% Mizar problem: t2_mesfunc4,mesfunc4,171,25 fof(t2_mesfunc4,conjecture,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) => ! [C] : ( m3_measure1(C,A,B) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ! [E] : ( ( v1_funct_1(E) & m2_relset_1(E,A,k3_supinf_1) ) => ! [F] : ( ( v1_prob_2(F) & m2_finseq_1(F,B) ) => ! [G] : ( m2_finseq_1(G,k3_supinf_1) => ! [H] : ( m2_finseq_1(H,k3_supinf_1) => ( ( r2_mesfunc2(A,B,E) & ! [I] : ( r2_hidden(I,k1_relat_1(E)) => r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,E,I)) ) & r1_mesfunc3(A,B,E,F,G) & k4_finseq_1(H) = k4_finseq_1(F) & ! [I] : ( m2_subset_1(I,k1_numbers,k5_numbers) => ( r2_hidden(I,k4_finseq_1(H)) => k4_mesfunc1(k5_numbers,H,I) = k2_extreal1(k4_mesfunc1(k5_numbers,G,I),k4_mesfunc1(k5_numbers,k1_partfun1(k5_numbers,B,B,k3_supinf_1,F,C),I)) ) ) & k3_finseq_1(F) = D ) => ( k1_relat_1(E) = k1_xboole_0 | k1_mesfunc3(A,B,C,E) = k5_convfun1(H) ) ) ) ) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_finseq_1,cc1_measure1,cc1_nat_1,cc1_relset_1,cc1_supinf_1,cc1_xreal_0,cc2_measure1,cc2_nat_1,cc2_supinf_1,cc2_xreal_0,cc3_measure1,cc3_nat_1,cc3_xreal_0,cc4_measure1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,commutativity_k1_nat_1,commutativity_k2_extreal1,commutativity_k2_supinf_2,commutativity_k2_tarski,commutativity_k2_xboole_0,commutativity_k2_xcmplx_0,commutativity_k3_xboole_0,commutativity_k5_subset_1,connectedness_r1_supinf_1,connectedness_r1_xreal_0,d10_xboole_0,d11_measure1,d15_finseq_1,d1_mesfunc2,d1_mesfunc3,d1_tarski,d1_xboole_0,d2_mesfunc3,d2_tarski,d2_xboole_0,d3_finseq_1,d3_prob_2,d3_tarski,d3_xboole_0,d4_finseq_1,d4_funct_1,d4_tarski,d5_convfun1,d5_funct_1,d5_mesfunc2,d5_real_1,d5_tarski,d7_finseq_1,d7_xboole_0,d8_mesfunc1,dt_k10_finseq_1,dt_k10_mesfunc1,dt_k11_supinf_1,dt_k12_finseq_1,dt_k13_binarith,dt_k16_finseq_1,dt_k19_supinf_2,dt_k1_card_1,dt_k1_extreal1,dt_k1_finseq_1,dt_k1_funct_1,dt_k1_measure1,dt_k1_mesfunc3,dt_k1_nat_1,dt_k1_numbers,dt_k1_partfun1,dt_k1_relat_1,dt_k1_supinf_1,dt_k1_supinf_2,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_extreal1,dt_k2_finseq_1,dt_k2_partfun1,dt_k2_relat_1,dt_k2_supinf_1,dt_k2_supinf_2,dt_k2_tarski,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_finseq_1,dt_k3_supinf_1,dt_k3_supinf_2,dt_k3_tarski,dt_k3_xboole_0,dt_k4_extreal1,dt_k4_finseq_1,dt_k4_finseq_2,dt_k4_measure6,dt_k4_mesfunc1,dt_k4_supinf_1,dt_k4_tarski,dt_k5_convfun1,dt_k5_finseq_1,dt_k5_finseqop,dt_k5_finsop_1,dt_k5_finsub_1,dt_k5_measure1,dt_k5_measure6,dt_k5_numbers,dt_k5_ordinal2,dt_k5_relat_1,dt_k5_relset_1,dt_k5_setfam_1,dt_k5_subset_1,dt_k5_supinf_1,dt_k7_finseq_1,dt_k7_funct_2,dt_k7_relat_1,dt_k8_finseq_1,dt_k9_finseq_1,dt_m1_finseq_1,dt_m1_finseq_2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_finseq_2,dt_m2_relset_1,dt_m2_subset_1,dt_m3_measure1,existence_m1_finseq_1,existence_m1_finseq_2,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_finseq_2,existence_m2_relset_1,existence_m2_subset_1,existence_m3_measure1,fc10_xreal_0,fc11_finseq_1,fc11_xreal_0,fc12_finseq_1,fc12_xreal_0,fc13_finseq_1,fc14_finseq_1,fc15_finseq_1,fc17_finseq_1,fc1_finseq_1,fc1_nat_1,fc1_ordinal2,fc1_supinf_1,fc2_finseq_1,fc3_finseq_1,fc3_nat_1,fc3_xreal_0,fc4_finseq_1,fc4_nat_1,fc5_finseq_1,fc7_finseq_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,idempotence_k2_xboole_0,idempotence_k3_xboole_0,idempotence_k5_subset_1,involutiveness_k3_supinf_2,rc1_finseq_1,rc1_measure1,rc1_nat_1,rc1_xreal_0,rc2_finseq_1,rc2_nat_1,rc2_xreal_0,rc3_finseq_1,rc3_measure1,rc3_nat_1,rc3_supinf_1,rc3_xreal_0,rc4_finseq_1,rc4_measure1,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,redefinition_k11_supinf_1,redefinition_k12_finseq_1,redefinition_k13_binarith,redefinition_k1_measure1,redefinition_k1_nat_1,redefinition_k1_partfun1,redefinition_k2_extreal1,redefinition_k2_finseq_1,redefinition_k2_partfun1,redefinition_k3_finseq_1,redefinition_k4_finseq_1,redefinition_k4_measure6,redefinition_k4_mesfunc1,redefinition_k4_supinf_1,redefinition_k5_finseqop,redefinition_k5_finsop_1,redefinition_k5_measure1,redefinition_k5_measure6,redefinition_k5_numbers,redefinition_k5_relset_1,redefinition_k5_setfam_1,redefinition_k5_subset_1,redefinition_k5_supinf_1,redefinition_k7_funct_2,redefinition_k8_finseq_1,redefinition_k9_finseq_1,redefinition_m2_finseq_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_supinf_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,s1_finseq_2__e12_2_2_8_1_2_3_1_1__mesfunc4,s1_finseq_2__e17_2_2_8_1_1_7_1_2__mesfunc4,s1_finseq_2__e20_2_2_8_1_1_7_1_1__mesfunc4,s1_nat_1__e10_2_2_8_1_1__mesfunc4,s1_nat_1__e3_2__mesfunc4,s5_finseq_1__e13_2_2_8_1_1_7_1_1__mesfunc4,s5_finseq_1__e13_2_2_8_1_1_7_1_2__mesfunc4,s5_finseq_1__e17_2_2_8_1_2_3_1_2__mesfunc4,s5_finseq_1__e22_2_2_8_1_2_3_1_2__mesfunc4,s5_finseq_1__e39_2_2_8_1_2_3_1_2__mesfunc4,s5_finseq_1__e3_2_2_8_1_1_7_1_1__mesfunc4,s5_finseq_1__e3_2_2_8_1_2_3_1_1__mesfunc4,s5_finseq_1__e6_2_2_8_1_1_7_1_2__mesfunc4,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,symmetry_r1_xboole_0,t11_finseq_5,t12_funct_1,t12_relset_1,t147_finseq_2,t17_finseq_1,t18_extreal1,t18_finseq_1,t18_supinf_2,t18_xboole_1,t19_finseq_2,t19_supinf_2,t1_arithm,t1_boole,t1_numerals,t1_real,t1_subset,t20_extreal2,t21_measure1,t22_extreal1,t22_supinf_1,t23_finseq_1,t23_funct_1,t25_finseq_1,t26_finseq_1,t27_finseq_1,t27_finseq_3,t28_xboole_1,t29_nat_1,t2_boole,t2_real,t2_subset,t2_xboole_1,t2_xreal_1,t2_zfmisc_1,t31_finseq_3,t34_mesfunc2,t35_finseq_1,t36_mesfunc2,t37_nat_1,t38_nat_1,t3_finseq_1,t3_real,t3_subset,t3_xboole_0,t3_xboole_1,t42_extreal1,t44_extreal2,t44_finseq_1,t45_extreal2,t45_measure1,t46_measure1,t4_finseq_1,t4_real,t4_subset,t55_finseq_1,t56_finseq_1,t58_extreal2,t59_finseq_1,t5_convfun1,t5_finseq_1,t5_real,t5_subset,t61_finseq_1,t62_measure1,t63_xboole_1,t65_relat_1,t65_xboole_1,t68_funct_1,t6_boole,t6_finseq_1,t6_real,t70_funct_1,t70_xboole_1,t76_funct_1,t79_finseq_1,t7_boole,t7_convfun1,t7_finseq_1,t7_real,t8_boole,t8_finseq_1,t8_real,t92_zfmisc_1,t93_zfmisc_1,t95_zfmisc_1,t96_zfmisc_1]), [file(mesfunc4,t2_mesfunc4)]). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc1_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ) ), file(finseq_1,cc1_finseq_1), []). fof(cc1_measure1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ( ( v1_finsub_1(B) & v1_prob_1(B,A) ) => v2_finsub_1(B) ) ) ), file(measure1,cc1_measure1), []). fof(cc1_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(nat_1,cc1_nat_1), []). 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(cc1_supinf_1,axiom,( ! [A] : ( m1_subset_1(A,k3_supinf_1) => v3_supinf_1(A) ) ), file(supinf_1,cc1_supinf_1), []). fof(cc1_xreal_0,axiom,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), []). fof(cc2_measure1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ( ( v2_finsub_1(B) & v1_prob_1(B,A) ) => v1_finsub_1(B) ) ) ), file(measure1,cc2_measure1), []). fof(cc2_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), []). fof(cc2_supinf_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_supinf_1(A) ) ) ), file(supinf_1,cc2_supinf_1), []). fof(cc2_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). fof(cc3_measure1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) ) => ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) ) ) ) ), file(measure1,cc3_measure1), []). fof(cc3_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), []). fof(cc3_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), []). fof(cc4_measure1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) ) => ( ~ v1_xboole_0(B) & v1_finsub_1(B) ) ) ) ), file(measure1,cc4_measure1), []). fof(cc4_xreal_0,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) ) ) ), file(xreal_0,cc4_xreal_0), []). fof(cc5_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & v3_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) ) ), file(xreal_0,cc5_xreal_0), []). fof(cc6_xreal_0,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_xreal_0(A) ) ) ), file(xreal_0,cc6_xreal_0), []). fof(cc7_xreal_0,axiom,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), []). fof(cc8_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_0), []). fof(commutativity_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k1_nat_1(B,A) ) ), file(nat_1,k1_nat_1), []). fof(commutativity_k2_extreal1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k3_supinf_1) & m1_subset_1(B,k3_supinf_1) ) => k2_extreal1(A,B) = k2_extreal1(B,A) ) ), file(extreal1,k2_extreal1), []). fof(commutativity_k2_supinf_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k3_supinf_1) & m1_subset_1(B,k3_supinf_1) ) => k2_supinf_2(A,B) = k2_supinf_2(B,A) ) ), file(supinf_2,k2_supinf_2), []). 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(commutativity_k2_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ), file(xcmplx_0,k2_xcmplx_0), []). fof(commutativity_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ), file(xboole_0,k3_xboole_0), []). fof(commutativity_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,C) = k5_subset_1(A,C,B) ) ), file(subset_1,k5_subset_1), []). fof(connectedness_r1_supinf_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k3_supinf_1) & m1_subset_1(B,k3_supinf_1) ) => ( r1_supinf_1(A,B) | r1_supinf_1(B,A) ) ) ), file(supinf_1,r1_supinf_1), []). fof(connectedness_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( r1_xreal_0(A,B) | r1_xreal_0(B,A) ) ) ), file(xreal_0,r1_xreal_0), []). fof(d10_xboole_0,axiom,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), []). fof(d11_measure1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,B,k3_supinf_1) & m2_relset_1(C,B,k3_supinf_1) ) => ( m3_measure1(C,A,B) <=> ( v6_supinf_2(C,B) & k1_funct_1(C,k1_xboole_0) = k1_supinf_2 & ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k5_numbers,B) & v1_prob_2(D) & m2_relset_1(D,k5_numbers,B) ) => k19_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,D,C)) = k1_funct_1(C,k5_setfam_1(A,k5_measure1(A,B,D))) ) ) ) ) ) ), file(measure1,d11_measure1), []). fof(d15_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ! [C] : ( v4_ordinal2(C) => k16_finseq_1(A,B,C) = k7_relat_1(B,k2_finseq_1(C)) ) ) ), file(finseq_1,d15_finseq_1), []). fof(d1_mesfunc2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & m2_relset_1(B,A,k3_supinf_1) ) => ( r1_mesfunc2(A,B) <=> ! [C] : ( m1_subset_1(C,A) => ~ ( r2_hidden(C,k1_relat_1(B)) & r1_supinf_1(k5_measure6,k4_extreal1(k4_mesfunc1(A,B,C))) ) ) ) ) ) ), file(mesfunc2,d1_mesfunc2), []). fof(d1_mesfunc3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) => ! [C] : ( ( v1_funct_1(C) & m2_relset_1(C,A,k3_supinf_1) ) => ! [D] : ( ( v1_prob_2(D) & m2_finseq_1(D,B) ) => ! [E] : ( m2_finseq_1(E,k3_supinf_1) => ( r1_mesfunc3(A,B,C,D,E) <=> ( k1_relat_1(C) = k3_tarski(k2_relat_1(D)) & k5_finsop_1(D) = k5_finsop_1(E) & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ( r2_hidden(F,k5_finsop_1(D)) => ! [G] : ( r2_hidden(G,k1_funct_1(D,F)) => k4_mesfunc1(A,C,G) = k4_mesfunc1(k5_numbers,E,F) ) ) ) ) ) ) ) ) ) ) ), file(mesfunc3,d1_mesfunc3), []). fof(d1_tarski,axiom,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), []). fof(d1_xboole_0,axiom,( ! [A] : ( A = k1_xboole_0 <=> ! [B] : ~ r2_hidden(B,A) ) ), file(xboole_0,d1_xboole_0), []). fof(d2_mesfunc3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) => ! [C] : ( m3_measure1(C,A,B) => ! [D] : ( ( v1_funct_1(D) & m2_relset_1(D,A,k3_supinf_1) ) => ( ( r2_mesfunc2(A,B,D) & ! [E] : ( r2_hidden(E,k1_relat_1(D)) => r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,D,E)) ) ) => ( k1_relat_1(D) = k1_xboole_0 | ! [E] : ( m1_subset_1(E,k3_supinf_1) => ( E = k1_mesfunc3(A,B,C,D) <=> ? [F] : ( v1_prob_2(F) & m2_finseq_1(F,B) & ? [G] : ( m2_finseq_1(G,k3_supinf_1) & ? [H] : ( m2_finseq_1(H,k3_supinf_1) & r1_mesfunc3(A,B,D,F,G) & k4_mesfunc1(k5_numbers,G,1) = k1_supinf_2 & ! [I] : ( m2_subset_1(I,k1_numbers,k5_numbers) => ( ( r1_xreal_0(2,I) & r2_hidden(I,k5_finsop_1(G)) ) => ( ~ r1_supinf_1(k4_mesfunc1(k5_numbers,G,I),k1_supinf_2) & ~ r1_supinf_1(k4_supinf_1,k4_mesfunc1(k5_numbers,G,I)) ) ) ) & k5_finsop_1(H) = k5_finsop_1(F) & ! [I] : ( m2_subset_1(I,k1_numbers,k5_numbers) => ( r2_hidden(I,k5_finsop_1(H)) => k4_mesfunc1(k5_numbers,H,I) = k2_extreal1(k4_mesfunc1(k5_numbers,G,I),k4_mesfunc1(k5_numbers,k5_finseqop(B,k3_supinf_1,F,C),I)) ) ) & E = k5_convfun1(H) ) ) ) ) ) ) ) ) ) ) ) ), file(mesfunc3,d2_mesfunc3), []). fof(d2_tarski,axiom,( ! [A,B,C] : ( C = k2_tarski(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( D = A | D = B ) ) ) ), file(tarski,d2_tarski), []). fof(d2_xboole_0,axiom,( ! [A,B,C] : ( C = k2_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) | r2_hidden(D,B) ) ) ) ), file(xboole_0,d2_xboole_0), []). fof(d3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( B = k3_finseq_1(A) <=> k2_finseq_1(B) = k1_relat_1(A) ) ) ) ), file(finseq_1,d3_finseq_1), []). fof(d3_prob_2,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v1_prob_2(A) <=> ! [B,C] : ( B != C => r1_xboole_0(k1_funct_1(A,B),k1_funct_1(A,C)) ) ) ) ), file(prob_2,d3_prob_2), []). fof(d3_tarski,axiom,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), []). fof(d3_xboole_0,axiom,( ! [A,B,C] : ( C = k3_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & r2_hidden(D,B) ) ) ) ), file(xboole_0,d3_xboole_0), []). fof(d4_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( m1_finseq_1(B,A) <=> r1_tarski(k2_relat_1(B),A) ) ) ), file(finseq_1,d4_finseq_1), []). fof(d4_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : ( ( r2_hidden(B,k1_relat_1(A)) => ( C = k1_funct_1(A,B) <=> r2_hidden(k4_tarski(B,C),A) ) ) & ( ~ r2_hidden(B,k1_relat_1(A)) => ( C = k1_funct_1(A,B) <=> C = k1_xboole_0 ) ) ) ) ), file(funct_1,d4_funct_1), []). fof(d4_tarski,axiom,( ! [A,B] : ( B = k3_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(C,D) & r2_hidden(D,A) ) ) ) ), file(tarski,d4_tarski), []). fof(d5_convfun1,axiom,( ! [A] : ( m2_finseq_1(A,k3_supinf_1) => ! [B] : ( m1_subset_1(B,k3_supinf_1) => ( B = k5_convfun1(A) <=> ? [C] : ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k3_supinf_1) & m2_relset_1(C,k5_numbers,k3_supinf_1) & B = k4_mesfunc1(k5_numbers,C,k3_finseq_1(A)) & k4_mesfunc1(k5_numbers,C,0) = k1_supinf_2 & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(k3_finseq_1(A),D) => k4_mesfunc1(k5_numbers,C,k1_nat_1(D,1)) = k2_supinf_2(k4_mesfunc1(k5_numbers,C,D),k4_mesfunc1(k5_numbers,A,k1_nat_1(D,1))) ) ) ) ) ) ) ), file(convfun1,d5_convfun1), []). fof(d5_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( B = k2_relat_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) ) ) ) ), file(funct_1,d5_funct_1), []). fof(d5_mesfunc2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) => ! [C] : ( ( v1_funct_1(C) & m2_relset_1(C,A,k3_supinf_1) ) => ( r2_mesfunc2(A,B,C) <=> ( r1_mesfunc2(A,C) & ? [D] : ( v1_prob_2(D) & m2_finseq_1(D,B) & k1_relat_1(C) = k3_tarski(k2_relat_1(D)) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ! [F] : ( m1_subset_1(F,A) => ! [G] : ( m1_subset_1(G,A) => ( ( r2_hidden(E,k4_finseq_1(D)) & r2_hidden(F,k1_funct_1(D,E)) & r2_hidden(G,k1_funct_1(D,E)) ) => k4_mesfunc1(A,C,F) = k4_mesfunc1(A,C,G) ) ) ) ) ) ) ) ) ) ) ), file(mesfunc2,d5_mesfunc2), []). fof(d5_real_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) <=> ~ ( r1_xreal_0(B,A) & B != A ) ) ) ) ), file(real_1,d5_real_1), []). 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_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ( C = k7_finseq_1(A,B) <=> ( k4_finseq_1(C) = k2_finseq_1(k1_nat_1(k3_finseq_1(A),k3_finseq_1(B))) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k4_finseq_1(A)) => k1_funct_1(C,D) = k1_funct_1(A,D) ) ) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k4_finseq_1(B)) => k1_funct_1(C,k1_nat_1(k3_finseq_1(A),D)) = k1_funct_1(B,D) ) ) ) ) ) ) ) ), file(finseq_1,d7_finseq_1), []). fof(d7_xboole_0,axiom,( ! [A,B] : ( r1_xboole_0(A,B) <=> k3_xboole_0(A,B) = k1_xboole_0 ) ), file(xboole_0,d7_xboole_0), []). fof(d8_mesfunc1,axiom,( k10_mesfunc1 = 1 ), file(mesfunc1,d8_mesfunc1), []). fof(dt_k10_finseq_1,axiom,( $true ), file(finseq_1,k10_finseq_1), []). fof(dt_k10_mesfunc1,axiom,( m1_subset_1(k10_mesfunc1,k3_supinf_1) ), file(mesfunc1,k10_mesfunc1), []). fof(dt_k11_supinf_1,axiom,( ! [A] : ( m1_subset_1(A,k3_supinf_1) => m1_subset_1(k11_supinf_1(A),k1_zfmisc_1(k3_supinf_1)) ) ), file(supinf_1,k11_supinf_1), []). fof(dt_k12_finseq_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m2_finseq_1(k12_finseq_1(A,B),A) ) ), file(finseq_1,k12_finseq_1), []). fof(dt_k13_binarith,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m2_finseq_2(k13_binarith(A,B),A,k4_finseq_2(1,A)) ) ), file(binarith,k13_binarith), []). fof(dt_k16_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & v4_ordinal2(C) ) => m2_finseq_1(k16_finseq_1(A,B,C),A) ) ), file(finseq_1,k16_finseq_1), []). fof(dt_k19_supinf_2,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k3_supinf_1) & m1_relset_1(A,k5_numbers,k3_supinf_1) ) => m1_subset_1(k19_supinf_2(A),k3_supinf_1) ) ), file(supinf_2,k19_supinf_2), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). fof(dt_k1_extreal1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k3_supinf_1) & m1_subset_1(B,k3_supinf_1) ) => m1_subset_1(k1_extreal1(A,B),k3_supinf_1) ) ), file(extreal1,k1_extreal1), []). fof(dt_k1_finseq_1,axiom,( $true ), file(finseq_1,k1_finseq_1), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_measure1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & v1_funct_2(B,A,k3_supinf_1) & m1_relset_1(B,A,k3_supinf_1) & m1_subset_1(C,A) ) => m1_subset_1(k1_measure1(A,B,C),k3_supinf_1) ) ), file(measure1,k1_measure1), []). fof(dt_k1_mesfunc3,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) & m3_measure1(C,A,B) & v1_funct_1(D) & m1_relset_1(D,A,k3_supinf_1) ) => m1_subset_1(k1_mesfunc3(A,B,C,D),k3_supinf_1) ) ), file(mesfunc3,k1_mesfunc3), []). fof(dt_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k1_nat_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_partfun1,axiom,( ! [A,B,C,D,E,F] : ( ( v1_funct_1(E) & m1_relset_1(E,A,B) & v1_funct_1(F) & m1_relset_1(F,C,D) ) => ( v1_funct_1(k1_partfun1(A,B,C,D,E,F)) & m2_relset_1(k1_partfun1(A,B,C,D,E,F),A,D) ) ) ), file(partfun1,k1_partfun1), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_supinf_1,axiom,( $true ), file(supinf_1,k1_supinf_1), []). fof(dt_k1_supinf_2,axiom,( m1_subset_1(k1_supinf_2,k3_supinf_1) ), file(supinf_2,k1_supinf_2), []). 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_extreal1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k3_supinf_1) & m1_subset_1(B,k3_supinf_1) ) => m1_subset_1(k2_extreal1(A,B),k3_supinf_1) ) ), file(extreal1,k2_extreal1), []). fof(dt_k2_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => m1_subset_1(k2_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(finseq_1,k2_finseq_1), []). fof(dt_k2_partfun1,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => ( v1_funct_1(k2_partfun1(A,B,C,D)) & m2_relset_1(k2_partfun1(A,B,C,D),A,B) ) ) ), file(partfun1,k2_partfun1), []). fof(dt_k2_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), []). fof(dt_k2_supinf_1,axiom,( $true ), file(supinf_1,k2_supinf_1), []). fof(dt_k2_supinf_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k3_supinf_1) & m1_subset_1(B,k3_supinf_1) ) => m1_subset_1(k2_supinf_2(A,B),k3_supinf_1) ) ), file(supinf_2,k2_supinf_2), []). 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_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m2_subset_1(k3_finseq_1(A),k1_numbers,k5_numbers) ) ), file(finseq_1,k3_finseq_1), []). fof(dt_k3_supinf_1,axiom,( $true ), file(supinf_1,k3_supinf_1), []). fof(dt_k3_supinf_2,axiom,( ! [A] : ( m1_subset_1(A,k3_supinf_1) => m1_subset_1(k3_supinf_2(A),k3_supinf_1) ) ), file(supinf_2,k3_supinf_2), []). fof(dt_k3_tarski,axiom,( $true ), file(tarski,k3_tarski), []). fof(dt_k3_xboole_0,axiom,( $true ), file(xboole_0,k3_xboole_0), []). fof(dt_k4_extreal1,axiom,( ! [A] : ( m1_subset_1(A,k3_supinf_1) => m1_subset_1(k4_extreal1(A),k3_supinf_1) ) ), file(extreal1,k4_extreal1), []). fof(dt_k4_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m1_subset_1(k4_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(finseq_1,k4_finseq_1), []). fof(dt_k4_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => m1_finseq_2(k4_finseq_2(A,B),B) ) ), file(finseq_2,k4_finseq_2), []). fof(dt_k4_measure6,axiom,( m1_subset_1(k4_measure6,k3_supinf_1) ), file(measure6,k4_measure6), []). fof(dt_k4_mesfunc1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & m1_relset_1(B,A,k3_supinf_1) ) => m1_subset_1(k4_mesfunc1(A,B,C),k3_supinf_1) ) ), file(mesfunc1,k4_mesfunc1), []). fof(dt_k4_supinf_1,axiom,( m1_subset_1(k4_supinf_1,k3_supinf_1) ), file(supinf_1,k4_supinf_1), []). fof(dt_k4_tarski,axiom,( $true ), file(tarski,k4_tarski), []). fof(dt_k5_convfun1,axiom,( ! [A] : ( m1_finseq_1(A,k3_supinf_1) => m1_subset_1(k5_convfun1(A),k3_supinf_1) ) ), file(convfun1,k5_convfun1), []). fof(dt_k5_finseq_1,axiom,( $true ), file(finseq_1,k5_finseq_1), []). fof(dt_k5_finseqop,axiom,( ! [A,B,C,D] : ( ( m1_finseq_1(C,A) & v1_funct_1(D) & v1_funct_2(D,A,B) & m1_relset_1(D,A,B) ) => m2_finseq_1(k5_finseqop(A,B,C,D),B) ) ), file(finseqop,k5_finseqop), []). fof(dt_k5_finsop_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m1_subset_1(k5_finsop_1(A),k5_finsub_1(k5_numbers)) ) ), file(finsop_1,k5_finsop_1), []). fof(dt_k5_finsub_1,axiom,( ! [A] : v4_finsub_1(k5_finsub_1(A)) ), file(finsub_1,k5_finsub_1), []). fof(dt_k5_measure1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,B) & m1_relset_1(C,k5_numbers,B) ) => ( ~ v1_xboole_0(k5_measure1(A,B,C)) & m1_subset_1(k5_measure1(A,B,C),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ), file(measure1,k5_measure1), []). fof(dt_k5_measure6,axiom,( m1_subset_1(k5_measure6,k3_supinf_1) ), file(measure6,k5_measure6), []). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), []). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), []). fof(dt_k5_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => v1_relat_1(k5_relat_1(A,B)) ) ), file(relat_1,k5_relat_1), []). fof(dt_k5_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k5_relset_1(A,B,C),k1_zfmisc_1(B)) ) ), file(relset_1,k5_relset_1), []). fof(dt_k5_setfam_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => m1_subset_1(k5_setfam_1(A,B),k1_zfmisc_1(A)) ) ), file(setfam_1,k5_setfam_1), []). fof(dt_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k5_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k5_subset_1), []). fof(dt_k5_supinf_1,axiom,( m1_subset_1(k5_supinf_1,k3_supinf_1) ), file(supinf_1,k5_supinf_1), []). fof(dt_k7_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,k7_finseq_1), []). fof(dt_k7_funct_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,A,B) & m1_relset_1(D,A,B) & v1_funct_1(E) & v1_funct_2(E,B,C) & m1_relset_1(E,B,C) ) => ( v1_funct_1(k7_funct_2(A,B,C,D,E)) & v1_funct_2(k7_funct_2(A,B,C,D,E),A,C) & m2_relset_1(k7_funct_2(A,B,C,D,E),A,C) ) ) ), file(funct_2,k7_funct_2), []). fof(dt_k7_relat_1,axiom,( ! [A,B] : ( v1_relat_1(A) => v1_relat_1(k7_relat_1(A,B)) ) ), file(relat_1,k7_relat_1), []). fof(dt_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => m2_finseq_1(k8_finseq_1(A,B,C),A) ) ), file(finseq_1,k8_finseq_1), []). fof(dt_k9_finseq_1,axiom,( ! [A] : ( v1_relat_1(k9_finseq_1(A)) & v1_funct_1(k9_finseq_1(A)) ) ), file(finseq_1,k9_finseq_1), []). fof(dt_m1_finseq_1,axiom,( ! [A,B] : ( m1_finseq_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) ) ), file(finseq_1,m1_finseq_1), []). fof(dt_m1_finseq_2,axiom,( $true ), file(finseq_2,m1_finseq_2), []). 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_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ( v1_funct_1(B) & v1_finseq_1(B) & m2_relset_1(B,k5_numbers,A) ) ) ), file(finseq_1,m2_finseq_1), []). fof(dt_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ! [C] : ( m2_finseq_2(C,A,B) => m2_finseq_1(C,A) ) ) ), file(finseq_2,m2_finseq_2), []). 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_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) => m1_subset_1(C,A) ) ) ), file(subset_1,m2_subset_1), []). fof(dt_m3_measure1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) => ! [C] : ( m3_measure1(C,A,B) => ( v1_funct_1(C) & v1_funct_2(C,B,k3_supinf_1) & m2_relset_1(C,B,k3_supinf_1) ) ) ) ), file(measure1,m3_measure1), []). fof(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), []). fof(existence_m1_finseq_2,axiom,( ! [A] : ? [B] : m1_finseq_2(B,A) ), file(finseq_2,m1_finseq_2), []). 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_finseq_1,axiom,( ! [A] : ? [B] : m2_finseq_1(B,A) ), file(finseq_1,m2_finseq_1), []). fof(existence_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ? [C] : m2_finseq_2(C,A,B) ) ), file(finseq_2,m2_finseq_2), []). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), []). fof(existence_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m2_subset_1(C,A,B) ) ), file(subset_1,m2_subset_1), []). fof(existence_m3_measure1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) => ? [C] : m3_measure1(C,A,B) ) ), file(measure1,m3_measure1), []). fof(fc10_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v2_xreal_0(k2_xcmplx_0(B,A)) & ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc10_xreal_0), []). fof(fc11_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_finset_1(A) ) => v1_finset_1(k2_relat_1(A)) ) ), file(finseq_1,fc11_finseq_1), []). fof(fc11_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) & v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc11_xreal_0), []). fof(fc12_finseq_1,axiom,( ! [A] : ( ~ v1_xboole_0(k5_finseq_1(A)) & v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) ) ), file(finseq_1,fc12_finseq_1), []). fof(fc12_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & ~ v2_xreal_0(k2_xcmplx_0(B,A)) & v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc12_xreal_0), []). fof(fc13_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( ~ v1_xboole_0(k7_finseq_1(A,B)) & v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finset_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,fc13_finseq_1), []). fof(fc14_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( ~ v1_xboole_0(k7_finseq_1(B,A)) & v1_relat_1(k7_finseq_1(B,A)) & v1_funct_1(k7_finseq_1(B,A)) & v1_finset_1(k7_finseq_1(B,A)) & v1_finseq_1(k7_finseq_1(B,A)) ) ) ), file(finseq_1,fc14_finseq_1), []). fof(fc15_finseq_1,axiom,( ! [A,B] : ( m1_finseq_1(B,A) => ( v1_xboole_0(k16_finseq_1(A,B,0)) & v1_relat_1(k16_finseq_1(A,B,0)) & v1_funct_1(k16_finseq_1(A,B,0)) & v2_funct_1(k16_finseq_1(A,B,0)) & v1_finset_1(k16_finseq_1(A,B,0)) & v1_finseq_1(k16_finseq_1(A,B,0)) ) ) ), file(finseq_1,fc15_finseq_1), []). fof(fc17_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) => v1_finset_1(k1_relat_1(A)) ) ), file(finseq_1,fc17_finseq_1), []). fof(fc1_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_finset_1(k1_finseq_1(A)) ) ), file(finseq_1,fc1_finseq_1), []). fof(fc1_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc1_nat_1), []). fof(fc1_ordinal2,axiom, ( v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & ~ v1_xboole_0(k5_ordinal2) ), file(ordinal2,fc1_ordinal2), []). fof(fc1_supinf_1,axiom,( ~ v1_xboole_0(k3_supinf_1) ), file(supinf_1,fc1_supinf_1), []). fof(fc2_finseq_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) ), file(finseq_1,fc2_finseq_1), []). fof(fc3_finseq_1,axiom,( ! [A] : ( v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) ) ), file(finseq_1,fc3_finseq_1), []). fof(fc3_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc3_nat_1), []). fof(fc3_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc3_xreal_0), []). fof(fc4_finseq_1,axiom,( ! [A] : ( v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) ) ), file(finseq_1,fc4_finseq_1), []). fof(fc4_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v4_ordinal2(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(nat_1,fc4_nat_1), []). fof(fc5_finseq_1,axiom,( ! [A,B] : ( v1_relat_1(k10_finseq_1(A,B)) & v1_funct_1(k10_finseq_1(A,B)) ) ), file(finseq_1,fc5_finseq_1), []). fof(fc7_finseq_1,axiom,( ! [A,B] : ( v1_relat_1(k10_finseq_1(A,B)) & v1_funct_1(k10_finseq_1(A,B)) & v1_finset_1(k10_finseq_1(A,B)) & v1_finseq_1(k10_finseq_1(A,B)) ) ), file(finseq_1,fc7_finseq_1), []). fof(fc7_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), []). fof(fc8_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc8_xreal_0), []). fof(fc9_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). fof(idempotence_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_xboole_0), []). fof(idempotence_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,B) = B ) ), file(subset_1,k5_subset_1), []). fof(involutiveness_k3_supinf_2,axiom,( ! [A] : ( m1_subset_1(A,k3_supinf_1) => k3_supinf_2(k3_supinf_2(A)) = A ) ), file(supinf_2,k3_supinf_2), []). fof(rc1_finseq_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc1_finseq_1), []). fof(rc1_measure1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) & ~ v1_xboole_0(B) & v1_finsub_1(B) & v2_finsub_1(B) & v1_prob_1(B,A) ) ), file(measure1,rc1_measure1), []). fof(rc1_nat_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc1_nat_1), []). fof(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). fof(rc2_finseq_1,axiom,( ! [A] : ? [B] : ( m1_relset_1(B,k5_numbers,A) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc2_finseq_1), []). fof(rc2_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), []). fof(rc2_xreal_0,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc2_xreal_0), []). fof(rc3_finseq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc3_finseq_1), []). fof(rc3_measure1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) & ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) ) ), file(measure1,rc3_measure1), []). fof(rc3_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), []). fof(rc3_supinf_1,axiom,( ? [A] : v3_supinf_1(A) ), file(supinf_1,rc3_supinf_1), []). fof(rc3_xreal_0,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & v3_xreal_0(A) ) ), file(xreal_0,rc3_xreal_0), []). fof(rc4_finseq_1,axiom,( ! [A] : ? [B] : ( m1_finseq_1(B,A) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc4_finseq_1), []). fof(rc4_measure1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) => ? [C] : ( m1_relset_1(C,k5_numbers,B) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,B) & v1_prob_2(C) ) ) ), file(measure1,rc4_measure1), []). fof(rc4_xreal_0,axiom,( ? [A] : ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc4_xreal_0), []). fof(rc6_finseq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc6_finseq_1), []). fof(rc7_finseq_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc7_finseq_1), []). fof(rc8_finseq_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc8_finseq_1), []). fof(redefinition_k11_supinf_1,axiom,( ! [A] : ( m1_subset_1(A,k3_supinf_1) => k11_supinf_1(A) = k1_tarski(A) ) ), file(supinf_1,k11_supinf_1), []). fof(redefinition_k12_finseq_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k12_finseq_1(A,B) = k5_finseq_1(B) ) ), file(finseq_1,k12_finseq_1), []). fof(redefinition_k13_binarith,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k13_binarith(A,B) = k5_finseq_1(B) ) ), file(binarith,k13_binarith), []). fof(redefinition_k1_measure1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & v1_funct_2(B,A,k3_supinf_1) & m1_relset_1(B,A,k3_supinf_1) & m1_subset_1(C,A) ) => k1_measure1(A,B,C) = k1_funct_1(B,C) ) ), file(measure1,k1_measure1), []). fof(redefinition_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k2_xcmplx_0(A,B) ) ), file(nat_1,k1_nat_1), []). fof(redefinition_k1_partfun1,axiom,( ! [A,B,C,D,E,F] : ( ( v1_funct_1(E) & m1_relset_1(E,A,B) & v1_funct_1(F) & m1_relset_1(F,C,D) ) => k1_partfun1(A,B,C,D,E,F) = k5_relat_1(E,F) ) ), file(partfun1,k1_partfun1), []). fof(redefinition_k2_extreal1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k3_supinf_1) & m1_subset_1(B,k3_supinf_1) ) => k2_extreal1(A,B) = k1_extreal1(A,B) ) ), file(extreal1,k2_extreal1), []). fof(redefinition_k2_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => k2_finseq_1(A) = k1_finseq_1(A) ) ), file(finseq_1,k2_finseq_1), []). fof(redefinition_k2_partfun1,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => k2_partfun1(A,B,C,D) = k7_relat_1(C,D) ) ), file(partfun1,k2_partfun1), []). fof(redefinition_k3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k3_finseq_1(A) = k1_card_1(A) ) ), file(finseq_1,k3_finseq_1), []). fof(redefinition_k4_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k4_finseq_1(A) = k1_relat_1(A) ) ), file(finseq_1,k4_finseq_1), []). fof(redefinition_k4_measure6,axiom,( k4_measure6 = k2_supinf_1 ), file(measure6,k4_measure6), []). fof(redefinition_k4_mesfunc1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & m1_relset_1(B,A,k3_supinf_1) ) => k4_mesfunc1(A,B,C) = k1_funct_1(B,C) ) ), file(mesfunc1,k4_mesfunc1), []). fof(redefinition_k4_supinf_1,axiom,( k4_supinf_1 = k1_supinf_1 ), file(supinf_1,k4_supinf_1), []). fof(redefinition_k5_finseqop,axiom,( ! [A,B,C,D] : ( ( m1_finseq_1(C,A) & v1_funct_1(D) & v1_funct_2(D,A,B) & m1_relset_1(D,A,B) ) => k5_finseqop(A,B,C,D) = k5_relat_1(C,D) ) ), file(finseqop,k5_finseqop), []). fof(redefinition_k5_finsop_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k5_finsop_1(A) = k1_relat_1(A) ) ), file(finsop_1,k5_finsop_1), []). fof(redefinition_k5_measure1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,B) & m1_relset_1(C,k5_numbers,B) ) => k5_measure1(A,B,C) = k2_relat_1(C) ) ), file(measure1,k5_measure1), []). fof(redefinition_k5_measure6,axiom,( k5_measure6 = k1_supinf_1 ), file(measure6,k5_measure6), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k5_relset_1(A,B,C) = k2_relat_1(C) ) ), file(relset_1,k5_relset_1), []). fof(redefinition_k5_setfam_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => k5_setfam_1(A,B) = k3_tarski(B) ) ), file(setfam_1,k5_setfam_1), []). fof(redefinition_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,C) = k3_xboole_0(B,C) ) ), file(subset_1,k5_subset_1), []). fof(redefinition_k5_supinf_1,axiom,( k5_supinf_1 = k2_supinf_1 ), file(supinf_1,k5_supinf_1), []). fof(redefinition_k7_funct_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,A,B) & m1_relset_1(D,A,B) & v1_funct_1(E) & v1_funct_2(E,B,C) & m1_relset_1(E,B,C) ) => k7_funct_2(A,B,C,D,E) = k5_relat_1(D,E) ) ), file(funct_2,k7_funct_2), []). fof(redefinition_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => k8_finseq_1(A,B,C) = k7_finseq_1(B,C) ) ), file(finseq_1,k8_finseq_1), []). fof(redefinition_k9_finseq_1,axiom,( ! [A] : k9_finseq_1(A) = k5_finseq_1(A) ), file(finseq_1,k9_finseq_1), []). fof(redefinition_m2_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) <=> m1_finseq_1(B,A) ) ), file(finseq_1,m2_finseq_1), []). fof(redefinition_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ! [C] : ( m2_finseq_2(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(finseq_2,m2_finseq_2), []). 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_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(subset_1,m2_subset_1), []). fof(reflexivity_r1_supinf_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k3_supinf_1) & m1_subset_1(B,k3_supinf_1) ) => r1_supinf_1(A,A) ) ), file(supinf_1,r1_supinf_1), []). fof(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(reflexivity_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), []). fof(s1_finseq_2__e12_2_2_8_1_2_3_1_1__mesfunc4,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) & m3_measure1(C,A,B) & m2_finseq_1(D,k3_supinf_1) & m2_finseq_1(E,k3_supinf_1) & m2_finseq_1(F,B) ) => ? [G] : ( m2_finseq_1(G,k3_supinf_1) & k3_finseq_1(G) = k3_finseq_1(E) & ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ( r2_hidden(H,k2_finseq_1(k3_finseq_1(E))) => k1_funct_1(G,H) = k2_extreal1(k4_mesfunc1(k5_numbers,D,H),k4_mesfunc1(k5_numbers,k1_partfun1(k5_numbers,B,B,k3_supinf_1,F,C),H)) ) ) ) ) ), file(mesfunc4,s1_finseq_2__e12_2_2_8_1_2_3_1_1__mesfunc4), []). fof(s1_finseq_2__e17_2_2_8_1_1_7_1_2__mesfunc4,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) & m3_measure1(C,A,B) & m2_finseq_1(D,B) & m2_finseq_1(E,k3_supinf_1) ) => ? [F] : ( m2_finseq_1(F,k3_supinf_1) & k3_finseq_1(F) = 2 & ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ( r2_hidden(G,k2_finseq_1(2)) => k1_funct_1(F,G) = k2_extreal1(k4_mesfunc1(k5_numbers,E,G),k4_mesfunc1(k5_numbers,k1_partfun1(k5_numbers,B,B,k3_supinf_1,D,C),G)) ) ) ) ) ), file(mesfunc4,s1_finseq_2__e17_2_2_8_1_1_7_1_2__mesfunc4), []). fof(s1_finseq_2__e20_2_2_8_1_1_7_1_1__mesfunc4,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) & m3_measure1(C,A,B) & m2_finseq_1(D,B) & m2_finseq_1(E,k3_supinf_1) ) => ? [F] : ( m2_finseq_1(F,k3_supinf_1) & k3_finseq_1(F) = 2 & ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ( r2_hidden(G,k2_finseq_1(2)) => k1_funct_1(F,G) = k2_extreal1(k4_mesfunc1(k5_numbers,E,G),k4_mesfunc1(k5_numbers,k1_partfun1(k5_numbers,B,B,k3_supinf_1,D,C),G)) ) ) ) ) ), file(mesfunc4,s1_finseq_2__e20_2_2_8_1_1_7_1_1__mesfunc4), []). fof(s1_nat_1__e10_2_2_8_1_1__mesfunc4,axiom,( ! [A,B] : ( ( m2_finseq_1(A,k3_supinf_1) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k3_supinf_1) & m2_relset_1(B,k5_numbers,k3_supinf_1) ) => ( ( ( ~ r1_xreal_0(k3_finseq_1(A),0) => k4_mesfunc1(k5_numbers,B,0) = k1_supinf_2 ) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( ~ r1_xreal_0(k3_finseq_1(A),C) => k4_mesfunc1(k5_numbers,B,C) = k1_supinf_2 ) => ( ~ r1_xreal_0(k3_finseq_1(A),k1_nat_1(C,1)) => k4_mesfunc1(k5_numbers,B,k1_nat_1(C,1)) = k1_supinf_2 ) ) ) ) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(k3_finseq_1(A),C) => k4_mesfunc1(k5_numbers,B,C) = k1_supinf_2 ) ) ) ) ), file(mesfunc4,s1_nat_1__e10_2_2_8_1_1__mesfunc4), []). fof(s1_nat_1__e3_2__mesfunc4,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) & m3_measure1(C,A,B) ) => ( ( ! [D] : ( ( v1_funct_1(D) & m2_relset_1(D,A,k3_supinf_1) ) => ! [E] : ( ( v1_prob_2(E) & m2_finseq_1(E,B) ) => ! [F] : ( m2_finseq_1(F,k3_supinf_1) => ! [G] : ( m2_finseq_1(G,k3_supinf_1) => ( ( r2_mesfunc2(A,B,D) & ! [H] : ( r2_hidden(H,k1_relat_1(D)) => r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,D,H)) ) & r1_mesfunc3(A,B,D,E,F) & k4_finseq_1(G) = k4_finseq_1(E) & ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ( r2_hidden(H,k4_finseq_1(G)) => k4_mesfunc1(k5_numbers,G,H) = k2_extreal1(k4_mesfunc1(k5_numbers,F,H),k4_mesfunc1(k5_numbers,k1_partfun1(k5_numbers,B,B,k3_supinf_1,E,C),H)) ) ) & k3_finseq_1(E) = 0 ) => ( k1_relat_1(D) = k1_xboole_0 | k1_mesfunc3(A,B,C,D) = k5_convfun1(G) ) ) ) ) ) ) & ! [I] : ( m2_subset_1(I,k1_numbers,k5_numbers) => ( ! [J] : ( ( v1_funct_1(J) & m2_relset_1(J,A,k3_supinf_1) ) => ! [K] : ( ( v1_prob_2(K) & m2_finseq_1(K,B) ) => ! [L] : ( m2_finseq_1(L,k3_supinf_1) => ! [M] : ( m2_finseq_1(M,k3_supinf_1) => ( ( r2_mesfunc2(A,B,J) & ! [N] : ( r2_hidden(N,k1_relat_1(J)) => r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,J,N)) ) & r1_mesfunc3(A,B,J,K,L) & k4_finseq_1(M) = k4_finseq_1(K) & ! [N] : ( m2_subset_1(N,k1_numbers,k5_numbers) => ( r2_hidden(N,k4_finseq_1(M)) => k4_mesfunc1(k5_numbers,M,N) = k2_extreal1(k4_mesfunc1(k5_numbers,L,N),k4_mesfunc1(k5_numbers,k1_partfun1(k5_numbers,B,B,k3_supinf_1,K,C),N)) ) ) & k3_finseq_1(K) = I ) => ( k1_relat_1(J) = k1_xboole_0 | k1_mesfunc3(A,B,C,J) = k5_convfun1(M) ) ) ) ) ) ) => ! [O] : ( ( v1_funct_1(O) & m2_relset_1(O,A,k3_supinf_1) ) => ! [P] : ( ( v1_prob_2(P) & m2_finseq_1(P,B) ) => ! [Q] : ( m2_finseq_1(Q,k3_supinf_1) => ! [R] : ( m2_finseq_1(R,k3_supinf_1) => ( ( r2_mesfunc2(A,B,O) & ! [S] : ( r2_hidden(S,k1_relat_1(O)) => r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,O,S)) ) & r1_mesfunc3(A,B,O,P,Q) & k4_finseq_1(R) = k4_finseq_1(P) & ! [S] : ( m2_subset_1(S,k1_numbers,k5_numbers) => ( r2_hidden(S,k4_finseq_1(R)) => k4_mesfunc1(k5_numbers,R,S) = k2_extreal1(k4_mesfunc1(k5_numbers,Q,S),k4_mesfunc1(k5_numbers,k1_partfun1(k5_numbers,B,B,k3_supinf_1,P,C),S)) ) ) & k3_finseq_1(P) = k1_nat_1(I,1) ) => ( k1_relat_1(O) = k1_xboole_0 | k1_mesfunc3(A,B,C,O) = k5_convfun1(R) ) ) ) ) ) ) ) ) ) => ! [I] : ( m2_subset_1(I,k1_numbers,k5_numbers) => ! [T] : ( ( v1_funct_1(T) & m2_relset_1(T,A,k3_supinf_1) ) => ! [U] : ( ( v1_prob_2(U) & m2_finseq_1(U,B) ) => ! [V] : ( m2_finseq_1(V,k3_supinf_1) => ! [W] : ( m2_finseq_1(W,k3_supinf_1) => ( ( r2_mesfunc2(A,B,T) & ! [X] : ( r2_hidden(X,k1_relat_1(T)) => r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,T,X)) ) & r1_mesfunc3(A,B,T,U,V) & k4_finseq_1(W) = k4_finseq_1(U) & ! [X] : ( m2_subset_1(X,k1_numbers,k5_numbers) => ( r2_hidden(X,k4_finseq_1(W)) => k4_mesfunc1(k5_numbers,W,X) = k2_extreal1(k4_mesfunc1(k5_numbers,V,X),k4_mesfunc1(k5_numbers,k1_partfun1(k5_numbers,B,B,k3_supinf_1,U,C),X)) ) ) & k3_finseq_1(U) = I ) => ( k1_relat_1(T) = k1_xboole_0 | k1_mesfunc3(A,B,C,T) = k5_convfun1(W) ) ) ) ) ) ) ) ) ) ), file(mesfunc4,s1_nat_1__e3_2__mesfunc4), []). fof(s5_finseq_1__e13_2_2_8_1_1_7_1_1__mesfunc4,axiom,( ! [A,B] : ( ( m2_subset_1(A,k1_numbers,k5_numbers) & m2_finseq_1(B,k3_supinf_1) ) => ( ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r2_hidden(C,k2_finseq_1(2)) & ! [D] : ( m1_subset_1(D,k3_supinf_1) => ~ ( ( C = 1 => D = k1_supinf_2 ) & ( C = 2 => D = k4_mesfunc1(k5_numbers,B,k1_nat_1(A,1)) ) ) ) ) ) => ? [C] : ( m2_finseq_1(C,k3_supinf_1) & k4_finseq_1(C) = k2_finseq_1(2) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k2_finseq_1(2)) => ( ( D = 1 => k1_funct_1(C,D) = k1_supinf_2 ) & ( D = 2 => k1_funct_1(C,D) = k4_mesfunc1(k5_numbers,B,k1_nat_1(A,1)) ) ) ) ) ) ) ) ), file(mesfunc4,s5_finseq_1__e13_2_2_8_1_1_7_1_1__mesfunc4), []). fof(s5_finseq_1__e13_2_2_8_1_1_7_1_2__mesfunc4,axiom, ( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r2_hidden(A,k2_finseq_1(2)) & ! [B] : ( m1_subset_1(B,k3_supinf_1) => ~ ( ( A = 1 => B = k1_supinf_2 ) & ( A = 2 => B = k10_mesfunc1 ) ) ) ) ) => ? [A] : ( m2_finseq_1(A,k3_supinf_1) & k4_finseq_1(A) = k2_finseq_1(2) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r2_hidden(B,k2_finseq_1(2)) => ( ( B = 1 => k1_funct_1(A,B) = k1_supinf_2 ) & ( B = 2 => k1_funct_1(A,B) = k10_mesfunc1 ) ) ) ) ) ), file(mesfunc4,s5_finseq_1__e13_2_2_8_1_1_7_1_2__mesfunc4), []). fof(s5_finseq_1__e17_2_2_8_1_2_3_1_2__mesfunc4,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) & m2_subset_1(C,k1_numbers,k5_numbers) & v1_prob_2(D) & m2_finseq_1(D,B) & v1_prob_2(E) & m2_finseq_1(E,B) ) => ( ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ~ ( r2_hidden(F,k2_finseq_1(k1_nat_1(k3_finseq_1(E),1))) & ! [G] : ( m1_subset_1(G,B) => ~ ( ( r1_xreal_0(F,k3_finseq_1(E)) => G = k1_funct_1(E,F) ) & ( F = k1_nat_1(k3_finseq_1(E),1) => G = k1_funct_1(D,k1_nat_1(C,1)) ) ) ) ) ) => ? [F] : ( m2_finseq_1(F,B) & k4_finseq_1(F) = k2_finseq_1(k1_nat_1(k3_finseq_1(E),1)) & ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ( r2_hidden(G,k2_finseq_1(k1_nat_1(k3_finseq_1(E),1))) => ( ( r1_xreal_0(G,k3_finseq_1(E)) => k1_funct_1(F,G) = k1_funct_1(E,G) ) & ( G = k1_nat_1(k3_finseq_1(E),1) => k1_funct_1(F,G) = k1_funct_1(D,k1_nat_1(C,1)) ) ) ) ) ) ) ) ), file(mesfunc4,s5_finseq_1__e17_2_2_8_1_2_3_1_2__mesfunc4), []). fof(s5_finseq_1__e22_2_2_8_1_2_3_1_2__mesfunc4,axiom,( ! [A,B,C] : ( ( m2_subset_1(A,k1_numbers,k5_numbers) & m2_finseq_1(B,k3_supinf_1) & m2_finseq_1(C,k3_supinf_1) ) => ( ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ~ ( r2_hidden(D,k2_finseq_1(k1_nat_1(k3_finseq_1(C),1))) & ! [E] : ( m1_subset_1(E,k3_supinf_1) => ~ ( ( r1_xreal_0(D,k3_finseq_1(C)) => E = k4_mesfunc1(k5_numbers,C,D) ) & ( D = k1_nat_1(k3_finseq_1(C),1) => E = k4_mesfunc1(k5_numbers,B,k1_nat_1(A,1)) ) ) ) ) ) => ? [D] : ( m2_finseq_1(D,k3_supinf_1) & k4_finseq_1(D) = k2_finseq_1(k1_nat_1(k3_finseq_1(C),1)) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( r2_hidden(E,k2_finseq_1(k1_nat_1(k3_finseq_1(C),1))) => ( ( r1_xreal_0(E,k3_finseq_1(C)) => k1_funct_1(D,E) = k4_mesfunc1(k5_numbers,C,E) ) & ( E = k1_nat_1(k3_finseq_1(C),1) => k1_funct_1(D,E) = k4_mesfunc1(k5_numbers,B,k1_nat_1(A,1)) ) ) ) ) ) ) ) ), file(mesfunc4,s5_finseq_1__e22_2_2_8_1_2_3_1_2__mesfunc4), []). fof(s5_finseq_1__e39_2_2_8_1_2_3_1_2__mesfunc4,axiom,( ! [A,B,C,D,E,F,G,H,I] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) & m3_measure1(C,A,B) & m2_subset_1(D,k1_numbers,k5_numbers) & v1_prob_2(E) & m2_finseq_1(E,B) & m2_finseq_1(F,k3_supinf_1) & m2_finseq_1(G,k3_supinf_1) & m2_finseq_1(H,k3_supinf_1) & m2_finseq_1(I,B) ) => ( ! [J] : ( m2_subset_1(J,k1_numbers,k5_numbers) => ~ ( r2_hidden(J,k2_finseq_1(k1_nat_1(k3_finseq_1(H),1))) & ! [K] : ( m1_subset_1(K,k3_supinf_1) => ~ ( ( r1_xreal_0(J,k3_finseq_1(H)) => K = k2_extreal1(k4_mesfunc1(k5_numbers,G,J),k4_mesfunc1(k5_numbers,k1_partfun1(k5_numbers,B,B,k3_supinf_1,I,C),J)) ) & ( J = k1_nat_1(k3_finseq_1(H),1) => K = k2_extreal1(k4_mesfunc1(k5_numbers,F,k1_nat_1(D,1)),k4_mesfunc1(k5_numbers,k1_partfun1(k5_numbers,B,B,k3_supinf_1,E,C),k1_nat_1(D,1))) ) ) ) ) ) => ? [J] : ( m2_finseq_1(J,k3_supinf_1) & k4_finseq_1(J) = k2_finseq_1(k1_nat_1(k3_finseq_1(H),1)) & ! [K] : ( m2_subset_1(K,k1_numbers,k5_numbers) => ( r2_hidden(K,k2_finseq_1(k1_nat_1(k3_finseq_1(H),1))) => ( ( r1_xreal_0(K,k3_finseq_1(H)) => k1_funct_1(J,K) = k2_extreal1(k4_mesfunc1(k5_numbers,G,K),k4_mesfunc1(k5_numbers,k1_partfun1(k5_numbers,B,B,k3_supinf_1,I,C),K)) ) & ( K = k1_nat_1(k3_finseq_1(H),1) => k1_funct_1(J,K) = k2_extreal1(k4_mesfunc1(k5_numbers,F,k1_nat_1(D,1)),k4_mesfunc1(k5_numbers,k1_partfun1(k5_numbers,B,B,k3_supinf_1,E,C),k1_nat_1(D,1))) ) ) ) ) ) ) ) ), file(mesfunc4,s5_finseq_1__e39_2_2_8_1_2_3_1_2__mesfunc4), []). fof(s5_finseq_1__e3_2_2_8_1_1_7_1_1__mesfunc4,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) & m2_subset_1(C,k1_numbers,k5_numbers) & v1_prob_2(D) & m2_finseq_1(D,B) & v1_prob_2(E) & m2_finseq_1(E,B) ) => ( ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ~ ( r2_hidden(F,k2_finseq_1(2)) & ! [G] : ( m1_subset_1(G,B) => ~ ( ( F = 1 => G = k3_tarski(k2_relat_1(E)) ) & ( F = 2 => G = k1_funct_1(D,k1_nat_1(C,1)) ) ) ) ) ) => ? [F] : ( m2_finseq_1(F,B) & k4_finseq_1(F) = k2_finseq_1(2) & ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ( r2_hidden(G,k2_finseq_1(2)) => ( ( G = 1 => k1_funct_1(F,G) = k3_tarski(k2_relat_1(E)) ) & ( G = 2 => k1_funct_1(F,G) = k1_funct_1(D,k1_nat_1(C,1)) ) ) ) ) ) ) ) ), file(mesfunc4,s5_finseq_1__e3_2_2_8_1_1_7_1_1__mesfunc4), []). fof(s5_finseq_1__e3_2_2_8_1_2_3_1_1__mesfunc4,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) & m2_subset_1(C,k1_numbers,k5_numbers) & v1_prob_2(D) & m2_finseq_1(D,B) & v1_prob_2(E) & m2_finseq_1(E,B) ) => ( ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ~ ( r2_hidden(F,k2_finseq_1(k3_finseq_1(E))) & ! [G] : ( m1_subset_1(G,B) => ~ ( ( F = 1 => G = k2_xboole_0(k1_funct_1(E,1),k1_funct_1(D,k1_nat_1(C,1))) ) & ( r1_xreal_0(2,F) => G = k1_funct_1(E,F) ) ) ) ) ) => ? [F] : ( m2_finseq_1(F,B) & k4_finseq_1(F) = k2_finseq_1(k3_finseq_1(E)) & ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ( r2_hidden(G,k2_finseq_1(k3_finseq_1(E))) => ( ( G = 1 => k1_funct_1(F,G) = k2_xboole_0(k1_funct_1(E,1),k1_funct_1(D,k1_nat_1(C,1))) ) & ( r1_xreal_0(2,G) => k1_funct_1(F,G) = k1_funct_1(E,G) ) ) ) ) ) ) ) ), file(mesfunc4,s5_finseq_1__e3_2_2_8_1_2_3_1_1__mesfunc4), []). fof(s5_finseq_1__e6_2_2_8_1_1_7_1_2__mesfunc4,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) & v1_prob_2(C) & m2_finseq_1(C,B) ) => ( ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ~ ( r2_hidden(D,k2_finseq_1(2)) & ! [E] : ( m1_subset_1(E,B) => ~ ( ( D = 1 => E = k3_tarski(k2_relat_1(C)) ) & ( D = 2 => E = k1_xboole_0 ) ) ) ) ) => ? [D] : ( m2_finseq_1(D,B) & k4_finseq_1(D) = k2_finseq_1(2) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( r2_hidden(E,k2_finseq_1(2)) => ( ( E = 1 => k1_funct_1(D,E) = k3_tarski(k2_relat_1(C)) ) & ( E = 2 => k1_funct_1(D,E) = k1_xboole_0 ) ) ) ) ) ) ) ), file(mesfunc4,s5_finseq_1__e6_2_2_8_1_1_7_1_2__mesfunc4), []). fof(spc0_boole,axiom,( v1_xboole_0(0) ), file(boole,spc0_boole), []). fof(spc1_boole,axiom,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), []). fof(spc1_numerals,axiom, ( v2_xreal_0(1) & m1_subset_1(1,k5_numbers) ), file(numerals,spc1_numerals), []). fof(spc2_boole,axiom,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), []). fof(spc2_numerals,axiom, ( v2_xreal_0(2) & m1_subset_1(2,k5_numbers) ), file(numerals,spc2_numerals), []). fof(symmetry_r1_xboole_0,axiom,( ! [A,B] : ( r1_xboole_0(A,B) => r1_xboole_0(B,A) ) ), file(xboole_0,r1_xboole_0), []). fof(t11_finseq_5,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ( ( r2_hidden(k1_nat_1(A,1),k4_finseq_1(B)) & C = k7_relat_1(B,k2_finseq_1(A)) ) => k7_relat_1(B,k2_finseq_1(k1_nat_1(A,1))) = k7_finseq_1(C,k9_finseq_1(k1_funct_1(B,k1_nat_1(A,1)))) ) ) ) ) ), file(finseq_5,t11_finseq_5), []). fof(t12_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r2_hidden(A,k1_relat_1(B)) => r2_hidden(k1_funct_1(B,A),k2_relat_1(B)) ) ) ), file(funct_1,t12_funct_1), []). fof(t12_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( r1_tarski(k1_relat_1(C),A) & r1_tarski(k2_relat_1(C),B) ) ) ), file(relset_1,t12_relset_1), []). fof(t147_finseq_2,axiom,( ! [A,B] : k2_relat_1(k10_finseq_1(A,B)) = k2_tarski(A,B) ), file(finseq_2,t147_finseq_2), []). fof(t17_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( ( k4_finseq_1(A) = k4_finseq_1(B) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r2_hidden(C,k4_finseq_1(A)) => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) ) => A = B ) ) ) ), file(finseq_1,t17_finseq_1), []). fof(t18_extreal1,axiom,( ! [A] : ( m1_subset_1(A,k3_supinf_1) => ! [B] : ( m1_subset_1(B,k1_numbers) => ( A = B => ( ~ ( ~ r1_xreal_0(B,0) & r1_supinf_1(A,k1_supinf_2) ) & ~ ( ~ r1_supinf_1(A,k1_supinf_2) & r1_xreal_0(B,0) ) ) ) ) ) ), file(extreal1,t18_extreal1), []). fof(t18_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( ( k3_finseq_1(A) = k3_finseq_1(B) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,C) & r1_xreal_0(C,k3_finseq_1(A)) ) => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) ) => A = B ) ) ) ), file(finseq_1,t18_finseq_1), []). fof(t18_supinf_2,axiom,( ! [A] : ( m1_subset_1(A,k3_supinf_1) => ( k2_supinf_2(A,k1_supinf_2) = A & k2_supinf_2(k1_supinf_2,A) = A ) ) ), file(supinf_2,t18_supinf_2), []). fof(t18_xboole_1,axiom,( ! [A,B,C] : ( r1_tarski(A,k3_xboole_0(B,C)) => r1_tarski(A,B) ) ), file(xboole_1,t18_xboole_1), []). fof(t19_finseq_2,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => k3_finseq_1(k7_finseq_1(B,k9_finseq_1(A))) = k1_nat_1(k3_finseq_1(B),1) ) ), file(finseq_2,t19_finseq_2), []). fof(t19_supinf_2,axiom, ( ~ r1_supinf_1(k1_supinf_2,k5_supinf_1) & ~ r1_supinf_1(k4_supinf_1,k1_supinf_2) ), file(supinf_2,t19_supinf_2), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). fof(t1_boole,axiom,( ! [A] : k2_xboole_0(A,k1_xboole_0) = A ), file(boole,t1_boole), []). fof(t1_numerals,axiom,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), []). fof(t1_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t20_extreal2,axiom, ( ~ r1_supinf_1(k5_measure6,k10_mesfunc1) & ~ r1_supinf_1(k10_mesfunc1,k4_measure6) ), file(extreal2,t20_extreal2), []). fof(t21_measure1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & v2_finsub_1(B) & v1_prob_1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) => ( r2_hidden(k1_xboole_0,B) & r2_hidden(A,B) ) ) ), file(measure1,t21_measure1), []). fof(t22_extreal1,axiom,( ! [A] : ( m1_subset_1(A,k3_supinf_1) => ! [B] : ( m1_subset_1(B,k3_supinf_1) => ( k2_extreal1(A,B) = k1_supinf_2 <=> ( A = k1_supinf_2 | B = k1_supinf_2 ) ) ) ) ), file(extreal1,t22_extreal1), []). fof(t22_supinf_1,axiom,( ! [A] : ( m1_subset_1(A,k3_supinf_1) => ! [B] : ( m1_subset_1(B,k3_supinf_1) => ( ( r1_supinf_1(A,B) & r1_supinf_1(B,A) ) => A = B ) ) ) ), file(supinf_1,t22_supinf_1), []). fof(t23_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B,C] : ( m2_finseq_1(C,B) => m2_finseq_1(k7_relat_1(C,k2_finseq_1(A)),B) ) ) ), file(finseq_1,t23_finseq_1), []). fof(t23_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(A,k1_relat_1(B)) => k1_funct_1(k5_relat_1(B,C),A) = k1_funct_1(C,k1_funct_1(B,A)) ) ) ) ), file(funct_1,t23_funct_1), []). fof(t25_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( k3_finseq_1(A) = 0 <=> A = k1_xboole_0 ) ) ), file(finseq_1,t25_finseq_1), []). fof(t26_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( A = k1_xboole_0 <=> k4_finseq_1(A) = k1_xboole_0 ) ) ), file(finseq_1,t26_finseq_1), []). fof(t27_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( A = k1_xboole_0 <=> k2_relat_1(A) = k1_xboole_0 ) ) ), file(finseq_1,t27_finseq_1), []). fof(t27_finseq_3,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( v4_ordinal2(B) => ( r2_hidden(B,k4_finseq_1(A)) <=> ( r1_xreal_0(1,B) & r1_xreal_0(B,k3_finseq_1(A)) ) ) ) ) ), file(finseq_3,t27_finseq_3), []). fof(t28_xboole_1,axiom,( ! [A,B] : ( r1_tarski(A,B) => k3_xboole_0(A,B) = A ) ), file(xboole_1,t28_xboole_1), []). fof(t29_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => r1_xreal_0(A,k2_xcmplx_0(A,B)) ) ) ), file(nat_1,t29_nat_1), []). fof(t2_boole,axiom,( ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ), file(boole,t2_boole), []). fof(t2_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v3_xreal_0(B) ) => v3_xreal_0(A) ) ) ) ), file(real,t2_real), []). 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_xboole_1,axiom,( ! [A] : r1_tarski(k1_xboole_0,A) ), file(xboole_1,t2_xboole_1), []). fof(t2_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,C) ) => r1_xreal_0(A,C) ) ) ) ) ), file(xreal_1,t2_xreal_1), []). fof(t2_zfmisc_1,axiom,( k3_tarski(k1_xboole_0) = k1_xboole_0 ), file(zfmisc_1,t2_zfmisc_1), []). fof(t31_finseq_3,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( k3_finseq_1(A) = k3_finseq_1(B) <=> k4_finseq_1(A) = k4_finseq_1(B) ) ) ) ), file(finseq_3,t31_finseq_3), []). fof(t34_mesfunc2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( ( v1_prob_2(C) & m2_finseq_1(C,B) ) => r2_hidden(k3_tarski(k2_relat_1(C)),B) ) ) ) ) ), file(mesfunc2,t34_mesfunc2), []). fof(t35_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => k3_finseq_1(k7_finseq_1(A,B)) = k1_nat_1(k3_finseq_1(A),k3_finseq_1(B)) ) ) ), file(finseq_1,t35_finseq_1), []). fof(t36_mesfunc2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( v1_relat_1(D) => ( ( v1_prob_2(D) & m2_finseq_1(D,B) ) => ( v1_prob_2(k7_relat_1(D,k2_finseq_1(C))) & m2_finseq_1(k7_relat_1(D,k2_finseq_1(C)),B) ) ) ) ) ) ) ), file(mesfunc2,t36_mesfunc2), []). fof(t37_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ! [C] : ( v4_ordinal2(C) => ( r1_xreal_0(A,B) => r1_xreal_0(A,k2_xcmplx_0(B,C)) ) ) ) ) ), file(nat_1,t37_nat_1), []). fof(t38_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ~ r1_xreal_0(k2_xcmplx_0(B,1),A) <=> r1_xreal_0(A,B) ) ) ) ), file(nat_1,t38_nat_1), []). fof(t3_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r2_hidden(A,k2_finseq_1(B)) <=> ( r1_xreal_0(1,A) & r1_xreal_0(A,B) ) ) ) ) ), file(finseq_1,t3_finseq_1), []). fof(t3_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v3_xreal_0(A) & v3_xreal_0(B) ) ) ) ), file(real,t3_real), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t3_xboole_0,axiom,( ! [A,B] : ( ~ ( ~ r1_xboole_0(A,B) & ! [C] : ~ ( r2_hidden(C,A) & r2_hidden(C,B) ) ) & ~ ( ? [C] : ( r2_hidden(C,A) & r2_hidden(C,B) ) & r1_xboole_0(A,B) ) ) ), file(xboole_0,t3_xboole_0), []). fof(t3_xboole_1,axiom,( ! [A] : ( r1_tarski(A,k1_xboole_0) => A = k1_xboole_0 ) ), file(xboole_1,t3_xboole_1), []). fof(t42_extreal1,axiom,( ! [A] : ( m1_subset_1(A,k3_supinf_1) => ! [B] : ( m1_subset_1(B,k3_supinf_1) => ! [C] : ( m1_subset_1(C,k3_supinf_1) => ( ( r1_supinf_1(A,B) & r1_supinf_1(k1_supinf_2,C) ) => r1_supinf_1(k2_extreal1(A,C),k2_extreal1(B,C)) ) ) ) ) ), file(extreal1,t42_extreal1), []). fof(t44_extreal2,axiom,( ! [A] : ( m1_subset_1(A,k3_supinf_1) => ! [B] : ( m1_subset_1(B,k3_supinf_1) => ( ~ ( ( ( ~ r1_supinf_1(A,k1_supinf_2) & ~ r1_supinf_1(k1_supinf_2,B) ) | ( ~ r1_supinf_1(k1_supinf_2,A) & ~ r1_supinf_1(B,k1_supinf_2) ) ) & r1_supinf_1(k1_supinf_2,k2_extreal1(A,B)) ) & ~ ( ~ r1_supinf_1(k1_supinf_2,k2_extreal1(A,B)) & ~ ( ~ r1_supinf_1(A,k1_supinf_2) & ~ r1_supinf_1(k1_supinf_2,B) ) & ~ ( ~ r1_supinf_1(k1_supinf_2,A) & ~ r1_supinf_1(B,k1_supinf_2) ) ) ) ) ) ), file(extreal2,t44_extreal2), []). fof(t44_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => k2_relat_1(k7_finseq_1(A,B)) = k2_xboole_0(k2_relat_1(A),k2_relat_1(B)) ) ) ), file(finseq_1,t44_finseq_1), []). fof(t45_extreal2,axiom,( ! [A] : ( m1_subset_1(A,k3_supinf_1) => ! [B] : ( m1_subset_1(B,k3_supinf_1) => ( ( ( ~ ( ~ r1_supinf_1(k1_supinf_2,A) & r1_supinf_1(A,k1_supinf_2) ) & ~ ( ~ r1_supinf_1(k1_supinf_2,B) & r1_supinf_1(B,k1_supinf_2) ) ) | ( ~ ( ~ r1_supinf_1(A,k1_supinf_2) & r1_supinf_1(k1_supinf_2,A) ) & ~ ( ~ r1_supinf_1(B,k1_supinf_2) & r1_supinf_1(k1_supinf_2,B) ) ) ) <=> r1_supinf_1(k1_supinf_2,k2_extreal1(A,B)) ) ) ) ), file(extreal2,t45_extreal2), []). fof(t45_measure1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) => ( r2_hidden(k1_xboole_0,B) & r2_hidden(A,B) ) ) ), file(measure1,t45_measure1), []). fof(t46_measure1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) => ! [C,D] : ( ( r2_hidden(C,B) & r2_hidden(D,B) ) => ( r2_hidden(k2_xboole_0(C,D),B) & r2_hidden(k3_xboole_0(C,D),B) ) ) ) ), file(measure1,t46_measure1), []). fof(t4_finseq_1,axiom, ( k2_finseq_1(0) = k1_xboole_0 & k2_finseq_1(1) = k1_tarski(1) & k2_finseq_1(2) = k2_tarski(1,2) ), file(finseq_1,t4_finseq_1), []). fof(t4_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v2_xreal_0(B) & v2_xreal_0(A) ) ) ) ), file(real,t4_real), []). 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(t55_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( B = k9_finseq_1(A) <=> ( k4_finseq_1(B) = k2_finseq_1(1) & k2_relat_1(B) = k1_tarski(A) ) ) ) ), file(finseq_1,t55_finseq_1), []). fof(t56_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( B = k9_finseq_1(A) <=> ( k3_finseq_1(B) = 1 & k2_relat_1(B) = k1_tarski(A) ) ) ) ), file(finseq_1,t56_finseq_1), []). fof(t58_extreal2,axiom,( ! [A] : ( m1_subset_1(A,k3_supinf_1) => ! [B] : ( m1_subset_1(B,k3_supinf_1) => ( ~ r1_supinf_1(B,k4_extreal1(A)) => ( ~ r1_supinf_1(A,k3_supinf_2(B)) & ~ r1_supinf_1(B,A) ) ) ) ) ), file(extreal2,t58_extreal2), []). fof(t59_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => k1_funct_1(k7_finseq_1(B,k9_finseq_1(A)),k1_nat_1(k3_finseq_1(B),1)) = A ) ), file(finseq_1,t59_finseq_1), []). fof(t5_convfun1,axiom,( ! [A] : ( m1_subset_1(A,k3_supinf_1) => k5_convfun1(k13_binarith(k3_supinf_1,A)) = A ) ), file(convfun1,t5_convfun1), []). fof(t5_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( A = 0 | r2_hidden(A,k2_finseq_1(A)) ) ) ), file(finseq_1,t5_finseq_1), []). fof(t5_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), []). 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(t61_finseq_1,axiom,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ( C = k10_finseq_1(A,B) <=> ( k3_finseq_1(C) = 2 & k1_funct_1(C,1) = A & k1_funct_1(C,2) = B ) ) ) ), file(finseq_1,t61_finseq_1), []). fof(t62_measure1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & v1_prob_1(B,A) & v1_measure1(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) => ! [C] : ( m3_measure1(C,A,B) => ! [D] : ( m2_subset_1(D,k1_zfmisc_1(A),B) => ! [E] : ( m2_subset_1(E,k1_zfmisc_1(A),B) => ( r1_tarski(D,E) => r1_supinf_1(k1_measure1(B,C,D),k1_measure1(B,C,E)) ) ) ) ) ) ), file(measure1,t62_measure1), []). fof(t63_xboole_1,axiom,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_xboole_0(B,C) ) => r1_xboole_0(A,C) ) ), file(xboole_1,t63_xboole_1), []). fof(t65_relat_1,axiom,( ! [A] : ( v1_relat_1(A) => ( k1_relat_1(A) = k1_xboole_0 <=> k2_relat_1(A) = k1_xboole_0 ) ) ), file(relat_1,t65_relat_1), []). fof(t65_xboole_1,axiom,( ! [A] : r1_xboole_0(A,k1_xboole_0) ), file(xboole_1,t65_xboole_1), []). fof(t68_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( B = k7_relat_1(C,A) <=> ( k1_relat_1(B) = k3_xboole_0(k1_relat_1(C),A) & ! [D] : ( r2_hidden(D,k1_relat_1(B)) => k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) ) ) ), file(funct_1,t68_funct_1), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => r2_hidden(k2_xcmplx_0(A,1),k2_finseq_1(k2_xcmplx_0(A,1))) ) ), file(finseq_1,t6_finseq_1), []). fof(t6_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), []). fof(t70_funct_1,axiom,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(B,k1_relat_1(k7_relat_1(C,A))) => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ), file(funct_1,t70_funct_1), []). fof(t70_xboole_1,axiom,( ! [A,B,C] : ( ~ ( ~ r1_xboole_0(A,k2_xboole_0(B,C)) & r1_xboole_0(A,B) & r1_xboole_0(A,C) ) & ~ ( ~ ( r1_xboole_0(A,B) & r1_xboole_0(A,C) ) & r1_xboole_0(A,k2_xboole_0(B,C)) ) ) ), file(xboole_1,t70_xboole_1), []). fof(t76_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r1_tarski(k1_relat_1(k7_relat_1(B,A)),k1_relat_1(B)) & r1_tarski(k2_relat_1(k7_relat_1(B,A)),k2_relat_1(B)) ) ) ), file(funct_1,t76_funct_1), []). fof(t79_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ! [C] : ( v4_ordinal2(C) => ( r1_xreal_0(k3_finseq_1(B),C) => k16_finseq_1(A,B,C) = B ) ) ) ), file(finseq_1,t79_finseq_1), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_convfun1,axiom,( ! [A] : ( m2_finseq_1(A,k3_supinf_1) => ! [B] : ( m2_finseq_1(B,k3_supinf_1) => ~ ( ~ r2_hidden(k4_measure6,k5_relset_1(k5_numbers,k3_supinf_1,A)) & ~ r2_hidden(k4_measure6,k5_relset_1(k5_numbers,k3_supinf_1,B)) & k5_convfun1(k8_finseq_1(k3_supinf_1,A,B)) != k2_supinf_2(k5_convfun1(A),k5_convfun1(B)) ) ) ) ), file(convfun1,t7_convfun1), []). fof(t7_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r1_xreal_0(A,B) <=> r1_tarski(k2_finseq_1(A),k2_finseq_1(B)) ) ) ) ), file(finseq_1,t7_finseq_1), []). fof(t7_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v2_xreal_0(A) & ~ v3_xreal_0(B) ) ) ) ), file(real,t7_real), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( k2_finseq_1(A) = k2_finseq_1(B) => A = B ) ) ) ), file(finseq_1,t8_finseq_1), []). fof(t8_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v3_xreal_0(B) & ~ v2_xreal_0(A) ) ) ) ), file(real,t8_real), []). fof(t92_zfmisc_1,axiom,( ! [A,B] : ( r2_hidden(A,B) => r1_tarski(A,k3_tarski(B)) ) ), file(zfmisc_1,t92_zfmisc_1), []). fof(t93_zfmisc_1,axiom,( ! [A,B] : k3_tarski(k2_tarski(A,B)) = k2_xboole_0(A,B) ), file(zfmisc_1,t93_zfmisc_1), []). fof(t95_zfmisc_1,axiom,( ! [A,B] : ( r1_tarski(A,B) => r1_tarski(k3_tarski(A),k3_tarski(B)) ) ), file(zfmisc_1,t95_zfmisc_1), []). fof(t96_zfmisc_1,axiom,( ! [A,B] : k3_tarski(k2_xboole_0(A,B)) = k2_xboole_0(k3_tarski(A),k3_tarski(B)) ), file(zfmisc_1,t96_zfmisc_1), []).