% Mizar problem: t25_projpl_1,projpl_1,751,5 
fof(t25_projpl_1, conjecture,  (! [A] :  ( (v6_incsp_1(A) &  (v1_incproj(A) &  (v2_incproj(A) &  (v3_incproj(A) &  (v4_incproj(A) &  (v5_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, u2_incsp_1(A)) =>  (r1_incsp_1(A, B, D) =>  (D=E |  (r1_incsp_1(A, C, D) |  (B=k2_projpl_1(A, D, E) |  (r1_incsp_1(A, k2_projpl_1(A, k1_projpl_1(A, C, B), E), E) &  ~ (r1_incsp_1(A, k2_projpl_1(A, k1_projpl_1(A, C, B), E), D)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
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(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_projpl_1, axiom,  (! [A, B, C] :  ( ( (v6_incsp_1(A) &  (v1_incproj(A) &  (v2_incproj(A) &  (v3_incproj(A) &  (v4_incproj(A) & l1_incsp_1(A)) ) ) ) )  &  (m1_subset_1(B, u1_incsp_1(A)) & m1_subset_1(C, u1_incsp_1(A))) )  => m1_subset_1(k1_projpl_1(A, B, C), u2_incsp_1(A))) ) ).
fof(dt_k1_zfmisc_1, axiom, $true).
fof(dt_k2_projpl_1, axiom,  (! [A, B, C] :  ( ( (v6_incsp_1(A) &  (v1_incproj(A) &  (v2_incproj(A) &  (v3_incproj(A) &  (v4_incproj(A) &  (v5_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(k2_projpl_1(A, B, C), u1_incsp_1(A))) ) ).
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_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(t16_projpl_1, 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)) =>  ( ~ (B=C)  =>  (r1_incsp_1(A, B, k1_projpl_1(A, B, C)) &  (r1_incsp_1(A, C, k1_projpl_1(A, B, C)) &  (k1_projpl_1(A, B, C)=k1_projpl_1(A, C, B) &  ( (r1_incsp_1(A, B, D) & r1_incsp_1(A, C, D))  => D=k1_projpl_1(A, B, C)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t23_projpl_1, axiom,  (! [A] :  (l1_incsp_1(A) =>  ( (v6_incsp_1(A) &  (v1_incproj(A) &  (v2_incproj(A) &  (v3_incproj(A) &  (v4_incproj(A) &  (v5_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, 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)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t24_projpl_1, axiom,  (! [A] :  ( (v6_incsp_1(A) &  (v1_incproj(A) &  (v2_incproj(A) &  (v3_incproj(A) &  (v4_incproj(A) &  (v5_incproj(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)) =>  ( ~ (C=D)  =>  (r1_incsp_1(A, k2_projpl_1(A, C, D), C) &  (r1_incsp_1(A, k2_projpl_1(A, C, D), D) &  (k2_projpl_1(A, C, D)=k2_projpl_1(A, D, C) &  ( (r1_incsp_1(A, B, C) & r1_incsp_1(A, B, D))  => B=k2_projpl_1(A, C, D)) ) ) ) ) ) ) ) ) ) ) ) ) ).
