% Mizar problem: t4_projpl_1,projpl_1,142,5 
fof(t4_projpl_1, conjecture,  (! [A] :  (l1_incsp_1(A) =>  ( (v6_incsp_1(A) &  (v1_incproj(A) &  (v2_incproj(A) &  (v3_incproj(A) &  (v4_incproj(A) & l1_incsp_1(A)) ) ) ) )  <=>  (v1_projpl_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)) & r4_incsp_1(A, k7_domain_1(u1_incsp_1(A), B, C), D)) ) ) ) ) )  &  ( ~ ( (! [B] :  (m1_subset_1(B, u1_incsp_1(A)) =>  (! [C] :  (m1_subset_1(C, u2_incsp_1(A)) => r1_incsp_1(A, B, C)) ) ) ) )  &  ( (! [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)) &  (r1_zfmisc_1(C, D, E) & r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), C, D, E), B)) ) ) ) ) ) ) ) )  &  (! [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, u2_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)) =>  ~ ( (r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), B, C, F), G) &  (r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), D, E, F), H) &  (r4_incsp_1(A, k7_domain_1(u1_incsp_1(A), B, D), I) &  (r4_incsp_1(A, k7_domain_1(u1_incsp_1(A), C, E), J) &  ( ~ (r1_incsp_1(A, F, I))  &  ( ~ (r1_incsp_1(A, F, J))  &  ( ~ (G=H)  &  (! [K] :  (m1_subset_1(K, u1_incsp_1(A)) =>  ~ (r2_projpl_1(A, K, I, J)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
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(d2_projpl_1, axiom,  (! [A] :  (l1_incsp_1(A) =>  (! [B] :  (m1_subset_1(B, u1_incsp_1(A)) =>  (! [C] :  (m1_subset_1(C, u2_incsp_1(A)) =>  (! [D] :  (m1_subset_1(D, u2_incsp_1(A)) =>  (r2_projpl_1(A, B, C, D) <=>  (r1_incsp_1(A, B, C) & r1_incsp_1(A, B, D)) ) ) ) ) ) ) ) ) ) ).
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(d5_zfmisc_1, axiom,  (! [A] :  (! [B] :  (! [C] :  (r1_zfmisc_1(A, B, C) <=>  ( ~ (A=B)  &  ( ~ (A=C)  &  ~ (B=C) ) ) ) ) ) ) ).
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(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_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(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(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(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)) ) ) ) ) ) ) ) ) ) ) ) ) ).
