% Mizar problem: t67_xxreal_2,xxreal_2,1603,5 
fof(t67_xxreal_2, conjecture,  (! [A] :  (v2_membered(A) =>  (! [B] :  (v2_membered(B) => r1_xxreal_0(k1_xxreal_2(k3_xboole_0(A, B)), k3_xxreal_0(k1_xxreal_2(A), k1_xxreal_2(B)))) ) ) ) ).
fof(commutativity_k3_xboole_0, axiom,  (! [A, B] : k3_xboole_0(A, B)=k3_xboole_0(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(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(d3_xxreal_2, axiom,  (! [A] :  (v2_membered(A) =>  (! [B] :  (v1_xxreal_0(B) =>  (B=k1_xxreal_2(A) <=>  (m1_xxreal_2(B, A) &  (! [C] :  (m1_xxreal_2(C, A) => r1_xxreal_0(B, C)) ) ) ) ) ) ) ) ).
fof(dt_k1_xxreal_2, axiom,  (! [A] :  (v2_membered(A) => v1_xxreal_0(k1_xxreal_2(A))) ) ).
fof(dt_k3_xboole_0, axiom, $true).
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_m1_xxreal_2, axiom,  (! [A] :  (v2_membered(A) =>  (! [B] :  (m1_xxreal_2(B, A) => v1_xxreal_0(B)) ) ) ) ).
fof(existence_m1_xxreal_2, axiom,  (! [A] :  (v2_membered(A) =>  (? [B] : m1_xxreal_2(B, A)) ) ) ).
fof(fc33_membered, axiom,  (! [A, B] :  (v2_membered(A) => v2_membered(k3_xboole_0(A, B))) ) ).
fof(fc34_membered, axiom,  (! [A, B] :  (v2_membered(A) => v2_membered(k3_xboole_0(B, A))) ) ).
fof(fc5_xxreal_0, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => v1_xxreal_0(k3_xxreal_0(A, B))) ) ).
fof(idempotence_k3_xboole_0, axiom,  (! [A, B] : k3_xboole_0(A, A)=A) ).
fof(idempotence_k3_xxreal_0, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => k3_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)) ) ).
fof(t65_xxreal_2, axiom,  (! [A] :  (v2_membered(A) =>  (! [B] :  (v2_membered(B) =>  (! [C] :  (m1_xxreal_2(C, A) =>  (! [D] :  (m1_xxreal_2(D, B) => m1_xxreal_2(k3_xxreal_0(C, D), k3_xboole_0(A, B))) ) ) ) ) ) ) ) ).
