% Mizar problem: t9_xxreal_2,xxreal_2,331,5 
fof(t9_xxreal_2, conjecture,  (! [A] :  (v2_membered(A) =>  (! [B] :  (v2_membered(B) => k2_xxreal_2(k2_xboole_0(A, B))=k3_xxreal_0(k2_xxreal_2(A), k2_xxreal_2(B))) ) ) ) ).
fof(cc14_membered, axiom,  (! [A] :  (v2_membered(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v2_membered(B)) ) ) ) ).
fof(cc1_finsub_1, axiom,  (! [A] :  (v4_finsub_1(A) =>  (v1_finsub_1(A) & v3_finsub_1(A)) ) ) ).
fof(cc2_finsub_1, axiom,  (! [A] :  ( (v1_finsub_1(A) & v3_finsub_1(A))  => v4_finsub_1(A)) ) ).
fof(cc7_membered, axiom,  (! [A] :  (v2_membered(A) =>  (! [B] :  (m1_subset_1(B, A) => v1_xxreal_0(B)) ) ) ) ).
fof(commutativity_k2_xboole_0, axiom,  (! [A, B] : k2_xboole_0(A, B)=k2_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(d4_xxreal_2, axiom,  (! [A] :  (v2_membered(A) =>  (! [B] :  (v1_xxreal_0(B) =>  (B=k2_xxreal_2(A) <=>  (m2_xxreal_2(B, A) &  (! [C] :  (m2_xxreal_2(C, A) => r1_xxreal_0(C, B)) ) ) ) ) ) ) ) ).
fof(dt_k1_zfmisc_1, axiom, $true).
fof(dt_k2_xboole_0, axiom, $true).
fof(dt_k2_xxreal_2, axiom,  (! [A] :  (v2_membered(A) => v1_xxreal_0(k2_xxreal_2(A))) ) ).
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_subset_1, axiom, $true).
fof(dt_m2_xxreal_2, axiom,  (! [A] :  (v2_membered(A) =>  (! [B] :  (m2_xxreal_2(B, A) => v1_xxreal_0(B)) ) ) ) ).
fof(existence_m1_subset_1, axiom,  (! [A] :  (? [B] : m1_subset_1(B, A)) ) ).
fof(existence_m2_xxreal_2, axiom,  (! [A] :  (v2_membered(A) =>  (? [B] : m2_xxreal_2(B, A)) ) ) ).
fof(fc1_finsub_1, axiom,  (! [A] : v4_finsub_1(k1_zfmisc_1(A))) ).
fof(fc26_membered, axiom,  (! [A, B] :  ( (v2_membered(A) & v2_membered(B))  => v2_membered(k2_xboole_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(idempotence_k2_xboole_0, axiom,  (! [A, B] : k2_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_tarski, axiom,  (! [A, B] : r1_tarski(A, A)) ).
fof(reflexivity_r1_xxreal_0, axiom,  (! [A, B] :  ( (v1_xxreal_0(A) & v1_xxreal_0(B))  => r1_xxreal_0(A, A)) ) ).
fof(t20_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(A, C))  => r1_xxreal_0(A, k3_xxreal_0(B, C))) ) ) ) ) ) ) ).
fof(t3_subset, axiom,  (! [A] :  (! [B] :  (m1_subset_1(A, k1_zfmisc_1(B)) <=> r1_tarski(A, B)) ) ) ).
fof(t5_xxreal_2, axiom,  (! [A] :  (v2_membered(A) =>  (! [B] :  (v2_membered(B) =>  (r1_tarski(B, A) =>  (! [C] :  (m2_xxreal_2(C, A) => m2_xxreal_2(C, B)) ) ) ) ) ) ) ).
fof(t7_xboole_1, axiom,  (! [A] :  (! [B] : r1_tarski(A, k2_xboole_0(A, B))) ) ).
fof(t7_xxreal_2, axiom,  (! [A] :  (v2_membered(A) =>  (! [B] :  (v2_membered(B) =>  (! [C] :  (m2_xxreal_2(C, A) =>  (! [D] :  (m2_xxreal_2(D, B) => m2_xxreal_2(k3_xxreal_0(C, D), k2_xboole_0(A, B))) ) ) ) ) ) ) ) ).
