% Mizar problem: t40_cqc_the3,cqc_the3,579,5 
fof(t40_cqc_the3, conjecture,  (! [A] :  (m1_qc_lang1(A) =>  (! [B] :  (m2_subset_1(B, k9_qc_lang1(A), k3_cqc_lang(A)) =>  (! [C] :  (m2_subset_1(C, k9_qc_lang1(A), k3_cqc_lang(A)) =>  (! [D] :  (m1_subset_1(D, k1_zfmisc_1(k3_cqc_lang(A))) =>  (r3_cqc_the1(A, D, k8_cqc_lang(A, B, C)) => r3_cqc_the1(A, k4_subset_1(k3_cqc_lang(A), D, k6_domain_1(k3_cqc_lang(A), B)), C)) ) ) ) ) ) ) ) ) ).
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(cc10_membered, axiom,  (! [A] :  (v5_membered(A) =>  (! [B] :  (m1_subset_1(B, A) => v1_int_1(B)) ) ) ) ).
fof(cc11_membered, axiom,  (! [A] :  (v6_membered(A) =>  (! [B] :  (m1_subset_1(B, A) => v7_ordinal1(B)) ) ) ) ).
fof(cc12_membered, axiom,  (! [A] :  (v1_xboole_0(A) => v6_membered(A)) ) ).
fof(cc13_membered, axiom,  (! [A] :  (v1_membered(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v1_membered(B)) ) ) ) ).
fof(cc14_membered, axiom,  (! [A] :  (v2_membered(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v2_membered(B)) ) ) ) ).
fof(cc15_membered, axiom,  (! [A] :  (v3_membered(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v3_membered(B)) ) ) ) ).
fof(cc16_membered, axiom,  (! [A] :  (v4_membered(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v4_membered(B)) ) ) ) ).
fof(cc17_membered, axiom,  (! [A] :  (v5_membered(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v5_membered(B)) ) ) ) ).
fof(cc18_membered, axiom,  (! [A] :  (v6_membered(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v6_membered(B)) ) ) ) ).
fof(cc19_membered, axiom,  (! [A] :  (v1_xboole_0(A) => v7_membered(A)) ) ).
fof(cc1_membered, axiom,  (! [A] :  (v6_membered(A) => v5_membered(A)) ) ).
fof(cc1_subset_1, axiom,  (! [A] :  (v1_xboole_0(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v1_xboole_0(B)) ) ) ) ).
fof(cc2_membered, axiom,  (! [A] :  (v5_membered(A) => v4_membered(A)) ) ).
fof(cc2_subset_1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) =>  ( ~ (v1_subset_1(B, A))  =>  ~ (v1_xboole_0(B)) ) ) ) ) ) ).
fof(cc2_xxreal_0, axiom,  (! [A] :  (v7_ordinal1(A) => v1_xxreal_0(A)) ) ).
fof(cc3_membered, axiom,  (! [A] :  (v4_membered(A) => v3_membered(A)) ) ).
fof(cc3_subset_1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) =>  (v1_xboole_0(B) => v1_subset_1(B, A)) ) ) ) ) ).
fof(cc4_membered, axiom,  (! [A] :  (v3_membered(A) => v2_membered(A)) ) ).
fof(cc4_subset_1, axiom,  (! [A] :  (v1_xboole_0(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) =>  ~ (v1_subset_1(B, A)) ) ) ) ) ).
fof(cc5_membered, axiom,  (! [A] :  (v3_membered(A) => v1_membered(A)) ) ).
fof(cc6_membered, axiom,  (! [A] :  (v1_membered(A) =>  (! [B] :  (m1_subset_1(B, A) => v1_xcmplx_0(B)) ) ) ) ).
fof(cc7_membered, axiom,  (! [A] :  (v2_membered(A) =>  (! [B] :  (m1_subset_1(B, A) => v1_xxreal_0(B)) ) ) ) ).
fof(cc8_membered, axiom,  (! [A] :  (v3_membered(A) =>  (! [B] :  (m1_subset_1(B, A) => v1_xreal_0(B)) ) ) ) ).
fof(cc9_membered, axiom,  (! [A] :  (v4_membered(A) =>  (! [B] :  (m1_subset_1(B, A) => v1_rat_1(B)) ) ) ) ).
fof(commutativity_k2_xboole_0, axiom,  (! [A, B] : k2_xboole_0(A, B)=k2_xboole_0(B, A)) ).
fof(commutativity_k4_subset_1, axiom,  (! [A, B, C] :  ( (m1_subset_1(B, k1_zfmisc_1(A)) & m1_subset_1(C, k1_zfmisc_1(A)))  => k4_subset_1(A, B, C)=k4_subset_1(A, C, B)) ) ).
fof(d1_tarski, axiom,  (! [A] :  (! [B] :  (B=k1_tarski(A) <=>  (! [C] :  (r2_hidden(C, B) <=> C=A) ) ) ) ) ).
fof(d3_tarski, axiom,  (! [A] :  (! [B] :  (r1_tarski(A, B) <=>  (! [C] :  (r2_hidden(C, A) => r2_hidden(C, B)) ) ) ) ) ).
fof(d3_xboole_0, axiom,  (! [A] :  (! [B] :  (! [C] :  (C=k2_xboole_0(A, B) <=>  (! [D] :  (r2_hidden(D, C) <=>  (r2_hidden(D, A) | r2_hidden(D, B)) ) ) ) ) ) ) ).
fof(dt_k1_tarski, axiom, $true).
fof(dt_k1_xboole_0, axiom, $true).
fof(dt_k1_zfmisc_1, axiom, $true).
fof(dt_k2_qc_lang2, axiom,  (! [A, B, C] :  ( (m1_qc_lang1(A) &  (m1_subset_1(B, k9_qc_lang1(A)) & m1_subset_1(C, k9_qc_lang1(A))) )  => m1_subset_1(k2_qc_lang2(A, B, C), k9_qc_lang1(A))) ) ).
fof(dt_k2_xboole_0, axiom, $true).
fof(dt_k3_cqc_lang, axiom,  (! [A] :  (m1_qc_lang1(A) => m1_subset_1(k3_cqc_lang(A), k1_zfmisc_1(k9_qc_lang1(A)))) ) ).
fof(dt_k4_subset_1, axiom,  (! [A, B, C] :  ( (m1_subset_1(B, k1_zfmisc_1(A)) & m1_subset_1(C, k1_zfmisc_1(A)))  => m1_subset_1(k4_subset_1(A, B, C), k1_zfmisc_1(A))) ) ).
fof(dt_k6_domain_1, axiom,  (! [A, B] :  ( ( ~ (v1_xboole_0(A))  & m1_subset_1(B, A))  => m1_subset_1(k6_domain_1(A, B), k1_zfmisc_1(A))) ) ).
fof(dt_k8_cqc_lang, axiom,  (! [A, B, C] :  ( (m1_qc_lang1(A) &  (m1_subset_1(B, k3_cqc_lang(A)) & m1_subset_1(C, k3_cqc_lang(A))) )  => m2_subset_1(k8_cqc_lang(A, B, C), k9_qc_lang1(A), k3_cqc_lang(A))) ) ).
fof(dt_k9_qc_lang1, axiom,  (! [A] :  (m1_qc_lang1(A) =>  ~ (v1_xboole_0(k9_qc_lang1(A))) ) ) ).
fof(dt_m1_qc_lang1, axiom, $true).
fof(dt_m1_subset_1, axiom, $true).
fof(dt_m2_subset_1, axiom,  (! [A, B] :  ( ( ~ (v1_xboole_0(A))  &  ( ~ (v1_xboole_0(B))  & m1_subset_1(B, k1_zfmisc_1(A))) )  =>  (! [C] :  (m2_subset_1(C, A, B) => m1_subset_1(C, A)) ) ) ) ).
fof(existence_m1_qc_lang1, axiom,  (? [A] : m1_qc_lang1(A)) ).
fof(existence_m1_subset_1, axiom,  (! [A] :  (? [B] : m1_subset_1(B, A)) ) ).
fof(existence_m2_subset_1, axiom,  (! [A, B] :  ( ( ~ (v1_xboole_0(A))  &  ( ~ (v1_xboole_0(B))  & m1_subset_1(B, k1_zfmisc_1(A))) )  =>  (? [C] : m2_subset_1(C, A, B)) ) ) ).
fof(fc10_membered, axiom,  (! [A] :  (v1_rat_1(A) => v4_membered(k1_tarski(A))) ) ).
fof(fc11_membered, axiom,  (! [A] :  (v1_int_1(A) => v5_membered(k1_tarski(A))) ) ).
fof(fc12_membered, axiom,  (! [A] :  (v7_ordinal1(A) => v6_membered(k1_tarski(A))) ) ).
fof(fc1_lukasi_1, axiom,  (! [A, B, C, D] :  ( (m1_qc_lang1(A) &  (m1_subset_1(B, k3_cqc_lang(A)) &  (m1_subset_1(C, k3_cqc_lang(A)) & m1_subset_1(D, k3_cqc_lang(A))) ) )  => v2_cqc_the1(k2_qc_lang2(A, k8_cqc_lang(A, B, C), k8_cqc_lang(A, k8_cqc_lang(A, C, D), k8_cqc_lang(A, B, D))), A)) ) ).
fof(fc1_subset_1, axiom,  (! [A] :  ~ (v1_xboole_0(k1_zfmisc_1(A))) ) ).
fof(fc25_membered, axiom,  (! [A, B] :  ( (v1_membered(A) & v1_membered(B))  => v1_membered(k2_xboole_0(A, B))) ) ).
fof(fc26_membered, axiom,  (! [A, B] :  ( (v2_membered(A) & v2_membered(B))  => v2_membered(k2_xboole_0(A, B))) ) ).
fof(fc27_membered, axiom,  (! [A, B] :  ( (v3_membered(A) & v3_membered(B))  => v3_membered(k2_xboole_0(A, B))) ) ).
fof(fc28_membered, axiom,  (! [A, B] :  ( (v4_membered(A) & v4_membered(B))  => v4_membered(k2_xboole_0(A, B))) ) ).
fof(fc29_membered, axiom,  (! [A, B] :  ( (v5_membered(A) & v5_membered(B))  => v5_membered(k2_xboole_0(A, B))) ) ).
fof(fc2_cqc_lang, axiom,  (! [A] :  (m1_qc_lang1(A) =>  ~ (v1_xboole_0(k3_cqc_lang(A))) ) ) ).
fof(fc2_lukasi_1, axiom,  (! [A, B] :  ( (m1_qc_lang1(A) & m1_subset_1(B, k3_cqc_lang(A)))  => v2_cqc_the1(k2_qc_lang2(A, B, B), A)) ) ).
fof(fc30_membered, axiom,  (! [A, B] :  ( (v6_membered(A) & v6_membered(B))  => v6_membered(k2_xboole_0(A, B))) ) ).
fof(fc3_lukasi_1, axiom,  (! [A, B, C] :  ( (m1_qc_lang1(A) &  (m1_subset_1(B, k3_cqc_lang(A)) & m1_subset_1(C, k3_cqc_lang(A))) )  => v2_cqc_the1(k2_qc_lang2(A, B, k8_cqc_lang(A, C, B)), A)) ) ).
fof(fc4_lukasi_1, axiom,  (! [A, B, C, D] :  ( (m1_qc_lang1(A) &  (m1_subset_1(B, k3_cqc_lang(A)) &  (m1_subset_1(C, k3_cqc_lang(A)) & m1_subset_1(D, k3_cqc_lang(A))) ) )  => v2_cqc_the1(k2_qc_lang2(A, k8_cqc_lang(A, D, k8_cqc_lang(A, C, B)), k8_cqc_lang(A, C, k8_cqc_lang(A, D, B))), A)) ) ).
fof(fc5_lukasi_1, axiom,  (! [A, B, C] :  ( (m1_qc_lang1(A) &  (m1_subset_1(B, k3_cqc_lang(A)) & m1_subset_1(C, k3_cqc_lang(A))) )  => v2_cqc_the1(k2_qc_lang2(A, B, k8_cqc_lang(A, k8_cqc_lang(A, B, C), C)), A)) ) ).
fof(fc6_lukasi_1, axiom,  (! [A, B, C] :  ( (m1_qc_lang1(A) &  (m1_subset_1(B, k3_cqc_lang(A)) & m1_subset_1(C, k3_cqc_lang(A))) )  => v2_cqc_the1(k2_qc_lang2(A, k8_cqc_lang(A, B, k8_cqc_lang(A, B, C)), k8_cqc_lang(A, B, C)), A)) ) ).
fof(fc7_lukasi_1, axiom,  (! [A, B, C, D] :  ( (m1_qc_lang1(A) &  (m1_subset_1(B, k3_cqc_lang(A)) &  (m1_subset_1(C, k3_cqc_lang(A)) & m1_subset_1(D, k3_cqc_lang(A))) ) )  => v2_cqc_the1(k2_qc_lang2(A, k8_cqc_lang(A, B, k8_cqc_lang(A, C, D)), k8_cqc_lang(A, k8_cqc_lang(A, B, C), k8_cqc_lang(A, B, D))), A)) ) ).
fof(fc7_membered, axiom,  (! [A] :  (v1_xcmplx_0(A) => v1_membered(k1_tarski(A))) ) ).
fof(fc8_lukasi_1, axiom,  (! [A, B, C, D] :  ( (m1_qc_lang1(A) &  (m1_subset_1(B, k3_cqc_lang(A)) &  (m1_subset_1(C, k3_cqc_lang(A)) & m1_subset_1(D, k3_cqc_lang(A))) ) )  => v2_cqc_the1(k2_qc_lang2(A, k8_cqc_lang(A, k8_cqc_lang(A, B, C), D), k8_cqc_lang(A, C, D)), A)) ) ).
fof(fc8_membered, axiom,  (! [A] :  (v1_xxreal_0(A) => v2_membered(k1_tarski(A))) ) ).
fof(fc9_lukasi_1, axiom,  (! [A, B, C, D] :  ( (m1_qc_lang1(A) &  (m1_subset_1(B, k3_cqc_lang(A)) &  (m1_subset_1(C, k3_cqc_lang(A)) & m1_subset_1(D, k3_cqc_lang(A))) ) )  => v2_cqc_the1(k2_qc_lang2(A, k8_cqc_lang(A, B, C), k8_cqc_lang(A, k8_cqc_lang(A, D, B), k8_cqc_lang(A, D, C))), A)) ) ).
fof(fc9_membered, axiom,  (! [A] :  (v1_xreal_0(A) => v3_membered(k1_tarski(A))) ) ).
fof(idempotence_k2_xboole_0, axiom,  (! [A, B] : k2_xboole_0(A, A)=A) ).
fof(idempotence_k4_subset_1, axiom,  (! [A, B, C] :  ( (m1_subset_1(B, k1_zfmisc_1(A)) & m1_subset_1(C, k1_zfmisc_1(A)))  => k4_subset_1(A, B, B)=B) ) ).
fof(rc1_membered, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  & v6_membered(A)) ) ).
fof(rc1_subset_1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (? [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) &  ~ (v1_xboole_0(B)) ) ) ) ) ).
fof(rc1_xxreal_0, axiom,  (? [A] : v1_xxreal_0(A)) ).
fof(rc2_membered, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  & v6_membered(A)) ) ).
fof(rc2_subset_1, axiom,  (! [A] :  (? [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) & v1_xboole_0(B)) ) ) ).
fof(rc2_xxreal_0, axiom,  (? [A] : v1_xxreal_0(A)) ).
fof(rc3_membered, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  &  (v6_membered(A) & v7_membered(A)) ) ) ).
fof(rc3_subset_1, axiom,  (! [A] :  (? [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) &  ~ (v1_subset_1(B, A)) ) ) ) ).
fof(rc4_subset_1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (? [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) & v1_subset_1(B, A)) ) ) ) ).
fof(redefinition_k4_subset_1, axiom,  (! [A, B, C] :  ( (m1_subset_1(B, k1_zfmisc_1(A)) & m1_subset_1(C, k1_zfmisc_1(A)))  => k4_subset_1(A, B, C)=k2_xboole_0(B, C)) ) ).
fof(redefinition_k6_domain_1, axiom,  (! [A, B] :  ( ( ~ (v1_xboole_0(A))  & m1_subset_1(B, A))  => k6_domain_1(A, B)=k1_tarski(B)) ) ).
fof(redefinition_k8_cqc_lang, axiom,  (! [A, B, C] :  ( (m1_qc_lang1(A) &  (m1_subset_1(B, k3_cqc_lang(A)) & m1_subset_1(C, k3_cqc_lang(A))) )  => k8_cqc_lang(A, B, C)=k2_qc_lang2(A, B, C)) ) ).
fof(redefinition_m2_subset_1, axiom,  (! [A, B] :  ( ( ~ (v1_xboole_0(A))  &  ( ~ (v1_xboole_0(B))  & m1_subset_1(B, k1_zfmisc_1(A))) )  =>  (! [C] :  (m2_subset_1(C, A, B) <=> m1_subset_1(C, B)) ) ) ) ).
fof(redefinition_r2_tarski, axiom,  (! [A, B] :  (r2_tarski(A, B) <=> r2_hidden(A, B)) ) ).
fof(reflexivity_r1_tarski, axiom,  (! [A, B] : r1_tarski(A, A)) ).
fof(t1_boole, axiom,  (! [A] : k2_xboole_0(A, k1_xboole_0)=A) ).
fof(t1_cqc_the3, axiom,  (! [A] :  (m1_qc_lang1(A) =>  (! [B] :  (m2_subset_1(B, k9_qc_lang1(A), k3_cqc_lang(A)) =>  (! [C] :  (m1_subset_1(C, k1_zfmisc_1(k3_cqc_lang(A))) =>  (r2_tarski(B, C) => r3_cqc_the1(A, C, B)) ) ) ) ) ) ) ).
fof(t1_subset, axiom,  (! [A] :  (! [B] :  (r2_tarski(A, B) => m1_subset_1(A, B)) ) ) ).
fof(t2_subset, axiom,  (! [A] :  (! [B] :  (m1_subset_1(A, B) =>  (v1_xboole_0(B) | r2_tarski(A, B)) ) ) ) ).
fof(t3_subset, axiom,  (! [A] :  (! [B] :  (m1_subset_1(A, k1_zfmisc_1(B)) <=> r1_tarski(A, B)) ) ) ).
fof(t4_cqc_the3, axiom,  (! [A] :  (m1_qc_lang1(A) =>  (! [B] :  (m2_subset_1(B, k9_qc_lang1(A), k3_cqc_lang(A)) =>  (! [C] :  (m1_subset_1(C, k1_zfmisc_1(k3_cqc_lang(A))) =>  (! [D] :  (m1_subset_1(D, k1_zfmisc_1(k3_cqc_lang(A))) =>  ( (r3_cqc_the1(A, C, B) & r1_tarski(C, D))  => r3_cqc_the1(A, D, B)) ) ) ) ) ) ) ) ) ).
fof(t4_subset, axiom,  (! [A] :  (! [B] :  (! [C] :  ( (r2_tarski(A, B) & m1_subset_1(B, k1_zfmisc_1(C)))  => m1_subset_1(A, C)) ) ) ) ).
fof(t55_cqc_the1, axiom,  (! [A] :  (m1_qc_lang1(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(k3_cqc_lang(A))) =>  (! [C] :  (m2_subset_1(C, k9_qc_lang1(A), k3_cqc_lang(A)) =>  (! [D] :  (m2_subset_1(D, k9_qc_lang1(A), k3_cqc_lang(A)) =>  ( (r3_cqc_the1(A, B, C) & r3_cqc_the1(A, B, k8_cqc_lang(A, C, D)))  => r3_cqc_the1(A, B, D)) ) ) ) ) ) ) ) ) ).
fof(t5_subset, axiom,  (! [A] :  (! [B] :  (! [C] :  ~ ( (r2_tarski(A, B) &  (m1_subset_1(B, k1_zfmisc_1(C)) & v1_xboole_0(C)) ) ) ) ) ) ).
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(t7_xboole_1, axiom,  (! [A] :  (! [B] : r1_tarski(A, k2_xboole_0(A, B))) ) ).
fof(t8_boole, axiom,  (! [A] :  (! [B] :  ~ ( (v1_xboole_0(A) &  ( ~ (A=B)  & v1_xboole_0(B)) ) ) ) ) ).
