% Mizar problem: t33_euler_2,euler_2,759,15 fof(t33_euler_2,conjecture,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r2_int_2(A,B) => ( A = 0 | r1_xreal_0(B,1) | k4_nat_1(k3_euler_2(A,k1_euler_1(B)),B) = 1 ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_finseq_1,cc1_funct_1,cc1_relset_1,cc1_xreal_0,cc2_funct_1,cc2_int_1,cc2_xreal_0,cc3_int_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,commutativity_k11_binop_2,commutativity_k1_nat_1,commutativity_k2_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_int_2,commutativity_k3_xcmplx_0,commutativity_k6_nat_1,connectedness_r1_xreal_0,d10_xboole_0,d13_finseq_1,d15_finseq_1,d1_euler_1,d1_euler_2,d1_rfinseq,d2_nat_1,d3_finseq_1,d3_tarski,d5_funct_1,d6_int_2,d8_funct_1,d8_int_1,d9_int_1,dt_k10_relat_1,dt_k10_wsierp_1,dt_k11_binop_2,dt_k12_finseq_1,dt_k14_finseq_1,dt_k16_finseq_1,dt_k16_rvsum_1,dt_k1_card_1,dt_k1_euler_1,dt_k1_euler_2,dt_k1_finseq_1,dt_k1_finseq_3,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_euler_2,dt_k2_finseq_1,dt_k2_nat_1,dt_k2_newton,dt_k2_relat_1,dt_k2_seq_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_euler_2,dt_k3_finseq_1,dt_k3_int_2,dt_k3_nat_1,dt_k3_wsierp_1,dt_k3_xcmplx_0,dt_k4_card_1,dt_k4_finseq_1,dt_k4_nat_1,dt_k4_wsierp_1,dt_k5_finseq_1,dt_k5_int_1,dt_k5_numbers,dt_k5_ordinal2,dt_k6_finseq_1,dt_k6_int_1,dt_k6_nat_1,dt_k6_xcmplx_0,dt_k7_finseq_1,dt_k7_relat_1,dt_k9_finseq_1,dt_k9_rvsum_1,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_subset_1,dt_m1_trees_4,dt_m2_finseq_1,dt_m2_relset_1,dt_m2_subset_1,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m1_trees_4,existence_m2_finseq_1,existence_m2_relset_1,existence_m2_subset_1,fc10_xreal_0,fc11_finseq_1,fc11_xreal_0,fc12_finseq_1,fc12_xreal_0,fc13_finseq_1,fc14_finseq_1,fc15_finseq_1,fc15_xreal_0,fc16_xreal_0,fc17_finseq_1,fc17_xreal_0,fc18_finseq_1,fc18_xreal_0,fc19_xreal_0,fc1_finseq_1,fc1_int_1,fc1_numbers,fc1_ordinal2,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc23_xreal_0,fc24_xreal_0,fc2_finseq_1,fc2_int_1,fc3_finseq_1,fc3_xreal_0,fc4_finseq_1,fc4_funct_1,fc4_int_1,fc4_xreal_0,fc5_xreal_0,fc6_int_1,fc7_int_1,fc7_xreal_0,fc8_int_1,fc8_xreal_0,fc9_int_1,fc9_xreal_0,fraenkel_a_1_0_euler_2,fraenkel_a_1_1_euler_1,fraenkel_a_2_0_euler_2,idempotence_k6_nat_1,l18_euler_2,l1_euler_2,l4_euler_2,l7_euler_2,rc1_finseq_1,rc1_funct_1,rc1_int_1,rc1_xreal_0,rc2_finseq_1,rc2_funct_1,rc2_int_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_xreal_0,rc4_finseq_1,rc4_funct_1,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,redefinition_k10_wsierp_1,redefinition_k11_binop_2,redefinition_k12_finseq_1,redefinition_k1_euler_2,redefinition_k1_nat_1,redefinition_k2_finseq_1,redefinition_k2_nat_1,redefinition_k2_seq_1,redefinition_k3_euler_2,redefinition_k3_finseq_1,redefinition_k3_wsierp_1,redefinition_k4_card_1,redefinition_k4_finseq_1,redefinition_k4_wsierp_1,redefinition_k5_numbers,redefinition_k9_finseq_1,redefinition_m1_trees_4,redefinition_m2_finseq_1,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_rfinseq,reflexivity_r1_tarski,reflexivity_r1_xreal_0,reflexivity_r2_int_1,s1_nat_1__e3_25_11__euler_2,s1_nat_1__e3_25_13__euler_2,spc0_boole,spc1_boole,spc1_numerals,symmetry_r1_int_2,symmetry_r1_rfinseq,symmetry_r2_int_2,t101_newton,t11_euler_1,t11_euler_2,t11_newton,t121_finseq_3,t124_rvsum_1,t12_prepower,t133_rvsum_1,t13_euler_1,t13_finset_1,t142_funct_1,t15_euler_1,t16_euler_1,t16_int_1,t16_rfinseq,t1_arithm,t1_euler_2,t1_numerals,t1_real,t1_subset,t1_xreal_1,t20_rfinseq,t23_finseq_2,t25_euler_2,t27_euler_2,t27_finseq_3,t29_nat_1,t2_arithm,t2_euler_1,t2_real,t2_subset,t2_tarski,t2_xreal_1,t38_nat_1,t39_nat_1,t3_arithm,t3_finseq_1,t3_real,t3_subset,t40_int_2,t41_int_2,t44_finseq_3,t46_nat_1,t4_arithm,t4_real,t4_subset,t5_euler_2,t5_real,t5_subset,t64_newton,t66_finseq_3,t66_rvsum_1,t6_boole,t6_euler_1,t6_euler_2,t6_newton,t6_real,t76_nat_1,t77_newton,t78_newton,t79_finseq_1,t79_newton,t7_boole,t7_euler_2,t7_real,t80_finseq_1,t80_finseq_5,t80_newton,t8_boole,t8_real,t8_xreal_1,t92_real_1,t97_finseq_3,t99_finseq_3,t9_newton]), [file(euler_2,t33_euler_2)]). 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_funct_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_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_xreal_0,axiom,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), []). fof(cc2_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), []). fof(cc2_int_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) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), []). fof(cc2_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). fof(cc3_int_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_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_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), []). 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_k11_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k11_binop_2(A,B) = k11_binop_2(B,A) ) ), file(binop_2,k11_binop_2), []). 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_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k2_nat_1(A,B) = k2_nat_1(B,A) ) ), file(nat_1,k2_nat_1), []). 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_int_2,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => k3_int_2(A,B) = k3_int_2(B,A) ) ), file(int_2,k3_int_2), []). fof(commutativity_k3_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k3_xcmplx_0(A,B) = k3_xcmplx_0(B,A) ) ), file(xcmplx_0,k3_xcmplx_0), []). fof(commutativity_k6_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k6_nat_1(A,B) = k6_nat_1(B,A) ) ), file(nat_1,k6_nat_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(d13_finseq_1,axiom,( ! [A] : ( ? [B] : ( v4_ordinal2(B) & r1_tarski(A,k2_finseq_1(B)) ) => ! [B] : ( m2_finseq_1(B,k5_numbers) => ( B = k14_finseq_1(A) <=> ( k2_relat_1(B) = A & ! [C] : ( v4_ordinal2(C) => ! [D] : ( v4_ordinal2(D) => ! [E] : ( v4_ordinal2(E) => ! [F] : ( v4_ordinal2(F) => ~ ( r1_xreal_0(1,C) & ~ r1_xreal_0(D,C) & r1_xreal_0(D,k3_finseq_1(B)) & E = k1_funct_1(B,C) & F = k1_funct_1(B,D) & r1_xreal_0(F,E) ) ) ) ) ) ) ) ) ) ), file(finseq_1,d13_finseq_1), []). 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_euler_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k1_euler_1(A) = k1_card_1(a_1_1_euler_1(A)) ) ), file(euler_1,d1_euler_1), []). fof(d1_euler_2,axiom,( ! [A] : ( m1_trees_4(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m1_trees_4(C,k1_numbers,k5_numbers) => ( C = k2_euler_2(A,B) <=> ( k3_finseq_1(C) = k3_finseq_1(A) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k4_finseq_1(A)) => k3_wsierp_1(C,D) = k4_nat_1(k3_wsierp_1(A,D),B) ) ) ) ) ) ) ) ), file(euler_2,d1_euler_2), []). fof(d1_rfinseq,axiom,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ( r1_rfinseq(A,B) <=> ! [C] : k1_card_1(k10_relat_1(A,k1_tarski(C))) = k1_card_1(k10_relat_1(B,k1_tarski(C))) ) ) ) ), file(rfinseq,d1_rfinseq), []). fof(d2_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( C = k4_nat_1(A,B) <=> ~ ( ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ~ ( A = k2_xcmplx_0(k3_xcmplx_0(B,D),C) & ~ r1_xreal_0(B,C) ) ) & ~ ( C = 0 & B = 0 ) ) ) ) ) ) ), file(nat_1,d2_nat_1), []). 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_tarski,axiom,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), []). 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(d6_int_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r2_int_2(A,B) <=> k6_nat_1(A,B) = 1 ) ) ) ), file(int_2,d6_int_2), []). fof(d8_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v2_funct_1(A) <=> ! [B,C] : ( ( r2_hidden(B,k1_relat_1(A)) & r2_hidden(C,k1_relat_1(A)) & k1_funct_1(A,B) = k1_funct_1(A,C) ) => B = C ) ) ) ), file(funct_1,d8_funct_1), []). fof(d8_int_1,axiom,( ! [A] : ( v1_int_1(A) => ! [B] : ( v1_int_1(B) => ( ( B != 0 => k6_int_1(A,B) = k6_xcmplx_0(A,k3_xcmplx_0(k5_int_1(A,B),B)) ) & ( B = 0 => k6_int_1(A,B) = 0 ) ) ) ) ), file(int_1,d8_int_1), []). fof(d9_int_1,axiom,( ! [A] : ( v1_int_1(A) => ! [B] : ( v1_int_1(B) => ( r2_int_1(A,B) <=> ? [C] : ( v1_int_1(C) & B = k3_xcmplx_0(A,C) ) ) ) ) ), file(int_1,d9_int_1), []). fof(dt_k10_relat_1,axiom,( $true ), file(relat_1,k10_relat_1), []). fof(dt_k10_wsierp_1,axiom,( ! [A] : ( m1_finseq_1(A,k5_numbers) => m2_subset_1(k10_wsierp_1(A),k1_numbers,k5_numbers) ) ), file(wsierp_1,k10_wsierp_1), []). fof(dt_k11_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k11_binop_2(A,B),k1_numbers) ) ), file(binop_2,k11_binop_2), []). 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_k14_finseq_1,axiom,( ! [A] : m2_finseq_1(k14_finseq_1(A),k5_numbers) ), file(finseq_1,k14_finseq_1), []). 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_k16_rvsum_1,axiom,( ! [A] : ( m1_finseq_1(A,k1_numbers) => m1_subset_1(k16_rvsum_1(A),k1_numbers) ) ), file(rvsum_1,k16_rvsum_1), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). fof(dt_k1_euler_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m2_subset_1(k1_euler_1(A),k1_numbers,k5_numbers) ) ), file(euler_1,k1_euler_1), []). fof(dt_k1_euler_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_finseq_1(B,k5_numbers) ) => m1_trees_4(k1_euler_2(A,B),k1_numbers,k5_numbers) ) ), file(euler_2,k1_euler_2), []). fof(dt_k1_finseq_1,axiom,( $true ), file(finseq_1,k1_finseq_1), []). fof(dt_k1_finseq_3,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(k1_finseq_3(A,B)) & v1_funct_1(k1_finseq_3(A,B)) & v1_finseq_1(k1_finseq_3(A,B)) ) ) ), file(finseq_3,k1_finseq_3), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). 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_relat_1,axiom,( $true ), file(relat_1,k1_relat_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_euler_2,axiom,( ! [A,B] : ( ( m1_finseq_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m1_trees_4(k2_euler_2(A,B),k1_numbers,k5_numbers) ) ), file(euler_2,k2_euler_2), []). 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_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k2_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k2_nat_1), []). fof(dt_k2_newton,axiom,( $true ), file(newton,k2_newton), []). fof(dt_k2_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), []). fof(dt_k2_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) ) => m1_subset_1(k2_seq_1(A,B,C,D),k1_numbers) ) ), file(seq_1,k2_seq_1), []). 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_euler_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k3_euler_2(A,B),k1_numbers,k5_numbers) ) ), file(euler_2,k3_euler_2), []). 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_int_2,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => v1_int_1(k3_int_2(A,B)) ) ), file(int_2,k3_int_2), []). fof(dt_k3_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => m2_subset_1(k3_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k3_nat_1), []). fof(dt_k3_wsierp_1,axiom,( ! [A,B] : ( ( m1_finseq_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k3_wsierp_1(A,B),k1_numbers,k5_numbers) ) ), file(wsierp_1,k3_wsierp_1), []). fof(dt_k3_xcmplx_0,axiom,( $true ), file(xcmplx_0,k3_xcmplx_0), []). fof(dt_k4_card_1,axiom,( ! [A] : ( v1_finset_1(A) => m2_subset_1(k4_card_1(A),k1_numbers,k5_numbers) ) ), file(card_1,k4_card_1), []). 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_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => m2_subset_1(k4_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k4_nat_1), []). fof(dt_k4_wsierp_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) & m1_finseq_1(C,B) & m1_finseq_1(D,B) ) => m1_trees_4(k4_wsierp_1(A,B,C,D),A,B) ) ), file(wsierp_1,k4_wsierp_1), []). fof(dt_k5_finseq_1,axiom,( $true ), file(finseq_1,k5_finseq_1), []). fof(dt_k5_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => v1_int_1(k5_int_1(A,B)) ) ), file(int_1,k5_int_1), []). 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_k6_finseq_1,axiom,( ! [A] : ( v1_xboole_0(k6_finseq_1(A)) & m2_finseq_1(k6_finseq_1(A),A) ) ), file(finseq_1,k6_finseq_1), []). fof(dt_k6_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => v1_int_1(k6_int_1(A,B)) ) ), file(int_1,k6_int_1), []). fof(dt_k6_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k6_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k6_nat_1), []). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), []). 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_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_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_k9_rvsum_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & m1_finseq_1(B,k1_numbers) ) => m2_finseq_1(k9_rvsum_1(A,B),k1_numbers) ) ), file(rvsum_1,k9_rvsum_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_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_m1_trees_4,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m1_trees_4(C,A,B) => m2_finseq_1(C,A) ) ) ), file(trees_4,m1_trees_4), []). 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_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(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_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_m1_trees_4,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m1_trees_4(C,A,B) ) ), file(trees_4,m1_trees_4), []). fof(existence_m2_finseq_1,axiom,( ! [A] : ? [B] : m2_finseq_1(B,A) ), file(finseq_1,m2_finseq_1), []). 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(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(fc15_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc15_xreal_0), []). fof(fc16_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc16_xreal_0), []). 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(fc17_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v2_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc17_xreal_0), []). fof(fc18_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) => v1_finset_1(k10_relat_1(A,B)) ) ), file(finseq_1,fc18_finseq_1), []). fof(fc18_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) & v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc18_xreal_0), []). fof(fc19_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v2_xreal_0(k6_xcmplx_0(A,B)) & v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc19_xreal_0), []). fof(fc1_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_finset_1(k1_finseq_1(A)) ) ), file(finseq_1,fc1_finseq_1), []). fof(fc1_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_int_1(k2_xcmplx_0(A,B)) ) ) ), file(int_1,fc1_int_1), []). fof(fc1_numbers,axiom,( ~ v1_xboole_0(k1_numbers) ), file(numbers,fc1_numbers), []). 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(fc20_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v2_xreal_0(k6_xcmplx_0(B,A)) & ~ v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc20_xreal_0), []). fof(fc21_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v2_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc21_xreal_0), []). fof(fc22_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & ~ v2_xreal_0(k3_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc22_xreal_0), []). fof(fc23_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc23_xreal_0), []). fof(fc24_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc24_xreal_0), []). 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(fc2_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & v1_int_1(k3_xcmplx_0(A,B)) ) ) ), file(int_1,fc2_int_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_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_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v1_relat_1(k7_relat_1(A,B)) & v1_funct_1(k7_relat_1(A,B)) ) ) ), file(funct_1,fc4_funct_1), []). fof(fc4_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc4_int_1), []). fof(fc4_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc4_xreal_0), []). fof(fc5_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc5_xreal_0), []). fof(fc6_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_int_1(k2_xcmplx_0(B,A)) ) ) ), file(int_1,fc6_int_1), []). fof(fc7_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & v1_int_1(k3_xcmplx_0(B,A)) ) ) ), file(int_1,fc7_int_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_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v1_int_1(k6_xcmplx_0(B,A)) ) ) ), file(int_1,fc8_int_1), []). 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_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc9_int_1), []). 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(fraenkel_a_1_0_euler_2,axiom,( ! [A,B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r2_hidden(A,a_1_0_euler_2(B)) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & A = C & r2_int_2(B,C) & r1_xreal_0(1,C) & r1_xreal_0(C,B) ) ) ) ), file(euler_2,a_1_0_euler_2), []). fof(fraenkel_a_1_1_euler_1,axiom,( ! [A,B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r2_hidden(A,a_1_1_euler_1(B)) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & A = C & r2_int_2(B,C) & r1_xreal_0(1,C) & r1_xreal_0(C,B) ) ) ) ), file(euler_1,a_1_1_euler_1), []). fof(fraenkel_a_2_0_euler_2,axiom,( ! [A,B,C] : ( ( m2_subset_1(B,k1_numbers,k5_numbers) & m2_subset_1(C,k1_numbers,k5_numbers) ) => ( r2_hidden(A,a_2_0_euler_2(B,C)) <=> ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & A = D & ? [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) & D = k4_nat_1(k2_nat_1(B,E),C) & r2_hidden(E,a_1_0_euler_2(C)) ) ) ) ) ), file(euler_2,a_2_0_euler_2), []). fof(idempotence_k6_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k6_nat_1(A,A) = A ) ), file(nat_1,k6_nat_1), []). fof(l18_euler_2,axiom,( ! [A] : ( m1_trees_4(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k10_wsierp_1(k4_wsierp_1(k1_numbers,k5_numbers,A,k12_finseq_1(k5_numbers,B))) = k2_nat_1(k10_wsierp_1(A),B) ) ) ), file(euler_2,l18_euler_2), []). fof(l1_euler_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k3_int_2(A,B) = k6_nat_1(A,B) ) ) ), file(euler_2,l1_euler_2), []). fof(l4_euler_2,axiom,( ! [A] : ( v1_int_1(A) => k3_int_2(1,A) = 1 ) ), file(euler_2,l4_euler_2), []). fof(l7_euler_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( v1_int_1(B) => ( ( r1_xreal_0(k6_xcmplx_0(1,A),B) & r1_xreal_0(B,k6_xcmplx_0(A,1)) & r2_int_1(A,B) ) => ( r1_xreal_0(A,1) | B = 0 ) ) ) ) ), file(euler_2,l7_euler_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_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), []). fof(rc1_int_1,axiom,( ? [A] : ( m1_subset_1(A,k1_numbers) & v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) ) ), file(int_1,rc1_int_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_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), []). fof(rc2_int_1,axiom,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_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_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_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_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc4_funct_1), []). 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_k10_wsierp_1,axiom,( ! [A] : ( m1_finseq_1(A,k5_numbers) => k10_wsierp_1(A) = k16_rvsum_1(A) ) ), file(wsierp_1,k10_wsierp_1), []). fof(redefinition_k11_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k11_binop_2(A,B) = k3_xcmplx_0(A,B) ) ), file(binop_2,k11_binop_2), []). 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_k1_euler_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_finseq_1(B,k5_numbers) ) => k1_euler_2(A,B) = k9_rvsum_1(A,B) ) ), file(euler_2,k1_euler_2), []). 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_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_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k2_nat_1(A,B) = k3_xcmplx_0(A,B) ) ), file(nat_1,k2_nat_1), []). fof(redefinition_k2_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) ) => k2_seq_1(A,B,C,D) = k1_funct_1(C,D) ) ), file(seq_1,k2_seq_1), []). fof(redefinition_k3_euler_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k3_euler_2(A,B) = k2_newton(A,B) ) ), file(euler_2,k3_euler_2), []). 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_k3_wsierp_1,axiom,( ! [A,B] : ( ( m1_finseq_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k3_wsierp_1(A,B) = k1_funct_1(A,B) ) ), file(wsierp_1,k3_wsierp_1), []). fof(redefinition_k4_card_1,axiom,( ! [A] : ( v1_finset_1(A) => k4_card_1(A) = k1_card_1(A) ) ), file(card_1,k4_card_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_wsierp_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) & m1_finseq_1(C,B) & m1_finseq_1(D,B) ) => k4_wsierp_1(A,B,C,D) = k7_finseq_1(C,D) ) ), file(wsierp_1,k4_wsierp_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k9_finseq_1,axiom,( ! [A] : k9_finseq_1(A) = k5_finseq_1(A) ), file(finseq_1,k9_finseq_1), []). fof(redefinition_m1_trees_4,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m1_trees_4(C,A,B) <=> m1_finseq_1(C,B) ) ) ), file(trees_4,m1_trees_4), []). 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_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_rfinseq,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => r1_rfinseq(A,A) ) ), file(rfinseq,r1_rfinseq), []). 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(reflexivity_r2_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => r2_int_1(A,A) ) ), file(int_1,r2_int_1), []). fof(s1_nat_1__e3_25_11__euler_2,axiom,( ! [A,B,C,D] : ( ( m2_subset_1(A,k1_numbers,k5_numbers) & m2_subset_1(B,k1_numbers,k5_numbers) & m1_trees_4(C,k1_numbers,k5_numbers) & m1_trees_4(D,k1_numbers,k5_numbers) ) => ( ( ( r1_xreal_0(0,k3_finseq_1(D)) => k4_nat_1(k10_wsierp_1(k16_finseq_1(k5_numbers,D,0)),B) = k4_nat_1(k2_nat_1(k3_euler_2(A,k3_finseq_1(k16_finseq_1(k5_numbers,D,0))),k10_wsierp_1(k16_finseq_1(k5_numbers,C,0))),B) ) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( ( r1_xreal_0(E,k3_finseq_1(D)) => k4_nat_1(k10_wsierp_1(k16_finseq_1(k5_numbers,D,E)),B) = k4_nat_1(k2_nat_1(k3_euler_2(A,k3_finseq_1(k16_finseq_1(k5_numbers,D,E))),k10_wsierp_1(k16_finseq_1(k5_numbers,C,E))),B) ) => ( r1_xreal_0(k1_nat_1(E,1),k3_finseq_1(D)) => k4_nat_1(k10_wsierp_1(k16_finseq_1(k5_numbers,D,k1_nat_1(E,1))),B) = k4_nat_1(k2_nat_1(k3_euler_2(A,k3_finseq_1(k16_finseq_1(k5_numbers,D,k1_nat_1(E,1)))),k10_wsierp_1(k16_finseq_1(k5_numbers,C,k1_nat_1(E,1)))),B) ) ) ) ) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( r1_xreal_0(E,k3_finseq_1(D)) => k4_nat_1(k10_wsierp_1(k16_finseq_1(k5_numbers,D,E)),B) = k4_nat_1(k2_nat_1(k3_euler_2(A,k3_finseq_1(k16_finseq_1(k5_numbers,D,E))),k10_wsierp_1(k16_finseq_1(k5_numbers,C,E))),B) ) ) ) ) ), file(euler_2,s1_nat_1__e3_25_11__euler_2), []). fof(s1_nat_1__e3_25_13__euler_2,axiom,( ! [A,B] : ( ( m2_subset_1(A,k1_numbers,k5_numbers) & m1_trees_4(B,k1_numbers,k5_numbers) ) => ( ( ( r1_xreal_0(0,k3_finseq_1(B)) => r2_int_2(k10_wsierp_1(k16_finseq_1(k5_numbers,B,0)),A) ) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r1_xreal_0(C,k3_finseq_1(B)) => r2_int_2(k10_wsierp_1(k16_finseq_1(k5_numbers,B,C)),A) ) => ( r1_xreal_0(k1_nat_1(C,1),k3_finseq_1(B)) => r2_int_2(k10_wsierp_1(k16_finseq_1(k5_numbers,B,k1_nat_1(C,1))),A) ) ) ) ) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r1_xreal_0(C,k3_finseq_1(B)) => r2_int_2(k10_wsierp_1(k16_finseq_1(k5_numbers,B,C)),A) ) ) ) ) ), file(euler_2,s1_nat_1__e3_25_13__euler_2), []). 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(symmetry_r1_int_2,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( r1_int_2(A,B) => r1_int_2(B,A) ) ) ), file(int_2,r1_int_2), []). fof(symmetry_r1_rfinseq,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => ( r1_rfinseq(A,B) => r1_rfinseq(B,A) ) ) ), file(rfinseq,r1_rfinseq), []). fof(symmetry_r2_int_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => ( r2_int_2(A,B) => r2_int_2(B,A) ) ) ), file(int_2,r2_int_2), []). fof(t101_newton,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( k5_int_1(A,B) = k3_nat_1(A,B) & k6_int_1(A,B) = k4_nat_1(A,B) ) ) ) ), file(newton,t101_newton), []). fof(t11_euler_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r2_int_2(A,B) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ? [D] : ( v1_int_1(D) & ? [E] : ( v1_int_1(E) & k2_xcmplx_0(k3_xcmplx_0(D,A),k3_xcmplx_0(E,B)) = C ) ) ) ) ) ) ), file(euler_1,t11_euler_1), []). fof(t11_euler_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k4_nat_1(k2_nat_1(A,B),C) = k4_nat_1(k2_nat_1(k4_nat_1(A,C),k4_nat_1(B,C)),C) ) ) ) ), file(euler_2,t11_euler_2), []). fof(t11_newton,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v1_xreal_0(B) => k2_newton(B,k2_xcmplx_0(A,1)) = k3_xcmplx_0(k2_newton(B,A),B) ) ) ), file(newton,t11_newton), []). fof(t121_finseq_3,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B,C] : ( m2_finseq_1(C,B) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r1_xreal_0(D,A) => k1_funct_1(k16_finseq_1(B,C,A),D) = k1_funct_1(C,D) ) ) ) ) ), file(finseq_3,t121_finseq_3), []). fof(t124_rvsum_1,axiom,( k16_rvsum_1(k6_finseq_1(k1_numbers)) = 1 ), file(rvsum_1,t124_rvsum_1), []). fof(t12_prepower,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v4_ordinal2(B) => ~ ( 0 != A & 0 = k2_newton(A,B) ) ) ) ), file(prepower,t12_prepower), []). fof(t133_rvsum_1,axiom,( ! [A] : ( m2_finseq_1(A,k1_numbers) => ( ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & r2_hidden(B,k4_finseq_1(A)) & k2_seq_1(k5_numbers,k1_numbers,A,B) = 0 ) <=> k16_rvsum_1(A) = 0 ) ) ), file(rvsum_1,t133_rvsum_1), []). fof(t13_euler_1,axiom,( ! [A] : ( v1_int_1(A) => ! [B] : ( v1_int_1(B) => ! [C] : ( v1_int_1(C) => k6_int_1(k2_xcmplx_0(A,k3_xcmplx_0(B,C)),C) = k6_int_1(A,C) ) ) ) ), file(euler_1,t13_euler_1), []). fof(t13_finset_1,axiom,( ! [A,B] : ( ( r1_tarski(A,B) & v1_finset_1(B) ) => v1_finset_1(A) ) ), file(finset_1,t13_finset_1), []). fof(t142_funct_1,axiom,( ! [A,B] : ( v1_relat_1(B) => ( r2_hidden(A,k2_relat_1(B)) <=> k10_relat_1(B,k1_tarski(A)) != k1_xboole_0 ) ) ), file(funct_1,t142_funct_1), []). fof(t15_euler_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r2_int_2(A,C) & r2_int_2(B,C) ) => ( A = 0 | B = 0 | C = 0 | r2_int_2(k2_nat_1(A,B),C) ) ) ) ) ) ), file(euler_1,t15_euler_1), []). fof(t16_euler_1,axiom,( ! [A] : ( v1_int_1(A) => ! [B] : ( v1_int_1(B) => ! [C] : ( v1_int_1(C) => ~ ( A != 0 & B != 0 & ~ r1_xreal_0(C,0) & k3_int_2(k3_xcmplx_0(C,A),k3_xcmplx_0(C,B)) != k3_xcmplx_0(C,k3_int_2(A,B)) ) ) ) ) ), file(euler_1,t16_euler_1), []). fof(t16_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( r1_xreal_0(0,A) => r2_hidden(A,k5_numbers) ) ) ), file(int_1,t16_int_1), []). fof(t16_rfinseq,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) ) => ( r1_rfinseq(A,B) => ( k3_finseq_1(A) = k3_finseq_1(B) & k4_finseq_1(A) = k4_finseq_1(B) ) ) ) ) ), file(rfinseq,t16_rfinseq), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). fof(t1_euler_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r1_int_2(A,B) <=> r2_int_2(A,B) ) ) ) ), file(euler_2,t1_euler_2), []). 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(t1_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,A) ) => A = B ) ) ) ), file(xreal_1,t1_xreal_1), []). fof(t20_rfinseq,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( ( k3_finseq_1(B) = k1_nat_1(C,1) & D = k1_funct_1(B,k1_nat_1(C,1)) ) => B = k7_finseq_1(k16_finseq_1(A,B,C),k9_finseq_1(D)) ) ) ) ) ), file(rfinseq,t20_rfinseq), []). fof(t23_finseq_2,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) ) => ( ( B = k7_relat_1(C,k2_finseq_1(A)) & r1_xreal_0(k3_finseq_1(C),A) ) => C = B ) ) ) ) ), file(finseq_2,t23_finseq_2), []). fof(t25_euler_2,axiom,( ! [A] : ( m1_trees_4(A,k1_numbers,k5_numbers) => ! [B] : ( m1_trees_4(B,k1_numbers,k5_numbers) => ( r1_rfinseq(A,B) => k10_wsierp_1(A) = k10_wsierp_1(B) ) ) ) ), file(euler_2,t25_euler_2), []). fof(t27_euler_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( k4_nat_1(k2_nat_1(A,C),B) = k4_nat_1(C,B) & r2_int_2(B,C) ) => ( A = 0 | r1_xreal_0(B,1) | C = 0 | k4_nat_1(A,B) = 1 ) ) ) ) ) ), file(euler_2,t27_euler_2), []). 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(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_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,0) = 0 ) ), file(arithm,t2_arithm), []). fof(t2_euler_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r2_int_2(A,A) <=> A = 1 ) ) ), file(euler_1,t2_euler_1), []). 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_tarski,axiom,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), []). 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(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(t39_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( ~ r1_xreal_0(1,A) => A = 0 ) ) ), file(nat_1,t39_nat_1), []). fof(t3_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(1,A) = A ) ), file(arithm,t3_arithm), []). 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(t40_int_2,axiom,( ! [A] : ( v1_int_1(A) => ! [B] : ( v1_int_1(B) => ! [C] : ( v1_int_1(C) => ( ( r2_int_1(A,k3_xcmplx_0(B,C)) & r1_int_2(B,A) ) => r2_int_1(A,C) ) ) ) ) ), file(int_2,t40_int_2), []). fof(t41_int_2,axiom,( ! [A] : ( v1_int_1(A) => ! [B] : ( v1_int_1(B) => ! [C] : ( v1_int_1(C) => ( ( r1_int_2(A,B) & r1_int_2(C,B) ) => r1_int_2(k3_xcmplx_0(A,C),B) ) ) ) ) ), file(int_2,t41_int_2), []). fof(t44_finseq_3,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v1_finset_1(B) => ( r1_tarski(B,k2_finseq_1(A)) => k3_finseq_1(k14_finseq_1(B)) = k4_card_1(B) ) ) ) ), file(finseq_3,t44_finseq_3), []). fof(t46_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(A,k4_nat_1(B,A)) ) ) ) ), file(nat_1,t46_nat_1), []). fof(t4_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), []). 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(t5_euler_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r2_int_2(A,C) & r2_int_2(B,C) ) => ( A = 0 | B = 0 | C = 0 | r2_int_2(C,k4_nat_1(k2_nat_1(A,B),C)) ) ) ) ) ) ), file(euler_2,t5_euler_2), []). 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(t64_newton,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k6_nat_1(A,1) = 1 ) ), file(newton,t64_newton), []). fof(t66_finseq_3,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : k3_finseq_1(k1_finseq_3(A,B)) = k6_xcmplx_0(k3_finseq_1(A),k4_card_1(k10_relat_1(A,B))) ) ), file(finseq_3,t66_finseq_3), []). fof(t66_rvsum_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m2_finseq_1(C,k1_numbers) => k2_seq_1(k5_numbers,k1_numbers,k9_rvsum_1(B,C),A) = k11_binop_2(B,k2_seq_1(k5_numbers,k1_numbers,C,A)) ) ) ) ), file(rvsum_1,t66_rvsum_1), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_euler_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( k6_nat_1(A,B) = 1 => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k6_nat_1(k2_nat_1(A,C),k2_nat_1(B,C)) = C ) ) ) ) ), file(euler_1,t6_euler_1), []). fof(t6_euler_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ( r2_int_2(A,C) & r2_int_2(D,A) & C = k4_nat_1(k2_nat_1(D,B),A) ) => ( r1_xreal_0(A,1) | B = 0 | r2_int_2(A,B) ) ) ) ) ) ) ), file(euler_2,t6_euler_2), []). fof(t6_newton,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( m2_finseq_1(B,k1_numbers) => k3_finseq_1(k9_rvsum_1(A,B)) = k3_finseq_1(B) ) ) ), file(newton,t6_newton), []). 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(t76_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ~ r1_xreal_0(A,B) => k4_nat_1(B,A) = B ) ) ) ), file(nat_1,t76_nat_1), []). fof(t77_newton,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(B,0) => k4_nat_1(A,B) = k6_xcmplx_0(A,k2_nat_1(B,k3_nat_1(A,B))) ) ) ) ), file(newton,t77_newton), []). fof(t78_newton,axiom,( ! [A] : ( v1_int_1(A) => ! [B] : ( v1_int_1(B) => ( r1_xreal_0(0,A) => r1_xreal_0(0,k6_int_1(B,A)) ) ) ) ), file(newton,t78_newton), []). 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(t79_newton,axiom,( ! [A] : ( v1_int_1(A) => ! [B] : ( v1_int_1(B) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(A,k6_int_1(B,A)) ) ) ) ), file(newton,t79_newton), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_euler_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k4_nat_1(k4_nat_1(A,B),B) = k4_nat_1(A,B) ) ) ), file(euler_2,t7_euler_2), []). 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(t80_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ! [C] : ( v4_ordinal2(C) => ( r1_xreal_0(C,k3_finseq_1(B)) => k3_finseq_1(k16_finseq_1(A,B,C)) = C ) ) ) ), file(finseq_1,t80_finseq_1), []). fof(t80_finseq_5,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r1_xreal_0(C,D) => ( k16_finseq_1(A,k16_finseq_1(A,B,C),D) = k16_finseq_1(A,B,C) & k16_finseq_1(A,k16_finseq_1(A,B,D),C) = k16_finseq_1(A,B,C) ) ) ) ) ) ) ), file(finseq_5,t80_finseq_5), []). fof(t80_newton,axiom,( ! [A] : ( v1_int_1(A) => ! [B] : ( v1_int_1(B) => ( A != 0 => B = k2_xcmplx_0(k3_xcmplx_0(k5_int_1(B,A),A),k6_int_1(B,A)) ) ) ) ), file(newton,t80_newton), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). 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(t8_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(k2_xcmplx_0(A,C),k2_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t8_xreal_1), []). fof(t92_real_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( ( r1_xreal_0(A,B) & r1_xreal_0(C,D) ) => r1_xreal_0(k6_xcmplx_0(A,D),k6_xcmplx_0(B,C)) ) & ~ ( ( ( ~ r1_xreal_0(B,A) & r1_xreal_0(C,D) ) | ( r1_xreal_0(A,B) & ~ r1_xreal_0(D,C) ) ) & r1_xreal_0(k6_xcmplx_0(B,C),k6_xcmplx_0(A,D)) ) ) ) ) ) ) ), file(real_1,t92_real_1), []). fof(t97_finseq_3,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v2_funct_1(A) & r2_hidden(B,k2_relat_1(A)) ) => k3_finseq_1(k1_finseq_3(A,k1_tarski(B))) = k6_xcmplx_0(k3_finseq_1(A),1) ) ) ), file(finseq_3,t97_finseq_3), []). fof(t99_finseq_3,axiom,( ! [A,B] : ( v4_ordinal2(B) => ( r1_tarski(A,k2_finseq_1(B)) => v2_funct_1(k14_finseq_1(A)) ) ) ), file(finseq_3,t99_finseq_3), []). fof(t9_newton,axiom,( ! [A] : ( v1_xreal_0(A) => k2_newton(A,0) = 1 ) ), file(newton,t9_newton), []).