% Mizar problem: t67_funct_1,funct_1,1796,5 
fof(t67_funct_1, conjecture,  (! [A] :  (! [B] :  (! [C] :  ( (v1_relat_1(C) & v1_funct_1(C))  => k7_relat_1(k6_relat_1(B, C), A)=k3_xboole_0(B, k7_relat_1(C, A))) ) ) ) ).
fof(antisymmetry_r2_hidden, axiom,  (! [A, B] :  (r2_hidden(A, B) =>  ~ (r2_hidden(B, A)) ) ) ).
fof(asymmetry_r2_tarski, axiom,  (! [A, B] :  (r2_tarski(A, B) =>  ~ (r2_tarski(B, A)) ) ) ).
fof(cc1_funct_1, axiom,  (! [A] :  (v1_xboole_0(A) => v1_funct_1(A)) ) ).
fof(cc1_relat_1, axiom,  (! [A] :  (v1_xboole_0(A) => v1_relat_1(A)) ) ).
fof(cc1_zfmisc_1, axiom,  (! [A] :  (v1_xboole_0(A) => v1_zfmisc_1(A)) ) ).
fof(cc2_funct_1, axiom,  (! [A] :  ( (v1_xboole_0(A) &  (v1_relat_1(A) & v1_funct_1(A)) )  =>  (v1_relat_1(A) &  (v1_funct_1(A) & v2_funct_1(A)) ) ) ) ).
fof(cc2_zfmisc_1, axiom,  (! [A] :  ( ~ (v1_zfmisc_1(A))  =>  ~ (v1_xboole_0(A)) ) ) ).
fof(cc3_relat_1, axiom,  (! [A] :  ( (v1_xboole_0(A) & v1_relat_1(A))  =>  (v1_relat_1(A) & v3_relat_1(A)) ) ) ).
fof(cc4_relat_1, axiom,  (! [A] :  ( (v1_xboole_0(A) & v1_relat_1(A))  =>  (v1_relat_1(A) & v2_relat_1(A)) ) ) ).
fof(commutativity_k3_xboole_0, axiom,  (! [A, B] : k3_xboole_0(A, B)=k3_xboole_0(B, A)) ).
fof(d4_xboole_0, axiom,  (! [A] :  (! [B] :  (! [C] :  (C=k3_xboole_0(A, B) <=>  (! [D] :  (r2_hidden(D, C) <=>  (r2_hidden(D, A) & r2_hidden(D, B)) ) ) ) ) ) ) ).
fof(d6_funct_1, axiom,  (! [A] :  ( (v1_relat_1(A) & v1_funct_1(A))  =>  (! [B] :  (! [C] :  (C=k7_relat_1(A, B) <=>  (! [D] :  (r2_hidden(D, C) <=>  (? [E] :  (r2_hidden(E, k9_xtuple_0(A)) &  (r2_hidden(E, B) & D=k1_funct_1(A, E)) ) ) ) ) ) ) ) ) ) ).
fof(dt_k1_funct_1, axiom, $true).
fof(dt_k1_xboole_0, axiom, $true).
fof(dt_k3_xboole_0, axiom, $true).
fof(dt_k6_relat_1, axiom,  (! [A, B] :  (v1_relat_1(B) => v1_relat_1(k6_relat_1(A, B))) ) ).
fof(dt_k7_relat_1, axiom, $true).
fof(dt_k9_xtuple_0, axiom, $true).
fof(dt_m1_subset_1, axiom, $true).
fof(existence_m1_subset_1, axiom,  (! [A] :  (? [B] : m1_subset_1(B, A)) ) ).
fof(fc10_relat_1, axiom,  (! [A] :  (v1_xboole_0(A) => v1_xboole_0(k9_xtuple_0(A))) ) ).
fof(fc18_relat_1, axiom,  (! [A, B] :  ( (v1_relat_1(A) & v1_xboole_0(B))  =>  (v1_xboole_0(k6_relat_1(B, A)) & v1_relat_1(k6_relat_1(B, A))) ) ) ).
fof(fc19_relat_1, axiom,  (! [A, B] :  ( (v1_relat_1(A) & v1_xboole_0(B))  => v1_xboole_0(k7_relat_1(A, B))) ) ).
fof(fc1_relat_1, axiom,  (! [A, B] :  (v1_relat_1(A) => v1_relat_1(k3_xboole_0(A, B))) ) ).
fof(fc1_xboole_0, axiom, v1_xboole_0(k1_xboole_0)).
fof(fc20_relat_1, axiom,  (! [A, B] :  ( (v1_xboole_0(A) & v1_relat_1(A))  => v1_xboole_0(k7_relat_1(A, B))) ) ).
fof(fc24_relat_1, axiom,  (! [A] :  ( (v1_zfmisc_1(A) & v1_relat_1(A))  => v1_zfmisc_1(k9_xtuple_0(A))) ) ).
fof(fc8_relat_1, axiom,  (! [A] :  ( ( ~ (v1_xboole_0(A))  & v1_relat_1(A))  =>  ~ (v1_xboole_0(k9_xtuple_0(A))) ) ) ).
fof(fc9_funct_1, axiom,  (! [A, B] :  ( (v1_relat_1(B) & v1_funct_1(B))  =>  (v1_relat_1(k6_relat_1(A, B)) & v1_funct_1(k6_relat_1(A, B))) ) ) ).
fof(idempotence_k3_xboole_0, axiom,  (! [A, B] : k3_xboole_0(A, A)=A) ).
fof(rc1_funct_1, axiom,  (? [A] :  (v1_relat_1(A) & v1_funct_1(A)) ) ).
fof(rc1_relat_1, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  & v1_relat_1(A)) ) ).
fof(rc1_xboole_0, axiom,  (? [A] : v1_xboole_0(A)) ).
fof(rc1_zfmisc_1, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  & v1_zfmisc_1(A)) ) ).
fof(rc2_funct_1, axiom,  (? [A] :  (v1_relat_1(A) &  (v1_funct_1(A) & v2_funct_1(A)) ) ) ).
fof(rc2_relat_1, axiom,  (? [A] :  (v1_relat_1(A) & v2_relat_1(A)) ) ).
fof(rc2_xboole_0, axiom,  (? [A] :  ~ (v1_xboole_0(A)) ) ).
fof(rc2_zfmisc_1, axiom,  (? [A] :  ~ (v1_zfmisc_1(A)) ) ).
fof(rd7_relat_1, axiom,  (! [A, B] :  (v1_relat_1(B) => k6_relat_1(A, k6_relat_1(A, B))=k6_relat_1(A, B)) ) ).
fof(redefinition_r2_tarski, axiom,  (! [A, B] :  (r2_tarski(A, B) <=> r2_hidden(A, B)) ) ).
fof(t1_subset, axiom,  (! [A] :  (! [B] :  (r2_tarski(A, B) => m1_subset_1(A, B)) ) ) ).
fof(t2_boole, axiom,  (! [A] : k3_xboole_0(A, k1_xboole_0)=k1_xboole_0) ).
fof(t2_subset, axiom,  (! [A] :  (! [B] :  (m1_subset_1(A, B) =>  (v1_xboole_0(B) | r2_tarski(A, B)) ) ) ) ).
fof(t2_tarski, axiom,  (! [A] :  (! [B] :  ( (! [C] :  (r2_hidden(C, A) <=> r2_hidden(C, B)) )  => A=B) ) ) ).
fof(t53_funct_1, axiom,  (! [A] :  (! [B] :  ( (v1_relat_1(B) & v1_funct_1(B))  =>  (! [C] :  ( (v1_relat_1(C) & v1_funct_1(C))  =>  (C=k6_relat_1(A, B) <=>  ( (! [D] :  (r2_hidden(D, k9_xtuple_0(C)) <=>  (r2_hidden(D, k9_xtuple_0(B)) & r2_tarski(k1_funct_1(B, D), A)) ) )  &  (! [D] :  (r2_hidden(D, k9_xtuple_0(C)) => k1_funct_1(C, D)=k1_funct_1(B, D)) ) ) ) ) ) ) ) ) ).
fof(t6_boole, axiom,  (! [A] :  (v1_xboole_0(A) => A=k1_xboole_0) ) ).
fof(t7_boole, axiom,  (! [A] :  (! [B] :  ~ ( (r2_tarski(A, B) & v1_xboole_0(B)) ) ) ) ).
fof(t8_boole, axiom,  (! [A] :  (! [B] :  ~ ( (v1_xboole_0(A) &  ( ~ (A=B)  & v1_xboole_0(B)) ) ) ) ) ).
