% Mizar problem: t23_projred2,projred2,1363,5 
fof(t23_projred2, conjecture,  (! [A] :  ( (v6_incsp_1(A) &  (v1_incproj(A) &  (v2_incproj(A) &  (v3_incproj(A) &  (v4_incproj(A) &  (v5_incproj(A) &  (v9_incproj(A) & l1_incsp_1(A)) ) ) ) ) ) )  =>  (! [B] :  (m1_subset_1(B, u1_incsp_1(A)) =>  (! [C] :  (m1_subset_1(C, u1_incsp_1(A)) =>  (! [D] :  (m1_subset_1(D, u1_incsp_1(A)) =>  (! [E] :  (m1_subset_1(E, u2_incsp_1(A)) =>  (! [F] :  (m1_subset_1(F, u2_incsp_1(A)) =>  (! [G] :  (m1_subset_1(G, u2_incsp_1(A)) =>  (! [H] :  (m1_subset_1(H, u2_incsp_1(A)) =>  ~ ( ( ~ (r1_incsp_1(A, B, E))  &  ( ~ (r1_incsp_1(A, C, F))  &  ( ~ (r1_incsp_1(A, B, G))  &  ( ~ (r1_incsp_1(A, C, G))  &  ( ~ (B=C)  &  (r1_incsp_1(A, B, H) &  (r1_incsp_1(A, C, H) &  (r1_incsp_1(A, D, H) &  ( ~ (r1_incsp_1(A, D, E))  &  ( ~ (D=C)  &  ( ~ (r1_projred2(A, E, F, G))  &  (! [I] :  (m1_subset_1(I, u2_incsp_1(A)) =>  ~ ( (r1_projred2(A, E, G, I) &  ( ~ (r1_incsp_1(A, C, I))  &  ( ~ (r1_incsp_1(A, D, I))  & k3_relat_1(k1_projred1(A, E, G, B), k1_projred1(A, G, F, C))=k3_relat_1(k1_projred1(A, E, I, D), k1_projred1(A, I, F, 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(cc1_funct_1, axiom,  (! [A] :  (v1_xboole_0(A) => v1_funct_1(A)) ) ).
fof(cc1_relset_1, axiom,  (! [A, B] :  (! [C] :  (m1_subset_1(C, k1_zfmisc_1(k2_zfmisc_1(A, B))) => v1_relat_1(C)) ) ) ).
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_relset_1, axiom,  (! [A, B] :  (! [C] :  (m1_subset_1(C, k1_zfmisc_1(k2_zfmisc_1(A, B))) =>  (v4_relat_1(C, A) & v5_relat_1(C, 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(cc3_relset_1, axiom,  (! [A, B] :  (v1_xboole_0(A) =>  (! [C] :  (m1_subset_1(C, k1_zfmisc_1(k2_zfmisc_1(A, B))) => v1_xboole_0(C)) ) ) ) ).
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(cc4_relset_1, axiom,  (! [A, B] :  (v1_xboole_0(A) =>  (! [C] :  (m1_subset_1(C, k1_zfmisc_1(k2_zfmisc_1(B, A))) => v1_xboole_0(C)) ) ) ) ).
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_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_funct_1, axiom,  (! [A] :  (v1_xboole_0(A) => v4_funct_1(A)) ) ).
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_funct_1, axiom,  (! [A] :  (v4_funct_1(A) =>  (! [B] :  (m1_subset_1(B, k1_zfmisc_1(A)) => v4_funct_1(B)) ) ) ) ).
fof(d1_projred2, axiom,  (! [A] :  ( (v6_incsp_1(A) &  (v1_incproj(A) &  (v2_incproj(A) &  (v3_incproj(A) &  (v4_incproj(A) & l1_incsp_1(A)) ) ) ) )  =>  (! [B] :  (m1_subset_1(B, u2_incsp_1(A)) =>  (! [C] :  (m1_subset_1(C, u2_incsp_1(A)) =>  (! [D] :  (m1_subset_1(D, u2_incsp_1(A)) =>  (r1_projred2(A, B, C, D) <=>  (? [E] :  (m1_subset_1(E, u1_incsp_1(A)) &  (r1_incsp_1(A, E, B) &  (r1_incsp_1(A, E, C) & r1_incsp_1(A, E, D)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d9_incproj, axiom,  (! [A] :  ( (v6_incsp_1(A) &  (v1_incproj(A) &  (v2_incproj(A) &  (v3_incproj(A) &  (v4_incproj(A) & l1_incsp_1(A)) ) ) ) )  =>  (v5_incproj(A) <=>  (! [B] :  (m1_subset_1(B, u2_incsp_1(A)) =>  (! [C] :  (m1_subset_1(C, u2_incsp_1(A)) =>  (? [D] :  (m1_subset_1(D, u1_incsp_1(A)) &  (r1_incsp_1(A, D, B) & r1_incsp_1(A, D, C)) ) ) ) ) ) ) ) ) ) ).
fof(dt_k1_projred1, axiom,  (! [A, B, C, D] :  ( ( (v6_incsp_1(A) &  (v1_incproj(A) &  (v2_incproj(A) &  (v3_incproj(A) &  (v4_incproj(A) &  (v5_incproj(A) &  (v9_incproj(A) & l1_incsp_1(A)) ) ) ) ) ) )  &  (m1_subset_1(B, u2_incsp_1(A)) &  (m1_subset_1(C, u2_incsp_1(A)) & m1_subset_1(D, u1_incsp_1(A))) ) )  =>  (v1_funct_1(k1_projred1(A, B, C, D)) & m1_subset_1(k1_projred1(A, B, C, D), k1_zfmisc_1(k2_zfmisc_1(u1_incsp_1(A), u1_incsp_1(A))))) ) ) ).
fof(dt_k1_xboole_0, axiom, $true).
fof(dt_k1_zfmisc_1, axiom, $true).
fof(dt_k2_zfmisc_1, axiom, $true).
fof(dt_k3_relat_1, axiom,  (! [A, B] : v1_relat_1(k3_relat_1(A, B))) ).
fof(dt_l1_incsp_1, axiom, $true).
fof(dt_m1_subset_1, axiom, $true).
fof(dt_u1_incsp_1, axiom,  (! [A] :  (l1_incsp_1(A) =>  ~ (v1_xboole_0(u1_incsp_1(A))) ) ) ).
fof(dt_u2_incsp_1, axiom,  (! [A] :  (l1_incsp_1(A) =>  ~ (v1_xboole_0(u2_incsp_1(A))) ) ) ).
fof(existence_l1_incsp_1, axiom,  (? [A] : l1_incsp_1(A)) ).
fof(existence_m1_subset_1, axiom,  (! [A] :  (? [B] : m1_subset_1(B, A)) ) ).
fof(fc2_funct_1, axiom,  (! [A, B] :  ( ( (v1_relat_1(A) & v1_funct_1(A))  &  (v1_relat_1(B) & v1_funct_1(B)) )  =>  (v1_relat_1(k3_relat_1(A, B)) & v1_funct_1(k3_relat_1(A, B))) ) ) ).
fof(fc7_funct_1, axiom,  (! [A, B] :  ( ( (v1_relat_1(A) &  (v1_funct_1(A) & v2_funct_1(A)) )  &  (v1_relat_1(B) &  (v1_funct_1(B) & v2_funct_1(B)) ) )  =>  (v1_relat_1(k3_relat_1(A, B)) & v2_funct_1(k3_relat_1(A, B))) ) ) ).
fof(l28_projred2, axiom,  (! [A] :  ( (v6_incsp_1(A) &  (v1_incproj(A) &  (v2_incproj(A) &  (v3_incproj(A) &  (v4_incproj(A) &  (v5_incproj(A) &  (v9_incproj(A) & l1_incsp_1(A)) ) ) ) ) ) )  =>  (! [B] :  (m1_subset_1(B, u1_incsp_1(A)) =>  (! [C] :  (m1_subset_1(C, u1_incsp_1(A)) =>  (! [D] :  (m1_subset_1(D, u1_incsp_1(A)) =>  (! [E] :  (m1_subset_1(E, u1_incsp_1(A)) =>  (! [F] :  (m1_subset_1(F, u2_incsp_1(A)) =>  (! [G] :  (m1_subset_1(G, u2_incsp_1(A)) =>  (! [H] :  (m1_subset_1(H, u2_incsp_1(A)) =>  (! [I] :  (m1_subset_1(I, u2_incsp_1(A)) =>  ~ ( ( ~ (r1_incsp_1(A, B, F))  &  ( ~ (r1_incsp_1(A, C, G))  &  ( ~ (r1_incsp_1(A, B, H))  &  ( ~ (r1_incsp_1(A, C, H))  &  (r1_incsp_1(A, D, F) &  (r1_incsp_1(A, D, H) &  ( ~ (B=C)  &  (r1_incsp_1(A, B, I) &  (r1_incsp_1(A, C, I) &  (r1_incsp_1(A, E, I) &  ( ~ (r1_incsp_1(A, E, F))  &  ( ~ (E=C)  &  ( ~ (r1_projred2(A, F, G, H))  &  ( ~ (r1_projred2(A, G, H, I))  &  (! [J] :  (m1_subset_1(J, u2_incsp_1(A)) =>  ~ ( (r1_incsp_1(A, D, J) &  ( ~ (r1_incsp_1(A, C, J))  &  ( ~ (r1_incsp_1(A, E, J))  & k3_relat_1(k1_projred1(A, F, H, B), k1_projred1(A, H, G, C))=k3_relat_1(k1_projred1(A, F, J, E), k1_projred1(A, J, G, C))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(l29_projred2, axiom,  (! [A] :  ( (v6_incsp_1(A) &  (v1_incproj(A) &  (v2_incproj(A) &  (v3_incproj(A) &  (v4_incproj(A) &  (v5_incproj(A) &  (v9_incproj(A) & l1_incsp_1(A)) ) ) ) ) ) )  =>  (! [B] :  (m1_subset_1(B, u1_incsp_1(A)) =>  (! [C] :  (m1_subset_1(C, u1_incsp_1(A)) =>  (! [D] :  (m1_subset_1(D, u1_incsp_1(A)) =>  (! [E] :  (m1_subset_1(E, u1_incsp_1(A)) =>  (! [F] :  (m1_subset_1(F, u2_incsp_1(A)) =>  (! [G] :  (m1_subset_1(G, u2_incsp_1(A)) =>  (! [H] :  (m1_subset_1(H, u2_incsp_1(A)) =>  (! [I] :  (m1_subset_1(I, u2_incsp_1(A)) =>  ~ ( ( ~ (r1_incsp_1(A, B, F))  &  ( ~ (r1_incsp_1(A, C, G))  &  ( ~ (r1_incsp_1(A, B, H))  &  ( ~ (r1_incsp_1(A, C, H))  &  (r1_incsp_1(A, D, F) &  (r1_incsp_1(A, D, H) &  ( ~ (B=C)  &  (r1_incsp_1(A, B, I) &  (r1_incsp_1(A, C, I) &  (r1_incsp_1(A, E, I) &  ( ~ (r1_incsp_1(A, E, F))  &  ( ~ (E=C)  &  ( ~ (r1_projred2(A, F, G, H))  &  (r1_projred2(A, G, H, I) &  (! [J] :  (m1_subset_1(J, u2_incsp_1(A)) =>  ~ ( (r1_incsp_1(A, D, J) &  ( ~ (r1_incsp_1(A, C, J))  &  ( ~ (r1_incsp_1(A, E, J))  & k3_relat_1(k1_projred1(A, F, H, B), k1_projred1(A, H, G, C))=k3_relat_1(k1_projred1(A, F, J, E), k1_projred1(A, J, G, C))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(rc1_funct_1, axiom,  (? [A] :  (v1_relat_1(A) & v1_funct_1(A)) ) ).
fof(rc1_relset_1, axiom,  (! [A, B] :  (? [C] :  (m1_subset_1(C, k1_zfmisc_1(k2_zfmisc_1(A, B))) &  (v1_xboole_0(C) &  (v1_relat_1(C) &  (v4_relat_1(C, A) & v5_relat_1(C, B)) ) ) ) ) ) ).
fof(rc2_funct_1, axiom,  (? [A] :  (v1_relat_1(A) &  (v1_funct_1(A) & v2_funct_1(A)) ) ) ).
fof(rc5_funct_1, axiom,  (? [A] :  (v1_relat_1(A) &  (v1_funct_1(A) &  ~ (v3_funct_1(A)) ) ) ) ).
fof(rc6_funct_1, axiom,  (! [A, B] :  (? [C] :  (v1_relat_1(C) &  (v4_relat_1(C, A) &  (v5_relat_1(C, B) & v1_funct_1(C)) ) ) ) ) ).
fof(rc7_funct_1, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  & v4_funct_1(A)) ) ).
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_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_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(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)) ) ) ) ) ).
