% Mizar problem: t12_absvalue,absvalue,88,26 
fof(t12_absvalue,conjecture,(
    ! [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) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[cc1_real_1,cc2_xreal_0,connectedness_r1_xreal_0,d1_absvalue,dt_k16_complex1,dt_k18_complex1,dt_k1_numbers,dt_k1_real_1,dt_k4_xcmplx_0,dt_m1_subset_1,existence_m1_subset_1,fc1_xreal_0,involutiveness_k1_real_1,involutiveness_k4_xcmplx_0,l3_absvalue,projectivity_k16_complex1,projectivity_k18_complex1,rc1_xreal_0,redefinition_k18_complex1,redefinition_k1_real_1,reflexivity_r1_xreal_0,t1_xreal_1,t26_xreal_1,t2_xreal_1]),
    [file(absvalue,t12_absvalue)]).

fof(cc1_real_1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ( v1_xreal_0(A)
        & v1_xcmplx_0(A) ) ) ),
    file(real_1,cc1_real_1),
    []).

fof(cc2_xreal_0,axiom,(
    ! [A] : 
      ( v1_xreal_0(A)
     => v1_xcmplx_0(A) ) ),
    file(xreal_0,cc2_xreal_0),
    []).

fof(connectedness_r1_xreal_0,axiom,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( r1_xreal_0(A,B)
        | r1_xreal_0(B,A) ) ) ),
    file(xreal_0,r1_xreal_0),
    []).

fof(d1_absvalue,axiom,(
    ! [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),
    []).

fof(dt_k16_complex1,axiom,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => v1_xcmplx_0(k16_complex1(A)) ) ),
    file(complex1,k16_complex1),
    []).

fof(dt_k18_complex1,axiom,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => m1_subset_1(k18_complex1(A),k1_numbers) ) ),
    file(complex1,k18_complex1),
    []).

fof(dt_k1_numbers,axiom,(
    $true ),
    file(numbers,k1_numbers),
    []).

fof(dt_k1_real_1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => m1_subset_1(k1_real_1(A),k1_numbers) ) ),
    file(real_1,k1_real_1),
    []).

fof(dt_k4_xcmplx_0,axiom,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => v1_xcmplx_0(k4_xcmplx_0(A)) ) ),
    file(xcmplx_0,k4_xcmplx_0),
    []).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    []).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    []).

fof(fc1_xreal_0,axiom,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ( v1_xcmplx_0(k4_xcmplx_0(A))
        & v1_xreal_0(k4_xcmplx_0(A)) ) ) ),
    file(xreal_0,fc1_xreal_0),
    []).

fof(involutiveness_k1_real_1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => k1_real_1(k1_real_1(A)) = A ) ),
    file(real_1,k1_real_1),
    []).

fof(involutiveness_k4_xcmplx_0,axiom,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ),
    file(xcmplx_0,k4_xcmplx_0),
    []).

fof(l3_absvalue,axiom,(
    ! [A] : 
      ( v1_xreal_0(A)
     => r1_xreal_0(0,k18_complex1(A)) ) ),
    file(absvalue,l3_absvalue),
    []).

fof(projectivity_k16_complex1,axiom,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k16_complex1(k16_complex1(A)) = k16_complex1(A) ) ),
    file(complex1,k16_complex1),
    []).

fof(projectivity_k18_complex1,axiom,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k18_complex1(k18_complex1(A)) = k18_complex1(A) ) ),
    file(complex1,k18_complex1),
    []).

fof(rc1_xreal_0,axiom,(
    ? [A] : 
      ( v1_xcmplx_0(A)
      & v1_xreal_0(A) ) ),
    file(xreal_0,rc1_xreal_0),
    []).

fof(redefinition_k18_complex1,axiom,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k18_complex1(A) = k16_complex1(A) ) ),
    file(complex1,k18_complex1),
    []).

fof(redefinition_k1_real_1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => k1_real_1(A) = k4_xcmplx_0(A) ) ),
    file(real_1,k1_real_1),
    []).

fof(reflexivity_r1_xreal_0,axiom,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => r1_xreal_0(A,A) ) ),
    file(xreal_0,r1_xreal_0),
    []).

fof(t1_xreal_1,axiom,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(A,B)
              & r1_xreal_0(B,A) )
           => A = B ) ) ) ),
    file(xreal_1,t1_xreal_1),
    []).

fof(t26_xreal_1,axiom,(
    ! [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),
    []).

fof(t2_xreal_1,axiom,(
    ! [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),
    []).