fof(t32_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k7_cqc_lang)) => ( r5_calcul_1(A,B) => r7_calcul_1(B,A) ) ) ) ), inference(mizar_proof,[status(thm)],[d10_calcul_1,d9_calcul_1,d12_calcul_1,t2_xreal_1,t29_nat_1,t2_xreal_1,l48_calcul_1,d8_calcul_1,d7_calcul_1,t15_calcul_1,d7_calcul_1,t30_calcul_1,d7_calcul_1,t38_nat_1,t2_xreal_1,t16_calcul_1,d7_calcul_1,t38_nat_1,t2_xreal_1,t2_xreal_1,t2_xreal_1,t5_calcul_1,t18_calcul_1,d7_calcul_1,t38_nat_1,t2_xreal_1,t2_xreal_1,t2_xreal_1,t5_calcul_1,t19_calcul_1,d7_calcul_1,t38_nat_1,t2_xreal_1,t2_xreal_1,t2_xreal_1,t5_calcul_1,t20_calcul_1,d7_calcul_1,t38_nat_1,t2_xreal_1,t5_calcul_1,t21_calcul_1,d7_calcul_1,t38_nat_1,t2_xreal_1,t5_calcul_1,t22_calcul_1,d7_calcul_1,t38_nat_1,t2_xreal_1,t5_calcul_1,t25_calcul_1,d7_calcul_1,t38_nat_1,t2_xreal_1,t5_calcul_1,t29_calcul_1,t31_calcul_1,t26_nat_1,s1_nat_1,d8_calcul_1,l47_calcul_1,d11_calcul_1,d11_calcul_1,d14_calcul_1,d15_calcul_1]), [file(calcul_1,t32_calcul_1),interesting(1.00)]). fof(t63_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => v1_finset_1(k3_calcul_1(A)) ) ), inference(mizar_proof,[status(thm)],[t12_funct_1,s2_funct_1,d2_finseq_1,d1_finset_1,t46_relat_1,t47_relat_1,d5_funct_1,t12_funct_1,t47_relat_1,t2_tarski,d1_finset_1,t20_cqc_sim1,t25_finset_1,t62_calcul_1]), [file(calcul_1,t63_calcul_1),interesting(0.92)]). fof(t68_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k15_cqc_lang(B,A)))) <=> r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k10_cqc_lang(k16_cqc_lang(B,k10_cqc_lang(A)))))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t66_calcul_1,t54_calcul_1,d5_qc_lang2,d5_qc_lang2,t55_calcul_1,t67_calcul_1]), [file(calcul_1,t68_calcul_1),interesting(0.91)]). fof(t54_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,B,k12_finseq_1(k7_cqc_lang,A))) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,B,k12_finseq_1(k7_cqc_lang,k10_cqc_lang(k10_cqc_lang(A))))) ) ) ) ), inference(mizar_proof,[status(thm)],[t5_calcul_1,t45_finseq_1,d9_finseq_1,t14_calcul_1,t61_finseq_1,t35_finseq_1,t8_xreal_1,t29_nat_1,t27_finseq_3,d3_calcul_1,t33_calcul_1,t5_calcul_1,t45_finseq_1,d9_finseq_1,t14_calcul_1,t61_finseq_1,t35_finseq_1,t10_calcul_1,d3_calcul_1,t33_calcul_1,t10_calcul_1,t3_calcul_1,t5_calcul_1,t44_calcul_1,t5_calcul_1,t45_finseq_1,d9_finseq_1,t14_calcul_1,t61_finseq_1,t35_finseq_1,t10_calcul_1,d3_calcul_1,t33_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t9_calcul_1,t37_calcul_1,t10_calcul_1,t45_calcul_1,t5_calcul_1]), [file(calcul_1,t54_calcul_1),interesting(0.87)]). fof(t33_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ( r1_calcul_1(k7_cqc_lang,k2_calcul_1(k7_cqc_lang,A),k1_calcul_1(k7_cqc_lang,A)) => r4_calcul_1(A) ) ) ), inference(mizar_proof,[status(thm)],[t55_finseq_1,d1_tarski,d6_calcul_1,t43_cqc_the1,t106_zfmisc_1,d3_tarski,d4_finseq_1,t56_finseq_1,t1_xreal_1,t57_finseq_1,t7_mcart_1,d7_calcul_1,d8_calcul_1,t57_finseq_1,t57_finseq_1,t7_mcart_1,d9_calcul_1]), [file(calcul_1,t33_calcul_1),interesting(0.82)]). fof(t55_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,B,k12_finseq_1(k7_cqc_lang,k10_cqc_lang(k10_cqc_lang(A))))) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,B,k12_finseq_1(k7_cqc_lang,A))) ) ) ) ), inference(mizar_proof,[status(thm)],[t5_calcul_1,t45_finseq_1,d9_finseq_1,t14_calcul_1,t61_finseq_1,t35_finseq_1,t10_calcul_1,d3_calcul_1,t33_calcul_1,t5_calcul_1,t45_finseq_1,d9_finseq_1,t14_calcul_1,t61_finseq_1,t35_finseq_1,t8_xreal_1,t29_nat_1,t27_finseq_3,d3_calcul_1,t33_calcul_1,t10_calcul_1,t3_calcul_1,t5_calcul_1,t44_calcul_1,t5_calcul_1,t45_finseq_1,d9_finseq_1,t14_calcul_1,t61_finseq_1,t35_finseq_1,t10_calcul_1,d3_calcul_1,t33_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t9_calcul_1,t37_calcul_1,t10_calcul_1,t45_calcul_1,t5_calcul_1]), [file(calcul_1,t55_calcul_1),interesting(0.81)]). fof(t65_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ~ ! [B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => r2_hidden(B,k3_calcul_1(A)) ) ) ), inference(mizar_proof,[status(thm)],[t63_calcul_1,t64_calcul_1,t2_tarski]), [file(calcul_1,t65_calcul_1),interesting(0.81)]). fof(t64_calcul_1,theorem, ( k1_card_1(k2_qc_lang1) = k3_card_1(0) & ~ v1_finset_1(k2_qc_lang1) ), inference(mizar_proof,[status(thm)],[t11_card_2,d2_qc_lang1,t38_card_1,t15_card_4,t81_card_4,t1_card_4]), [file(calcul_1,t64_calcul_1),interesting(0.78)]). fof(t62_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => k3_calcul_1(A) = k3_tarski(a_1_0_calcul_1(A)) ) ), inference(mizar_proof,[status(thm)],[d5_calcul_1,d4_tarski,d4_tarski,d5_calcul_1,t2_tarski]), [file(calcul_1,t62_calcul_1),interesting(0.76)]). fof(t67_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k15_cqc_lang(B,k10_cqc_lang(k10_cqc_lang(A)))))) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k15_cqc_lang(B,A)))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t5_calcul_1,t65_calcul_1,t42_calcul_1,t57_calcul_1,t57_calcul_1,t55_calcul_1,t59_calcul_1,d2_xboole_0,t5_calcul_1,t60_calcul_1,t43_calcul_1,t5_calcul_1]), [file(calcul_1,t67_calcul_1),interesting(0.75)]). fof(t66_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k15_cqc_lang(B,A)))) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k15_cqc_lang(B,k10_cqc_lang(k10_cqc_lang(A)))))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t5_calcul_1,t65_calcul_1,t42_calcul_1,t54_calcul_1,t57_calcul_1,t57_calcul_1,t59_calcul_1,d2_xboole_0,t5_calcul_1,t60_calcul_1,t16_qc_lang3,t11_qc_lang3,t11_qc_lang3,t16_qc_lang3,t43_calcul_1,t5_calcul_1]), [file(calcul_1,t66_calcul_1),interesting(0.74)]). fof(t51_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,A))) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k13_cqc_lang(A,B)))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t5_calcul_1,t59_finseq_1,t56_finseq_1,t35_finseq_1,t10_calcul_1,d3_calcul_1,t33_calcul_1,t40_calcul_1,t49_calcul_1,d3_qc_lang2,t10_calcul_1,t45_calcul_1,t5_calcul_1]), [file(calcul_1,t51_calcul_1),interesting(0.72)]). fof(t52_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,A))) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k13_cqc_lang(B,A)))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t5_calcul_1,t59_finseq_1,t56_finseq_1,t35_finseq_1,t10_calcul_1,d3_calcul_1,t33_calcul_1,t41_calcul_1,t49_calcul_1,d3_qc_lang2,t10_calcul_1,t45_calcul_1,t5_calcul_1]), [file(calcul_1,t52_calcul_1),interesting(0.72)]). fof(t60_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => k3_calcul_1(k12_finseq_1(k7_cqc_lang,A)) = k24_qc_lang1(A) ) ), inference(mizar_proof,[status(thm)],[d5_calcul_1,t55_finseq_1,t4_finseq_1,d1_tarski,t57_finseq_1,t4_finseq_1,d1_tarski,t55_finseq_1,t57_finseq_1,d5_calcul_1,t2_tarski]), [file(calcul_1,t60_calcul_1),interesting(0.72)]). fof(t59_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => k3_calcul_1(k8_finseq_1(k7_cqc_lang,A,B)) = k4_subset_1(k2_qc_lang1,k3_calcul_1(A),k3_calcul_1(B)) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d5_calcul_1,d7_finseq_1,d5_calcul_1,t7_xboole_1,d7_finseq_1,d5_calcul_1,t7_xboole_1,t38_finseq_1,d10_xboole_0,d3_tarski,d5_calcul_1,t39_finseq_1,d7_finseq_1,d5_calcul_1,d5_calcul_1,t41_finseq_1,d7_finseq_1,d5_calcul_1,d2_xboole_0]), [file(calcul_1,t59_calcul_1),interesting(0.71)]). fof(t45_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => ( ( r1_xreal_0(1,k3_finseq_1(B)) & r4_calcul_1(B) & r4_calcul_1(k8_finseq_1(k7_cqc_lang,B,k12_finseq_1(k7_cqc_lang,A))) ) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,B),k12_finseq_1(k7_cqc_lang,A))) ) ) ) ), inference(mizar_proof,[status(thm)],[t5_calcul_1,t5_calcul_1,t8_calcul_1,t36_calcul_1,t59_finseq_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t19_finseq_2,t10_calcul_1,d3_calcul_1,t33_calcul_1,t5_calcul_1,t5_calcul_1,t44_calcul_1,t8_xreal_1,t56_finseq_1,t35_finseq_1,t38_nat_1,t5_calcul_1,t10_calcul_1,t8_xreal_1,t56_finseq_1,t35_finseq_1,t38_nat_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t37_calcul_1,t5_calcul_1,t5_calcul_1]), [file(calcul_1,t45_calcul_1),interesting(0.67)]). fof(t15_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ( r1_calcul_1(k7_cqc_lang,k2_calcul_1(k7_cqc_lang,A),k1_calcul_1(k7_cqc_lang,A)) => r10_calcul_1(k1_calcul_1(k7_cqc_lang,A),k2_calcul_1(k7_cqc_lang,A)) ) ) ), inference(mizar_proof,[status(thm)],[d3_calcul_1,d15_calcul_1,d14_calcul_1,t12_funct_1,d11_calcul_1]), [file(calcul_1,t15_calcul_1),interesting(0.66)]). fof(t8_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => r2_calcul_1(A,k8_finseq_1(k7_cqc_lang,A,B)) ) ) ), inference(mizar_proof,[status(thm)],[d4_calcul_1,t19_finseq_1,d3_finseq_1,t33_finseq_1,t7_calcul_1]), [file(calcul_1,t8_calcul_1),interesting(0.66)]). fof(l47_calcul_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( A != k1_xboole_0 => r1_xreal_0(1,k3_finseq_1(A)) ) ) ), inference(mizar_proof,[status(thm)],[t8_graph_5]), [file(calcul_1,l47_calcul_1),interesting(0.64)]). fof(t4_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ~ ( ~ r1_xreal_0(k3_finseq_1(A),1) & r1_xreal_0(k3_finseq_1(k1_calcul_1(k7_cqc_lang,A)),0) ) ) ), inference(mizar_proof,[status(thm)],[t2_calcul_1,t38_nat_1]), [file(calcul_1,t4_calcul_1),interesting(0.61)]). fof(t57_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => ! [C] : ( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1) => k4_substut2(k10_cqc_lang(A),B,C) = k10_cqc_lang(k4_substut2(A,B,C)) ) ) ) ), inference(mizar_proof,[status(thm)],[d2_substut2,t7_mcart_1,d20_substut1,t29_substut1,d2_substut2]), [file(calcul_1,t57_calcul_1),interesting(0.58)]). fof(t58_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( ? [D] : ( m2_subset_1(D,k1_qc_lang1,k2_qc_lang1) & r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k4_substut2(A,B,D)))) ) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k16_cqc_lang(B,A)))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t5_calcul_1,t59_finseq_1,t56_finseq_1,t35_finseq_1,t10_calcul_1,d3_calcul_1,t33_calcul_1,t42_calcul_1,t57_calcul_1,t49_calcul_1,d5_qc_lang2,t10_calcul_1,t45_calcul_1,t5_calcul_1]), [file(calcul_1,t58_calcul_1),interesting(0.57)]). fof(t40_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( ( k11_cqc_lang(A,B) = k2_calcul_1(k7_cqc_lang,C) & r4_calcul_1(C) ) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,C),k12_finseq_1(k7_cqc_lang,A))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d9_calcul_1,d6_calcul_1,t43_cqc_the1,t55_finseq_1,d1_tarski,t106_zfmisc_1,d3_tarski,d4_finseq_1,d8_calcul_1,t48_finseq_1,t27_finseq_3,t27_finseq_3,d8_calcul_1,t34_calcul_1,t55_finseq_1,t4_finseq_1,d1_tarski,t57_finseq_1,t7_mcart_1,d7_finseq_1,t6_calcul_1,t38_nat_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t38_finseq_1,d8_calcul_1,t35_finseq_1,t56_finseq_1,t4_finseq_1,d1_tarski,t55_finseq_1,d7_finseq_1,t57_finseq_1,t7_mcart_1,d9_calcul_1]), [file(calcul_1,t40_calcul_1),interesting(0.56)]). fof(t41_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( ( k11_cqc_lang(A,B) = k2_calcul_1(k7_cqc_lang,C) & r4_calcul_1(C) ) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,C),k12_finseq_1(k7_cqc_lang,B))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d9_calcul_1,d6_calcul_1,t43_cqc_the1,t55_finseq_1,d1_tarski,t106_zfmisc_1,d3_tarski,d4_finseq_1,d8_calcul_1,t48_finseq_1,t27_finseq_3,t27_finseq_3,d8_calcul_1,t34_calcul_1,t55_finseq_1,t4_finseq_1,d1_tarski,t57_finseq_1,t7_mcart_1,d7_finseq_1,t6_calcul_1,t38_nat_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t38_finseq_1,d8_calcul_1,t35_finseq_1,t56_finseq_1,t4_finseq_1,d1_tarski,t55_finseq_1,d7_finseq_1,t57_finseq_1,t7_mcart_1,d9_calcul_1]), [file(calcul_1,t41_calcul_1),interesting(0.56)]). fof(t42_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => ! [C] : ( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1) => ! [D] : ( m2_finseq_1(D,k7_cqc_lang) => ( ( k2_calcul_1(k7_cqc_lang,D) = k15_cqc_lang(B,A) & r4_calcul_1(D) ) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,D),k12_finseq_1(k7_cqc_lang,k4_substut2(A,B,C)))) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d9_calcul_1,d6_calcul_1,t43_cqc_the1,t55_finseq_1,d1_tarski,t106_zfmisc_1,d3_tarski,d4_finseq_1,d8_calcul_1,t48_finseq_1,t27_finseq_3,t27_finseq_3,d8_calcul_1,t34_calcul_1,t55_finseq_1,t4_finseq_1,d1_tarski,t57_finseq_1,t7_mcart_1,d7_finseq_1,t6_calcul_1,t38_nat_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t38_finseq_1,d8_calcul_1,t35_finseq_1,t56_finseq_1,t4_finseq_1,d1_tarski,t55_finseq_1,d7_finseq_1,t57_finseq_1,t7_mcart_1,d9_calcul_1]), [file(calcul_1,t42_calcul_1),interesting(0.55)]). fof(t9_calcul_1,theorem,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ~ r1_xreal_0(k3_finseq_1(k7_finseq_1(k7_finseq_1(C,k9_finseq_1(A)),k9_finseq_1(B))),1) ) ), inference(mizar_proof,[status(thm)],[t35_finseq_1,t56_finseq_1,t35_finseq_1,t56_finseq_1,t29_nat_1,t38_nat_1]), [file(calcul_1,t9_calcul_1),interesting(0.52)]). fof(t34_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_1(B,k2_zfmisc_1(k4_calcul_1,k2_cqc_the1)) => ! [C] : ( m2_finseq_1(C,k2_zfmisc_1(k4_calcul_1,k2_cqc_the1)) => ( ( r1_xreal_0(1,A) & r1_xreal_0(A,k3_finseq_1(B)) ) => ( r3_calcul_1(B,A) <=> r3_calcul_1(k8_finseq_1(k2_zfmisc_1(k4_calcul_1,k2_cqc_the1),B,C),A) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t27_finseq_3,d7_finseq_1,t35_finseq_1,t29_nat_1,t2_xreal_1,d7_calcul_1,d7_calcul_1,d7_calcul_1,t2_xreal_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t2_xreal_1,t2_xreal_1,t3_finseq_1,d3_finseq_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t2_xreal_1,t2_xreal_1,t3_finseq_1,d3_finseq_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t2_xreal_1,t2_xreal_1,t3_finseq_1,d3_finseq_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t2_xreal_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t2_xreal_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t2_xreal_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t2_xreal_1,t27_finseq_3,d7_finseq_1,t31_calcul_1,d7_calcul_1,d7_calcul_1,d7_calcul_1,d7_calcul_1,t2_xreal_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t2_xreal_1,t2_xreal_1,t3_finseq_1,d3_finseq_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t2_xreal_1,t2_xreal_1,t3_finseq_1,d3_finseq_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t2_xreal_1,t2_xreal_1,t3_finseq_1,d3_finseq_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t2_xreal_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t2_xreal_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t2_xreal_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t2_xreal_1,t27_finseq_3,d7_finseq_1,t31_calcul_1,d7_calcul_1]), [file(calcul_1,t34_calcul_1),interesting(0.51)]). fof(t13_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => ( ~ r1_xreal_0(k3_finseq_1(A),0) => r2_calcul_1(A,k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,A),B),k12_finseq_1(k7_cqc_lang,k2_calcul_1(k7_cqc_lang,A)))) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_relat_1,t8_funct_1,t3_finseq_1,t45_finseq_1,t6_calcul_1,t2_xreal_1,t27_finseq_3,d1_tarski,t35_finseq_1,t35_finseq_1,t56_finseq_1,t29_nat_1,t27_finseq_3,d2_xboole_0,d3_tarski,t90_relat_1,t28_xboole_1,t12_calcul_1,t27_finseq_3,t2_calcul_1,t3_finseq_1,t2_tarski,t38_nat_1,t29_nat_1,t2_xreal_1,t3_finseq_1,t37_zfmisc_1,t3_finseq_1,t29_nat_1,t38_nat_1,d1_tarski,t2_xreal_1,t48_finseq_3,t50_finseq_3,t54_finseq_3,t54_finseq_2,d7_finseq_1,t56_finseq_2,t8_funct_1,d14_finseq_1,t23_funct_1,t45_finseq_1,t7_xboole_1,d3_finseq_1,t72_funct_1,t103_relat_1,t33_finseq_1,d3_finseq_1,t3_calcul_1,d7_finseq_1,t2_calcul_1,d2_calcul_1,t55_finseq_2,t59_finseq_1,t8_funct_1,d14_finseq_1,t23_funct_1,d1_tarski,t7_xboole_1,t72_funct_1,t35_finseq_1,t3_finseq_1,t55_finseq_1,d7_finseq_1,t57_finseq_1,t71_finseq_1,t46_relat_1,d14_finseq_1,t8_finseq_2,t8_funct_1,d4_calcul_1,d3_tarski]), [file(calcul_1,t13_calcul_1),interesting(0.50)]). fof(t17_calcul_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_valuat_1(B,A) => ! [C] : ( m2_fraenkel(C,k2_qc_lang1,A,k2_valuat_1(A)) => ! [D] : ( m2_finseq_1(D,k7_cqc_lang) => ( ~ r1_xreal_0(k3_finseq_1(D),0) => ( ( r9_calcul_1(k1_calcul_1(k7_cqc_lang,D),A,B,C) & r1_valuat_1(A,k2_calcul_1(k7_cqc_lang,D),B,C) ) <=> r9_calcul_1(D,A,B,C) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d14_calcul_1,d11_calcul_1,d14_calcul_1,t3_calcul_1,d2_xboole_0,d11_calcul_1,d1_tarski,d14_calcul_1,d11_calcul_1,t7_xboole_1,t3_calcul_1,d11_calcul_1,d14_calcul_1,d2_calcul_1,t38_nat_1,t27_finseq_3,t12_funct_1,d11_calcul_1]), [file(calcul_1,t17_calcul_1),interesting(0.46)]). fof(t26_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m2_fraenkel(C,k2_qc_lang1,B,k2_valuat_1(B)) => ! [D] : ( m1_subset_1(D,B) => ! [E] : ( r1_tarski(E,k2_qc_lang1) => ( r2_hidden(A,E) | k8_relset_1(k2_qc_lang1,B,k1_sublemma(B,C,k13_sublemma(B,D,A)),E) = k8_relset_1(k2_qc_lang1,B,C,E) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t64_sublemma,t64_sublemma,t49_sublemma,t68_funct_1,t68_funct_1,t9_funct_1]), [file(calcul_1,t26_calcul_1),interesting(0.46)]). fof(t23_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_valuat_1(C,B) => ! [D] : ( m2_fraenkel(D,k2_qc_lang1,B,k2_valuat_1(B)) => ! [E] : ( m1_subset_1(E,k1_substut1) => ( r1_sublemma(k2_substut2(A,E),B,D,C) <=> r1_valuat_1(B,A,C,D) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_sublemma,t7_mcart_1]), [file(calcul_1,t23_calcul_1),interesting(0.45)]). fof(l48_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_1(B,k2_zfmisc_1(k4_calcul_1,k2_cqc_the1)) => ( ( r1_xreal_0(1,A) & r1_xreal_0(A,k3_finseq_1(B)) ) => m2_finseq_1(k1_mcart_1(k1_funct_1(B,A)),k7_cqc_lang) ) ) ) ), inference(mizar_proof,[status(thm)],[t27_finseq_3,d5_funct_1,t10_mcart_1,d6_calcul_1]), [file(calcul_1,l48_calcul_1),interesting(0.44)]). fof(t35_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_1(B,k2_zfmisc_1(k4_calcul_1,k2_cqc_the1)) => ! [C] : ( m2_finseq_1(C,k2_zfmisc_1(k4_calcul_1,k2_cqc_the1)) => ( ( r1_xreal_0(1,A) & r1_xreal_0(A,k3_finseq_1(B)) & r3_calcul_1(B,A) ) => r3_calcul_1(k8_finseq_1(k2_zfmisc_1(k4_calcul_1,k2_cqc_the1),C,B),k1_nat_1(A,k3_finseq_1(C))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t27_finseq_3,d7_finseq_1,t8_xreal_1,t35_finseq_1,t37_nat_1,d7_calcul_1,d7_calcul_1,d7_calcul_1,t37_nat_1,t8_xreal_1,t2_xreal_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t37_nat_1,t8_xreal_1,t37_nat_1,t8_xreal_1,t2_xreal_1,t2_xreal_1,t3_finseq_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t37_nat_1,t8_xreal_1,t37_nat_1,t8_xreal_1,t2_xreal_1,t2_xreal_1,t3_finseq_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t37_nat_1,t8_xreal_1,t37_nat_1,t8_xreal_1,t2_xreal_1,t2_xreal_1,t3_finseq_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t37_nat_1,t8_xreal_1,t2_xreal_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t37_nat_1,t8_xreal_1,t2_xreal_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t37_nat_1,t8_xreal_1,t2_xreal_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t37_nat_1,t8_xreal_1,t2_xreal_1,t27_finseq_3,d7_finseq_1,t31_calcul_1,d7_calcul_1]), [file(calcul_1,t35_calcul_1),interesting(0.40)]). fof(t43_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => ! [C] : ( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1) => ! [D] : ( m2_finseq_1(D,k7_cqc_lang) => ( ( k2_calcul_1(k7_cqc_lang,D) = k4_substut2(A,B,C) & r4_calcul_1(D) ) => ( r2_hidden(C,k3_calcul_1(k1_calcul_1(k7_cqc_lang,D))) | r2_hidden(C,k24_qc_lang1(k15_cqc_lang(B,A))) | r4_calcul_1(k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,D),k12_finseq_1(k7_cqc_lang,k15_cqc_lang(B,A)))) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d9_calcul_1,d6_calcul_1,t43_cqc_the1,t55_finseq_1,d1_tarski,t106_zfmisc_1,d3_tarski,d4_finseq_1,d8_calcul_1,t48_finseq_1,t27_finseq_3,t27_finseq_3,d8_calcul_1,t34_calcul_1,t55_finseq_1,t4_finseq_1,d1_tarski,t57_finseq_1,t7_mcart_1,d7_finseq_1,t6_calcul_1,t38_nat_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t38_finseq_1,d8_calcul_1,t35_finseq_1,t56_finseq_1,t4_finseq_1,d1_tarski,t55_finseq_1,d7_finseq_1,t57_finseq_1,t7_mcart_1,d9_calcul_1]), [file(calcul_1,t43_calcul_1),interesting(0.37)]). fof(t3_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ( ~ r1_xreal_0(k3_finseq_1(A),0) => ( A = k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,A),k12_finseq_1(k7_cqc_lang,k2_calcul_1(k7_cqc_lang,A))) & k5_relset_1(k5_numbers,k7_cqc_lang,A) = k4_subset_1(k7_cqc_lang,k5_relset_1(k5_numbers,k7_cqc_lang,k1_calcul_1(k7_cqc_lang,A)),k6_domain_1(k7_cqc_lang,k2_calcul_1(k7_cqc_lang,A))) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_calcul_1,t56_finseq_1,t35_finseq_1,t3_finseq_1,t27_finseq_3,d1_calcul_1,d3_finseq_1,t72_funct_1,d7_finseq_1,t2_calcul_1,d2_calcul_1,t57_finseq_1,t3_finseq_1,t55_finseq_1,d7_finseq_1,t26_nat_1,t10_finseq_2,t44_finseq_1,t55_finseq_1,t10_finseq_2]), [file(calcul_1,t3_calcul_1),interesting(0.28)]). fof(t29_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => ! [C] : ( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1) => ! [D] : ( m2_finseq_1(D,k7_cqc_lang) => ( ( k2_calcul_1(k7_cqc_lang,D) = k4_substut2(A,B,C) & r10_calcul_1(k1_calcul_1(k7_cqc_lang,D),k2_calcul_1(k7_cqc_lang,D)) ) => ( r2_hidden(C,k3_calcul_1(k1_calcul_1(k7_cqc_lang,D))) | r2_hidden(C,k24_qc_lang1(k15_cqc_lang(B,A))) | r10_calcul_1(k1_calcul_1(k7_cqc_lang,D),k15_cqc_lang(B,A)) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d15_calcul_1,t26_calcul_1,t27_calcul_1,d15_calcul_1,t24_calcul_1,t50_sublemma,t28_calcul_1,t70_sublemma,t51_sublemma]), [file(calcul_1,t29_calcul_1),interesting(0.28)]). fof(t14_calcul_1,theorem,( ! [A,B,C] : ( m2_finseq_1(C,k7_cqc_lang) => ( r2_hidden(1,k1_relat_1(k10_finseq_1(A,B))) & r2_hidden(2,k1_relat_1(k10_finseq_1(A,B))) & k1_funct_1(k7_finseq_1(C,k10_finseq_1(A,B)),k1_nat_1(k3_finseq_1(C),1)) = A & k1_funct_1(k7_finseq_1(C,k10_finseq_1(A,B)),k1_nat_1(k3_finseq_1(C),2)) = B ) ) ), inference(mizar_proof,[status(thm)],[t61_finseq_1,t27_finseq_3,d7_finseq_1,t61_finseq_1,t27_finseq_3,d7_finseq_1,t61_finseq_1,t27_finseq_3]), [file(calcul_1,t14_calcul_1),interesting(0.26)]). fof(t28_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => ! [C] : ( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1) => ! [D] : ( ~ v1_xboole_0(D) => ! [E] : ( m2_fraenkel(E,k2_qc_lang1,D,k2_valuat_1(D)) => ! [F] : ( m1_subset_1(F,D) => ( ~ r2_hidden(B,k24_qc_lang1(k15_cqc_lang(C,A))) => k8_relset_1(k2_qc_lang1,D,k1_sublemma(D,k1_sublemma(D,E,k13_sublemma(D,F,B)),k13_sublemma(D,F,C)),k24_qc_lang1(A)) = k8_relset_1(k2_qc_lang1,D,k1_sublemma(D,E,k13_sublemma(D,F,C)),k24_qc_lang1(A)) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_sublemma,d1_sublemma,t15_funct_4,t16_qc_lang3,t64_zfmisc_1,t59_sublemma,t17_zfmisc_1,t36_funct_4,t15_funct_4,d1_sublemma,d1_sublemma,t26_calcul_1,d1_sublemma]), [file(calcul_1,t28_calcul_1),interesting(0.25)]). fof(t19_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( ( k1_calcul_1(k7_cqc_lang,B) = k1_calcul_1(k7_cqc_lang,C) & k10_cqc_lang(A) = k2_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,B)) & k10_cqc_lang(k2_calcul_1(k7_cqc_lang,B)) = k2_calcul_1(k7_cqc_lang,C) & r10_calcul_1(k1_calcul_1(k7_cqc_lang,B),k2_calcul_1(k7_cqc_lang,B)) & r10_calcul_1(k1_calcul_1(k7_cqc_lang,C),k2_calcul_1(k7_cqc_lang,C)) ) => ( r1_xreal_0(k3_finseq_1(B),1) | r10_calcul_1(k1_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,B)),A) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t4_calcul_1,d15_calcul_1,d15_calcul_1,t28_valuat_1,t17_calcul_1,t28_valuat_1]), [file(calcul_1,t19_calcul_1),interesting(0.24)]). fof(t24_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => ! [C] : ( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1) => ! [D] : ( ~ v1_xboole_0(D) => ! [E] : ( m1_valuat_1(E,D) => ! [F] : ( m2_fraenkel(F,k2_qc_lang1,D,k2_valuat_1(D)) => ( r1_valuat_1(D,k4_substut2(A,B,C),E,F) <=> ? [G] : ( m1_subset_1(G,D) & k8_funct_2(k2_qc_lang1,D,F,C) = G & r1_valuat_1(D,A,E,k1_sublemma(D,F,k13_sublemma(D,G,B))) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t93_sublemma,t23_calcul_1,d2_sublemma,t7_mcart_1,d2_substut1,d1_substut2,t59_sublemma,t23_funcop_1,d9_sublemma,d2_substut2]), [file(calcul_1,t24_calcul_1),interesting(0.19)]). fof(t61_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang) => ! [C] : ( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1) => ! [D] : ( m2_subset_1(D,k1_qc_lang1,k2_qc_lang1) => ! [E] : ( m2_finseq_1(E,k7_cqc_lang) => ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,E,k12_finseq_1(k7_cqc_lang,k4_substut2(A,C,D))),k12_finseq_1(k7_cqc_lang,B))) => ( r2_hidden(D,k3_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,E,k12_finseq_1(k7_cqc_lang,k16_cqc_lang(C,A))),k12_finseq_1(k7_cqc_lang,B)))) | r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,E,k12_finseq_1(k7_cqc_lang,k16_cqc_lang(C,A))),k12_finseq_1(k7_cqc_lang,B))) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t46_calcul_1,t57_calcul_1,t59_calcul_1,d2_xboole_0,t60_calcul_1,t59_calcul_1,t11_qc_lang3,d2_xboole_0,t60_calcul_1,d2_xboole_0,t59_calcul_1,t5_calcul_1,t60_calcul_1,t23_qc_lang3,t11_qc_lang3,t16_qc_lang3,t43_calcul_1,t5_calcul_1,t48_calcul_1,d5_qc_lang2]), [file(calcul_1,t61_calcul_1),interesting(0.13)]). fof(t38_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( ( k1_calcul_1(k7_cqc_lang,B) = k1_calcul_1(k7_cqc_lang,C) & k2_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,B)) = k10_cqc_lang(A) & k10_cqc_lang(k2_calcul_1(k7_cqc_lang,B)) = k2_calcul_1(k7_cqc_lang,C) & r4_calcul_1(B) & r4_calcul_1(C) ) => ( r1_xreal_0(k3_finseq_1(B),1) | r4_calcul_1(k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,B)),k12_finseq_1(k7_cqc_lang,A))) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d9_calcul_1,d9_calcul_1,d6_calcul_1,t43_cqc_the1,t55_finseq_1,d1_tarski,t106_zfmisc_1,d3_tarski,d4_finseq_1,d8_calcul_1,t48_finseq_1,t48_finseq_1,t27_finseq_3,t27_finseq_3,d8_calcul_1,t34_calcul_1,t6_calcul_1,t2_xreal_1,t34_calcul_1,t27_finseq_3,d8_calcul_1,t35_calcul_1,t8_xreal_1,t29_nat_1,t2_xreal_1,t35_finseq_1,t34_calcul_1,t38_finseq_1,t55_finseq_1,t4_finseq_1,d1_tarski,t57_finseq_1,t7_mcart_1,d7_finseq_1,d8_calcul_1,t6_calcul_1,t27_finseq_3,d7_finseq_1,t27_finseq_3,d7_finseq_1,t6_calcul_1,t38_nat_1,t6_calcul_1,t27_finseq_3,d7_finseq_1,t35_finseq_1,t6_calcul_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t38_finseq_1,d8_calcul_1,t35_finseq_1,t56_finseq_1,t4_finseq_1,d1_tarski,t55_finseq_1,d7_finseq_1,t57_finseq_1,t7_mcart_1,d9_calcul_1]), [file(calcul_1,t38_calcul_1),interesting(0.11)]). fof(t11_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(A,0) => k3_finseq_1(k14_finseq_1(k4_subset_1(k5_numbers,k2_finseq_1(B),k6_domain_1(k5_numbers,k1_nat_1(B,A))))) = k1_nat_1(B,1) ) ) ) ), inference(mizar_proof,[status(thm)],[t11_finseq_3,t54_card_2,t78_finseq_1,t38_nat_1,t29_nat_1,t2_xreal_1,t3_finseq_1,t37_zfmisc_1,t19_finseq_3,t8_xboole_1,t44_finseq_3]), [file(calcul_1,t11_calcul_1),interesting(0.05)]). fof(d10_calcul_1,definition,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k7_cqc_lang)) => ( r5_calcul_1(A,B) <=> ? [C] : ( m2_finseq_1(C,k7_cqc_lang) & r1_tarski(k5_relset_1(k5_numbers,k7_cqc_lang,k1_calcul_1(k7_cqc_lang,C)),B) & k2_calcul_1(k7_cqc_lang,C) = A & r4_calcul_1(C) ) ) ) ) ), file(calcul_1,d10_calcul_1), [interesting(0.00)]). fof(d9_calcul_1,definition,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ( r4_calcul_1(A) <=> ? [B] : ( m2_finseq_1(B,k2_zfmisc_1(k4_calcul_1,k2_cqc_the1)) & v1_calcul_1(B) & A = k1_mcart_1(k1_funct_1(B,k3_finseq_1(B))) ) ) ) ), file(calcul_1,d9_calcul_1), [interesting(0.00)]). fof(d12_calcul_1,definition,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang)) => ! [B] : ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang) => ( r7_calcul_1(A,B) <=> ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( m1_valuat_1(D,C) => ! [E] : ( m2_fraenkel(E,k2_qc_lang1,C,k2_valuat_1(C)) => ( r6_calcul_1(A,C,D,E) => r1_valuat_1(C,B,D,E) ) ) ) ) ) ) ) ), file(calcul_1,d12_calcul_1), [interesting(0.00)]). fof(t2_xreal_1,theorem,( ! [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), [interesting(0.00)]). fof(t29_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => r1_xreal_0(A,k2_xcmplx_0(A,B)) ) ) ), file(nat_1,t29_nat_1), [interesting(0.00)]). fof(t27_finseq_3,theorem,( ! [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), [interesting(0.00)]). fof(d5_funct_1,definition,( ! [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), [interesting(0.00)]). fof(t10_mcart_1,theorem,( ! [A,B,C] : ( r2_hidden(A,k2_zfmisc_1(B,C)) => ( r2_hidden(k1_mcart_1(A),B) & r2_hidden(k2_mcart_1(A),C) ) ) ), file(mcart_1,t10_mcart_1), [interesting(0.00)]). fof(d6_calcul_1,definition,( ! [A] : ( A = k4_calcul_1 <=> ! [B] : ( r2_hidden(B,A) <=> m2_finseq_1(B,k7_cqc_lang) ) ) ), file(calcul_1,d6_calcul_1), [interesting(0.00)]). fof(d8_calcul_1,definition,( ! [A] : ( m2_finseq_1(A,k2_zfmisc_1(k4_calcul_1,k2_cqc_the1)) => ( v1_calcul_1(A) <=> ( A != k1_xboole_0 & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,B) & r1_xreal_0(B,k3_finseq_1(A)) ) => r3_calcul_1(A,B) ) ) ) ) ) ), file(calcul_1,d8_calcul_1), [interesting(0.00)]). fof(d7_calcul_1,definition,( ! [A] : ( m2_finseq_1(A,k2_zfmisc_1(k4_calcul_1,k2_cqc_the1)) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( k2_mcart_1(k1_funct_1(A,B)) = 0 => ( r3_calcul_1(A,B) <=> ? [C] : ( m2_finseq_1(C,k7_cqc_lang) & r1_calcul_1(k7_cqc_lang,k2_calcul_1(k7_cqc_lang,C),k1_calcul_1(k7_cqc_lang,C)) & k1_mcart_1(k1_funct_1(A,B)) = C ) ) ) & ( k2_mcart_1(k1_funct_1(A,B)) = 1 => ( r3_calcul_1(A,B) <=> ? [C] : ( m2_finseq_1(C,k7_cqc_lang) & k1_mcart_1(k1_funct_1(A,B)) = k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k9_cqc_lang)) ) ) ) & ( k2_mcart_1(k1_funct_1(A,B)) = 2 => ( r3_calcul_1(A,B) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & ? [D] : ( m2_finseq_1(D,k7_cqc_lang) & ? [E] : ( m2_finseq_1(E,k7_cqc_lang) & r1_xreal_0(1,C) & ~ r1_xreal_0(B,C) & r2_calcul_1(k1_calcul_1(k7_cqc_lang,D),k1_calcul_1(k7_cqc_lang,E)) & k2_calcul_1(k7_cqc_lang,D) = k2_calcul_1(k7_cqc_lang,E) & k1_mcart_1(k1_funct_1(A,C)) = D & k1_mcart_1(k1_funct_1(A,B)) = E ) ) ) ) ) & ( k2_mcart_1(k1_funct_1(A,B)) = 3 => ( r3_calcul_1(A,B) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & ? [E] : ( m2_finseq_1(E,k7_cqc_lang) & ? [F] : ( m2_finseq_1(F,k7_cqc_lang) & r1_xreal_0(1,C) & ~ r1_xreal_0(B,C) & r1_xreal_0(1,D) & ~ r1_xreal_0(C,D) & ~ r1_xreal_0(k3_finseq_1(E),1) & ~ r1_xreal_0(k3_finseq_1(F),1) & k1_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,E)) = k1_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,F)) & k10_cqc_lang(k2_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,E))) = k2_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,F)) & k2_calcul_1(k7_cqc_lang,E) = k2_calcul_1(k7_cqc_lang,F) & E = k1_mcart_1(k1_funct_1(A,D)) & F = k1_mcart_1(k1_funct_1(A,C)) & k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,E)),k12_finseq_1(k7_cqc_lang,k2_calcul_1(k7_cqc_lang,E))) = k1_mcart_1(k1_funct_1(A,B)) ) ) ) ) ) ) & ( k2_mcart_1(k1_funct_1(A,B)) = 4 => ( r3_calcul_1(A,B) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & ? [E] : ( m2_finseq_1(E,k7_cqc_lang) & ? [F] : ( m2_finseq_1(F,k7_cqc_lang) & ? [G] : ( m2_subset_1(G,k8_qc_lang1,k7_cqc_lang) & r1_xreal_0(1,C) & ~ r1_xreal_0(B,C) & r1_xreal_0(1,D) & ~ r1_xreal_0(C,D) & ~ r1_xreal_0(k3_finseq_1(E),1) & k1_calcul_1(k7_cqc_lang,E) = k1_calcul_1(k7_cqc_lang,F) & k2_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,E)) = k10_cqc_lang(G) & k10_cqc_lang(k2_calcul_1(k7_cqc_lang,E)) = k2_calcul_1(k7_cqc_lang,F) & E = k1_mcart_1(k1_funct_1(A,D)) & F = k1_mcart_1(k1_funct_1(A,C)) & k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,E)),k12_finseq_1(k7_cqc_lang,G)) = k1_mcart_1(k1_funct_1(A,B)) ) ) ) ) ) ) ) & ( k2_mcart_1(k1_funct_1(A,B)) = 5 => ( r3_calcul_1(A,B) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & ? [E] : ( m2_finseq_1(E,k7_cqc_lang) & ? [F] : ( m2_finseq_1(F,k7_cqc_lang) & r1_xreal_0(1,C) & ~ r1_xreal_0(B,C) & r1_xreal_0(1,D) & ~ r1_xreal_0(C,D) & k1_calcul_1(k7_cqc_lang,E) = k1_calcul_1(k7_cqc_lang,F) & E = k1_mcart_1(k1_funct_1(A,D)) & F = k1_mcart_1(k1_funct_1(A,C)) & k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,E),k12_finseq_1(k7_cqc_lang,k11_cqc_lang(k2_calcul_1(k7_cqc_lang,E),k2_calcul_1(k7_cqc_lang,F)))) = k1_mcart_1(k1_funct_1(A,B)) ) ) ) ) ) ) & ( k2_mcart_1(k1_funct_1(A,B)) = 6 => ( r3_calcul_1(A,B) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & ? [D] : ( m2_finseq_1(D,k7_cqc_lang) & ? [E] : ( m2_subset_1(E,k8_qc_lang1,k7_cqc_lang) & ? [F] : ( m2_subset_1(F,k8_qc_lang1,k7_cqc_lang) & r1_xreal_0(1,C) & ~ r1_xreal_0(B,C) & k11_cqc_lang(E,F) = k2_calcul_1(k7_cqc_lang,D) & D = k1_mcart_1(k1_funct_1(A,C)) & k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,D),k12_finseq_1(k7_cqc_lang,E)) = k1_mcart_1(k1_funct_1(A,B)) ) ) ) ) ) ) & ( k2_mcart_1(k1_funct_1(A,B)) = 7 => ( r3_calcul_1(A,B) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & ? [D] : ( m2_finseq_1(D,k7_cqc_lang) & ? [E] : ( m2_subset_1(E,k8_qc_lang1,k7_cqc_lang) & ? [F] : ( m2_subset_1(F,k8_qc_lang1,k7_cqc_lang) & r1_xreal_0(1,C) & ~ r1_xreal_0(B,C) & k11_cqc_lang(E,F) = k2_calcul_1(k7_cqc_lang,D) & D = k1_mcart_1(k1_funct_1(A,C)) & k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,D),k12_finseq_1(k7_cqc_lang,F)) = k1_mcart_1(k1_funct_1(A,B)) ) ) ) ) ) ) & ( k2_mcart_1(k1_funct_1(A,B)) = 8 => ( r3_calcul_1(A,B) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & ? [D] : ( m2_finseq_1(D,k7_cqc_lang) & ? [E] : ( m2_subset_1(E,k8_qc_lang1,k7_cqc_lang) & ? [F] : ( m2_subset_1(F,k1_qc_lang1,k2_qc_lang1) & ? [G] : ( m2_subset_1(G,k1_qc_lang1,k2_qc_lang1) & r1_xreal_0(1,C) & ~ r1_xreal_0(B,C) & k2_calcul_1(k7_cqc_lang,D) = k15_cqc_lang(F,E) & D = k1_mcart_1(k1_funct_1(A,C)) & k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,D),k12_finseq_1(k7_cqc_lang,k4_substut2(E,F,G))) = k1_mcart_1(k1_funct_1(A,B)) ) ) ) ) ) ) ) & ( k2_mcart_1(k1_funct_1(A,B)) = 9 => ( r3_calcul_1(A,B) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & ? [D] : ( m2_finseq_1(D,k7_cqc_lang) & ? [E] : ( m2_subset_1(E,k8_qc_lang1,k7_cqc_lang) & ? [F] : ( m2_subset_1(F,k1_qc_lang1,k2_qc_lang1) & ? [G] : ( m2_subset_1(G,k1_qc_lang1,k2_qc_lang1) & r1_xreal_0(1,C) & ~ r1_xreal_0(B,C) & k2_calcul_1(k7_cqc_lang,D) = k4_substut2(E,F,G) & ~ r2_hidden(G,k3_calcul_1(k1_calcul_1(k7_cqc_lang,D))) & ~ r2_hidden(G,k24_qc_lang1(k15_cqc_lang(F,E))) & D = k1_mcart_1(k1_funct_1(A,C)) & k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,D),k12_finseq_1(k7_cqc_lang,k15_cqc_lang(F,E))) = k1_mcart_1(k1_funct_1(A,B)) ) ) ) ) ) ) ) ) ) ) ), file(calcul_1,d7_calcul_1), [interesting(0.00)]). fof(d3_calcul_1,definition,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_subset_1(B,A) => ! [C] : ( m2_finseq_1(C,A) => ( r1_calcul_1(A,B,C) <=> ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & r2_hidden(D,k4_relset_1(k5_numbers,A,C)) & k1_funct_1(C,D) = B ) ) ) ) ) ), file(calcul_1,d3_calcul_1), [interesting(0.00)]). fof(d15_calcul_1,definition,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang) => ( r10_calcul_1(A,B) <=> ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( m1_valuat_1(D,C) => ! [E] : ( m2_fraenkel(E,k2_qc_lang1,C,k2_valuat_1(C)) => ( r9_calcul_1(A,C,D,E) => r1_valuat_1(C,B,D,E) ) ) ) ) ) ) ) ), file(calcul_1,d15_calcul_1), [interesting(0.00)]). fof(d14_calcul_1,definition,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_valuat_1(C,B) => ! [D] : ( m2_fraenkel(D,k2_qc_lang1,B,k2_valuat_1(B)) => ( r9_calcul_1(A,B,C,D) <=> r6_calcul_1(k5_relset_1(k5_numbers,k7_cqc_lang,A),B,C,D) ) ) ) ) ) ), file(calcul_1,d14_calcul_1), [interesting(0.00)]). fof(t12_funct_1,theorem,( ! [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), [interesting(0.00)]). fof(d11_calcul_1,definition,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang)) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_valuat_1(C,B) => ! [D] : ( m2_fraenkel(D,k2_qc_lang1,B,k2_valuat_1(B)) => ( r6_calcul_1(A,B,C,D) <=> ! [E] : ( m2_subset_1(E,k8_qc_lang1,k7_cqc_lang) => ( r2_hidden(E,A) => r1_valuat_1(B,E,C,D) ) ) ) ) ) ) ) ), file(calcul_1,d11_calcul_1), [interesting(0.00)]). fof(t19_finseq_2,theorem,( ! [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), [interesting(0.00)]). fof(t59_finseq_1,theorem,( ! [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), [interesting(0.00)]). fof(d2_calcul_1,definition,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ( ~ r1_xreal_0(k3_finseq_1(B),0) => k2_calcul_1(A,B) = k1_funct_1(B,k3_finseq_1(B)) ) ) ) ), file(calcul_1,d2_calcul_1), [interesting(0.00)]). fof(t16_finseq_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ~ ( r2_hidden(A,B) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r2_hidden(C,k4_finseq_1(B)) & A = k4_tarski(C,k1_funct_1(B,C)) ) ) ) ) ), file(finseq_1,t16_finseq_1), [interesting(0.00)]). fof(t18_finseq_2,theorem,( ! [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(A,k4_finseq_1(B)) => r2_hidden(A,k4_finseq_1(k7_finseq_1(B,C))) ) ) ) ) ), file(finseq_2,t18_finseq_2), [interesting(0.00)]). fof(d7_finseq_1,definition,( ! [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), [interesting(0.00)]). fof(t8_funct_1,theorem,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(k4_tarski(A,B),C) <=> ( r2_hidden(A,k1_relat_1(C)) & B = k1_funct_1(C,A) ) ) ) ), file(funct_1,t8_funct_1), [interesting(0.00)]). fof(d3_tarski,definition,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), [interesting(0.00)]). fof(t64_grfunc_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r1_tarski(A,B) => k7_relat_1(B,k1_relat_1(A)) = A ) ) ) ), file(grfunc_1,t64_grfunc_1), [interesting(0.00)]). fof(d3_finseq_1,definition,( ! [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), [interesting(0.00)]). fof(d1_calcul_1,definition,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ! [C] : ( m2_finseq_1(C,A) => ( ( ~ r1_xreal_0(k3_finseq_1(B),0) => ( C = k1_calcul_1(A,B) <=> ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( k3_finseq_1(B) = k1_nat_1(D,1) => C = k8_relset_1(k5_numbers,A,B,k2_finseq_1(D)) ) ) ) ) & ( r1_xreal_0(k3_finseq_1(B),0) => ( C = k1_calcul_1(A,B) <=> C = k1_xboole_0 ) ) ) ) ) ) ), file(calcul_1,d1_calcul_1), [interesting(0.00)]). fof(t5_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => ( k2_calcul_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,B,k12_finseq_1(k7_cqc_lang,A))) = A & k1_calcul_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,B,k12_finseq_1(k7_cqc_lang,A))) = B ) ) ) ), inference(mizar_proof,[status(thm)],[t19_finseq_2,t59_finseq_1,d2_calcul_1,t16_finseq_1,t18_finseq_2,d7_finseq_1,t8_funct_1,d3_tarski,t64_grfunc_1,d3_finseq_1,d1_calcul_1]), [file(calcul_1,t5_calcul_1),interesting(0.00)]). fof(t44_valuat_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_fraenkel(B,k2_qc_lang1,A,k2_valuat_1(A)) => ! [C] : ( m1_valuat_1(C,A) => r1_valuat_1(A,k9_cqc_lang,C,B) ) ) ) ), file(valuat_1,t44_valuat_1), [interesting(0.00)]). fof(t30_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => r10_calcul_1(k1_calcul_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,A,k12_finseq_1(k7_cqc_lang,k9_cqc_lang))),k2_calcul_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,A,k12_finseq_1(k7_cqc_lang,k9_cqc_lang)))) ) ), inference(mizar_proof,[status(thm)],[d15_calcul_1,t5_calcul_1,t44_valuat_1]), [file(calcul_1,t30_calcul_1),interesting(0.00)]). fof(t38_nat_1,theorem,( ! [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), [interesting(0.00)]). fof(d4_calcul_1,definition,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => ( r2_calcul_1(A,B) <=> ? [C] : ( m1_subset_1(C,k1_zfmisc_1(k5_numbers)) & r1_tarski(A,k15_finseq_1(k8_relset_1(k5_numbers,k7_cqc_lang,B,C))) ) ) ) ) ), file(calcul_1,d4_calcul_1), [interesting(0.00)]). fof(t11_finseq_2,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ~ ( r2_hidden(A,k2_relat_1(B)) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r2_hidden(C,k4_finseq_1(B)) & k1_funct_1(B,C) = A ) ) ) ) ), file(finseq_2,t11_finseq_2), [interesting(0.00)]). fof(t25_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ( r1_tarski(A,B) => ( r1_tarski(k1_relat_1(A),k1_relat_1(B)) & r1_tarski(k2_relat_1(A),k2_relat_1(B)) ) ) ) ) ), file(relat_1,t25_relat_1), [interesting(0.00)]). fof(d14_finseq_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v2_finseq_1(A) ) => k15_finseq_1(A) = k5_relat_1(k14_finseq_1(k1_relat_1(A)),A) ) ), file(finseq_1,d14_finseq_1), [interesting(0.00)]). fof(t45_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => r1_tarski(k2_relat_1(k5_relat_1(A,B)),k2_relat_1(B)) ) ) ), file(relat_1,t45_relat_1), [interesting(0.00)]). fof(t99_relat_1,theorem,( ! [A,B] : ( v1_relat_1(B) => r1_tarski(k2_relat_1(k7_relat_1(B,A)),k2_relat_1(B)) ) ), file(relat_1,t99_relat_1), [interesting(0.00)]). fof(t1_xboole_1,theorem,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(B,C) ) => r1_tarski(A,C) ) ), file(xboole_1,t1_xboole_1), [interesting(0.00)]). fof(t1_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => ( r2_calcul_1(A,B) => ( r1_tarski(k5_relset_1(k5_numbers,k7_cqc_lang,A),k5_relset_1(k5_numbers,k7_cqc_lang,B)) & ? [C] : ( m1_subset_1(C,k1_zfmisc_1(k5_numbers)) & r1_tarski(k5_relset_1(k5_numbers,k7_cqc_lang,A),k5_relset_1(k5_numbers,k7_cqc_lang,k8_relset_1(k5_numbers,k7_cqc_lang,B,C))) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d4_calcul_1,t11_finseq_2,t25_relat_1,t8_funct_1,t8_funct_1,t12_funct_1,d14_finseq_1,t45_relat_1,d3_tarski,t99_relat_1,t1_xboole_1,d3_tarski]), [file(calcul_1,t1_calcul_1),interesting(0.00)]). fof(t16_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => ( ( r2_calcul_1(k1_calcul_1(k7_cqc_lang,A),k1_calcul_1(k7_cqc_lang,B)) & k2_calcul_1(k7_cqc_lang,A) = k2_calcul_1(k7_cqc_lang,B) & r10_calcul_1(k1_calcul_1(k7_cqc_lang,A),k2_calcul_1(k7_cqc_lang,A)) ) => r10_calcul_1(k1_calcul_1(k7_cqc_lang,B),k2_calcul_1(k7_cqc_lang,B)) ) ) ) ), inference(mizar_proof,[status(thm)],[d15_calcul_1,d14_calcul_1,t1_calcul_1,d11_calcul_1,d11_calcul_1,d14_calcul_1,d15_calcul_1]), [file(calcul_1,t16_calcul_1),interesting(0.00)]). fof(t22_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ~ ( A != 0 & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => A != k1_nat_1(B,1) ) ) ) ), file(nat_1,t22_nat_1), [interesting(0.00)]). fof(t68_funct_1,theorem,( ! [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), [interesting(0.00)]). fof(t7_finseq_1,theorem,( ! [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), [interesting(0.00)]). fof(t28_xboole_1,theorem,( ! [A,B] : ( r1_tarski(A,B) => k3_xboole_0(A,B) = A ) ), file(xboole_1,t28_xboole_1), [interesting(0.00)]). fof(t8_finseq_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( k2_finseq_1(A) = k2_finseq_1(B) => A = B ) ) ) ), file(finseq_1,t8_finseq_1), [interesting(0.00)]). fof(t2_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ( ~ r1_xreal_0(k3_finseq_1(A),0) => ( k1_nat_1(k3_finseq_1(k1_calcul_1(k7_cqc_lang,A)),1) = k3_finseq_1(A) & ~ r1_xreal_0(k3_finseq_1(A),k3_finseq_1(k1_calcul_1(k7_cqc_lang,A))) ) ) ) ), inference(mizar_proof,[status(thm)],[t22_nat_1,d1_calcul_1,t68_funct_1,d3_finseq_1,d3_finseq_1,t29_nat_1,t7_finseq_1,t28_xboole_1,t8_finseq_1,t8_finseq_1,t38_nat_1]), [file(calcul_1,t2_calcul_1),interesting(0.00)]). fof(t56_finseq_1,theorem,( ! [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), [interesting(0.00)]). fof(t35_finseq_1,theorem,( ! [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), [interesting(0.00)]). fof(t3_finseq_1,theorem,( ! [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), [interesting(0.00)]). fof(t72_funct_1,theorem,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(B,A) => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ), file(funct_1,t72_funct_1), [interesting(0.00)]). fof(t57_finseq_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( B = k9_finseq_1(A) <=> ( k3_finseq_1(B) = 1 & k1_funct_1(B,1) = A ) ) ) ), file(finseq_1,t57_finseq_1), [interesting(0.00)]). fof(t55_finseq_1,theorem,( ! [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), [interesting(0.00)]). fof(t26_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ~ ( r1_xreal_0(A,k2_xcmplx_0(B,1)) & ~ r1_xreal_0(A,B) & A != k2_xcmplx_0(B,1) ) ) ) ), file(nat_1,t26_nat_1), [interesting(0.00)]). fof(t10_finseq_2,theorem,( ! [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) ) => ( ( k3_finseq_1(B) = A & k3_finseq_1(C) = A & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k2_finseq_1(A)) => k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) => B = C ) ) ) ) ), file(finseq_2,t10_finseq_2), [interesting(0.00)]). fof(t44_finseq_1,theorem,( ! [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), [interesting(0.00)]). fof(d2_xboole_0,definition,( ! [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), [interesting(0.00)]). fof(d1_tarski,definition,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), [interesting(0.00)]). fof(t7_xboole_1,theorem,( ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ), file(xboole_1,t7_xboole_1), [interesting(0.00)]). fof(t28_valuat_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_fraenkel(B,k2_qc_lang1,A,k2_valuat_1(A)) => ! [C] : ( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang) => ! [D] : ( m1_valuat_1(D,A) => ( r1_valuat_1(A,k10_cqc_lang(C),D,B) <=> ~ r1_valuat_1(A,C,D,B) ) ) ) ) ) ), file(valuat_1,t28_valuat_1), [interesting(0.00)]). fof(t18_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => ( ( k1_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,A)) = k1_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,B)) & k10_cqc_lang(k2_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,A))) = k2_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,B)) & k2_calcul_1(k7_cqc_lang,A) = k2_calcul_1(k7_cqc_lang,B) & r10_calcul_1(k1_calcul_1(k7_cqc_lang,A),k2_calcul_1(k7_cqc_lang,A)) & r10_calcul_1(k1_calcul_1(k7_cqc_lang,B),k2_calcul_1(k7_cqc_lang,B)) ) => ( r1_xreal_0(k3_finseq_1(A),1) | r1_xreal_0(k3_finseq_1(B),1) | r10_calcul_1(k1_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,A)),k2_calcul_1(k7_cqc_lang,A)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t4_calcul_1,d15_calcul_1,t17_calcul_1,d15_calcul_1,t28_valuat_1,t17_calcul_1,d15_calcul_1]), [file(calcul_1,t18_calcul_1),interesting(0.00)]). fof(t29_valuat_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_fraenkel(B,k2_qc_lang1,A,k2_valuat_1(A)) => ! [C] : ( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang) => ! [D] : ( m2_subset_1(D,k8_qc_lang1,k7_cqc_lang) => ! [E] : ( m1_valuat_1(E,A) => ( r1_valuat_1(A,k11_cqc_lang(C,D),E,B) <=> ( r1_valuat_1(A,C,E,B) & r1_valuat_1(A,D,E,B) ) ) ) ) ) ) ) ), file(valuat_1,t29_valuat_1), [interesting(0.00)]). fof(t20_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => ( ( k1_calcul_1(k7_cqc_lang,A) = k1_calcul_1(k7_cqc_lang,B) & r10_calcul_1(k1_calcul_1(k7_cqc_lang,A),k2_calcul_1(k7_cqc_lang,A)) & r10_calcul_1(k1_calcul_1(k7_cqc_lang,B),k2_calcul_1(k7_cqc_lang,B)) ) => r10_calcul_1(k1_calcul_1(k7_cqc_lang,A),k11_cqc_lang(k2_calcul_1(k7_cqc_lang,A),k2_calcul_1(k7_cqc_lang,B))) ) ) ) ), inference(mizar_proof,[status(thm)],[d15_calcul_1,d15_calcul_1,t29_valuat_1]), [file(calcul_1,t20_calcul_1),interesting(0.00)]). fof(t21_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( ( k2_calcul_1(k7_cqc_lang,C) = k11_cqc_lang(A,B) & r10_calcul_1(k1_calcul_1(k7_cqc_lang,C),k11_cqc_lang(A,B)) ) => r10_calcul_1(k1_calcul_1(k7_cqc_lang,C),A) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d15_calcul_1,d15_calcul_1,t29_valuat_1]), [file(calcul_1,t21_calcul_1),interesting(0.00)]). fof(t22_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( ( k2_calcul_1(k7_cqc_lang,C) = k11_cqc_lang(A,B) & r10_calcul_1(k1_calcul_1(k7_cqc_lang,C),k11_cqc_lang(A,B)) ) => r10_calcul_1(k1_calcul_1(k7_cqc_lang,C),B) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d15_calcul_1,d15_calcul_1,t29_valuat_1]), [file(calcul_1,t22_calcul_1),interesting(0.00)]). fof(t51_sublemma,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( m1_valuat_1(D,C) => ! [E] : ( m2_fraenkel(E,k2_qc_lang1,C,k2_valuat_1(C)) => ( r1_valuat_1(C,k15_cqc_lang(B,A),D,E) <=> ! [F] : ( m1_subset_1(F,C) => r1_valuat_1(C,A,D,k1_sublemma(C,E,k13_sublemma(C,F,B))) ) ) ) ) ) ) ) ), file(sublemma,t51_sublemma), [interesting(0.00)]). fof(t93_sublemma,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_valuat_1(B,A) => ! [C] : ( m2_subset_1(C,k16_substut1,k38_substut1) => ! [D] : ( m2_fraenkel(D,k2_qc_lang1,A,k2_valuat_1(A)) => ( r1_valuat_1(A,k39_substut1(C),B,D) <=> r1_sublemma(C,A,k1_sublemma(A,D,k3_sublemma(C,A,D)),B) ) ) ) ) ) ), file(sublemma,t93_sublemma), [interesting(0.00)]). fof(d3_sublemma,definition,( ! [A] : ( m2_subset_1(A,k16_substut1,k38_substut1) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m2_fraenkel(C,k2_qc_lang1,B,k2_valuat_1(B)) => ! [D] : ( m1_valuat_1(D,B) => ( r1_sublemma(A,B,C,D) <=> r1_valuat_1(B,k2_sublemma(A),D,C) ) ) ) ) ) ), file(sublemma,d3_sublemma), [interesting(0.00)]). fof(t7_mcart_1,theorem,( ! [A,B] : ( k1_mcart_1(k4_tarski(A,B)) = A & k2_mcart_1(k4_tarski(A,B)) = B ) ), file(mcart_1,t7_mcart_1), [interesting(0.00)]). fof(d2_sublemma,definition,( ! [A] : ( m2_subset_1(A,k16_substut1,k38_substut1) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m2_fraenkel(C,k2_qc_lang1,B,k2_valuat_1(B)) => k3_sublemma(A,B,C) = k7_relset_1(k2_qc_lang1,k2_qc_lang1,k2_qc_lang1,B,k2_substut1(k19_substut1(A)),C) ) ) ) ), file(sublemma,d2_sublemma), [interesting(0.00)]). fof(d2_substut1,definition,( ! [A] : ( m1_subset_1(A,k1_substut1) => k2_substut1(A) = A ) ), file(substut1,d2_substut1), [interesting(0.00)]). fof(d1_substut2,definition,( ! [A] : ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1) => ! [B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => k3_substut2(A,B) = k3_cqc_lang(A,B) ) ) ), file(substut2,d1_substut2), [interesting(0.00)]). fof(t59_sublemma,theorem,( ! [A] : ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m2_fraenkel(C,k2_qc_lang1,B,k2_valuat_1(B)) => ! [D] : ( m1_subset_1(D,B) => ( k4_relset_1(k2_qc_lang1,B,C) = k2_qc_lang1 & k4_relset_1(k2_qc_lang1,B,k13_sublemma(B,D,A)) = k6_domain_1(k2_qc_lang1,A) ) ) ) ) ) ), file(sublemma,t59_sublemma), [interesting(0.00)]). fof(t23_funcop_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : ( r2_hidden(C,k1_relat_1(A)) => k5_relat_1(k2_funcop_1(B,C),A) = k2_funcop_1(B,k1_funct_1(A,C)) ) ) ), file(funcop_1,t23_funcop_1), [interesting(0.00)]). fof(d9_sublemma,definition,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_subset_1(B,A) => ! [C] : ( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1) => k13_sublemma(A,B,C) = k3_cqc_lang(C,B) ) ) ) ), file(sublemma,d9_sublemma), [interesting(0.00)]). fof(d2_substut2,definition,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => ! [C] : ( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1) => k4_substut2(A,B,C) = k39_substut1(k2_substut2(A,k3_substut2(B,C))) ) ) ) ), file(substut2,d2_substut2), [interesting(0.00)]). fof(t25_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( ( k2_calcul_1(k7_cqc_lang,C) = k15_cqc_lang(B,A) & r10_calcul_1(k1_calcul_1(k7_cqc_lang,C),k2_calcul_1(k7_cqc_lang,C)) ) => ! [D] : ( m2_subset_1(D,k1_qc_lang1,k2_qc_lang1) => r10_calcul_1(k1_calcul_1(k7_cqc_lang,C),k4_substut2(A,B,D)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d15_calcul_1,d15_calcul_1,t51_sublemma,t24_calcul_1]), [file(calcul_1,t25_calcul_1),interesting(0.00)]). fof(t64_sublemma,theorem,( ! [A] : ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m2_fraenkel(C,k2_qc_lang1,B,k2_valuat_1(B)) => ! [D] : ( m1_subset_1(D,B) => ! [E] : ( r1_tarski(E,k2_qc_lang1) => ( k4_relset_1(k2_qc_lang1,B,k8_relset_1(k2_qc_lang1,B,C,E)) = k4_relset_1(k2_qc_lang1,B,k8_relset_1(k2_qc_lang1,B,k1_sublemma(B,C,k13_sublemma(B,D,A)),E)) & k4_relset_1(k2_qc_lang1,B,k8_relset_1(k2_qc_lang1,B,C,E)) = E ) ) ) ) ) ) ), file(sublemma,t64_sublemma), [interesting(0.00)]). fof(t49_sublemma,theorem,( ! [A,B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( m2_fraenkel(D,k2_qc_lang1,C,k2_valuat_1(C)) => ! [E] : ( m1_subset_1(E,C) => ( B != A => k1_funct_1(k1_sublemma(C,D,k13_sublemma(C,E,B)),A) = k1_funct_1(D,A) ) ) ) ) ) ), file(sublemma,t49_sublemma), [interesting(0.00)]). fof(t9_funct_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( ( k1_relat_1(A) = k1_relat_1(B) & ! [C] : ( r2_hidden(C,k1_relat_1(A)) => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) => A = B ) ) ) ), file(funct_1,t9_funct_1), [interesting(0.00)]). fof(d5_calcul_1,definition,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k2_qc_lang1)) => ( B = k3_calcul_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & ? [E] : ( m2_subset_1(E,k8_qc_lang1,k7_cqc_lang) & r2_hidden(D,k4_relset_1(k5_numbers,k7_cqc_lang,A)) & E = k1_funct_1(A,D) & r2_hidden(C,k24_qc_lang1(E)) ) ) ) ) ) ) ), file(calcul_1,d5_calcul_1), [interesting(0.00)]). fof(t188_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ! [C,D] : ( ( r1_tarski(C,D) & k7_relat_1(A,D) = k7_relat_1(B,D) ) => k7_relat_1(A,C) = k7_relat_1(B,C) ) ) ) ), file(relat_1,t188_relat_1), [interesting(0.00)]). fof(t70_sublemma,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_valuat_1(B,A) => ! [C] : ( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang) => ! [D] : ( m2_fraenkel(D,k2_qc_lang1,A,k2_valuat_1(A)) => ! [E] : ( m2_fraenkel(E,k2_qc_lang1,A,k2_valuat_1(A)) => ( k8_relset_1(k2_qc_lang1,A,D,k24_qc_lang1(C)) = k8_relset_1(k2_qc_lang1,A,E,k24_qc_lang1(C)) => ( r1_valuat_1(A,C,B,D) <=> r1_valuat_1(A,C,B,E) ) ) ) ) ) ) ) ), file(sublemma,t70_sublemma), [interesting(0.00)]). fof(t27_calcul_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_valuat_1(B,A) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ! [D] : ( m2_fraenkel(D,k2_qc_lang1,A,k2_valuat_1(A)) => ! [E] : ( m2_fraenkel(E,k2_qc_lang1,A,k2_valuat_1(A)) => ( k8_relset_1(k2_qc_lang1,A,D,k3_calcul_1(C)) = k8_relset_1(k2_qc_lang1,A,E,k3_calcul_1(C)) => ( r9_calcul_1(C,A,B,D) <=> r9_calcul_1(C,A,B,E) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d5_calcul_1,d3_tarski,t188_relat_1,t70_sublemma,d14_calcul_1,d11_calcul_1,d14_calcul_1,t11_finseq_2,d11_calcul_1,d14_calcul_1,d11_calcul_1,d14_calcul_1,t11_finseq_2,d11_calcul_1]), [file(calcul_1,t27_calcul_1),interesting(0.00)]). fof(t50_sublemma,theorem,( ! [A] : ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1) => ! [B] : ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1) => ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( m2_fraenkel(D,k2_qc_lang1,C,k2_valuat_1(C)) => ! [E] : ( m1_subset_1(E,C) => ( A = B => k8_funct_2(k2_qc_lang1,C,k1_sublemma(C,D,k13_sublemma(C,E,A)),B) = E ) ) ) ) ) ) ), file(sublemma,t50_sublemma), [interesting(0.00)]). fof(d1_sublemma,definition,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_fraenkel(B,k2_qc_lang1,A,k2_valuat_1(A)) => ! [C] : ( ( v1_funct_1(C) & m2_relset_1(C,k2_qc_lang1,A) ) => k1_sublemma(A,B,C) = k1_funct_4(B,C) ) ) ) ), file(sublemma,d1_sublemma), [interesting(0.00)]). fof(t15_funct_4,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => k1_funct_4(k1_funct_4(A,B),C) = k1_funct_4(A,k1_funct_4(B,C)) ) ) ) ), file(funct_4,t15_funct_4), [interesting(0.00)]). fof(t16_qc_lang3,theorem,( ! [A] : ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1) => ! [B] : ( m1_subset_1(B,k8_qc_lang1) => k24_qc_lang1(k14_qc_lang1(A,B)) = k6_subset_1(k2_qc_lang1,k24_qc_lang1(B),k23_qc_lang1(A)) ) ) ), file(qc_lang3,t16_qc_lang3), [interesting(0.00)]). fof(t64_zfmisc_1,theorem,( ! [A,B,C] : ( r2_hidden(A,k4_xboole_0(B,k1_tarski(C))) <=> ( r2_hidden(A,B) & A != C ) ) ), file(zfmisc_1,t64_zfmisc_1), [interesting(0.00)]). fof(t17_zfmisc_1,theorem,( ! [A,B] : ( A != B => r1_xboole_0(k1_tarski(A),k1_tarski(B)) ) ), file(zfmisc_1,t17_zfmisc_1), [interesting(0.00)]). fof(t36_funct_4,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B)) => k1_funct_4(A,B) = k1_funct_4(B,A) ) ) ) ), file(funct_4,t36_funct_4), [interesting(0.00)]). fof(d3_cqc_the1,definition,( k2_cqc_the1 = a_0_0_cqc_the1 ), file(cqc_the1,d3_cqc_the1), [interesting(0.00)]). fof(t10_cqc_the1,theorem,( ! [A] : ( v4_ordinal2(A) => ~ ( r1_xreal_0(A,9) & A != 0 & A != 1 & A != 2 & A != 3 & A != 4 & A != 5 & A != 6 & A != 7 & A != 8 & A != 9 ) ) ), file(cqc_the1,t10_cqc_the1), [interesting(0.00)]). fof(t31_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_1(B,k2_zfmisc_1(k4_calcul_1,k2_cqc_the1)) => ~ ( r1_xreal_0(1,A) & r1_xreal_0(A,k3_finseq_1(B)) & k2_mcart_1(k1_funct_1(B,A)) != 0 & k2_mcart_1(k1_funct_1(B,A)) != 1 & k2_mcart_1(k1_funct_1(B,A)) != 2 & k2_mcart_1(k1_funct_1(B,A)) != 3 & k2_mcart_1(k1_funct_1(B,A)) != 4 & k2_mcart_1(k1_funct_1(B,A)) != 5 & k2_mcart_1(k1_funct_1(B,A)) != 6 & k2_mcart_1(k1_funct_1(B,A)) != 7 & k2_mcart_1(k1_funct_1(B,A)) != 8 & k2_mcart_1(k1_funct_1(B,A)) != 9 ) ) ) ), inference(mizar_proof,[status(thm)],[t27_finseq_3,d5_funct_1,d3_cqc_the1,t10_mcart_1,t10_cqc_the1]), [file(calcul_1,t31_calcul_1),interesting(0.00)]). fof(s1_nat_1,theorem, ( ( p1_s1_nat_1(0) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( p1_s1_nat_1(A) => p1_s1_nat_1(k1_nat_1(A,1)) ) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => p1_s1_nat_1(A) ) ), file(nat_1,s1_nat_1), [interesting(0.00)]). fof(t8_graph_5,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( A != k1_xboole_0 <=> r1_xreal_0(1,k3_finseq_1(A)) ) ) ), file(graph_5,t8_graph_5), [interesting(0.00)]). fof(t10_calcul_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( r1_xreal_0(1,k3_finseq_1(k7_finseq_1(B,k9_finseq_1(A)))) & r2_hidden(k3_finseq_1(k7_finseq_1(B,k9_finseq_1(A))),k1_relat_1(k7_finseq_1(B,k9_finseq_1(A)))) ) ) ), inference(mizar_proof,[status(thm)],[t19_finseq_2,t29_nat_1,t27_finseq_3]), [file(calcul_1,t10_calcul_1),interesting(0.00)]). fof(d1_relat_1,definition,( ! [A] : ( v1_relat_1(A) <=> ! [B] : ~ ( r2_hidden(B,A) & ! [C,D] : B != k4_tarski(C,D) ) ) ), file(relat_1,d1_relat_1), [interesting(0.00)]). fof(t45_finseq_1,theorem,( ! [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) ) => k7_finseq_1(k7_finseq_1(A,B),C) = k7_finseq_1(A,k7_finseq_1(B,C)) ) ) ) ), file(finseq_1,t45_finseq_1), [interesting(0.00)]). fof(t37_nat_1,theorem,( ! [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), [interesting(0.00)]). fof(t25_finseq_1,theorem,( ! [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), [interesting(0.00)]). fof(t8_xreal_1,theorem,( ! [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), [interesting(0.00)]). fof(t6_calcul_1,theorem,( ! [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_xreal_0(k3_finseq_1(A),k3_finseq_1(k7_finseq_1(A,B))) & r1_xreal_0(k3_finseq_1(B),k3_finseq_1(k7_finseq_1(A,B))) & ( A != k1_xboole_0 => ( r1_xreal_0(1,k3_finseq_1(A)) & ~ r1_xreal_0(k3_finseq_1(k7_finseq_1(B,A)),k3_finseq_1(B)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t35_finseq_1,t37_nat_1,t25_finseq_1,t38_nat_1,t8_xreal_1,t35_finseq_1,t38_nat_1]), [file(calcul_1,t6_calcul_1),interesting(0.00)]). fof(t90_relat_1,theorem,( ! [A,B] : ( v1_relat_1(B) => k1_relat_1(k7_relat_1(B,A)) = k3_xboole_0(k1_relat_1(B),A) ) ), file(relat_1,t90_relat_1), [interesting(0.00)]). fof(t11_finseq_3,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r2_hidden(k2_xcmplx_0(A,B),k2_finseq_1(A)) => B = 0 ) ) ) ), file(finseq_3,t11_finseq_3), [interesting(0.00)]). fof(t54_card_2,theorem,( ! [A,B] : ( v1_finset_1(B) => ( ~ r2_hidden(A,B) => k4_card_1(k2_xboole_0(B,k1_tarski(A))) = k1_nat_1(k4_card_1(B),1) ) ) ), file(card_2,t54_card_2), [interesting(0.00)]). fof(t78_finseq_1,theorem, ( ! [A] : ( v4_ordinal2(A) => k4_card_1(k2_finseq_1(A)) = A ) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( k4_card_1(A) = A & k4_card_1(k1_card_1(A)) = A ) ) ), file(finseq_1,t78_finseq_1), [interesting(0.00)]). fof(t37_zfmisc_1,theorem,( ! [A,B] : ( r1_tarski(k1_tarski(A),B) <=> r2_hidden(A,B) ) ), file(zfmisc_1,t37_zfmisc_1), [interesting(0.00)]). fof(t19_finseq_3,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => r1_tarski(k2_finseq_1(A),k2_finseq_1(k2_xcmplx_0(A,B))) ) ) ), file(finseq_3,t19_finseq_3), [interesting(0.00)]). fof(t8_xboole_1,theorem,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(C,B) ) => r1_tarski(k2_xboole_0(A,C),B) ) ), file(xboole_1,t8_xboole_1), [interesting(0.00)]). fof(t44_finseq_3,theorem,( ! [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), [interesting(0.00)]). fof(t12_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(A,0) => k4_relset_1(k5_numbers,k5_numbers,k14_finseq_1(k4_subset_1(k5_numbers,k2_finseq_1(B),k6_domain_1(k5_numbers,k1_nat_1(B,A))))) = k2_finseq_1(k1_nat_1(B,1)) ) ) ) ), inference(mizar_proof,[status(thm)],[t11_calcul_1,d3_finseq_1]), [file(calcul_1,t12_calcul_1),interesting(0.00)]). fof(t2_tarski,theorem,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), [interesting(0.00)]). fof(t48_finseq_3,theorem,( ! [A,B,C] : ( v4_ordinal2(C) => ! [D] : ( v4_ordinal2(D) => ( ( r1_tarski(A,k2_finseq_1(C)) & r1_tarski(B,k2_finseq_1(D)) ) => ( ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ~ ( r2_hidden(E,A) & r2_hidden(F,B) & r1_xreal_0(F,E) ) ) ) <=> k14_finseq_1(k2_xboole_0(A,B)) = k8_finseq_1(k5_numbers,k14_finseq_1(A),k14_finseq_1(B)) ) ) ) ) ), file(finseq_3,t48_finseq_3), [interesting(0.00)]). fof(t50_finseq_3,theorem,( ! [A] : ( v4_ordinal2(A) => ( 0 != A => k14_finseq_1(k1_tarski(A)) = k9_finseq_1(A) ) ) ), file(finseq_3,t50_finseq_3), [interesting(0.00)]). fof(t54_finseq_3,theorem,( ! [A] : ( v4_ordinal2(A) => k14_finseq_1(k2_finseq_1(A)) = k1_finseq_2(A) ) ), file(finseq_3,t54_finseq_3), [interesting(0.00)]). fof(t54_finseq_2,theorem,( ! [A] : ( v4_ordinal2(A) => k4_finseq_1(k1_finseq_2(A)) = k2_finseq_1(A) ) ), file(finseq_2,t54_finseq_2), [interesting(0.00)]). fof(t56_finseq_2,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r2_hidden(A,k2_finseq_1(B)) => k1_funct_1(k1_finseq_2(B),A) = A ) ) ) ), file(finseq_2,t56_finseq_2), [interesting(0.00)]). fof(t23_funct_1,theorem,( ! [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), [interesting(0.00)]). fof(t103_relat_1,theorem,( ! [A,B,C] : ( v1_relat_1(C) => ( r1_tarski(A,B) => k7_relat_1(k7_relat_1(C,B),A) = k7_relat_1(C,A) ) ) ), file(relat_1,t103_relat_1), [interesting(0.00)]). fof(t33_finseq_1,theorem,( ! [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) ) => A = k7_relat_1(k7_finseq_1(A,B),k4_finseq_1(A)) ) ) ), file(finseq_1,t33_finseq_1), [interesting(0.00)]). fof(t55_finseq_2,theorem,( ! [A] : ( v4_ordinal2(A) => k3_finseq_1(k1_finseq_2(A)) = A ) ), file(finseq_2,t55_finseq_2), [interesting(0.00)]). fof(t71_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v2_finseq_1(A) ) => k2_relat_1(k14_finseq_1(k1_relat_1(A))) = k1_relat_1(A) ) ), file(finseq_1,t71_finseq_1), [interesting(0.00)]). fof(t46_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ( r1_tarski(k2_relat_1(A),k1_relat_1(B)) => k1_relat_1(k5_relat_1(A,B)) = k1_relat_1(A) ) ) ) ), file(relat_1,t46_relat_1), [interesting(0.00)]). fof(t8_finseq_2,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r2_hidden(A,k2_finseq_1(k1_nat_1(B,1))) & ~ r2_hidden(A,k2_finseq_1(B)) & A != k1_nat_1(B,1) ) ) ) ), file(finseq_2,t8_finseq_2), [interesting(0.00)]). fof(t43_cqc_the1,theorem, ( r2_hidden(0,k2_cqc_the1) & r2_hidden(1,k2_cqc_the1) & r2_hidden(2,k2_cqc_the1) & r2_hidden(3,k2_cqc_the1) & r2_hidden(4,k2_cqc_the1) & r2_hidden(5,k2_cqc_the1) & r2_hidden(6,k2_cqc_the1) & r2_hidden(7,k2_cqc_the1) & r2_hidden(8,k2_cqc_the1) & r2_hidden(9,k2_cqc_the1) ), file(cqc_the1,t43_cqc_the1), [interesting(0.00)]). fof(t106_zfmisc_1,theorem,( ! [A,B,C,D] : ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D)) <=> ( r2_hidden(A,C) & r2_hidden(B,D) ) ) ), file(zfmisc_1,t106_zfmisc_1), [interesting(0.00)]). fof(d4_finseq_1,definition,( ! [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), [interesting(0.00)]). fof(t48_finseq_1,theorem,( ! [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) ) => ( k7_finseq_1(A,B) = k1_xboole_0 => ( A = k1_xboole_0 & B = k1_xboole_0 ) ) ) ) ), file(finseq_1,t48_finseq_1), [interesting(0.00)]). fof(t4_finseq_1,theorem, ( 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), [interesting(0.00)]). fof(t38_finseq_1,theorem,( ! [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] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r2_hidden(C,k4_finseq_1(k7_finseq_1(A,B))) & ~ r2_hidden(C,k4_finseq_1(A)) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ~ ( r2_hidden(D,k4_finseq_1(B)) & C = k1_nat_1(k3_finseq_1(A),D) ) ) ) ) ) ) ), file(finseq_1,t38_finseq_1), [interesting(0.00)]). fof(t36_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => ( ( r2_calcul_1(k1_calcul_1(k7_cqc_lang,A),k1_calcul_1(k7_cqc_lang,B)) & k2_calcul_1(k7_cqc_lang,A) = k2_calcul_1(k7_cqc_lang,B) & r4_calcul_1(A) ) => r4_calcul_1(B) ) ) ) ), inference(mizar_proof,[status(thm)],[d9_calcul_1,d6_calcul_1,t43_cqc_the1,t55_finseq_1,d1_tarski,t106_zfmisc_1,d3_tarski,d4_finseq_1,d8_calcul_1,t48_finseq_1,t27_finseq_3,t27_finseq_3,d8_calcul_1,t34_calcul_1,t55_finseq_1,t4_finseq_1,d1_tarski,t57_finseq_1,t7_mcart_1,d7_finseq_1,t6_calcul_1,t38_nat_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t38_finseq_1,d8_calcul_1,t35_finseq_1,t56_finseq_1,t4_finseq_1,d1_tarski,t55_finseq_1,d7_finseq_1,t57_finseq_1,t7_mcart_1,d9_calcul_1]), [file(calcul_1,t36_calcul_1),interesting(0.00)]). fof(d9_finseq_1,definition,( ! [A,B] : k10_finseq_1(A,B) = k7_finseq_1(k9_finseq_1(A),k9_finseq_1(B)) ), file(finseq_1,d9_finseq_1), [interesting(0.00)]). fof(t61_finseq_1,theorem,( ! [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), [interesting(0.00)]). fof(t41_finseq_1,theorem,( ! [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(A,k4_finseq_1(B)) => r2_hidden(k1_nat_1(k3_finseq_1(C),A),k4_finseq_1(k7_finseq_1(C,B))) ) ) ) ) ), file(finseq_1,t41_finseq_1), [interesting(0.00)]). fof(t1_xreal_1,theorem,( ! [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), [interesting(0.00)]). fof(t19_finseq_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( v1_relat_1(k7_relat_1(B,k2_finseq_1(A))) & v1_funct_1(k7_relat_1(B,k2_finseq_1(A))) & v1_finseq_1(k7_relat_1(B,k2_finseq_1(A))) ) ) ) ), file(finseq_1,t19_finseq_1), [interesting(0.00)]). fof(t24_graph_2,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v2_finseq_1(A) ) => ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k15_finseq_1(A) = A ) ) ), file(graph_2,t24_graph_2), [interesting(0.00)]). fof(t7_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => k15_finseq_1(k8_relset_1(k5_numbers,k7_cqc_lang,k8_finseq_1(k7_cqc_lang,A,B),k4_relset_1(k5_numbers,k7_cqc_lang,A))) = k8_relset_1(k5_numbers,k7_cqc_lang,k8_finseq_1(k7_cqc_lang,A,B),k4_relset_1(k5_numbers,k7_cqc_lang,A)) ) ) ), inference(mizar_proof,[status(thm)],[t33_finseq_1,t24_graph_2]), [file(calcul_1,t7_calcul_1),interesting(0.00)]). fof(t44_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => ( ( r4_calcul_1(B) & r4_calcul_1(k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,B),k12_finseq_1(k7_cqc_lang,k10_cqc_lang(k2_calcul_1(k7_cqc_lang,B))))) ) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,B),k12_finseq_1(k7_cqc_lang,A))) ) ) ) ), inference(mizar_proof,[status(thm)],[t5_calcul_1,t5_calcul_1,t8_calcul_1,t36_calcul_1,t5_calcul_1,t8_calcul_1,t5_calcul_1,t5_calcul_1,t36_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t9_calcul_1,t38_calcul_1,t5_calcul_1]), [file(calcul_1,t44_calcul_1),interesting(0.00)]). fof(t37_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => ( ( k1_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,A)) = k1_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,B)) & k10_cqc_lang(k2_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,A))) = k2_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,B)) & k2_calcul_1(k7_cqc_lang,A) = k2_calcul_1(k7_cqc_lang,B) & r4_calcul_1(A) & r4_calcul_1(B) ) => ( r1_xreal_0(k3_finseq_1(A),1) | r1_xreal_0(k3_finseq_1(B),1) | r4_calcul_1(k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,A)),k12_finseq_1(k7_cqc_lang,k2_calcul_1(k7_cqc_lang,A)))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d9_calcul_1,d9_calcul_1,d6_calcul_1,t43_cqc_the1,t55_finseq_1,d1_tarski,t106_zfmisc_1,d3_tarski,d4_finseq_1,d8_calcul_1,t48_finseq_1,t48_finseq_1,t27_finseq_3,t27_finseq_3,d8_calcul_1,t34_calcul_1,t6_calcul_1,t2_xreal_1,t34_calcul_1,t27_finseq_3,d8_calcul_1,t35_calcul_1,t8_xreal_1,t29_nat_1,t2_xreal_1,t35_finseq_1,t34_calcul_1,t38_finseq_1,t55_finseq_1,t4_finseq_1,d1_tarski,t57_finseq_1,t7_mcart_1,d7_finseq_1,d8_calcul_1,t6_calcul_1,t27_finseq_3,d7_finseq_1,t27_finseq_3,d7_finseq_1,t6_calcul_1,t38_nat_1,t6_calcul_1,t27_finseq_3,d7_finseq_1,t35_finseq_1,t6_calcul_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t38_finseq_1,d8_calcul_1,t35_finseq_1,t56_finseq_1,t4_finseq_1,d1_tarski,t55_finseq_1,d7_finseq_1,t57_finseq_1,t7_mcart_1,d9_calcul_1]), [file(calcul_1,t37_calcul_1),interesting(0.00)]). fof(t47_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k10_cqc_lang(A))),k12_finseq_1(k7_cqc_lang,k10_cqc_lang(B)))) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,B)),k12_finseq_1(k7_cqc_lang,A))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t5_calcul_1,t5_calcul_1,t5_calcul_1,t10_calcul_1,t13_calcul_1,t36_calcul_1,t5_calcul_1,t45_finseq_1,d9_finseq_1,t14_calcul_1,t5_calcul_1,t14_calcul_1,t41_finseq_1,d3_calcul_1,t33_calcul_1,t5_calcul_1,t5_calcul_1,t10_calcul_1,t3_calcul_1,t44_calcul_1,t5_calcul_1,t45_finseq_1,d9_finseq_1,t14_calcul_1,t5_calcul_1,t14_calcul_1,t41_finseq_1,d3_calcul_1,t33_calcul_1,t9_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t37_calcul_1]), [file(calcul_1,t47_calcul_1),interesting(0.00)]). fof(t46_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,A)),k12_finseq_1(k7_cqc_lang,B))) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k10_cqc_lang(B))),k12_finseq_1(k7_cqc_lang,k10_cqc_lang(A)))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t5_calcul_1,t5_calcul_1,t5_calcul_1,t10_calcul_1,t13_calcul_1,t36_calcul_1,t5_calcul_1,t45_finseq_1,d9_finseq_1,t14_calcul_1,t5_calcul_1,t14_calcul_1,t41_finseq_1,d3_calcul_1,t5_calcul_1,t33_calcul_1,t44_calcul_1,t5_calcul_1,t45_finseq_1,d9_finseq_1,t14_calcul_1,t5_calcul_1,t14_calcul_1,t41_finseq_1,d3_calcul_1,t33_calcul_1,t9_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t37_calcul_1]), [file(calcul_1,t46_calcul_1),interesting(0.00)]). fof(t39_calcul_1,theorem,( ! [A] : ( m2_finseq_1(A,k7_cqc_lang) => ! [B] : ( m2_finseq_1(B,k7_cqc_lang) => ( ( k1_calcul_1(k7_cqc_lang,A) = k1_calcul_1(k7_cqc_lang,B) & r4_calcul_1(A) & r4_calcul_1(B) ) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,k1_calcul_1(k7_cqc_lang,A),k12_finseq_1(k7_cqc_lang,k11_cqc_lang(k2_calcul_1(k7_cqc_lang,A),k2_calcul_1(k7_cqc_lang,B))))) ) ) ) ), inference(mizar_proof,[status(thm)],[d9_calcul_1,d9_calcul_1,d6_calcul_1,t43_cqc_the1,t55_finseq_1,d1_tarski,t106_zfmisc_1,d3_tarski,d4_finseq_1,d8_calcul_1,t48_finseq_1,t48_finseq_1,t27_finseq_3,t27_finseq_3,d8_calcul_1,t34_calcul_1,t6_calcul_1,t2_xreal_1,t34_calcul_1,t27_finseq_3,d8_calcul_1,t35_calcul_1,t8_xreal_1,t29_nat_1,t2_xreal_1,t35_finseq_1,t34_calcul_1,t38_finseq_1,t55_finseq_1,t4_finseq_1,d1_tarski,t57_finseq_1,t7_mcart_1,d7_finseq_1,d8_calcul_1,t6_calcul_1,t27_finseq_3,d7_finseq_1,t27_finseq_3,d7_finseq_1,t6_calcul_1,t38_nat_1,t6_calcul_1,t27_finseq_3,d7_finseq_1,t35_finseq_1,t6_calcul_1,t27_finseq_3,d7_finseq_1,d7_calcul_1,t38_finseq_1,d8_calcul_1,t35_finseq_1,t56_finseq_1,t4_finseq_1,d1_tarski,t55_finseq_1,d7_finseq_1,t57_finseq_1,t7_mcart_1,d9_calcul_1]), [file(calcul_1,t39_calcul_1),interesting(0.00)]). fof(t48_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k10_cqc_lang(A))),k12_finseq_1(k7_cqc_lang,B))) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k10_cqc_lang(B))),k12_finseq_1(k7_cqc_lang,A))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t5_calcul_1,t5_calcul_1,t5_calcul_1,t10_calcul_1,t13_calcul_1,t36_calcul_1,t5_calcul_1,t45_finseq_1,d9_finseq_1,t14_calcul_1,t5_calcul_1,t14_calcul_1,t41_finseq_1,d3_calcul_1,t5_calcul_1,t33_calcul_1,t44_calcul_1,t5_calcul_1,t45_finseq_1,d9_finseq_1,t14_calcul_1,t5_calcul_1,t14_calcul_1,t41_finseq_1,d3_calcul_1,t33_calcul_1,t9_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t37_calcul_1,t5_calcul_1]), [file(calcul_1,t48_calcul_1),interesting(0.00)]). fof(d3_qc_lang2,definition,( ! [A] : ( m1_subset_1(A,k8_qc_lang1) => ! [B] : ( m1_subset_1(B,k8_qc_lang1) => k3_qc_lang2(A,B) = k12_qc_lang1(k13_qc_lang1(k12_qc_lang1(A),k12_qc_lang1(B))) ) ) ), file(qc_lang2,d3_qc_lang2), [interesting(0.00)]). fof(t50_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang) => ! [C] : ( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang) => ! [D] : ( m2_finseq_1(D,k7_cqc_lang) => ( ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,D,k12_finseq_1(k7_cqc_lang,A)),k12_finseq_1(k7_cqc_lang,B))) & r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,D,k12_finseq_1(k7_cqc_lang,C)),k12_finseq_1(k7_cqc_lang,B))) ) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,D,k12_finseq_1(k7_cqc_lang,k13_cqc_lang(A,C))),k12_finseq_1(k7_cqc_lang,B))) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t46_calcul_1,t5_calcul_1,t5_calcul_1,t39_calcul_1,t48_calcul_1,d3_qc_lang2]), [file(calcul_1,t50_calcul_1),interesting(0.00)]). fof(t49_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,A)),k12_finseq_1(k7_cqc_lang,k10_cqc_lang(B)))) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,B)),k12_finseq_1(k7_cqc_lang,k10_cqc_lang(A)))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t5_calcul_1,t5_calcul_1,t5_calcul_1,t10_calcul_1,t13_calcul_1,t36_calcul_1,t5_calcul_1,t45_finseq_1,d9_finseq_1,t14_calcul_1,t5_calcul_1,t14_calcul_1,t41_finseq_1,d3_calcul_1,t33_calcul_1,t5_calcul_1,t5_calcul_1,t10_calcul_1,t3_calcul_1,t44_calcul_1,t5_calcul_1,t45_finseq_1,d9_finseq_1,t14_calcul_1,t5_calcul_1,t14_calcul_1,t41_finseq_1,d3_calcul_1,t33_calcul_1,t9_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t5_calcul_1,t37_calcul_1]), [file(calcul_1,t49_calcul_1),interesting(0.00)]). fof(t53_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang) => ! [C] : ( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang) => ! [D] : ( m2_finseq_1(D,k7_cqc_lang) => ( ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,D,k12_finseq_1(k7_cqc_lang,A)),k12_finseq_1(k7_cqc_lang,B))) & r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,D,k12_finseq_1(k7_cqc_lang,C)),k12_finseq_1(k7_cqc_lang,B))) ) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,D,k12_finseq_1(k7_cqc_lang,k13_cqc_lang(A,C))),k12_finseq_1(k7_cqc_lang,B))) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t46_calcul_1,t5_calcul_1,t39_calcul_1,t48_calcul_1,d3_qc_lang2]), [file(calcul_1,t53_calcul_1),interesting(0.00)]). fof(d2_qc_lang2,definition,( ! [A] : ( m1_subset_1(A,k8_qc_lang1) => ! [B] : ( m1_subset_1(B,k8_qc_lang1) => k2_qc_lang2(A,B) = k12_qc_lang1(k13_qc_lang1(A,k12_qc_lang1(B))) ) ) ), file(qc_lang2,d2_qc_lang2), [interesting(0.00)]). fof(t56_calcul_1,theorem,( ! [A] : ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang) => ! [B] : ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang) => ! [C] : ( m2_finseq_1(C,k7_cqc_lang) => ( ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,k12_cqc_lang(A,B)))) & r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,A))) ) => r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,B))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t5_calcul_1,t5_calcul_1,t8_calcul_1,t36_calcul_1,t5_calcul_1,t59_finseq_1,t56_finseq_1,t35_finseq_1,t10_calcul_1,d3_calcul_1,t33_calcul_1,t10_calcul_1,t5_calcul_1,t3_calcul_1,t44_calcul_1,t5_calcul_1,t59_finseq_1,t56_finseq_1,t35_finseq_1,t10_calcul_1,d3_calcul_1,t33_calcul_1,t53_calcul_1,d3_qc_lang2,t48_calcul_1,t5_calcul_1,t40_calcul_1,t41_calcul_1,t55_calcul_1,t5_calcul_1,t39_calcul_1,t5_calcul_1,t48_calcul_1,d2_qc_lang2,t10_calcul_1,t45_calcul_1,t5_calcul_1]), [file(calcul_1,t56_calcul_1),interesting(0.00)]). fof(d20_substut1,definition,( ! [A] : ( m1_subset_1(A,k16_substut1) => k20_substut1(A) = k1_domain_1(k8_qc_lang1,k1_substut1,k12_qc_lang1(k18_substut1(A)),k19_substut1(A)) ) ), file(substut1,d20_substut1), [interesting(0.00)]). fof(t29_substut1,theorem,( ! [A] : ( m1_subset_1(A,k16_substut1) => k37_substut1(k20_substut1(A)) = k12_qc_lang1(k37_substut1(A)) ) ), file(substut1,t29_substut1), [interesting(0.00)]). fof(d5_qc_lang2,definition,( ! [A] : ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1) => ! [B] : ( m1_subset_1(B,k8_qc_lang1) => k5_qc_lang2(A,B) = k12_qc_lang1(k14_qc_lang1(A,k12_qc_lang1(B))) ) ) ), file(qc_lang2,d5_qc_lang2), [interesting(0.00)]). fof(d10_xboole_0,definition,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), [interesting(0.00)]). fof(t39_finseq_1,theorem,( ! [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_tarski(k4_finseq_1(A),k4_finseq_1(k7_finseq_1(A,B))) ) ) ), file(finseq_1,t39_finseq_1), [interesting(0.00)]). fof(t11_qc_lang3,theorem,( ! [A] : ( m1_subset_1(A,k8_qc_lang1) => k24_qc_lang1(k12_qc_lang1(A)) = k24_qc_lang1(A) ) ), file(qc_lang3,t11_qc_lang3), [interesting(0.00)]). fof(t23_qc_lang3,theorem,( ! [A] : ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1) => ! [B] : ( m1_subset_1(B,k8_qc_lang1) => k24_qc_lang1(k5_qc_lang2(A,B)) = k6_subset_1(k2_qc_lang1,k24_qc_lang1(B),k23_qc_lang1(A)) ) ) ), file(qc_lang3,t23_qc_lang3), [interesting(0.00)]). fof(s2_funct_1,theorem, ( ( ! [A,B,C] : ( ( r2_hidden(A,f1_s2_funct_1) & p1_s2_funct_1(A,B) & p1_s2_funct_1(A,C) ) => B = C ) & ! [A] : ~ ( r2_hidden(A,f1_s2_funct_1) & ! [B] : ~ p1_s2_funct_1(A,B) ) ) => ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & k1_relat_1(A) = f1_s2_funct_1 & ! [B] : ( r2_hidden(B,f1_s2_funct_1) => p1_s2_funct_1(B,k1_funct_1(A,B)) ) ) ), file(funct_1,s2_funct_1), [interesting(0.00)]). fof(d2_finseq_1,definition,( ! [A] : ( v1_relat_1(A) => ( v1_finseq_1(A) <=> ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & k1_relat_1(A) = k2_finseq_1(B) ) ) ) ), file(finseq_1,d2_finseq_1), [interesting(0.00)]). fof(d1_finset_1,definition,( ! [A] : ( v1_finset_1(A) <=> ? [B] : ( v1_relat_1(B) & v1_funct_1(B) & k2_relat_1(B) = A & r2_hidden(k1_relat_1(B),k5_ordinal2) ) ) ), file(finset_1,d1_finset_1), [interesting(0.00)]). fof(t47_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ( r1_tarski(k1_relat_1(A),k2_relat_1(B)) => k2_relat_1(k5_relat_1(B,A)) = k2_relat_1(A) ) ) ) ), file(relat_1,t47_relat_1), [interesting(0.00)]). fof(t20_cqc_sim1,theorem,( ! [A] : ( m1_subset_1(A,k8_qc_lang1) => v1_finset_1(k24_qc_lang1(A)) ) ), file(cqc_sim1,t20_cqc_sim1), [interesting(0.00)]). fof(t25_finset_1,theorem,( ! [A] : ( ( v1_finset_1(A) & ! [B] : ( r2_hidden(B,A) => v1_finset_1(B) ) ) <=> v1_finset_1(k3_tarski(A)) ) ), file(finset_1,t25_finset_1), [interesting(0.00)]). fof(d4_tarski,definition,( ! [A,B] : ( B = k3_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(C,D) & r2_hidden(D,A) ) ) ) ), file(tarski,d4_tarski), [interesting(0.00)]). fof(t11_card_2,theorem,( ! [A,B] : ( r2_wellord2(k2_zfmisc_1(A,B),k2_zfmisc_1(B,A)) & k1_card_1(k2_zfmisc_1(A,B)) = k1_card_1(k2_zfmisc_1(B,A)) ) ), file(card_2,t11_card_2), [interesting(0.00)]). fof(d2_qc_lang1,definition,( k2_qc_lang1 = k2_zfmisc_1(k1_tarski(4),k5_numbers) ), file(qc_lang1,d2_qc_lang1), [interesting(0.00)]). fof(t38_card_1,theorem,( k3_card_1(0) = k1_card_1(k5_numbers) ), file(card_1,t38_card_1), [interesting(0.00)]). fof(t15_card_4,theorem,( ~ v1_finset_1(k5_numbers) ), file(card_4,t15_card_4), [interesting(0.00)]). fof(t81_card_4,theorem,( ! [A,B] : ( v1_finset_1(B) => ( v1_finset_1(A) | B = k1_xboole_0 | ( r2_wellord2(k2_zfmisc_1(A,B),A) & k1_card_1(k2_zfmisc_1(A,B)) = k1_card_1(A) ) ) ) ), file(card_4,t81_card_4), [interesting(0.00)]). fof(t1_card_4,theorem,( ! [A] : ( v1_finset_1(A) <=> v1_finset_1(k1_card_1(A)) ) ), file(card_4,t1_card_4), [interesting(0.00)]).