% Mizar problem: t9_absvalue,absvalue,121,5 
fof(t9_absvalue, conjecture,  (! [A] :  (v1_xreal_0(A) =>  (! [B] :  (v1_xreal_0(B) =>  (! [C] :  (v1_xreal_0(C) =>  (! [D] :  (v1_xreal_0(D) =>  ( (r1_xxreal_0(k9_complex1(A), C) & r1_xxreal_0(k9_complex1(B), D))  => r1_xxreal_0(k9_complex1(k2_xcmplx_0(A, B)), k2_xcmplx_0(C, D))) ) ) ) ) ) ) ) ) ).
fof(cc3_xreal_0, axiom,  (! [A] :  (v1_xreal_0(A) => v1_xcmplx_0(A)) ) ).
fof(cc4_xreal_0, axiom,  (! [A] :  (v1_xreal_0(A) => v1_xxreal_0(A)) ) ).
fof(commutativity_k2_xcmplx_0, axiom,  (! [A, B] :  ( (v1_xcmplx_0(A) & v1_xcmplx_0(B))  => k2_xcmplx_0(A, B)=k2_xcmplx_0(B, A)) ) ).
fof(connectedness_r1_xxreal_0, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  =>  (r1_xxreal_0(A, B) | r1_xxreal_0(B, A)) ) ) ).
fof(dt_k2_xcmplx_0, axiom, $true).
fof(dt_k9_complex1, axiom,  (! [A] :  (v1_xcmplx_0(A) => v1_xreal_0(k9_complex1(A))) ) ).
fof(fc5_xreal_0, axiom,  (! [A, B] :  ( (v1_xreal_0(A) & v1_xreal_0(B))  => v1_xreal_0(k2_xcmplx_0(A, B))) ) ).
fof(projectivity_k9_complex1, axiom,  (! [A] :  (v1_xcmplx_0(A) => k9_complex1(k9_complex1(A))=k9_complex1(A)) ) ).
fof(rc1_xreal_0, axiom,  (? [A] : v1_xreal_0(A)) ).
fof(rc1_xxreal_0, axiom,  (? [A] : v1_xxreal_0(A)) ).
fof(rc2_xreal_0, axiom,  (? [A] : v1_xreal_0(A)) ).
fof(rc2_xxreal_0, axiom,  (? [A] : v1_xxreal_0(A)) ).
fof(reflexivity_r1_xxreal_0, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => r1_xxreal_0(A, A)) ) ).
fof(spc6_arithm, axiom,  (! [A, B, C] :  ( (v1_xcmplx_0(A) &  (v1_xcmplx_0(B) & v1_xcmplx_0(C)) )  => k2_xcmplx_0(k2_xcmplx_0(A, B), C)=k2_xcmplx_0(A, k2_xcmplx_0(B, C))) ) ).
fof(t2_xxreal_0, axiom,  (! [A] :  (v1_xxreal_0(A) =>  (! [B] :  (v1_xxreal_0(B) =>  (! [C] :  (v1_xxreal_0(C) =>  ( (r1_xxreal_0(A, B) & r1_xxreal_0(B, C))  => r1_xxreal_0(A, C)) ) ) ) ) ) ) ).
fof(t56_complex1, axiom,  (! [A] :  (v1_xcmplx_0(A) =>  (! [B] :  (v1_xcmplx_0(B) => r1_xxreal_0(k9_complex1(k2_xcmplx_0(A, B)), k2_xcmplx_0(k9_complex1(A), k9_complex1(B)))) ) ) ) ).
fof(t7_xreal_1, axiom,  (! [A] :  (v1_xreal_0(A) =>  (! [B] :  (v1_xreal_0(B) =>  (! [C] :  (v1_xreal_0(C) =>  (! [D] :  (v1_xreal_0(D) =>  ( (r1_xxreal_0(A, B) & r1_xxreal_0(C, D))  => r1_xxreal_0(k2_xcmplx_0(A, C), k2_xcmplx_0(B, D))) ) ) ) ) ) ) ) ) ).
