% Mizar problem: t51_afproj,afproj,1623,13 fof(t51_afproj,conjecture,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ( v10_incproj(k13_afproj(A)) => v4_aff_2(A) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_incsp_1,antisymmetry_r2_hidden,cc1_partfun1,cc1_relset_1,commutativity_k2_aff_1,commutativity_k2_tarski,commutativity_k2_xboole_0,d10_afproj,d13_afproj,d13_incproj,d14_afproj,d19_incproj,d1_afproj,d1_incsp_1,d2_afproj,d3_aff_1,d3_afproj,d4_aff_2,d4_afproj,d5_incproj,d5_tarski,d7_afproj,d8_afproj,d8_incproj,d9_afproj,d9_incproj,dt_g1_incsp_1,dt_k10_afproj,dt_k11_afproj,dt_k12_afproj,dt_k13_afproj,dt_k14_afproj,dt_k1_aff_1,dt_k1_afproj,dt_k1_domain_1,dt_k1_mcart_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_aff_1,dt_k2_afproj,dt_k2_mcart_1,dt_k2_tarski,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k3_afproj,dt_k4_afproj,dt_k4_tarski,dt_k5_afproj,dt_k6_afproj,dt_k7_afproj,dt_k7_eqrel_1,dt_k8_afproj,dt_k8_eqrel_1,dt_k9_afproj,dt_l1_analoaf,dt_l1_incsp_1,dt_l1_struct_0,dt_m1_eqrel_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_incsp_1,dt_u1_struct_0,dt_u2_incsp_1,dt_u3_incsp_1,existence_l1_analoaf,existence_l1_incsp_1,existence_l1_struct_0,existence_m1_eqrel_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,fc1_afproj,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc2_subset_1,fc2_xboole_0,fc3_subset_1,fc3_xboole_0,fc4_subset_1,fraenkel_a_1_0_afproj,fraenkel_a_1_1_afproj,fraenkel_a_1_2_afproj,fraenkel_a_1_3_afproj,free_g1_incsp_1,idempotence_k2_xboole_0,rc1_incproj,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rc3_struct_0,rc5_struct_0,redefinition_k1_domain_1,redefinition_k2_aff_1,redefinition_k8_eqrel_1,redefinition_m2_relset_1,redefinition_r4_aff_1,reflexivity_r1_tarski,spc1_boole,spc2_boole,symmetry_r4_aff_1,t11_afproj,t1_boole,t1_subset,t20_afproj,t21_afproj,t23_afproj,t24_afproj,t26_aff_1,t26_afproj,t27_afproj,t28_afproj,t2_subset,t2_tarski,t30_aff_1,t30_afproj,t3_subset,t40_aff_4,t41_afproj,t4_subset,t50_afproj,t51_aff_1,t53_aff_1,t58_aff_1,t59_aff_1,t5_subset,t65_aff_1,t6_boole,t7_boole,t7_mcart_1,t8_aff_4,t8_boole,t9_aff_4]), [file(afproj,t51_afproj)]). fof(abstractness_v1_incsp_1,axiom,( ! [A] : ( l1_incsp_1(A) => ( v1_incsp_1(A) => A = g1_incsp_1(u1_incsp_1(A),u2_incsp_1(A),u3_incsp_1(A)) ) ) ), file(incsp_1,v1_incsp_1), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc1_partfun1,axiom,( ! [A] : ( ( v1_relat_1(A) & v3_relat_2(A) & v8_relat_2(A) ) => ( v1_relat_1(A) & v1_relat_2(A) ) ) ), file(partfun1,cc1_partfun1), []). fof(cc1_relset_1,axiom,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) => v1_relat_1(C) ) ), file(relset_1,cc1_relset_1), []). fof(commutativity_k2_aff_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k2_aff_1(A,B,C) = k2_aff_1(A,C,B) ) ), file(aff_1,k2_aff_1), []). fof(commutativity_k2_tarski,axiom,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), []). fof(commutativity_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ), file(xboole_0,k2_xboole_0), []). fof(d10_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => k10_afproj(A) = k2_xboole_0(k2_zfmisc_1(k1_afproj(A),k1_tarski(1)),k2_zfmisc_1(k8_afproj(A),k1_tarski(2))) ) ), file(afproj,d10_afproj), []). fof(d13_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => k13_afproj(A) = g1_incsp_1(k9_afproj(A),k10_afproj(A),k11_afproj(A)) ) ), file(afproj,d13_afproj), []). fof(d13_incproj,axiom,( ! [A] : ( l1_incsp_1(A) => ( v5_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) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), file(incproj,d13_incproj), []). fof(d14_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => k14_afproj(A) = g1_incsp_1(k7_afproj(A),k8_afproj(A),k12_afproj(A)) ) ), file(afproj,d14_afproj), []). fof(d19_incproj,axiom,( ! [A] : ( ( v1_incproj(A) & v2_incproj(A) & v3_incproj(A) & v4_incproj(A) & v5_incproj(A) & l1_incsp_1(A) ) => ( v10_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,u1_incsp_1(A)) => ! [J] : ( m1_subset_1(J,u1_incsp_1(A)) => ! [K] : ( m1_subset_1(K,u1_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,u2_incsp_1(A)) => ! [R] : ( m1_subset_1(R,u2_incsp_1(A)) => ! [S] : ( m1_subset_1(S,u2_incsp_1(A)) => ! [T] : ( m1_subset_1(T,u2_incsp_1(A)) => ~ ( r4_incproj(A,B,C,L,D) & r4_incproj(A,B,F,M,E) & r4_incproj(A,B,H,N,G) & r4_incproj(A,H,F,O,K) & r4_incproj(A,H,I,P,D) & r4_incproj(A,F,J,Q,D) & r4_incproj(A,K,E,R,G) & r4_incproj(A,C,I,S,G) & r4_incproj(A,C,J,T,E) & r1_incproj(L,M,N) & B != D & B != F & B != H & B != C & B != E & B != G & D != C & F != E & H != G & ! [U] : ( m1_subset_1(U,u2_incsp_1(A)) => ~ r4_incproj(A,I,J,U,K) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), file(incproj,d19_incproj), []). fof(d1_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => k1_afproj(A) = a_1_0_afproj(A) ) ), file(afproj,d1_afproj), []). fof(d1_incsp_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)) => ( r1_incsp_1(A,B,C) <=> r2_hidden(k1_domain_1(u1_incsp_1(A),u2_incsp_1(A),B,C),u3_incsp_1(A)) ) ) ) ) ), file(incsp_1,d1_incsp_1), []). fof(d2_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => k2_afproj(A) = a_1_1_afproj(A) ) ), file(afproj,d2_afproj), []). fof(d3_aff_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_aff_1(B,A) <=> ? [C] : ( m1_subset_1(C,u1_struct_0(A)) & ? [D] : ( m1_subset_1(D,u1_struct_0(A)) & C != D & B = k2_aff_1(A,C,D) ) ) ) ) ) ), file(aff_1,d3_aff_1), []). fof(d3_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => k3_afproj(A) = a_1_2_afproj(A) ) ), file(afproj,d3_afproj), []). fof(d4_aff_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ( v4_aff_2(A) <=> ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ! [F] : ( m1_subset_1(F,u1_struct_0(A)) => ! [G] : ( m1_subset_1(G,u1_struct_0(A)) => ! [H] : ( m1_subset_1(H,u1_struct_0(A)) => ! [I] : ( m1_subset_1(I,u1_struct_0(A)) => ! [J] : ( m1_subset_1(J,u1_struct_0(A)) => ! [K] : ( m1_subset_1(K,u1_struct_0(A)) => ( ( r2_hidden(E,B) & r2_hidden(E,C) & r2_hidden(E,D) & r2_hidden(F,B) & r2_hidden(I,B) & r2_hidden(G,C) & r2_hidden(J,C) & r2_hidden(H,D) & r2_hidden(K,D) & v1_aff_1(B,A) & v1_aff_1(C,A) & v1_aff_1(D,A) & r2_analoaf(A,F,G,I,J) & r2_analoaf(A,F,H,I,K) ) => ( E = F | E = G | E = H | B = C | B = D | r2_analoaf(A,G,H,J,K) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), file(aff_2,d4_aff_2), []). fof(d4_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => k4_afproj(A) = a_1_3_afproj(A) ) ), file(afproj,d4_afproj), []). fof(d5_incproj,axiom,( ! [A,B,C] : ( r1_incproj(A,B,C) <=> ( A != B & B != C & C != A ) ) ), file(incproj,d5_incproj), []). fof(d5_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(d7_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => k7_afproj(A) = k8_eqrel_1(k1_afproj(A),k3_afproj(A)) ) ), file(afproj,d7_afproj), []). fof(d8_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => k8_afproj(A) = k8_eqrel_1(k2_afproj(A),k4_afproj(A)) ) ), file(afproj,d8_afproj), []). fof(d8_incproj,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,u2_incsp_1(A)) => ! [E] : ( m1_subset_1(E,u1_incsp_1(A)) => ( r4_incproj(A,B,C,D,E) <=> ( r1_incsp_1(A,B,D) & r1_incsp_1(A,C,D) & r1_incsp_1(A,E,D) ) ) ) ) ) ) ) ), file(incproj,d8_incproj), []). fof(d9_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => k9_afproj(A) = k2_xboole_0(u1_struct_0(A),k7_afproj(A)) ) ), file(afproj,d9_afproj), []). fof(d9_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 ) ) ) ) ) ) ) ), file(incproj,d9_incproj), []). fof(dt_g1_incsp_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_relset_1(C,A,B) ) => ( v1_incsp_1(g1_incsp_1(A,B,C)) & l1_incsp_1(g1_incsp_1(A,B,C)) ) ) ), file(incsp_1,g1_incsp_1), []). fof(dt_k10_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ~ v1_xboole_0(k10_afproj(A)) ) ), file(afproj,k10_afproj), []). fof(dt_k11_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => m2_relset_1(k11_afproj(A),k9_afproj(A),k10_afproj(A)) ) ), file(afproj,k11_afproj), []). fof(dt_k12_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => m2_relset_1(k12_afproj(A),k7_afproj(A),k8_afproj(A)) ) ), file(afproj,k12_afproj), []). fof(dt_k13_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ( v1_incsp_1(k13_afproj(A)) & l1_incsp_1(k13_afproj(A)) ) ) ), file(afproj,k13_afproj), []). fof(dt_k14_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ( v1_incsp_1(k14_afproj(A)) & l1_incsp_1(k14_afproj(A)) ) ) ), file(afproj,k14_afproj), []). fof(dt_k1_aff_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k1_aff_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ), file(aff_1,k1_aff_1), []). fof(dt_k1_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => m1_subset_1(k1_afproj(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ), file(afproj,k1_afproj), []). fof(dt_k1_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => m1_subset_1(k1_domain_1(A,B,C,D),k2_zfmisc_1(A,B)) ) ), file(domain_1,k1_domain_1), []). fof(dt_k1_mcart_1,axiom,( $true ), file(mcart_1,k1_mcart_1), []). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k2_aff_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k2_aff_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ), file(aff_1,k2_aff_1), []). fof(dt_k2_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => m1_subset_1(k2_afproj(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ), file(afproj,k2_afproj), []). fof(dt_k2_mcart_1,axiom,( $true ), file(mcart_1,k2_mcart_1), []). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(dt_k2_xboole_0,axiom,( $true ), file(xboole_0,k2_xboole_0), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k3_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ( v3_relat_2(k3_afproj(A)) & v8_relat_2(k3_afproj(A)) & v1_partfun1(k3_afproj(A),k1_afproj(A),k1_afproj(A)) & m2_relset_1(k3_afproj(A),k1_afproj(A),k1_afproj(A)) ) ) ), file(afproj,k3_afproj), []). fof(dt_k4_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ( v3_relat_2(k4_afproj(A)) & v8_relat_2(k4_afproj(A)) & v1_partfun1(k4_afproj(A),k2_afproj(A),k2_afproj(A)) & m2_relset_1(k4_afproj(A),k2_afproj(A),k2_afproj(A)) ) ) ), file(afproj,k4_afproj), []). fof(dt_k4_tarski,axiom,( $true ), file(tarski,k4_tarski), []). fof(dt_k5_afproj,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => m1_subset_1(k5_afproj(A,B),k1_zfmisc_1(k1_afproj(A))) ) ), file(afproj,k5_afproj), []). fof(dt_k6_afproj,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => m1_subset_1(k6_afproj(A,B),k1_zfmisc_1(k2_afproj(A))) ) ), file(afproj,k6_afproj), []). fof(dt_k7_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ~ v1_xboole_0(k7_afproj(A)) ) ), file(afproj,k7_afproj), []). fof(dt_k7_eqrel_1,axiom,( ! [A,B] : ( ( v3_relat_2(B) & v8_relat_2(B) & v1_partfun1(B,A,A) & m1_relset_1(B,A,A) ) => m1_subset_1(k7_eqrel_1(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ), file(eqrel_1,k7_eqrel_1), []). fof(dt_k8_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ~ v1_xboole_0(k8_afproj(A)) ) ), file(afproj,k8_afproj), []). fof(dt_k8_eqrel_1,axiom,( ! [A,B] : ( ( v3_relat_2(B) & v8_relat_2(B) & v1_partfun1(B,A,A) & m1_relset_1(B,A,A) ) => m1_eqrel_1(k8_eqrel_1(A,B),A) ) ), file(eqrel_1,k8_eqrel_1), []). fof(dt_k9_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ~ v1_xboole_0(k9_afproj(A)) ) ), file(afproj,k9_afproj), []). fof(dt_l1_analoaf,axiom,( ! [A] : ( l1_analoaf(A) => l1_struct_0(A) ) ), file(analoaf,l1_analoaf), []). fof(dt_l1_incsp_1,axiom,( $true ), file(incsp_1,l1_incsp_1), []). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), []). fof(dt_m1_eqrel_1,axiom,( ! [A,B] : ( m1_eqrel_1(B,A) => m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ), file(eqrel_1,m1_eqrel_1), []). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), []). fof(dt_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(relset_1,m2_relset_1), []). fof(dt_u1_incsp_1,axiom,( ! [A] : ( l1_incsp_1(A) => ~ v1_xboole_0(u1_incsp_1(A)) ) ), file(incsp_1,u1_incsp_1), []). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), []). fof(dt_u2_incsp_1,axiom,( ! [A] : ( l1_incsp_1(A) => ~ v1_xboole_0(u2_incsp_1(A)) ) ), file(incsp_1,u2_incsp_1), []). fof(dt_u3_incsp_1,axiom,( ! [A] : ( l1_incsp_1(A) => m2_relset_1(u3_incsp_1(A),u1_incsp_1(A),u2_incsp_1(A)) ) ), file(incsp_1,u3_incsp_1), []). fof(existence_l1_analoaf,axiom,( ? [A] : l1_analoaf(A) ), file(analoaf,l1_analoaf), []). fof(existence_l1_incsp_1,axiom,( ? [A] : l1_incsp_1(A) ), file(incsp_1,l1_incsp_1), []). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). fof(existence_m1_eqrel_1,axiom,( ! [A] : ? [B] : m1_eqrel_1(B,A) ), file(eqrel_1,m1_eqrel_1), []). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), []). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), []). fof(fc1_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ( v1_incsp_1(k13_afproj(A)) & v1_incproj(k13_afproj(A)) & v2_incproj(k13_afproj(A)) & v3_incproj(k13_afproj(A)) & v4_incproj(k13_afproj(A)) & v5_incproj(k13_afproj(A)) ) ) ), file(afproj,fc1_afproj), []). fof(fc1_struct_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(u1_struct_0(A)) ) ), file(struct_0,fc1_struct_0), []). fof(fc1_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), []). fof(fc1_xboole_0,axiom,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), []). fof(fc2_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). fof(fc2_xboole_0,axiom,( ! [A,B] : ( ~ v1_xboole_0(A) => ~ v1_xboole_0(k2_xboole_0(A,B)) ) ), file(xboole_0,fc2_xboole_0), []). fof(fc3_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), []). fof(fc3_xboole_0,axiom,( ! [A,B] : ( ~ v1_xboole_0(A) => ~ v1_xboole_0(k2_xboole_0(B,A)) ) ), file(xboole_0,fc3_xboole_0), []). fof(fc4_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), []). fof(fraenkel_a_1_0_afproj,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & ~ v3_realset2(B) & v1_diraf(B) & l1_analoaf(B) ) => ( r2_hidden(A,a_1_0_afproj(B)) <=> ? [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) & A = C & v1_aff_1(C,B) ) ) ) ), file(afproj,a_1_0_afproj), []). fof(fraenkel_a_1_1_afproj,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & ~ v3_realset2(B) & v1_diraf(B) & l1_analoaf(B) ) => ( r2_hidden(A,a_1_1_afproj(B)) <=> ? [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) & A = C & v1_aff_4(C,B) ) ) ) ), file(afproj,a_1_1_afproj), []). fof(fraenkel_a_1_2_afproj,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & ~ v3_realset2(B) & v1_diraf(B) & l1_analoaf(B) ) => ( r2_hidden(A,a_1_2_afproj(B)) <=> ? [C,D] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) & A = k4_tarski(C,D) & v1_aff_1(C,B) & v1_aff_1(D,B) & r1_aff_4(B,C,D) ) ) ) ), file(afproj,a_1_2_afproj), []). fof(fraenkel_a_1_3_afproj,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & ~ v3_realset2(B) & v1_diraf(B) & l1_analoaf(B) ) => ( r2_hidden(A,a_1_3_afproj(B)) <=> ? [C,D] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) & A = k4_tarski(C,D) & v1_aff_4(C,B) & v1_aff_4(D,B) & r1_aff_4(B,C,D) ) ) ) ), file(afproj,a_1_3_afproj), []). fof(free_g1_incsp_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_relset_1(C,A,B) ) => ! [D,E,F] : ( g1_incsp_1(A,B,C) = g1_incsp_1(D,E,F) => ( A = D & B = E & C = F ) ) ) ), file(incsp_1,g1_incsp_1), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). fof(rc1_incproj,axiom,( ? [A] : ( l1_incsp_1(A) & v1_incsp_1(A) & v1_incproj(A) & v2_incproj(A) & v3_incproj(A) & v4_incproj(A) & v5_incproj(A) ) ), file(incproj,rc1_incproj), []). fof(rc1_subset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), []). fof(rc1_xboole_0,axiom,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), []). fof(rc2_subset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), []). fof(rc2_xboole_0,axiom,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), []). fof(rc3_struct_0,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), []). fof(rc5_struct_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) ) ) ), file(struct_0,rc5_struct_0), []). fof(redefinition_k1_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => k1_domain_1(A,B,C,D) = k4_tarski(C,D) ) ), file(domain_1,k1_domain_1), []). fof(redefinition_k2_aff_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k2_aff_1(A,B,C) = k1_aff_1(A,B,C) ) ), file(aff_1,k2_aff_1), []). fof(redefinition_k8_eqrel_1,axiom,( ! [A,B] : ( ( v3_relat_2(B) & v8_relat_2(B) & v1_partfun1(B,A,A) & m1_relset_1(B,A,A) ) => k8_eqrel_1(A,B) = k7_eqrel_1(A,B) ) ), file(eqrel_1,k8_eqrel_1), []). fof(redefinition_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), []). fof(redefinition_r4_aff_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) => ( r4_aff_1(A,B,C) <=> r3_aff_1(A,B,C) ) ) ), file(aff_1,r4_aff_1), []). fof(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(spc1_boole,axiom,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), []). fof(spc2_boole,axiom,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), []). fof(symmetry_r4_aff_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) => ( r4_aff_1(A,B,C) => r4_aff_1(A,C,B) ) ) ), file(aff_1,r4_aff_1), []). fof(t11_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( ( v1_aff_1(B,A) & v1_aff_1(C,A) ) => ( k5_afproj(A,B) = k5_afproj(A,C) <=> r4_aff_1(A,B,C) ) ) ) ) ) ), file(afproj,t11_afproj), []). fof(t1_boole,axiom,( ! [A] : k2_xboole_0(A,k1_xboole_0) = A ), file(boole,t1_boole), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t20_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,u1_incsp_1(k13_afproj(A))) <=> ~ ( ~ m1_subset_1(B,u1_struct_0(A)) & ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ~ ( B = k5_afproj(A,C) & v1_aff_1(C,A) ) ) ) ) ) ), file(afproj,t20_afproj), []). fof(t21_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,u1_incsp_1(k14_afproj(A))) <=> ? [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) & B = k5_afproj(A,C) & v1_aff_1(C,A) ) ) ) ), file(afproj,t21_afproj), []). fof(t23_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,u2_incsp_1(k13_afproj(A))) <=> ~ ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( ~ ( B = k4_tarski(C,1) & v1_aff_1(C,A) ) & ~ ( B = k4_tarski(k6_afproj(A,C),2) & v1_aff_4(C,A) ) ) ) ) ) ), file(afproj,t23_afproj), []). fof(t24_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,u2_incsp_1(k14_afproj(A))) <=> ? [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) & B = k6_afproj(A,C) & v1_aff_4(C,A) ) ) ) ), file(afproj,t24_afproj), []). fof(t26_aff_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ( r2_hidden(B,k2_aff_1(A,B,C)) & r2_hidden(C,k2_aff_1(A,B,C)) ) ) ) ) ), file(aff_1,t26_aff_1), []). fof(t26_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,u1_incsp_1(k13_afproj(A))) => ! [E] : ( m1_subset_1(E,u2_incsp_1(k13_afproj(A))) => ( ( B = D & k4_tarski(C,1) = E ) => ( r1_incsp_1(k13_afproj(A),D,E) <=> ( v1_aff_1(C,A) & r2_hidden(B,C) ) ) ) ) ) ) ) ) ), file(afproj,t26_afproj), []). fof(t27_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,u1_incsp_1(k13_afproj(A))) => ! [E] : ( m1_subset_1(E,u2_incsp_1(k13_afproj(A))) => ~ ( B = D & k4_tarski(k6_afproj(A,C),2) = E & v1_aff_4(C,A) & r1_incsp_1(k13_afproj(A),D,E) ) ) ) ) ) ) ), file(afproj,t27_afproj), []). fof(t28_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,u1_incsp_1(k13_afproj(A))) => ! [E] : ( m1_subset_1(E,u2_incsp_1(k13_afproj(A))) => ( ( D = k5_afproj(A,B) & k4_tarski(C,1) = E & v1_aff_1(B,A) & v1_aff_1(C,A) ) => ( r1_incsp_1(k13_afproj(A),D,E) <=> r1_aff_4(A,B,C) ) ) ) ) ) ) ) ), file(afproj,t28_afproj), []). fof(t2_subset,axiom,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), []). fof(t2_tarski,axiom,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), []). fof(t30_aff_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) => ~ ( v1_aff_1(D,A) & v1_aff_1(E,A) & r2_hidden(B,D) & r2_hidden(C,D) & r2_hidden(B,E) & r2_hidden(C,E) & B != C & D != E ) ) ) ) ) ) ), file(aff_1,t30_aff_1), []). fof(t30_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,u1_incsp_1(k13_afproj(A))) => ! [D] : ( m1_subset_1(D,u2_incsp_1(k13_afproj(A))) => ( ( v1_aff_1(B,A) & C = k5_afproj(A,B) & D = k4_tarski(B,1) ) => r1_incsp_1(k13_afproj(A),C,D) ) ) ) ) ) ), file(afproj,t30_afproj), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t40_aff_4,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( ( v1_aff_1(B,A) & v1_aff_1(C,A) ) => ( r4_aff_1(A,B,C) <=> r1_aff_4(A,B,C) ) ) ) ) ) ), file(aff_4,t40_aff_4), []). fof(t41_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,u1_incsp_1(k14_afproj(A))) => ! [C] : ( m1_subset_1(C,u1_incsp_1(k14_afproj(A))) => ! [D] : ( m1_subset_1(D,u2_incsp_1(k13_afproj(A))) => ~ ( B != C & r2_hidden(k4_tarski(B,D),u3_incsp_1(k13_afproj(A))) & r2_hidden(k4_tarski(C,D),u3_incsp_1(k13_afproj(A))) & ! [E] : ( m1_subset_1(E,u2_incsp_1(k14_afproj(A))) => D != k4_tarski(E,2) ) ) ) ) ) ) ), file(afproj,t41_afproj), []). fof(t4_subset,axiom,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), []). fof(t50_afproj,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ! [F] : ( m1_subset_1(F,u1_struct_0(A)) => ! [G] : ( m1_subset_1(G,u1_struct_0(A)) => ! [H] : ( m1_subset_1(H,u1_struct_0(A)) => ! [I] : ( m1_subset_1(I,u1_struct_0(A)) => ! [J] : ( m1_subset_1(J,u1_struct_0(A)) => ! [K] : ( m1_subset_1(K,u1_struct_0(A)) => ( ( r2_hidden(E,B) & r2_hidden(E,C) & r2_hidden(E,D) & r2_hidden(F,B) & r2_hidden(I,B) & r2_hidden(G,C) & r2_hidden(J,C) & r2_hidden(H,D) & r2_hidden(K,D) & v1_aff_1(B,A) & v1_aff_1(C,A) & v1_aff_1(D,A) & r2_analoaf(A,F,G,I,J) & r2_analoaf(A,F,H,I,K) ) => ( E = F | E = G | E = H | B = C | B = D | ( E != I & F != I ) | r2_analoaf(A,G,H,J,K) ) ) ) ) ) ) ) ) ) ) ) ) ) ), file(afproj,t50_afproj), []). fof(t51_aff_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( r3_aff_1(A,B,C) <=> ? [D] : ( m1_subset_1(D,u1_struct_0(A)) & ? [E] : ( m1_subset_1(E,u1_struct_0(A)) & ? [F] : ( m1_subset_1(F,u1_struct_0(A)) & ? [G] : ( m1_subset_1(G,u1_struct_0(A)) & D != E & F != G & r2_analoaf(A,D,E,F,G) & B = k2_aff_1(A,D,E) & C = k2_aff_1(A,F,G) ) ) ) ) ) ) ) ) ), file(aff_1,t51_aff_1), []). fof(t53_aff_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ! [F] : ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) => ! [G] : ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) => ( ( r2_hidden(B,F) & r2_hidden(C,F) & r2_hidden(D,G) & r2_hidden(E,G) & r3_aff_1(A,F,G) ) => r2_analoaf(A,B,C,D,E) ) ) ) ) ) ) ) ) ), file(aff_1,t53_aff_1), []). fof(t58_aff_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ( ~ ( ~ ( r4_aff_1(A,B,C) & r4_aff_1(A,C,D) ) & ~ ( r4_aff_1(A,B,C) & r4_aff_1(A,D,C) ) & ~ ( r4_aff_1(A,C,B) & r4_aff_1(A,C,D) ) & ~ ( r4_aff_1(A,C,B) & r4_aff_1(A,D,C) ) ) => r4_aff_1(A,B,D) ) ) ) ) ) ), file(aff_1,t58_aff_1), []). fof(t59_aff_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ( ( r4_aff_1(A,C,D) & r2_hidden(B,C) & r2_hidden(B,D) ) => C = D ) ) ) ) ) ), file(aff_1,t59_aff_1), []). fof(t5_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(t65_aff_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ! [F] : ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) => ( ( v1_aff_1(F,A) & r2_hidden(B,F) & r2_hidden(C,F) & r2_hidden(D,F) & r2_hidden(E,F) ) => r2_analoaf(A,B,C,D,E) ) ) ) ) ) ) ) ), file(aff_1,t65_aff_1), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_mcart_1,axiom,( ! [A,B] : ( k1_mcart_1(k4_tarski(A,B)) = A & k2_mcart_1(k4_tarski(A,B)) = B ) ), file(mcart_1,t7_mcart_1), []). fof(t8_aff_4,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ! [F] : ( m1_subset_1(F,u1_struct_0(A)) => ! [G] : ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) => ! [H] : ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(A))) => ( ( r2_hidden(B,G) & r2_hidden(B,H) & r2_hidden(C,G) & r2_hidden(D,G) & r2_hidden(E,H) & r2_hidden(F,H) & v1_aff_1(G,A) & v1_aff_1(H,A) & B = D ) => ( B = C | B = E | G = H | ( ~ r2_analoaf(A,C,E,D,F) & ~ r2_analoaf(A,E,C,F,D) ) | B = F ) ) ) ) ) ) ) ) ) ) ), file(aff_4,t8_aff_4), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t9_aff_4,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v1_diraf(A) & l1_analoaf(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ! [F] : ( m1_subset_1(F,u1_struct_0(A)) => ! [G] : ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) => ! [H] : ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(A))) => ( ( r2_hidden(B,G) & r2_hidden(B,H) & r2_hidden(C,G) & r2_hidden(D,G) & r2_hidden(E,H) & r2_hidden(F,H) & v1_aff_1(G,A) & v1_aff_1(H,A) & C = D ) => ( B = C | B = E | G = H | ( ~ r2_analoaf(A,C,E,D,F) & ~ r2_analoaf(A,E,C,F,D) ) | E = F ) ) ) ) ) ) ) ) ) ) ), file(aff_4,t9_aff_4), []).