% Mizar problem: t22_projpl_1,projpl_1,568,5 
fof(t22_projpl_1, conjecture,  (! [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, u2_incsp_1(A)) =>  (! [C] :  (m1_subset_1(C, u2_incsp_1(A)) =>  (? [D] :  (m1_subset_1(D, u1_incsp_1(A)) & r2_projpl_1(A, D, B, C)) ) ) ) ) ) ) )  =>  ( (! [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)) =>  ~ (r5_projpl_1(A, B, C, D, E)) ) ) ) ) ) ) ) )  |  (! [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)) ) ) ) ) ) ) ) ) ) ) ) ) ).
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(d3_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)) =>  (! [E] :  (m1_subset_1(E, u2_incsp_1(A)) =>  (r3_projpl_1(A, B, C, D, E) <=>  (r1_incsp_1(A, B, C) &  (r1_incsp_1(A, B, D) & r1_incsp_1(A, B, E)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d5_projpl_1, axiom,  (! [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)) =>  (r4_projpl_1(A, B, C, D) <=>  (? [E] :  (m1_subset_1(E, u2_incsp_1(A)) & r4_incsp_1(A, k8_domain_1(u1_incsp_1(A), B, C, D), E)) ) ) ) ) ) ) ) ) ) ) ).
fof(d6_projpl_1, axiom,  (! [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)) =>  (r5_projpl_1(A, B, C, D, E) <=>  ( ~ (r4_projpl_1(A, B, C, D))  &  ( ~ (r4_projpl_1(A, C, D, E))  &  ( ~ (r4_projpl_1(A, D, E, B))  &  ~ (r4_projpl_1(A, E, B, C)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
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(t13_projpl_1, axiom,  (! [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)) =>  (r5_projpl_1(A, B, C, D, E) =>  (r5_projpl_1(A, B, D, C, E) &  (r5_projpl_1(A, C, B, D, E) &  (r5_projpl_1(A, C, D, B, E) &  (r5_projpl_1(A, D, B, C, E) &  (r5_projpl_1(A, D, C, B, E) &  (r5_projpl_1(A, B, C, E, D) &  (r5_projpl_1(A, B, D, E, C) &  (r5_projpl_1(A, C, B, E, D) &  (r5_projpl_1(A, C, D, E, B) &  (r5_projpl_1(A, D, B, E, C) &  (r5_projpl_1(A, D, C, E, B) &  (r5_projpl_1(A, B, E, C, D) &  (r5_projpl_1(A, B, E, D, C) &  (r5_projpl_1(A, C, E, B, D) &  (r5_projpl_1(A, C, E, D, B) &  (r5_projpl_1(A, D, E, B, C) &  (r5_projpl_1(A, D, E, C, B) &  (r5_projpl_1(A, E, B, C, D) &  (r5_projpl_1(A, E, B, D, C) &  (r5_projpl_1(A, E, C, B, D) &  (r5_projpl_1(A, E, C, D, B) &  (r5_projpl_1(A, E, D, B, C) & r5_projpl_1(A, E, D, C, B)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
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(t20_projpl_1, axiom,  (! [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)) =>  ( (r5_projpl_1(A, B, C, D, E) &  (r4_incsp_1(A, k7_domain_1(u1_incsp_1(A), B, C), F) &  (r4_incsp_1(A, k7_domain_1(u1_incsp_1(A), B, D), G) & r4_incsp_1(A, k7_domain_1(u1_incsp_1(A), B, E), H)) ) )  => r1_zfmisc_1(F, G, H)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t21_projpl_1, axiom,  (! [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)) =>  ( (v1_projpl_1(A) &  (r3_projpl_1(A, B, G, H, I) &  (r1_zfmisc_1(G, H, I) &  (r2_projpl_1(A, C, F, G) &  (r2_projpl_1(A, D, F, H) & r2_projpl_1(A, E, F, I)) ) ) ) )  =>  (r1_incsp_1(A, B, F) | r1_zfmisc_1(C, D, E)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
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(t5_projpl_1, axiom,  (! [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)) =>  (r4_projpl_1(A, B, C, D) <=>  (? [E] :  (m1_subset_1(E, u2_incsp_1(A)) &  (r1_incsp_1(A, B, E) &  (r1_incsp_1(A, C, E) & r1_incsp_1(A, D, E)) ) ) ) ) ) ) ) ) ) ) ) ) ).
