% Mizar problem: l15_projred1,projred1,757,5 
fof(l15_projred1, conjecture,  (! [A] :  ( (v6_incsp_1(A) &  (v1_incproj(A) &  (v2_incproj(A) &  (v3_incproj(A) &  (v4_incproj(A) &  (v8_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, u1_incsp_1(A)) &  (? [G] :  (m1_subset_1(G, u1_incsp_1(A)) &  (? [H] :  (m1_subset_1(H, u1_incsp_1(A)) &  (? [I] :  (m1_subset_1(I, u2_incsp_1(A)) &  (? [J] :  (m1_subset_1(J, u2_incsp_1(A)) &  (? [K] :  (m1_subset_1(K, u2_incsp_1(A)) &  (? [L] :  (m1_subset_1(L, u2_incsp_1(A)) &  (? [M] :  (m1_subset_1(M, u2_incsp_1(A)) &  (? [N] :  (m1_subset_1(N, u2_incsp_1(A)) &  (? [O] :  (m1_subset_1(O, u2_incsp_1(A)) &  (? [P] :  (m1_subset_1(P, u2_incsp_1(A)) &  (? [Q] :  (m1_subset_1(Q, u1_incsp_1(A)) &  ( ~ (r1_incsp_1(A, C, M))  &  ( ~ (r1_incsp_1(A, D, M))  &  ( ~ (r1_incsp_1(A, B, L))  &  ( ~ (r1_incsp_1(A, E, L))  &  ( ~ (r1_incsp_1(A, B, N))  &  ( ~ (r1_incsp_1(A, D, N))  &  ( ~ (r1_incsp_1(A, C, O))  &  ( ~ (r1_incsp_1(A, E, O))  &  (r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), F, B, E), M) &  (r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), F, C, D), L) &  (r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), G, C, E), N) &  (r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), G, B, D), O) &  (r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), H, B, C), I) &  (r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), H, D, E), J) &  (r4_incsp_1(A, k7_domain_1(u1_incsp_1(A), F, G), K) &  ( ~ (r1_incsp_1(A, H, K))  &  (r1_incsp_1(A, G, P) &  (r1_incsp_1(A, H, P) &  (r2_zfmisc_1(K, P, N, O) &  (r1_incsp_1(A, Q, I) & r2_zfmisc_1(H, Q, 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(cc1_funct_1, axiom,  (! [A] :  (v1_xboole_0(A) => v1_funct_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(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_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_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(commutativity_k2_tarski, axiom,  (! [A, B] : k2_tarski(A, B)=k2_tarski(B, A)) ).
fof(commutativity_k7_domain_1, axiom,  (! [A, B, C] :  ( ( ~ (v1_xboole_0(A))  &  (m1_subset_1(B, A) & m1_subset_1(C, A)) )  => k7_domain_1(A, B, C)=k7_domain_1(A, C, B)) ) ).
fof(d12_incproj, axiom,  (! [A] :  (l1_incsp_1(A) =>  (v8_incproj(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, u1_incsp_1(A)) =>  (! [G] :  (m1_subset_1(G, u1_incsp_1(A)) =>  (! [H] :  (m1_subset_1(H, u1_incsp_1(A)) =>  (! [I] :  (m1_subset_1(I, u2_incsp_1(A)) =>  (! [J] :  (m1_subset_1(J, u2_incsp_1(A)) =>  (! [K] :  (m1_subset_1(K, u2_incsp_1(A)) =>  (! [L] :  (m1_subset_1(L, u2_incsp_1(A)) =>  (! [M] :  (m1_subset_1(M, u2_incsp_1(A)) =>  (! [N] :  (m1_subset_1(N, u2_incsp_1(A)) =>  (! [O] :  (m1_subset_1(O, u2_incsp_1(A)) =>  ~ ( ( ~ (r1_incsp_1(A, C, I))  &  ( ~ (r1_incsp_1(A, D, I))  &  ( ~ (r1_incsp_1(A, B, J))  &  ( ~ (r1_incsp_1(A, E, J))  &  ( ~ (r1_incsp_1(A, B, K))  &  ( ~ (r1_incsp_1(A, D, K))  &  ( ~ (r1_incsp_1(A, C, L))  &  ( ~ (r1_incsp_1(A, E, L))  &  (r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), F, B, E), I) &  (r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), F, C, D), J) &  (r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), G, C, E), K) &  (r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), G, B, D), L) &  (r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), H, B, C), M) &  (r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), H, D, E), N) &  (r4_incsp_1(A, k7_domain_1(u1_incsp_1(A), F, G), O) & r1_incsp_1(A, H, O)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d4_incproj, axiom,  (! [A] :  (l1_incsp_1(A) =>  (v1_incproj(A) <=>  (! [B] :  (m1_subset_1(B, u1_incsp_1(A)) =>  (! [C] :  (m1_subset_1(C, u1_incsp_1(A)) =>  (! [D] :  (m1_subset_1(D, u2_incsp_1(A)) =>  (! [E] :  (m1_subset_1(E, u2_incsp_1(A)) =>  ~ ( (r1_incsp_1(A, B, D) &  (r1_incsp_1(A, C, D) &  (r1_incsp_1(A, B, E) &  (r1_incsp_1(A, C, E) &  ( ~ (B=C)  &  ~ (D=E) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d5_incproj, axiom,  (! [A] :  (l1_incsp_1(A) =>  (v6_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, u2_incsp_1(A)) &  (r1_incsp_1(A, B, D) & r1_incsp_1(A, C, D)) ) ) ) ) ) ) ) ) ) ).
fof(d6_incproj, axiom,  (! [A] :  (l1_incsp_1(A) =>  (v2_incproj(A) <=>  ~ ( (! [B] :  (m1_subset_1(B, u1_incsp_1(A)) =>  (! [C] :  (m1_subset_1(C, u2_incsp_1(A)) => r1_incsp_1(A, B, C)) ) ) ) ) ) ) ) ).
fof(d6_zfmisc_1, axiom,  (! [A] :  (! [B] :  (! [C] :  (! [D] :  (r2_zfmisc_1(A, B, C, D) <=>  ( ~ (A=B)  &  ( ~ (A=C)  &  ( ~ (A=D)  &  ( ~ (B=C)  &  ( ~ (B=D)  &  ~ (C=D) ) ) ) ) ) ) ) ) ) ) ).
fof(d7_incproj, axiom,  (! [A] :  (l1_incsp_1(A) =>  (v3_incproj(A) <=>  (! [B] :  (m1_subset_1(B, u2_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)) &  ( ~ (C=D)  &  ( ~ (D=E)  &  ( ~ (E=C)  &  (r1_incsp_1(A, C, B) &  (r1_incsp_1(A, D, B) & r1_incsp_1(A, E, B)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d8_incproj, axiom,  (! [A] :  (l1_incsp_1(A) =>  (v4_incproj(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, u1_incsp_1(A)) =>  (! [G] :  (m1_subset_1(G, u1_incsp_1(A)) =>  (! [H] :  (m1_subset_1(H, u2_incsp_1(A)) =>  (! [I] :  (m1_subset_1(I, u2_incsp_1(A)) =>  (! [J] :  (m1_subset_1(J, u2_incsp_1(A)) =>  (! [K] :  (m1_subset_1(K, u2_incsp_1(A)) =>  ~ ( (r1_incsp_1(A, B, H) &  (r1_incsp_1(A, C, H) &  (r1_incsp_1(A, D, I) &  (r1_incsp_1(A, E, I) &  (r1_incsp_1(A, F, H) &  (r1_incsp_1(A, F, I) &  (r1_incsp_1(A, B, J) &  (r1_incsp_1(A, D, J) &  (r1_incsp_1(A, C, K) &  (r1_incsp_1(A, E, K) &  ( ~ (r1_incsp_1(A, F, J))  &  ( ~ (r1_incsp_1(A, F, K))  &  ( ~ (H=I)  &  (! [L] :  (m1_subset_1(L, u1_incsp_1(A)) =>  ~ ( (r1_incsp_1(A, L, J) & r1_incsp_1(A, L, K)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(dt_k1_enumset1, axiom, $true).
fof(dt_k1_xboole_0, axiom, $true).
fof(dt_k1_zfmisc_1, axiom, $true).
fof(dt_k2_tarski, axiom, $true).
fof(dt_k7_domain_1, axiom,  (! [A, B, C] :  ( ( ~ (v1_xboole_0(A))  &  (m1_subset_1(B, A) & m1_subset_1(C, A)) )  => m1_subset_1(k7_domain_1(A, B, C), k1_zfmisc_1(A))) ) ).
fof(dt_k8_domain_1, axiom,  (! [A, B, C, D] :  ( ( ~ (v1_xboole_0(A))  &  (m1_subset_1(B, A) &  (m1_subset_1(C, A) & m1_subset_1(D, A)) ) )  => m1_subset_1(k8_domain_1(A, B, C, D), k1_zfmisc_1(A))) ) ).
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(fc15_funct_1, axiom,  (! [A, B] :  ( ( (v1_relat_1(A) & v1_funct_1(A))  &  (v1_relat_1(B) & v1_funct_1(B)) )  => v4_funct_1(k2_tarski(A, B))) ) ).
fof(rc1_funct_1, axiom,  (? [A] :  (v1_relat_1(A) & v1_funct_1(A)) ) ).
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(rc7_funct_1, axiom,  (? [A] :  ( ~ (v1_xboole_0(A))  & v4_funct_1(A)) ) ).
fof(redefinition_k7_domain_1, axiom,  (! [A, B, C] :  ( ( ~ (v1_xboole_0(A))  &  (m1_subset_1(B, A) & m1_subset_1(C, A)) )  => k7_domain_1(A, B, C)=k2_tarski(B, C)) ) ).
fof(redefinition_k8_domain_1, axiom,  (! [A, B, C, D] :  ( ( ~ (v1_xboole_0(A))  &  (m1_subset_1(B, A) &  (m1_subset_1(C, A) & m1_subset_1(D, A)) ) )  => k8_domain_1(A, B, C, D)=k1_enumset1(B, C, D)) ) ).
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_incsp_1, axiom,  (! [A] :  (l1_incsp_1(A) =>  (! [B] :  (m1_subset_1(B, u2_incsp_1(A)) =>  (! [C] :  (m1_subset_1(C, u1_incsp_1(A)) =>  (! [D] :  (m1_subset_1(D, u1_incsp_1(A)) =>  (r4_incsp_1(A, k7_domain_1(u1_incsp_1(A), C, D), B) <=>  (r1_incsp_1(A, C, B) & r1_incsp_1(A, D, B)) ) ) ) ) ) ) ) ) ) ).
fof(t1_subset, axiom,  (! [A] :  (! [B] :  (r2_tarski(A, B) => m1_subset_1(A, B)) ) ) ).
fof(t2_incsp_1, axiom,  (! [A] :  (l1_incsp_1(A) =>  (! [B] :  (m1_subset_1(B, u2_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)) =>  (r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), C, D, E), B) <=>  (r1_incsp_1(A, C, B) &  (r1_incsp_1(A, D, B) & r1_incsp_1(A, E, 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)) ) ) ) ) ).
fof(t8_projred1, 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, u1_incsp_1(A)) =>  (! [C] :  (m1_subset_1(C, u1_incsp_1(A)) =>  (! [D] :  (m1_subset_1(D, u2_incsp_1(A)) =>  (? [E] :  (m1_subset_1(E, u1_incsp_1(A)) &  (r1_incsp_1(A, E, D) &  ( ~ (E=B)  &  ~ (E=C) ) ) ) ) ) ) ) ) ) ) ) ) ).
