% Mizar problem: t13_substlat,substlat,469,5 
fof(t13_substlat, conjecture,  (! [A] :  (! [B] :  (! [C] :  (m1_subset_1(C, k5_finsub_1(k4_partfun1(A, B))) =>  (! [D] :  (m1_subset_1(D, k5_finsub_1(k4_partfun1(A, B))) => k3_substlat(A, B, k1_finsub_1(k5_finsub_1(k4_partfun1(A, B)), k3_substlat(A, B, C), D))=k3_substlat(A, B, k1_finsub_1(k5_finsub_1(k4_partfun1(A, B)), C, D))) ) ) ) ) ) ).
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_finset_1, axiom,  (! [A] :  (v1_xboole_0(A) => v1_finset_1(A)) ) ).
fof(cc1_finsub_1, axiom,  (! [A] :  (v4_finsub_1(A) =>  (v1_finsub_1(A) & v3_finsub_1(A)) ) ) ).
fof(cc1_funct_1, axiom,  (! [A] :  (v1_xboole_0(A) => v1_funct_1(A)) ) ).
fof(cc1_substlat, axiom,  (! [A, B] :  (! [C] :  (m1_subset_1(C, k1_substlat(A, B)) => v1_finset_1(C)) ) ) ).
fof(cc2_finset_1, axiom,  (! [A] :  (v1_finset_1(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v1_finset_1(B)) ) ) ) ).
fof(cc2_finsub_1, axiom,  (! [A] :  ( (v1_finsub_1(A) & v3_finsub_1(A))  => v4_finsub_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_substlat, axiom,  (! [A, B] :  (! [C] :  (m1_subset_1(C, k1_substlat(A, B)) => v4_funct_1(C)) ) ) ).
fof(cc3_finsub_1, axiom,  (! [A] :  (! [B] :  (m1_subset_1(B, k5_finsub_1(A)) => v1_finset_1(B)) ) ) ).
fof(cc3_funct_1, axiom,  (! [A] :  ( (v1_relat_1(A) & v1_funct_1(A))  =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v1_funct_1(B)) ) ) ) ).
fof(cc4_finset_1, axiom,  (! [A] :  (v1_zfmisc_1(A) => v1_finset_1(A)) ) ).
fof(cc4_funct_1, axiom,  (! [A] :  ( (v1_xboole_0(A) &  (v1_relat_1(A) & v1_funct_1(A)) )  =>  (v1_relat_1(A) &  (v1_funct_1(A) & v3_funct_1(A)) ) ) ) ).
fof(cc5_finset_1, axiom,  (! [A] :  ( ~ (v1_finset_1(A))  =>  ~ (v1_zfmisc_1(A)) ) ) ).
fof(cc5_funct_1, axiom,  (! [A] :  ( (v1_relat_1(A) &  (v1_funct_1(A) &  ~ (v3_funct_1(A)) ) )  =>  ( ~ (v1_zfmisc_1(A))  &  (v1_relat_1(A) & v1_funct_1(A)) ) ) ) ).
fof(cc6_finset_1, axiom,  (! [A] :  (v1_xboole_0(A) => v5_finset_1(A)) ) ).
fof(cc6_funct_1, axiom,  (! [A] :  ( (v1_zfmisc_1(A) &  (v1_relat_1(A) & v1_funct_1(A)) )  =>  (v1_relat_1(A) &  (v1_funct_1(A) & v3_funct_1(A)) ) ) ) ).
fof(cc7_finset_1, axiom,  (! [A] :  (v5_finset_1(A) =>  (! [B] :  (m1_subset_1(B, A) => v1_finset_1(B)) ) ) ) ).
fof(cc7_funct_1, axiom,  (! [A] :  (v1_xboole_0(A) => v4_funct_1(A)) ) ).
fof(cc8_finset_1, axiom,  (! [A] :  (v5_finset_1(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v5_finset_1(B)) ) ) ) ).
fof(cc8_funct_1, axiom,  (! [A] :  (v4_funct_1(A) =>  (! [B] :  (m1_subset_1(B, A) =>  (v1_relat_1(B) & v1_funct_1(B)) ) ) ) ) ).
fof(cc9_finset_1, axiom,  (! [A] :  ( (v1_xboole_0(A) & v1_relat_1(A))  =>  (v1_relat_1(A) & v2_finset_1(A)) ) ) ).
fof(cc9_funct_1, axiom,  (! [A] :  (v4_funct_1(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v4_funct_1(B)) ) ) ) ).
fof(commutativity_k1_finsub_1, axiom,  (! [A, B, C] :  ( ( ( ~ (v1_xboole_0(A))  & v4_finsub_1(A))  &  (m1_subset_1(B, A) & m1_subset_1(C, A)) )  => k1_finsub_1(A, B, C)=k1_finsub_1(A, C, B)) ) ).
fof(commutativity_k2_xboole_0, axiom,  (! [A, B] : k2_xboole_0(A, B)=k2_xboole_0(B, A)) ).
fof(d10_xboole_0, axiom,  (! [A] :  (! [B] :  (A=B <=>  (r1_tarski(A, B) & r1_tarski(B, A)) ) ) ) ).
fof(d1_substlat, axiom,  (! [A] :  (! [B] : k1_substlat(A, B)=a_2_0_substlat(A, B)) ) ).
fof(d2_substlat, axiom,  (! [A] :  (! [B] :  (! [C] :  (m1_subset_1(C, k5_finsub_1(k4_partfun1(A, B))) => k3_substlat(A, B, C)=a_3_0_substlat(A, B, C)) ) ) ) ).
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_finsub_1, axiom,  (! [A, B, C] :  ( ( ( ~ (v1_xboole_0(A))  & v4_finsub_1(A))  &  (m1_subset_1(B, A) & m1_subset_1(C, A)) )  => m1_subset_1(k1_finsub_1(A, B, C), A)) ) ).
fof(dt_k1_substlat, axiom,  (! [A, B] : m1_subset_1(k1_substlat(A, B), k1_zfmisc_1(k5_finsub_1(k4_partfun1(A, B))))) ).
fof(dt_k1_xboole_0, axiom, $true).
fof(dt_k1_zfmisc_1, axiom, $true).
fof(dt_k2_xboole_0, axiom, $true).
fof(dt_k3_substlat, axiom,  (! [A, B, C] :  (m1_subset_1(C, k5_finsub_1(k4_partfun1(A, B))) => m2_subset_1(k3_substlat(A, B, C), k5_finsub_1(k4_partfun1(A, B)), k1_substlat(A, B))) ) ).
fof(dt_k4_partfun1, axiom, $true).
fof(dt_k5_finsub_1, axiom,  (! [A] : v4_finsub_1(k5_finsub_1(A))) ).
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_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(fc17_finset_1, axiom,  (! [A] :  (v1_finset_1(A) => v1_finset_1(k1_zfmisc_1(A))) ) ).
fof(fc1_finsub_1, axiom,  (! [A] : v4_finsub_1(k1_zfmisc_1(A))) ).
fof(fc1_partfun1, axiom,  (! [A, B] :  ~ (v1_xboole_0(k4_partfun1(A, B))) ) ).
fof(fc1_substlat, axiom,  (! [A, B] :  ~ (v1_xboole_0(k1_substlat(A, B))) ) ).
fof(fc1_xboole_0, axiom, v1_xboole_0(k1_xboole_0)).
fof(fc2_finsub_1, axiom,  (! [A] :  ( ~ (v1_xboole_0(k5_finsub_1(A)))  & v4_finsub_1(k5_finsub_1(A))) ) ).
fof(fc31_finset_1, axiom,  (! [A] :  (v1_finset_1(A) => v5_finset_1(k1_zfmisc_1(A))) ) ).
fof(fc33_finset_1, axiom,  (! [A, B] :  ( (v5_finset_1(A) & v5_finset_1(B))  => v5_finset_1(k2_xboole_0(A, B))) ) ).
fof(fc4_partfun1, axiom,  (! [A, B] : v4_funct_1(k4_partfun1(A, B))) ).
fof(fc4_xboole_0, axiom,  (! [A, B] :  ( ~ (v1_xboole_0(A))  =>  ~ (v1_xboole_0(k2_xboole_0(A, B))) ) ) ).
fof(fc5_xboole_0, axiom,  (! [A, B] :  ( ~ (v1_xboole_0(A))  =>  ~ (v1_xboole_0(k2_xboole_0(B, A))) ) ) ).
fof(fc9_finset_1, axiom,  (! [A, B] :  ( (v1_finset_1(A) & v1_finset_1(B))  => v1_finset_1(k2_xboole_0(A, B))) ) ).
fof(fraenkel_a_2_0_substlat, axiom,  (! [A, B, C] :  (r2_hidden(A, a_2_0_substlat(B, C)) <=>  (? [D] :  (m1_subset_1(D, k5_finsub_1(k4_partfun1(B, C))) &  (A=D &  ( (! [E] :  (r2_tarski(E, D) => v1_finset_1(E)) )  &  (! [E] :  (m1_subset_1(E, k4_partfun1(B, C)) =>  (! [F] :  (m1_subset_1(F, k4_partfun1(B, C)) =>  ( (r2_tarski(E, D) &  (r2_tarski(F, D) & r1_tarski(E, F)) )  => E=F) ) ) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_3_0_substlat, axiom,  (! [A, B, C, D] :  (m1_subset_1(D, k5_finsub_1(k4_partfun1(B, C))) =>  (r2_hidden(A, a_3_0_substlat(B, C, D)) <=>  (? [E] :  (m1_subset_1(E, k4_partfun1(B, C)) &  (A=E &  (v1_finset_1(E) &  (! [F] :  (m1_subset_1(F, k4_partfun1(B, C)) =>  ( (r2_tarski(F, D) & r1_tarski(F, E))  <=> F=E) ) ) ) ) ) ) ) ) ) ).
fof(idempotence_k1_finsub_1, axiom,  (! [A, B, C] :  ( ( ( ~ (v1_xboole_0(A))  & v4_finsub_1(A))  &  (m1_subset_1(B, A) & m1_subset_1(C, A)) )  => k1_finsub_1(A, B, B)=B) ) ).
fof(idempotence_k2_xboole_0, axiom,  (! [A, B] : k2_xboole_0(A, A)=A) ).
fof(l2_substlat, axiom,  (! [A] :  (! [B] :  (! [C] :  (! [D] :  ( (r2_tarski(D, k1_substlat(A, B)) & r2_tarski(C, D))  => v1_finset_1(C)) ) ) ) ) ).
fof(rc10_finset_1, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  &  (v1_relat_1(A) &  (v1_funct_1(A) & v2_finset_1(A)) ) ) ) ).
fof(rc1_finset_1, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  & v1_finset_1(A)) ) ).
fof(rc1_funct_1, axiom,  (? [A] :  (v1_relat_1(A) & v1_funct_1(A)) ) ).
fof(rc1_substlat, axiom,  (! [A, B] :  (? [C] :  (m1_subset_1(C, k1_substlat(A, B)) &  ~ (v1_xboole_0(C)) ) ) ) ).
fof(rc1_xboole_0, axiom,  (? [A] : v1_xboole_0(A)) ).
fof(rc2_finset_1, axiom,  (! [A] :  (? [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) & v1_finset_1(B)) ) ) ).
fof(rc2_funct_1, axiom,  (? [A] :  (v1_relat_1(A) &  (v1_funct_1(A) & v2_funct_1(A)) ) ) ).
fof(rc2_substlat, axiom,  (! [A, B] :  (? [C] :  (m1_subset_1(C, k4_partfun1(A, B)) &  (v1_relat_1(C) &  (v1_funct_1(C) & v1_finset_1(C)) ) ) ) ) ).
fof(rc2_xboole_0, axiom,  (? [A] :  ~ (v1_xboole_0(A)) ) ).
fof(rc3_finset_1, axiom,  (! [A] :  ( ~ (v1_xboole_0(A))  =>  (? [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) &  ( ~ (v1_xboole_0(B))  & v1_finset_1(B)) ) ) ) ) ).
fof(rc4_finset_1, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  &  (v1_relat_1(A) &  (v1_funct_1(A) & v1_finset_1(A)) ) ) ) ).
fof(rc5_funct_1, axiom,  (? [A] :  (v1_relat_1(A) &  (v1_funct_1(A) &  ~ (v3_funct_1(A)) ) ) ) ).
fof(rc7_funct_1, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  & v4_funct_1(A)) ) ).
fof(rc8_finset_1, axiom,  (! [A] :  ( ~ (v1_zfmisc_1(A))  =>  (? [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) &  ( ~ (v1_zfmisc_1(B))  & v1_finset_1(B)) ) ) ) ) ).
fof(rc9_finset_1, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  &  (v1_finset_1(A) & v5_finset_1(A)) ) ) ).
fof(redefinition_k1_finsub_1, axiom,  (! [A, B, C] :  ( ( ( ~ (v1_xboole_0(A))  & v4_finsub_1(A))  &  (m1_subset_1(B, A) & m1_subset_1(C, A)) )  => k1_finsub_1(A, B, C)=k2_xboole_0(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(t10_substlat, axiom,  (! [A] :  (! [B] :  (! [C] :  (m1_subset_1(C, k5_finsub_1(k4_partfun1(A, B))) =>  (! [D] :  (v1_finset_1(D) =>  ~ ( (r2_tarski(D, C) &  (! [E] :  ~ ( (r1_tarski(E, D) & r2_tarski(E, k3_substlat(A, B, C))) ) ) ) ) ) ) ) ) ) ) ).
fof(t12_substlat, axiom,  (! [A] :  (! [B] :  (! [C] :  (m1_subset_1(C, k5_finsub_1(k4_partfun1(A, B))) =>  (! [D] :  (m1_subset_1(D, k5_finsub_1(k4_partfun1(A, B))) => r1_tarski(k3_substlat(A, B, k1_finsub_1(k5_finsub_1(k4_partfun1(A, B)), C, D)), k1_finsub_1(k5_finsub_1(k4_partfun1(A, B)), k3_substlat(A, B, C), D))) ) ) ) ) ) ).
fof(t1_boole, axiom,  (! [A] : k2_xboole_0(A, k1_xboole_0)=A) ).
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(t2_tarski, axiom,  (! [A] :  (! [B] :  ( (! [C] :  (r2_hidden(C, A) <=> r2_hidden(C, B)) )  => A=B) ) ) ).
fof(t3_subset, axiom,  (! [A] :  (! [B] :  (m1_subset_1(A, k1_zfmisc_1(B)) <=> r1_tarski(A, 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(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(t6_substlat, axiom,  (! [A] :  (! [B] :  (! [C] :  (m1_subset_1(C, k5_finsub_1(k4_partfun1(A, B))) =>  (! [D] :  (r2_tarski(D, k3_substlat(A, B, C)) =>  (r2_tarski(D, C) &  (! [E] :  ( (r2_tarski(E, C) & r1_tarski(E, D))  => E=D) ) ) ) ) ) ) ) ) ).
fof(t7_boole, axiom,  (! [A] :  (! [B] :  ~ ( (r2_tarski(A, B) & v1_xboole_0(B)) ) ) ) ).
fof(t7_substlat, axiom,  (! [A] :  (! [B] :  (! [C] :  (m1_subset_1(C, k5_finsub_1(k4_partfun1(A, B))) =>  (! [D] :  (v1_finset_1(D) =>  ( (r2_tarski(D, C) &  (! [E] :  (v1_finset_1(E) =>  ( (r2_tarski(E, C) & r1_tarski(E, D))  => E=D) ) ) )  => r2_tarski(D, k3_substlat(A, B, C))) ) ) ) ) ) ) ).
fof(t8_boole, axiom,  (! [A] :  (! [B] :  ~ ( (v1_xboole_0(A) &  ( ~ (A=B)  & v1_xboole_0(B)) ) ) ) ) ).
fof(t8_substlat, axiom,  (! [A] :  (! [B] :  (! [C] :  (m1_subset_1(C, k5_finsub_1(k4_partfun1(A, B))) => r1_tarski(k3_substlat(A, B, C), C)) ) ) ) ).
fof(t9_xboole_1, axiom,  (! [A] :  (! [B] :  (! [C] :  (r1_tarski(A, B) => r1_tarski(k2_xboole_0(A, C), k2_xboole_0(B, C))) ) ) ) ).
