                          & r1_xreal_0(D,k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,E))) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t26_toprns_1,t10_xreal_1,t18_nat_1,t10_xreal_1,t10_xreal_1,t10_xreal_1,t2_xreal_1,t10_xreal_1,t2_xreal_1,t26_nat_1,t26_toprns_1,t10_xreal_1,t2_xreal_1,t10_xreal_1,t10_xreal_1,t26_nat_1,s1_nat_1]),
    [file(toprns_1,t39_toprns_1),interesting(0.46)]).

fof(t2_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
              & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
          <=> ( k1_relat_1(B) = k5_numbers
              & ! [C] : 
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => m1_subset_1(k1_funct_1(B,C),u1_struct_0(k15_euclid(A))) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t17_normsp_1,t17_normsp_1]),
    [file(toprns_1,t2_toprns_1),interesting(0.44)]).

fof(t9_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => k6_toprns_1(A,k2_toprns_1(B,A,C)) = k14_seq_1(k6_toprns_1(A,C),k18_complex1(B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d7_toprns_1,d3_toprns_1,t8_toprns_1,d7_toprns_1,t13_seq_1,t113_funct_2]),
    [file(toprns_1,t9_toprns_1),interesting(0.40)]).

fof(s1_zfrefle1,theorem,
    ( ! [A] : ~ ( r2_hidden(A,f1_s1_zfrefle1)
        & ! [B] : ~ p1_s1_zfrefle1(A,B) )
   => ? [A] : 
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & k1_relat_1(A) = f1_s1_zfrefle1
        & ! [B] : 
            ( r2_hidden(B,f1_s1_zfrefle1)
           => p1_s1_zfrefle1(B,k1_funct_1(A,B)) ) ) ),
    file(zfrefle1,s1_zfrefle1),
    [interesting(0.00)]).

fof(t17_normsp_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & l1_struct_0(B) )
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(B))
             => ( ( v1_funct_1(A)
                  & v1_funct_2(A,k5_numbers,u1_struct_0(B))
                  & m2_relset_1(A,k5_numbers,u1_struct_0(B)) )
              <=> ( k1_relat_1(A) = k5_numbers
                  & ! [D] : 
                      ( r2_hidden(D,k5_numbers)
                     => m1_subset_1(k1_funct_1(A,D),u1_struct_0(B)) ) ) ) ) ) ) ),
    file(normsp_1,t17_normsp_1),
    [interesting(0.00)]).

fof(s1_toprns_1,theorem,(
    ? [A] : 
      ( v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,u1_struct_0(k15_euclid(f1_s1_toprns_1)))
      & m2_relset_1(A,k5_numbers,u1_struct_0(k15_euclid(f1_s1_toprns_1)))
      & ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_normsp_1(k15_euclid(f1_s1_toprns_1),A,B) = f2_s1_toprns_1(B) ) ) ),
    inference(mizar_proof,[status(thm)],[s1_zfrefle1,t17_normsp_1]),
    [file(toprns_1,s1_toprns_1),interesting(0.00)]).

fof(d2_toprns_1,definition,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
                 => ( D = k1_toprns_1(A,B,C)
                  <=> ! [E] : 
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(k15_euclid(A),D,E) = k17_euclid(A,k2_normsp_1(k15_euclid(A),B,E),k2_normsp_1(k15_euclid(A),C,E)) ) ) ) ) ) ) ),
    file(toprns_1,d2_toprns_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(t5_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( ! [D] : 
                    ( r2_hidden(D,k5_numbers)
                   => k1_funct_1(B,D) = k1_funct_1(C,D) )
               => B = C ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t17_normsp_1,t9_funct_1]),
    [file(toprns_1,t5_toprns_1),interesting(0.00)]).

fof(t6_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_normsp_1(k15_euclid(A),B,D) = k2_normsp_1(k15_euclid(A),C,D) )
               => B = C ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t5_toprns_1]),
    [file(toprns_1,t6_toprns_1),interesting(0.00)]).

fof(t10_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => k1_toprns_1(A,B,C) = k1_toprns_1(A,C,B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_toprns_1,d2_toprns_1,t6_toprns_1]),
    [file(toprns_1,t10_toprns_1),interesting(0.00)]).

fof(d3_toprns_1,definition,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(B)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(B))) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(B)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(B))) )
                 => ( D = k2_toprns_1(A,B,C)
                  <=> ! [E] : 
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(k15_euclid(B),D,E) = k18_euclid(A,B,k2_normsp_1(k15_euclid(B),C,E)) ) ) ) ) ) ) ),
    file(toprns_1,d3_toprns_1),
    [interesting(0.00)]).

fof(t43_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => k19_euclid(A,B) = k18_euclid(k1_real_1(1),A,B) ) ) ),
    file(euclid,t43_euclid),
    [interesting(0.00)]).

fof(d4_toprns_1,definition,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( C = k3_toprns_1(A,B)
              <=> ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_normsp_1(k15_euclid(A),C,D) = k19_euclid(A,k2_normsp_1(k15_euclid(A),B,D)) ) ) ) ) ) ),
    file(toprns_1,d4_toprns_1),
    [interesting(0.00)]).

fof(t36_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] : 
                  ( v1_xreal_0(D)
                 => k18_euclid(D,A,k17_euclid(A,B,C)) = k17_euclid(A,k18_euclid(D,A,B),k18_euclid(D,A,C)) ) ) ) ) ),
    file(euclid,t36_euclid),
    [interesting(0.00)]).

fof(t13_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
                 => k2_toprns_1(B,A,k1_toprns_1(A,C,D)) = k1_toprns_1(A,k2_toprns_1(B,A,C),k2_toprns_1(B,A,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_toprns_1,d2_toprns_1,t36_euclid,d3_toprns_1,d3_toprns_1,d2_toprns_1,t6_toprns_1]),
    [file(toprns_1,t13_toprns_1),interesting(0.00)]).

fof(t30_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => k17_euclid(A,k17_euclid(A,B,C),D) = k17_euclid(A,B,k17_euclid(A,C,D)) ) ) ) ) ),
    file(euclid,t30_euclid),
    [interesting(0.00)]).

fof(t11_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
                 => k1_toprns_1(A,k1_toprns_1(A,B,C),D) = k1_toprns_1(A,B,k1_toprns_1(A,C,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_toprns_1,d2_toprns_1,t30_euclid,d2_toprns_1,d2_toprns_1,t6_toprns_1]),
    [file(toprns_1,t11_toprns_1),interesting(0.00)]).

fof(t16_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
                 => k4_toprns_1(A,B,k1_toprns_1(A,C,D)) = k4_toprns_1(A,k4_toprns_1(A,B,C),D) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t12_toprns_1,t13_toprns_1,t12_toprns_1,t12_toprns_1,t11_toprns_1]),
    [file(toprns_1,t16_toprns_1),interesting(0.00)]).

fof(t1_toprns_1,theorem,(
    $true ),
    file(toprns_1,t1_toprns_1),
    [interesting(0.00)]).

fof(t34_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( v1_xreal_0(C)
             => ! [D] : 
                  ( v1_xreal_0(D)
                 => k18_euclid(k3_xcmplx_0(C,D),A,B) = k18_euclid(C,A,k18_euclid(D,A,B)) ) ) ) ) ),
    file(euclid,t34_euclid),
    [interesting(0.00)]).

fof(t14_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( m1_subset_1(C,k1_numbers)
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
                 => k2_toprns_1(k4_real_1(B,C),A,D) = k2_toprns_1(B,A,k2_toprns_1(C,A,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_toprns_1,t34_euclid,d3_toprns_1,d3_toprns_1,t6_toprns_1]),
    [file(toprns_1,t14_toprns_1),interesting(0.00)]).

fof(t33_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ( k18_euclid(1,A,B) = B
            & k18_euclid(0,A,B) = k16_euclid(A) ) ) ) ),
    file(euclid,t33_euclid),
    [interesting(0.00)]).

fof(t17_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => k2_toprns_1(1,A,B) = B ) ) ),
    inference(mizar_proof,[status(thm)],[d3_toprns_1,t33_euclid,t6_toprns_1]),
    [file(toprns_1,t17_toprns_1),interesting(0.00)]).

fof(t20_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
                 => k4_toprns_1(A,B,k4_toprns_1(A,C,D)) = k1_toprns_1(A,k4_toprns_1(A,B,C),D) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t12_toprns_1,t15_toprns_1,t12_toprns_1,t12_toprns_1,t19_toprns_1,t11_toprns_1]),
    [file(toprns_1,t20_toprns_1),interesting(0.00)]).

fof(t21_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
                 => k1_toprns_1(A,B,k4_toprns_1(A,C,D)) = k4_toprns_1(A,k1_toprns_1(A,B,C),D) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t11_toprns_1]),
    [file(toprns_1,t21_toprns_1),interesting(0.00)]).

fof(d1_toprns_1,definition,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v1_toprns_1(B,A)
          <=> r1_tarski(k2_relat_1(B),k4_xboole_0(u1_struct_0(k15_euclid(A)),k1_struct_0(k15_euclid(A),k16_euclid(A)))) ) ) ) ),
    file(toprns_1,d1_toprns_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(d4_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k4_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & ~ r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d4_xboole_0),
    [interesting(0.00)]).

fof(d1_tarski,definition,(
    ! [A,B] : 
      ( B = k1_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> C = A ) ) ),
    file(tarski,d1_tarski),
    [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(t35_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( v1_xreal_0(C)
             => ~ ( k18_euclid(C,A,B) = k16_euclid(A)
                  & C != 0
                  & B != k16_euclid(A) ) ) ) ) ),
    file(euclid,t35_euclid),
    [interesting(0.00)]).

fof(t25_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => u1_struct_0(k15_euclid(A)) = k1_euclid(A) ) ),
    file(euclid,t25_euclid),
    [interesting(0.00)]).

fof(d6_toprns_1,definition,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,k1_numbers)
             => ( C = k5_toprns_1(A,B)
              <=> ? [D] : 
                    ( m2_finseq_1(D,k1_numbers)
                    & B = D
                    & C = k12_euclid(D) ) ) ) ) ) ),
    file(toprns_1,d6_toprns_1),
    [interesting(0.00)]).

fof(t11_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ( k12_euclid(B) = 0
           => B = k5_euclid(A) ) ) ) ),
    file(euclid,t11_euclid),
    [interesting(0.00)]).

fof(d9_euclid,definition,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k16_euclid(A) = k5_euclid(A) ) ),
    file(euclid,d9_euclid),
    [interesting(0.00)]).

fof(t47_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ( k20_euclid(A,B,C) = k16_euclid(A)
               => B = C ) ) ) ) ),
    file(euclid,t47_euclid),
    [interesting(0.00)]).

fof(t46_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => k20_euclid(A,B,B) = k16_euclid(A) ) ) ),
    file(euclid,t46_euclid),
    [interesting(0.00)]).

fof(t10_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k12_euclid(k5_euclid(A)) = 0 ) ),
    file(euclid,t10_euclid),
    [interesting(0.00)]).

fof(d13_euclid,definition,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ( D = k20_euclid(A,B,C)
                  <=> ! [E] : 
                        ( m2_finseq_2(E,k1_numbers,k1_euclid(A))
                       => ! [F] : 
                            ( m2_finseq_2(F,k1_numbers,k1_euclid(A))
                           => ( ( E = B
                                & F = C )
                             => D = k8_euclid(A,E,F) ) ) ) ) ) ) ) ) ),
    file(euclid,d13_euclid),
    [interesting(0.00)]).

fof(t16_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] : 
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => r1_xreal_0(k12_euclid(k8_euclid(A,B,C)),k3_real_1(k12_euclid(B),k12_euclid(C))) ) ) ) ),
    file(euclid,t16_euclid),
    [interesting(0.00)]).

fof(t31_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => r1_xreal_0(k5_toprns_1(A,k20_euclid(A,B,C)),k3_real_1(k5_toprns_1(A,B),k5_toprns_1(A,C))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_euclid,d13_euclid,d6_toprns_1,t16_euclid,d6_toprns_1,d6_toprns_1]),
    [file(toprns_1,t31_toprns_1),interesting(0.00)]).

fof(d10_euclid,definition,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ( D = k17_euclid(A,B,C)
                  <=> ! [E] : 
                        ( m2_finseq_2(E,k1_numbers,k1_euclid(A))
                       => ! [F] : 
                            ( m2_finseq_2(F,k1_numbers,k1_euclid(A))
                           => ( ( E = B
                                & F = C )
                             => D = k7_euclid(A,E,F) ) ) ) ) ) ) ) ) ),
    file(euclid,d10_euclid),
    [interesting(0.00)]).

fof(t17_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] : 
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => r1_xreal_0(k5_real_1(k12_euclid(B),k12_euclid(C)),k12_euclid(k7_euclid(A,B,C))) ) ) ) ),
    file(euclid,t17_euclid),
    [interesting(0.00)]).

fof(t32_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => r1_xreal_0(k5_real_1(k5_toprns_1(A,B),k5_toprns_1(A,C)),k5_toprns_1(A,k17_euclid(A,B,C))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_euclid,d10_euclid,d6_toprns_1,t17_euclid,d6_toprns_1,d6_toprns_1]),
    [file(toprns_1,t32_toprns_1),interesting(0.00)]).

fof(t20_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] : 
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => ~ ( B != C
                  & r1_xreal_0(k12_euclid(k8_euclid(A,B,C)),0) ) ) ) ) ),
    file(euclid,t20_euclid),
    [interesting(0.00)]).

fof(t45_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => k20_euclid(A,B,C) = k17_euclid(A,B,k19_euclid(A,C)) ) ) ) ),
    file(euclid,t45_euclid),
    [interesting(0.00)]).

fof(t31_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ( k17_euclid(A,k16_euclid(A),B) = B
            & k17_euclid(A,B,k16_euclid(A)) = B ) ) ) ),
    file(euclid,t31_euclid),
    [interesting(0.00)]).

fof(t49_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => k17_euclid(A,B,k20_euclid(A,C,D)) = k20_euclid(A,k17_euclid(A,B,C),D) ) ) ) ) ),
    file(euclid,t49_euclid),
    [interesting(0.00)]).

fof(t15_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] : 
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => r1_xreal_0(k12_euclid(k7_euclid(A,B,C)),k3_real_1(k12_euclid(B),k12_euclid(C))) ) ) ) ),
    file(euclid,t15_euclid),
    [interesting(0.00)]).

fof(t30_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => r1_xreal_0(k5_toprns_1(A,k17_euclid(A,B,C)),k3_real_1(k5_toprns_1(A,B),k5_toprns_1(A,C))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_euclid,d10_euclid,d6_toprns_1,t15_euclid,d6_toprns_1,d6_toprns_1]),
    [file(toprns_1,t30_toprns_1),interesting(0.00)]).

fof(t70_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ~ ( ~ r1_xreal_0(A,0)
                  & ~ r1_xreal_0(C,B)
                  & r1_xreal_0(k3_xcmplx_0(C,A),k3_xcmplx_0(B,A)) ) ) ) ) ),
    file(xreal_1,t70_xreal_1),
    [interesting(0.00)]).

fof(t12_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => r1_xreal_0(0,k12_euclid(B)) ) ) ),
    file(euclid,t12_euclid),
    [interesting(0.00)]).

fof(t66_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(0,C) )
               => r1_xreal_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) ) ) ) ) ),
    file(xreal_1,t66_xreal_1),
    [interesting(0.00)]).

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

fof(t36_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( m1_subset_1(C,k1_numbers)
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ! [E] : 
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
                     => ~ ( r1_xreal_0(0,B)
                          & ~ r1_xreal_0(k5_toprns_1(A,E),k5_toprns_1(A,D))
                          & ~ r1_xreal_0(C,B)
                          & r1_xreal_0(k4_real_1(k5_toprns_1(A,E),C),k4_real_1(k5_toprns_1(A,D),B)) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t70_xreal_1,t26_toprns_1,t66_xreal_1,t2_xreal_1]),
    [file(toprns_1,t36_toprns_1),interesting(0.00)]).

fof(t37_toprns_1,theorem,(
    $true ),
    file(toprns_1,t37_toprns_1),
    [interesting(0.00)]).

fof(t12_absvalue,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(k4_xcmplx_0(A),B)
              & r1_xreal_0(B,A) )
          <=> r1_xreal_0(k18_complex1(B),A) ) ) ) ),
    file(absvalue,t12_absvalue),
    [interesting(0.00)]).

fof(d1_absvalue,definition,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ( ( r1_xreal_0(0,A)
         => k16_complex1(A) = A )
        & ( ~ r1_xreal_0(0,A)
         => k16_complex1(A) = k4_xcmplx_0(A) ) ) ) ),
    file(absvalue,d1_absvalue),
    [interesting(0.00)]).

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

fof(t132_complex1,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => r1_xreal_0(0,k17_complex1(A)) ) ),
    file(complex1,t132_complex1),
    [interesting(0.00)]).

fof(t26_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
          <=> r1_xreal_0(k4_xcmplx_0(B),k4_xcmplx_0(A)) ) ) ) ),
    file(xreal_1,t26_xreal_1),
    [interesting(0.00)]).

fof(t38_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ( ( ~ r1_xreal_0(B,k1_real_1(k5_toprns_1(A,C)))
                  & ~ r1_xreal_0(k5_toprns_1(A,C),B) )
              <=> ~ r1_xreal_0(k5_toprns_1(A,C),k18_complex1(B)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t12_absvalue,d1_absvalue,d1_absvalue,d5_real_1,t12_absvalue,t12_absvalue,t132_complex1,t132_complex1,t26_xreal_1,d1_absvalue,d5_real_1,d1_absvalue,d5_real_1]),
    [file(toprns_1,t38_toprns_1),interesting(0.00)]).

fof(t40_toprns_1,theorem,(
    $true ),
    file(toprns_1,t40_toprns_1),
    [interesting(0.00)]).

fof(d9_toprns_1,definition,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v3_toprns_1(B,A)
          <=> ? [C] : 
                ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
                & ! [D] : 
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(D,0)
                        & ! [E] : 
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ? [F] : 
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                                & r1_xreal_0(E,F)
                                & r1_xreal_0(D,k5_toprns_1(A,k20_euclid(A,k2_normsp_1(k15_euclid(A),B,F),C))) ) ) ) ) ) ) ) ) ),
    file(toprns_1,d9_toprns_1),
    [interesting(0.00)]).

fof(d12_euclid,definition,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ( C = k19_euclid(A,B)
              <=> ! [D] : 
                    ( m2_finseq_2(D,k1_numbers,k1_euclid(A))
                   => ( D = B
                     => C = k6_euclid(A,D) ) ) ) ) ) ) ),
    file(euclid,d12_euclid),
    [interesting(0.00)]).

fof(t13_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => k12_euclid(k6_euclid(A,B)) = k12_euclid(B) ) ) ),
    file(euclid,t13_euclid),
    [interesting(0.00)]).

fof(t48_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ( k19_euclid(A,k20_euclid(A,B,C)) = k20_euclid(A,C,B)
                & k19_euclid(A,k20_euclid(A,B,C)) = k17_euclid(A,k19_euclid(A,B),C) ) ) ) ) ),
    file(euclid,t48_euclid),
    [interesting(0.00)]).

fof(t133_complex1,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => ( ~ ( A != 0
            & r1_xreal_0(k17_complex1(A),0) )
        & ~ ( ~ r1_xreal_0(k17_complex1(A),0)
            & A = 0 ) ) ) ),
    file(complex1,t133_complex1),
    [interesting(0.00)]).

fof(t73_real_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( ~ r1_xreal_0(A,0)
               => ( ~ ( ~ r1_xreal_0(C,B)
                      & r1_xreal_0(k7_xcmplx_0(C,A),k7_xcmplx_0(B,A)) )
                  & ~ ( ~ r1_xreal_0(k7_xcmplx_0(C,A),k7_xcmplx_0(B,A))
                      & r1_xreal_0(C,B) ) ) ) ) ) ) ),
    file(real_1,t73_real_1),
    [interesting(0.00)]).

fof(t53_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] : 
                  ( v1_xreal_0(D)
                 => k18_euclid(D,A,k20_euclid(A,B,C)) = k20_euclid(A,k18_euclid(D,A,B),k18_euclid(D,A,C)) ) ) ) ) ),
    file(euclid,t53_euclid),
    [interesting(0.00)]).

fof(d11_euclid,definition,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(B)))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(B)))
                 => ( D = k18_euclid(A,B,C)
                  <=> ! [E] : 
                        ( m2_finseq_2(E,k1_numbers,k1_euclid(B))
                       => ( E = C
                         => D = k9_euclid(B,A,E) ) ) ) ) ) ) ) ),
    file(euclid,d11_euclid),
    [interesting(0.00)]).

fof(t14_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => k12_euclid(k9_euclid(A,B,C)) = k4_real_1(k18_complex1(B),k12_euclid(C)) ) ) ) ),
    file(euclid,t14_euclid),
    [interesting(0.00)]).

fof(d9_xcmplx_0,definition,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => ! [B] : 
          ( v1_xcmplx_0(B)
         => k7_xcmplx_0(A,B) = k3_xcmplx_0(A,k5_xcmplx_0(B)) ) ) ),
    file(xcmplx_0,d9_xcmplx_0),
    [interesting(0.00)]).

fof(d7_xcmplx_0,definition,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => ! [B] : 
          ( v1_xcmplx_0(B)
         => ( ( A != 0
             => ( B = k5_xcmplx_0(A)
              <=> k3_xcmplx_0(A,B) = 1 ) )
            & ( A = 0
             => ( B = k5_xcmplx_0(A)
              <=> B = 0 ) ) ) ) ) ),
    file(xcmplx_0,d7_xcmplx_0),
    [interesting(0.00)]).

fof(t3_seq_2,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ( ~ r1_xreal_0(A,0)
       => ( ~ r1_xreal_0(k7_xcmplx_0(A,2),0)
          & ~ r1_xreal_0(k7_xcmplx_0(A,4),0) ) ) ) ),
    file(seq_2,t3_seq_2),
    [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(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(t50_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => k20_euclid(A,B,k17_euclid(A,C,D)) = k20_euclid(A,k20_euclid(A,B,C),D) ) ) ) ) ),
    file(euclid,t50_euclid),
    [interesting(0.00)]).

fof(t41_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( ( v3_toprns_1(B,A)
                  & v3_toprns_1(C,A) )
               => v3_toprns_1(k1_toprns_1(A,B,C),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_toprns_1,d9_toprns_1,d9_toprns_1,t3_seq_2,t37_nat_1,t2_xreal_1,t37_nat_1,t2_xreal_1,t10_xreal_1,d2_toprns_1,t49_euclid,t45_euclid,t30_euclid,t45_euclid,t50_euclid,t45_euclid,t30_euclid,t45_euclid,t30_toprns_1,t2_xreal_1]),
    [file(toprns_1,t41_toprns_1),interesting(0.00)]).

fof(d10_toprns_1,definition,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v3_toprns_1(B,A)
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
               => ( C = k7_toprns_1(A,B)
                <=> ! [D] : 
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,0)
                          & ! [E] : 
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ? [F] : 
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                  & r1_xreal_0(E,F)
                                  & r1_xreal_0(D,k5_toprns_1(A,k20_euclid(A,k2_normsp_1(k15_euclid(A),B,F),C))) ) ) ) ) ) ) ) ) ) ),
    file(toprns_1,d10_toprns_1),
    [interesting(0.00)]).

fof(t44_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( k19_euclid(A,k18_euclid(C,A,B)) = k18_euclid(k4_xcmplx_0(C),A,B)
                & k19_euclid(A,k18_euclid(C,A,B)) = k18_euclid(C,A,k19_euclid(A,B)) ) ) ) ) ),
    file(euclid,t44_euclid),
    [interesting(0.00)]).

fof(t49_toprns_1,theorem,(
    $true ),
    file(toprns_1,t49_toprns_1),
    [interesting(0.00)]).

fof(t18_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] : 
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => r1_xreal_0(k5_real_1(k12_euclid(B),k12_euclid(C)),k12_euclid(k8_euclid(A,B,C))) ) ) ) ),
    file(euclid,t18_euclid),
    [interesting(0.00)]).

fof(t33_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => r1_xreal_0(k5_real_1(k5_toprns_1(A,B),k5_toprns_1(A,C)),k5_toprns_1(A,k20_euclid(A,B,C))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_euclid,d13_euclid,d6_toprns_1,t18_euclid,d6_toprns_1,d6_toprns_1]),
    [file(toprns_1,t33_toprns_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(t18_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => r1_xreal_0(0,A) ) ),
    file(nat_1,t18_nat_1),
    [interesting(0.00)]).

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

fof(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(d8_toprns_1,definition,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v2_toprns_1(B,A)
          <=> ? [C] : 
                ( m1_subset_1(C,k1_numbers)
                & ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ~ r1_xreal_0(C,k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,D))) ) ) ) ) ) ),
    file(toprns_1,d8_toprns_1),
    [interesting(0.00)]).

fof(t51_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ~ ( v3_toprns_1(B,A)
              & k7_toprns_1(A,B) != k16_euclid(A)
              & ! [C] : 
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ? [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                      & r1_xreal_0(C,D)
                      & r1_xreal_0(k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,D)),k6_real_1(k5_toprns_1(A,k7_toprns_1(A,B)),2)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t26_toprns_1,t25_toprns_1,t3_seq_2,d10_toprns_1,t33_toprns_1,t28_toprns_1,t2_xreal_1,t8_xreal_1]),
    [file(toprns_1,t51_toprns_1),interesting(0.00)]).

fof(t7_toprns_1,theorem,(
    $true ),
    file(toprns_1,t7_toprns_1),
    [interesting(0.00)]).

fof(d7_toprns_1,definition,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( C = k6_toprns_1(A,B)
              <=> ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,C,D) = k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,D)) ) ) ) ) ) ),
    file(toprns_1,d7_toprns_1),
    [interesting(0.00)]).

fof(t13_seq_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( B = k14_seq_1(C,A)
              <=> ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,B,D) = k3_xcmplx_0(A,k2_seq_1(k5_numbers,k1_numbers,C,D)) ) ) ) ) ) ),
    file(seq_1,t13_seq_1),
    [interesting(0.00)]).

fof(t113_funct_2,theorem,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B) )
     => ! [D] : 
          ( ( v1_funct_1(D)
            & v1_funct_2(D,A,B)
            & m2_relset_1(D,A,B) )
         => ( ! [E] : 
                ( m1_subset_1(E,A)
               => k1_funct_1(C,E) = k1_funct_1(D,E) )
           => C = D ) ) ) ),
    file(funct_2,t113_funct_2),
    [interesting(0.00)]).