fof(l6_normsp_1,theorem,(
    m1_subset_1(k1_rlvect_1(c1),u1_struct_0(k1_rlsub_1(c1))) ),
    inference(mizar_proof,[status(thm)],[l3_normsp_1,d1_tarski]),
    [file(normsp_1,l6_normsp_1),interesting(1.00)]).

fof(l3_normsp_1,theorem,(
    u1_struct_0(k1_rlsub_1(c1)) = k1_struct_0(c1,k1_rlvect_1(c1)) ),
    inference(mizar_proof,[status(thm)],[d3_rlsub_1]),
    [file(normsp_1,l3_normsp_1),interesting(0.81)]).

fof(l5_normsp_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,u1_struct_0(k1_rlsub_1(c1)))
     => k2_seq_1(u1_struct_0(k1_rlsub_1(c1)),k1_numbers,c2,A) = 0 ) ),
    inference(mizar_proof,[status(thm)],[t13_funcop_1]),
    [file(normsp_1,l5_normsp_1),interesting(0.75)]).

fof(t19_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ? [C] : 
              ( v1_funct_1(C)
              & v1_funct_2(C,k5_numbers,u1_struct_0(A))
              & m2_relset_1(C,k5_numbers,u1_struct_0(A))
              & k2_relat_1(C) = k1_struct_0(A,B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t15_funct_1,d1_funct_2,t11_relset_1]),
    [file(normsp_1,t19_normsp_1),interesting(0.75)]).

fof(t39_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v4_normsp_1(B,A)
           => v4_seq_2(k7_normsp_1(A,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_normsp_1,d1_xreal_0,t13_normsp_1,t2_xreal_1,d10_normsp_1,d6_seq_2]),
    [file(normsp_1,t39_normsp_1),interesting(0.75)]).

fof(t8_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => r1_xreal_0(0,k1_normsp_1(A,B)) ) ) ),
    inference(mizar_proof,[status(thm)],[t28_rlvect_1,d2_normsp_1,t7_normsp_1,t138_xreal_1]),
    [file(normsp_1,t8_normsp_1),interesting(0.75)]).

fof(l7_normsp_1,theorem,(
    k2_seq_1(u1_struct_0(k1_rlsub_1(c1)),k1_numbers,c2,k1_rlvect_1(c1)) = 0 ),
    inference(mizar_proof,[status(thm)],[l6_normsp_1,l5_normsp_1]),
    [file(normsp_1,l7_normsp_1),interesting(0.72)]).

fof(t37_normsp_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v7_rlvect_1(B)
            & v2_normsp_1(B)
            & l1_normsp_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(B))
                & m2_relset_1(C,k5_numbers,u1_struct_0(B)) )
             => ( v4_normsp_1(C,B)
               => v4_normsp_1(k6_normsp_1(B,C,A),B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_normsp_1,d9_normsp_1,t23_rlvect_1,t23_rlvect_1,t26_rlvect_1,d2_normsp_1,t8_normsp_1,d8_normsp_1,t133_complex1,t133_complex1,t73_real_1,t48_rlvect_1,d2_normsp_1,d9_xcmplx_0,d7_xcmplx_0,t70_xreal_1,d8_normsp_1]),
    [file(normsp_1,t37_normsp_1),interesting(0.69)]).

fof(t27_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v5_seqm_3(B)
          <=> ? [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
                & k2_relat_1(B) = k1_struct_0(A,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_normsp_1,t20_normsp_1,t21_normsp_1,t22_normsp_1,t23_normsp_1,t24_normsp_1,d4_normsp_1]),
    [file(normsp_1,t27_normsp_1),interesting(0.64)]).

fof(t13_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => r1_xreal_0(k18_complex1(k5_real_1(k1_normsp_1(A,B),k1_normsp_1(A,C))),k1_normsp_1(A,k6_rlvect_1(A,B,C))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t12_normsp_1,t43_rlvect_1,t28_rlvect_1,t26_rlvect_1,d2_normsp_1,t22_xreal_1,t11_normsp_1,t26_xreal_1,t12_absvalue]),
    [file(normsp_1,t13_normsp_1),interesting(0.64)]).

fof(t36_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( v4_normsp_1(C,A)
               => v4_normsp_1(k5_normsp_1(A,C,B),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_normsp_1,d9_normsp_1,t26_rlvect_1,t28_rlvect_1,t43_rlvect_1,d6_rlvect_1,t47_rlvect_1,d7_normsp_1]),
    [file(normsp_1,t36_normsp_1),interesting(0.57)]).

fof(t5_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => k1_normsp_1(A,k1_rlvect_1(A)) = 0 ) ),
    inference(mizar_proof,[status(thm)],[d2_normsp_1]),
    [file(normsp_1,t5_normsp_1),interesting(0.52)]).

fof(t35_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( v4_normsp_1(B,A)
                  & v4_normsp_1(C,A) )
               => v4_normsp_1(k4_normsp_1(A,B,C),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_normsp_1,d9_normsp_1,d9_normsp_1,t3_seq_2,t37_nat_1,t2_xreal_1,t37_nat_1,t2_xreal_1,t10_xreal_1,d6_normsp_1,t43_rlvect_1,t41_rlvect_1,t41_rlvect_1,t43_rlvect_1,t7_normsp_1,t2_xreal_1]),
    [file(normsp_1,t35_normsp_1),interesting(0.50)]).

fof(t20_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ~ ( ? [C] : 
                  ( m1_subset_1(C,u1_struct_0(A))
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k8_funct_2(k5_numbers,u1_struct_0(A),B,D) = C ) )
              & ! [C] : 
                  ( m1_subset_1(C,u1_struct_0(A))
                 => k2_relat_1(B) != k1_struct_0(A,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_funct_1,d1_funct_2,d1_tarski,d3_tarski,d1_tarski,d1_funct_2,d5_funct_1,d3_tarski,d10_xboole_0]),
    [file(normsp_1,t20_normsp_1),interesting(0.46)]).

fof(t6_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => k1_normsp_1(A,k5_rlvect_1(A,B)) = k1_normsp_1(A,B) ) ) ),
    inference(mizar_proof,[status(thm)],[t29_rlvect_1,d2_normsp_1,d1_absvalue]),
    [file(normsp_1,t6_normsp_1),interesting(0.45)]).

fof(t34_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( v4_normsp_1(B,A)
                  & v4_normsp_1(C,A) )
               => v4_normsp_1(k3_normsp_1(A,B,C),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_normsp_1,d9_normsp_1,d9_normsp_1,t3_seq_2,t37_nat_1,t2_xreal_1,t37_nat_1,t2_xreal_1,t10_xreal_1,d5_normsp_1,d6_rlvect_1,d6_rlvect_1,t45_rlvect_1,d6_rlvect_1,d6_rlvect_1,d6_rlvect_1,d2_normsp_1,t2_xreal_1]),
    [file(normsp_1,t34_normsp_1),interesting(0.45)]).

fof(t45_normsp_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v7_rlvect_1(B)
            & v2_normsp_1(B)
            & l1_normsp_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(B))
                & m2_relset_1(C,k5_numbers,u1_struct_0(B)) )
             => ( v4_normsp_1(C,B)
               => k8_normsp_1(B,k6_normsp_1(B,C,A)) = k3_rlvect_1(B,k8_normsp_1(B,C),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t37_normsp_1,d11_normsp_1,t23_rlvect_1,t23_rlvect_1,t26_rlvect_1,d2_normsp_1,t8_normsp_1,d8_normsp_1,t133_complex1,t133_complex1,t73_real_1,d11_normsp_1,t48_rlvect_1,d2_normsp_1,d9_xcmplx_0,d7_xcmplx_0,t70_xreal_1,d8_normsp_1,d11_normsp_1]),
    [file(normsp_1,t45_normsp_1),interesting(0.36)]).

fof(l11_normsp_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,u1_struct_0(c3))
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(c3))
         => ! [C] : 
              ( m1_subset_1(C,k1_numbers)
             => ( ( k1_normsp_1(c3,A) = 0
                 => A = k1_rlvect_1(c3) )
                & ( A = k1_rlvect_1(c3)
                 => k1_normsp_1(c3,A) = 0 )
                & k1_normsp_1(c3,k3_rlvect_1(c3,A,C)) = k4_real_1(k18_complex1(C),k1_normsp_1(c3,A))
                & r1_xreal_0(k1_normsp_1(c3,k2_rlvect_1(c3,A,B)),k3_real_1(k1_normsp_1(c3,A),k1_normsp_1(c3,B))) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t19_rlsub_1,l3_normsp_1,l7_normsp_1,d1_tarski,l8_normsp_1,l9_normsp_1]),
    [file(normsp_1,l11_normsp_1),interesting(0.33)]).

fof(t21_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( ? [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
                & k2_relat_1(B) = k1_struct_0(A,C) )
           => ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,1)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_2,d5_funct_1,d1_tarski]),
    [file(normsp_1,t21_normsp_1),interesting(0.30)]).

fof(t44_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( v4_normsp_1(C,A)
               => k8_normsp_1(A,k5_normsp_1(A,C,B)) = k6_rlvect_1(A,k8_normsp_1(A,C),B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t36_normsp_1,d11_normsp_1,t26_rlvect_1,t28_rlvect_1,t43_rlvect_1,d6_rlvect_1,t47_rlvect_1,d7_normsp_1,d11_normsp_1]),
    [file(normsp_1,t44_normsp_1),interesting(0.20)]).

fof(t41_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( v4_normsp_1(C,A)
                  & k8_normsp_1(A,C) = B )
               => ( v4_seq_2(k7_normsp_1(A,k5_normsp_1(A,C,B)))
                  & k2_seq_2(k7_normsp_1(A,k5_normsp_1(A,C,B))) = 0 ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t36_normsp_1,t39_normsp_1,d1_xreal_0,d11_normsp_1,t26_rlvect_1,t13_normsp_1,t2_xreal_1,d2_normsp_1,d7_normsp_1,d10_normsp_1,d7_seq_2]),
    [file(normsp_1,t41_normsp_1),interesting(0.19)]).

fof(t25_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ? [B] : 
          ( v1_funct_1(B)
          & v1_funct_2(B,k5_numbers,u1_struct_0(A))
          & m2_relset_1(B,k5_numbers,u1_struct_0(A))
          & k2_relat_1(B) = k1_struct_0(A,k1_rlvect_1(A)) ) ) ),
    inference(mizar_proof,[status(thm)],[t19_normsp_1]),
    [file(normsp_1,t25_normsp_1),interesting(0.18)]).

fof(t43_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( v4_normsp_1(B,A)
                  & v4_normsp_1(C,A) )
               => k8_normsp_1(A,k4_normsp_1(A,B,C)) = k6_rlvect_1(A,k8_normsp_1(A,B),k8_normsp_1(A,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t35_normsp_1,t3_seq_2,d11_normsp_1,d11_normsp_1,t37_nat_1,t2_xreal_1,t37_nat_1,t2_xreal_1,t10_xreal_1,d6_normsp_1,t43_rlvect_1,t41_rlvect_1,t41_rlvect_1,t43_rlvect_1,t7_normsp_1,t2_xreal_1,d11_normsp_1]),
    [file(normsp_1,t43_normsp_1),interesting(0.11)]).

fof(d1_struct_0,definition,(
    ! [A] : 
      ( l1_struct_0(A)
     => ( v3_struct_0(A)
      <=> v1_xboole_0(u1_struct_0(A)) ) ) ),
    file(struct_0,d1_struct_0),
    [interesting(0.00)]).

fof(l10_normsp_1,theorem,
    ( ~ v3_struct_0(g1_normsp_1(u1_struct_0(k1_rlsub_1(c1)),u2_struct_0(k1_rlsub_1(c1)),u1_rlvect_1(k1_rlsub_1(c1)),u2_rlvect_1(k1_rlsub_1(c1)),c2))
    & l1_normsp_1(g1_normsp_1(u1_struct_0(k1_rlsub_1(c1)),u2_struct_0(k1_rlsub_1(c1)),u1_rlvect_1(k1_rlsub_1(c1)),u2_rlvect_1(k1_rlsub_1(c1)),c2)) ),
    inference(mizar_proof,[status(thm)],[d1_struct_0]),
    [file(normsp_1,l10_normsp_1),interesting(0.00)]).

fof(t19_rlsub_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] : 
          ( m1_rlsub_1(B,A)
         => k1_rlvect_1(B) = k1_rlvect_1(A) ) ) ),
    file(rlsub_1,t19_rlsub_1),
    [interesting(0.00)]).

fof(d3_rlsub_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] : 
          ( ( v2_rlvect_1(B)
            & m1_rlsub_1(B,A) )
         => ( B = k1_rlsub_1(A)
          <=> u1_struct_0(B) = k1_struct_0(A,k1_rlvect_1(A)) ) ) ) ),
    file(rlsub_1,d3_rlsub_1),
    [interesting(0.00)]).

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

fof(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(l8_normsp_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,u1_struct_0(k1_rlsub_1(c1)))
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => k2_seq_1(u1_struct_0(k1_rlsub_1(c1)),k1_numbers,c2,k3_rlvect_1(k1_rlsub_1(c1),A,B)) = k4_real_1(k18_complex1(B),k2_seq_1(u1_struct_0(k1_rlsub_1(c1)),k1_numbers,c2,A)) ) ) ),
    inference(mizar_proof,[status(thm)],[l3_normsp_1,l7_normsp_1,d1_tarski,l3_normsp_1,l7_normsp_1,d1_tarski]),
    [file(normsp_1,l8_normsp_1),interesting(0.00)]).

fof(l9_normsp_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,u1_struct_0(k1_rlsub_1(c1)))
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k1_rlsub_1(c1)))
         => r1_xreal_0(k2_seq_1(u1_struct_0(k1_rlsub_1(c1)),k1_numbers,c2,k4_rlvect_1(k1_rlsub_1(c1),A,B)),k3_real_1(k2_seq_1(u1_struct_0(k1_rlsub_1(c1)),k1_numbers,c2,A),k2_seq_1(u1_struct_0(k1_rlsub_1(c1)),k1_numbers,c2,B))) ) ) ),
    inference(mizar_proof,[status(thm)],[l3_normsp_1,d1_tarski,l3_normsp_1,l7_normsp_1,d1_tarski]),
    [file(normsp_1,l9_normsp_1),interesting(0.00)]).

fof(l2_normsp_1,theorem,(
    ? [A] : 
      ( ~ v3_struct_0(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_1(A)
      & v7_rlvect_1(A)
      & l2_rlvect_1(A) ) ),
    file(normsp_1,l2_normsp_1),
    [interesting(0.00)]).

fof(l42_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => m1_subset_1(k8_funct_2(k5_numbers,u1_struct_0(A),B,C),u1_struct_0(A)) ) ) ) ),
    file(normsp_1,l42_normsp_1),
    [interesting(0.00)]).

fof(t57_funcop_1,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(C,B)
     => ( v1_funct_1(k2_funcop_1(A,C))
        & v1_funct_2(k2_funcop_1(A,C),A,B)
        & m2_relset_1(k2_funcop_1(A,C),A,B) ) ) ),
    file(funcop_1,t57_funcop_1),
    [interesting(0.00)]).

fof(l4_normsp_1,theorem,
    ( v1_funct_1(k2_funcop_1(u1_struct_0(k1_rlsub_1(c1)),0))
    & v1_funct_2(k2_funcop_1(u1_struct_0(k1_rlsub_1(c1)),0),u1_struct_0(k1_rlsub_1(c1)),k1_numbers)
    & m2_relset_1(k2_funcop_1(u1_struct_0(k1_rlsub_1(c1)),0),u1_struct_0(k1_rlsub_1(c1)),k1_numbers) ),
    inference(mizar_proof,[status(thm)],[t57_funcop_1]),
    [file(normsp_1,l4_normsp_1),interesting(0.00)]).

fof(t10_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ( k2_rlvect_1(A,B,k1_rlvect_1(A)) = B
            & k2_rlvect_1(A,k1_rlvect_1(A),B) = B ) ) ) ),
    file(rlvect_1,t10_rlvect_1),
    [interesting(0.00)]).

fof(t16_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ( k2_rlvect_1(A,B,k5_rlvect_1(A,B)) = k1_rlvect_1(A)
            & k2_rlvect_1(A,k5_rlvect_1(A,B),B) = k1_rlvect_1(A) ) ) ) ),
    file(rlvect_1,t16_rlvect_1),
    [interesting(0.00)]).

fof(d6_rlvect_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v4_rlvect_1(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] : 
                    ( m1_subset_1(D,u1_struct_0(A))
                   => k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) ) ) ) ) ) ),
    file(rlvect_1,d6_rlvect_1),
    [interesting(0.00)]).

fof(d2_normsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_normsp_1(A) )
     => ( v2_normsp_1(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] : 
                    ( m1_subset_1(D,k1_numbers)
                   => ( ( k1_normsp_1(A,B) = 0
                       => B = k1_rlvect_1(A) )
                      & ( B = k1_rlvect_1(A)
                       => k1_normsp_1(A,B) = 0 )
                      & k1_normsp_1(A,k3_rlvect_1(A,B,D)) = k4_real_1(k18_complex1(D),k1_normsp_1(A,B))
                      & r1_xreal_0(k1_normsp_1(A,k2_rlvect_1(A,B,C)),k3_real_1(k1_normsp_1(A,B),k1_normsp_1(A,C))) ) ) ) ) ) ) ),
    file(normsp_1,d2_normsp_1),
    [interesting(0.00)]).

fof(t14_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => r1_xreal_0(k1_normsp_1(A,k6_rlvect_1(A,B,C)),k3_real_1(k1_normsp_1(A,k6_rlvect_1(A,B,D)),k1_normsp_1(A,k6_rlvect_1(A,D,C)))) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t10_rlvect_1,t16_rlvect_1,d6_rlvect_1,d6_rlvect_1,d2_normsp_1]),
    [file(normsp_1,t14_normsp_1),interesting(0.00)]).

fof(t28_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => k6_rlvect_1(A,B,B) = k1_rlvect_1(A) ) ) ),
    file(rlvect_1,t28_rlvect_1),
    [interesting(0.00)]).

fof(t35_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ( k6_rlvect_1(A,B,C) = k1_rlvect_1(A)
               => B = C ) ) ) ) ),
    file(rlvect_1,t35_rlvect_1),
    [interesting(0.00)]).

fof(t10_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ( k1_normsp_1(A,k6_rlvect_1(A,B,C)) = 0
              <=> B = C ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_normsp_1,t28_rlvect_1,t35_rlvect_1]),
    [file(normsp_1,t10_normsp_1),interesting(0.00)]).

fof(t15_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( B != C
                  & k1_normsp_1(A,k6_rlvect_1(A,B,C)) = 0 ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t10_normsp_1]),
    [file(normsp_1,t15_normsp_1),interesting(0.00)]).

fof(t16_normsp_1,theorem,(
    $true ),
    file(normsp_1,t16_normsp_1),
    [interesting(0.00)]).

fof(d1_funct_2,definition,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => ( ( ( B = k1_xboole_0
           => A = k1_xboole_0 )
         => ( v1_funct_2(C,A,B)
          <=> A = k4_relset_1(A,B,C) ) )
        & ( B = k1_xboole_0
         => ( A = k1_xboole_0
            | ( v1_funct_2(C,A,B)
            <=> C = k1_xboole_0 ) ) ) ) ) ),
    file(funct_2,d1_funct_2),
    [interesting(0.00)]).

fof(t12_relset_1,theorem,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => ( r1_tarski(k1_relat_1(C),A)
        & r1_tarski(k2_relat_1(C),B) ) ) ),
    file(relset_1,t12_relset_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(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(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(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)) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_2,t12_relset_1,d5_funct_1,d1_funct_2,d5_funct_1,d3_tarski,d1_funct_2,t11_relset_1]),
    [file(normsp_1,t17_normsp_1),interesting(0.00)]).

fof(t18_normsp_1,theorem,(
    $true ),
    file(normsp_1,t18_normsp_1),
    [interesting(0.00)]).

fof(t1_normsp_1,theorem,(
    $true ),
    file(normsp_1,t1_normsp_1),
    [interesting(0.00)]).

fof(t15_funct_1,theorem,(
    ! [A] : 
      ( A != k1_xboole_0
     => ! [B] : 
        ? [C] : 
          ( v1_relat_1(C)
          & v1_funct_1(C)
          & k1_relat_1(C) = A
          & k2_relat_1(C) = k1_tarski(B) ) ) ),
    file(funct_1,t15_funct_1),
    [interesting(0.00)]).

fof(t26_normsp_1,theorem,(
    $true ),
    file(normsp_1,t26_normsp_1),
    [interesting(0.00)]).

fof(d4_normsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v5_seqm_3(B)
          <=> ? [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
                & ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k8_funct_2(k5_numbers,u1_struct_0(A),B,D) = C ) ) ) ) ) ),
    file(normsp_1,d4_normsp_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(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(t22_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,1)) )
           => ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,D)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s1_nat_1]),
    [file(normsp_1,t22_normsp_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(t23_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,D)) ) )
           => ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,D) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t28_nat_1,t28_nat_1]),
    [file(normsp_1,t23_normsp_1),interesting(0.00)]).

fof(t24_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ~ ( ! [C] : 
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k8_funct_2(k5_numbers,u1_struct_0(A),B,C) = k8_funct_2(k5_numbers,u1_struct_0(A),B,D) ) )
              & ! [C] : 
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ~ ! [D] : 
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => k8_funct_2(k5_numbers,u1_struct_0(A),B,D) = C ) ) ) ) ) ),
    file(normsp_1,t24_normsp_1),
    [interesting(0.00)]).

fof(t28_normsp_1,theorem,(
    $true ),
    file(normsp_1,t28_normsp_1),
    [interesting(0.00)]).

fof(t29_normsp_1,theorem,(
    $true ),
    file(normsp_1,t29_normsp_1),
    [interesting(0.00)]).

fof(t2_normsp_1,theorem,(
    $true ),
    file(normsp_1,t2_normsp_1),
    [interesting(0.00)]).

fof(t30_normsp_1,theorem,(
    $true ),
    file(normsp_1,t30_normsp_1),
    [interesting(0.00)]).

fof(t31_normsp_1,theorem,(
    $true ),
    file(normsp_1,t31_normsp_1),
    [interesting(0.00)]).

fof(t32_normsp_1,theorem,(
    $true ),
    file(normsp_1,t32_normsp_1),
    [interesting(0.00)]).

fof(t33_normsp_1,theorem,(
    $true ),
    file(normsp_1,t33_normsp_1),
    [interesting(0.00)]).

fof(t38_normsp_1,theorem,(
    $true ),
    file(normsp_1,t38_normsp_1),
    [interesting(0.00)]).

fof(t3_normsp_1,theorem,(
    $true ),
    file(normsp_1,t3_normsp_1),
    [interesting(0.00)]).

fof(t40_normsp_1,theorem,(
    $true ),
    file(normsp_1,t40_normsp_1),
    [interesting(0.00)]).

fof(d9_normsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v4_normsp_1(B,A)
          <=> ? [C] : 
                ( m1_subset_1(C,u1_struct_0(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,k1_normsp_1(A,k6_rlvect_1(A,k2_normsp_1(A,B,F),C))) ) ) ) ) ) ) ) ) ),
    file(normsp_1,d9_normsp_1),
    [interesting(0.00)]).

fof(t26_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => k6_rlvect_1(A,B,k1_rlvect_1(A)) = B ) ) ),
    file(rlvect_1,t26_rlvect_1),
    [interesting(0.00)]).

fof(t43_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k6_rlvect_1(A,B,k6_rlvect_1(A,C,D)) = k4_rlvect_1(A,k6_rlvect_1(A,B,C),D) ) ) ) ) ),
    file(rlvect_1,t43_rlvect_1),
    [interesting(0.00)]).

fof(t47_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => k5_rlvect_1(A,k6_rlvect_1(A,B,C)) = k2_rlvect_1(A,C,k5_rlvect_1(A,B)) ) ) ) ),
    file(rlvect_1,t47_rlvect_1),
    [interesting(0.00)]).

fof(d7_normsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( D = k5_normsp_1(A,B,C)
                  <=> ! [E] : 
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k6_rlvect_1(A,k2_normsp_1(A,B,E),C) ) ) ) ) ) ) ),
    file(normsp_1,d7_normsp_1),
    [interesting(0.00)]).

fof(d1_xreal_0,definition,(
    ! [A] : 
      ( v1_xreal_0(A)
    <=> r2_hidden(A,k1_numbers) ) ),
    file(xreal_0,d1_xreal_0),
    [interesting(0.00)]).

fof(t22_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( r1_xreal_0(A,k2_xcmplx_0(B,C))
              <=> r1_xreal_0(k6_xcmplx_0(A,B),C) ) ) ) ) ),
    file(xreal_1,t22_xreal_1),
    [interesting(0.00)]).

fof(t12_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => r1_xreal_0(k5_real_1(k1_normsp_1(A,B),k1_normsp_1(A,C)),k1_normsp_1(A,k6_rlvect_1(A,B,C))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t43_rlvect_1,t28_rlvect_1,t26_rlvect_1,d2_normsp_1,t22_xreal_1]),
    [file(normsp_1,t12_normsp_1),interesting(0.00)]).

fof(t29_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => k5_rlvect_1(A,B) = k3_rlvect_1(A,B,k1_real_1(1)) ) ) ),
    file(rlvect_1,t29_rlvect_1),
    [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(t11_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => k1_normsp_1(A,k6_rlvect_1(A,B,C)) = k1_normsp_1(A,k6_rlvect_1(A,C,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t47_rlvect_1,t6_normsp_1]),
    [file(normsp_1,t11_normsp_1),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(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(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(d10_normsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( C = k7_normsp_1(A,B)
              <=> ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,C,D) = k1_normsp_1(A,k2_normsp_1(A,B,D)) ) ) ) ) ) ),
    file(normsp_1,d10_normsp_1),
    [interesting(0.00)]).

fof(d6_seq_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v4_seq_2(A)
      <=> ? [B] : 
            ( v1_xreal_0(B)
            & ! [C] : 
                ( v1_xreal_0(C)
               => ~ ( ~ r1_xreal_0(C,0)
                    & ! [D] : 
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ? [E] : 
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                            & r1_xreal_0(D,E)
                            & r1_xreal_0(C,k18_complex1(k6_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,E),B))) ) ) ) ) ) ) ) ),
    file(seq_2,d6_seq_2),
    [interesting(0.00)]).

fof(d11_normsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v4_normsp_1(B,A)
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ( C = k8_normsp_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,k1_normsp_1(A,k6_rlvect_1(A,k2_normsp_1(A,B,F),C))) ) ) ) ) ) ) ) ) ) ),
    file(normsp_1,d11_normsp_1),
    [interesting(0.00)]).

fof(d7_seq_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v4_seq_2(A)
       => ! [B] : 
            ( v1_xreal_0(B)
           => ( B = k1_seq_2(A)
            <=> ! [C] : 
                  ( v1_xreal_0(C)
                 => ~ ( ~ r1_xreal_0(C,0)
                      & ! [D] : 
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => ? [E] : 
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                              & r1_xreal_0(D,E)
                              & r1_xreal_0(C,k18_complex1(k6_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,E),B))) ) ) ) ) ) ) ) ) ),
    file(seq_2,d7_seq_2),
    [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(d5_normsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( D = k3_normsp_1(A,B,C)
                  <=> ! [E] : 
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k4_rlvect_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,C,E)) ) ) ) ) ) ) ),
    file(normsp_1,d5_normsp_1),
    [interesting(0.00)]).

fof(t45_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => k5_rlvect_1(A,k2_rlvect_1(A,B,C)) = k2_rlvect_1(A,k5_rlvect_1(A,C),k5_rlvect_1(A,B)) ) ) ) ),
    file(rlvect_1,t45_rlvect_1),
    [interesting(0.00)]).

fof(t42_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( v4_normsp_1(B,A)
                  & v4_normsp_1(C,A) )
               => k8_normsp_1(A,k3_normsp_1(A,B,C)) = k4_rlvect_1(A,k8_normsp_1(A,B),k8_normsp_1(A,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t34_normsp_1,t3_seq_2,d11_normsp_1,d11_normsp_1,t37_nat_1,t2_xreal_1,t37_nat_1,t2_xreal_1,t10_xreal_1,d5_normsp_1,d6_rlvect_1,d6_rlvect_1,t45_rlvect_1,d6_rlvect_1,d6_rlvect_1,d6_rlvect_1,d2_normsp_1,t2_xreal_1,d11_normsp_1]),
    [file(normsp_1,t42_normsp_1),interesting(0.00)]).

fof(d6_normsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( D = k4_normsp_1(A,B,C)
                  <=> ! [E] : 
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k6_rlvect_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,C,E)) ) ) ) ) ) ) ),
    file(normsp_1,d6_normsp_1),
    [interesting(0.00)]).

fof(t41_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k6_rlvect_1(A,B,k2_rlvect_1(A,C,D)) = k6_rlvect_1(A,k6_rlvect_1(A,B,D),C) ) ) ) ) ),
    file(rlvect_1,t41_rlvect_1),
    [interesting(0.00)]).

fof(t7_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_normsp_1(A)
        & l1_normsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => r1_xreal_0(k1_normsp_1(A,k6_rlvect_1(A,B,C)),k3_real_1(k1_normsp_1(A,B),k1_normsp_1(A,C))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_normsp_1,t6_normsp_1]),
    [file(normsp_1,t7_normsp_1),interesting(0.00)]).

fof(t23_rlvect_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v7_rlvect_1(B)
            & l2_rlvect_1(B) )
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(B))
             => ( ( A = 0
                  | C = k1_rlvect_1(B) )
               => k3_rlvect_1(B,C,A) = k1_rlvect_1(B) ) ) ) ) ),
    file(rlvect_1,t23_rlvect_1),
    [interesting(0.00)]).

fof(t138_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(0,A)
              & r1_xreal_0(0,B) )
           => r1_xreal_0(0,k7_xcmplx_0(A,B)) ) ) ) ),
    file(xreal_1,t138_xreal_1),
    [interesting(0.00)]).

fof(d8_normsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] : 
              ( m1_subset_1(C,k1_numbers)
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( D = k6_normsp_1(A,B,C)
                  <=> ! [E] : 
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k3_rlvect_1(A,k2_normsp_1(A,B,E),C) ) ) ) ) ) ) ),
    file(normsp_1,d8_normsp_1),
    [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(t48_rlvect_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v7_rlvect_1(B)
            & l2_rlvect_1(B) )
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(B))
                 => k3_rlvect_1(B,k6_rlvect_1(B,C,D),A) = k6_rlvect_1(B,k3_rlvect_1(B,C,A),k3_rlvect_1(B,D,A)) ) ) ) ) ),
    file(rlvect_1,t48_rlvect_1),
    [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(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(t4_normsp_1,theorem,(
    $true ),
    file(normsp_1,t4_normsp_1),
    [interesting(0.00)]).

fof(t9_normsp_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( ( ~ v3_struct_0(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_rlvect_1(C)
                & v2_normsp_1(C)
                & l1_normsp_1(C) )
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(C))
                 => ! [E] : 
                      ( m1_subset_1(E,u1_struct_0(C))
                     => r1_xreal_0(k1_normsp_1(C,k4_rlvect_1(C,k3_rlvect_1(C,D,A),k3_rlvect_1(C,E,B))),k3_real_1(k4_real_1(k18_complex1(A),k1_normsp_1(C,D)),k4_real_1(k18_complex1(B),k1_normsp_1(C,E)))) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_normsp_1,d2_normsp_1,d2_normsp_1]),
    [file(normsp_1,t9_normsp_1),interesting(0.00)]).