% Mizar problem: t39_extreal1,extreal1,1674,5 
fof(t39_extreal1, conjecture,  (! [A] :  (v1_xxreal_0(A) =>  (! [B] :  (v1_xxreal_0(B) => k1_xxreal_3(k3_xxreal_0(A, B), k4_xxreal_0(A, B))=k1_xxreal_3(A, B)) ) ) ) ).
fof(commutativity_k1_xxreal_3, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => k1_xxreal_3(A, B)=k1_xxreal_3(B, A)) ) ).
fof(commutativity_k3_xxreal_0, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => k3_xxreal_0(A, B)=k3_xxreal_0(B, A)) ) ).
fof(commutativity_k4_xxreal_0, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => k4_xxreal_0(A, B)=k4_xxreal_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(d10_xxreal_0, axiom,  (! [A] :  (v1_xxreal_0(A) =>  (! [B] :  (v1_xxreal_0(B) =>  ( (r1_xxreal_0(B, A) => k4_xxreal_0(A, B)=A)  &  ( ~ (r1_xxreal_0(B, A))  => k4_xxreal_0(A, B)=B) ) ) ) ) ) ).
fof(d9_xxreal_0, axiom,  (! [A] :  (v1_xxreal_0(A) =>  (! [B] :  (v1_xxreal_0(B) =>  ( (r1_xxreal_0(A, B) => k3_xxreal_0(A, B)=A)  &  ( ~ (r1_xxreal_0(A, B))  => k3_xxreal_0(A, B)=B) ) ) ) ) ) ).
fof(dt_k1_xxreal_3, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => v1_xxreal_0(k1_xxreal_3(A, B))) ) ).
fof(dt_k3_xxreal_0, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => v1_xxreal_0(k3_xxreal_0(A, B))) ) ).
fof(dt_k4_xxreal_0, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => v1_xxreal_0(k4_xxreal_0(A, B))) ) ).
fof(fc5_xxreal_0, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => v1_xxreal_0(k3_xxreal_0(A, B))) ) ).
fof(fc6_xxreal_0, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => v1_xxreal_0(k4_xxreal_0(A, B))) ) ).
fof(idempotence_k3_xxreal_0, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => k3_xxreal_0(A, A)=A) ) ).
fof(idempotence_k4_xxreal_0, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => k4_xxreal_0(A, A)=A) ) ).
fof(rc1_xxreal_0, axiom,  (? [A] : v1_xxreal_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)) ) ).
