fof(t33_algseq_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v3_struct_0(B) & l2_struct_0(B) ) => k2_normsp_1(B,k5_algseq_1(B,k1_rlvect_1(B)),A) = k1_rlvect_1(B) ) ) ), inference(mizar_proof,[status(thm)],[t19_nat_1,t31_algseq_1,t22_algseq_1,d6_algseq_1]), [file(algseq_1,t33_algseq_1),interesting(1.00)]). fof(t14_algseq_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( k2_algseq_1(A) = k2_algseq_1(B) => A = B ) ) ) ), inference(mizar_proof,[status(thm)],[t13_algseq_1,t1_xreal_1]), [file(algseq_1,t14_algseq_1),interesting(0.79)]). fof(t12_algseq_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => r2_hidden(A,k2_algseq_1(k1_nat_1(A,1))) ) ), inference(mizar_proof,[status(thm)],[t31_xreal_1]), [file(algseq_1,t12_algseq_1),interesting(0.78)]). fof(t13_algseq_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r1_xreal_0(A,B) <=> r1_tarski(k2_algseq_1(A),k2_algseq_1(B)) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,t10_algseq_1,t2_xreal_1,t10_algseq_1]), [file(algseq_1,t13_algseq_1),interesting(0.74)]). fof(t17_algseq_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k2_xboole_0(k2_algseq_1(A),k1_tarski(A)) = k2_algseq_1(k1_nat_1(A,1)) ) ), inference(mizar_proof,[status(thm)],[t9_xreal_1,t13_algseq_1,d3_tarski,d2_xboole_0,d1_tarski,t12_algseq_1,d10_xboole_0,d3_tarski,t10_algseq_1,t70_nat_1,d1_tarski,d2_xboole_0]), [file(algseq_1,t17_algseq_1),interesting(0.72)]). fof(t32_algseq_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l2_struct_0(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & v1_algseq_1(B,A) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ( B = k5_algseq_1(A,k1_rlvect_1(A)) <=> k4_algseq_1(A,B) = k1_xboole_0 ) ) ) ), inference(mizar_proof,[status(thm)],[l40_algseq_1,t11_algseq_1,t11_algseq_1,t14_algseq_1,t31_algseq_1]), [file(algseq_1,t32_algseq_1),interesting(0.72)]). fof(t34_algseq_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l2_struct_0(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & v1_algseq_1(B,A) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ( B = k5_algseq_1(A,k1_rlvect_1(A)) <=> k2_relat_1(B) = k1_struct_0(A,k1_rlvect_1(A)) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d5_funct_1,d1_funct_2,t33_algseq_1,d1_tarski,d10_xboole_0,d3_tarski,d1_tarski,d6_algseq_1,d1_funct_2,d5_funct_1,d1_funct_2,d5_funct_1,d1_tarski,d3_algseq_1,t18_nat_1,d4_algseq_1,t31_algseq_1]), [file(algseq_1,t34_algseq_1),interesting(0.72)]). fof(t31_algseq_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l2_struct_0(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & v1_algseq_1(B,A) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ( B = k5_algseq_1(A,k1_rlvect_1(A)) <=> k3_algseq_1(A,B) = 0 ) ) ) ), inference(mizar_proof,[status(thm)],[l40_algseq_1,l40_algseq_1,t18_nat_1,t28_algseq_1]), [file(algseq_1,t31_algseq_1),interesting(0.67)]). fof(t11_algseq_1,theorem, ( k2_algseq_1(0) = k1_xboole_0 & k2_algseq_1(1) = k1_tarski(0) & k2_algseq_1(2) = k2_tarski(0,1) ), inference(mizar_proof,[status(thm)],[l2_algseq_1,t10_algseq_1,t18_nat_1,t38_nat_1,t18_nat_1,d1_tarski,d1_tarski,t2_tarski,t71_nat_1,d2_tarski,d2_tarski,t2_tarski]), [file(algseq_1,t11_algseq_1),interesting(0.65)]). fof(l22_algseq_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l2_struct_0(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & v1_algseq_1(B,A) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & r1_algseq_1(A,B,C) ) ) ) ), inference(mizar_proof,[status(thm)],[d2_algseq_1,d3_algseq_1]), [file(algseq_1,l22_algseq_1),interesting(0.62)]). fof(l40_algseq_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l2_struct_0(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & v1_algseq_1(B,A) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ( B = k5_algseq_1(A,k1_rlvect_1(A)) => k3_algseq_1(A,B) = 0 ) ) ) ), inference(mizar_proof,[status(thm)],[d6_algseq_1,t25_algseq_1,d5_real_1,t39_nat_1]), [file(algseq_1,l40_algseq_1),interesting(0.58)]). fof(l23_algseq_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l2_struct_0(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & v1_algseq_1(B,A) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & r1_algseq_1(A,B,C) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r1_algseq_1(A,B,D) => r1_xreal_0(C,D) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[l22_algseq_1,s5_nat_1]), [file(algseq_1,l23_algseq_1),interesting(0.36)]). fof(t22_algseq_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v3_struct_0(B) & l2_struct_0(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,u1_struct_0(B)) & v1_algseq_1(C,B) & m2_relset_1(C,k5_numbers,u1_struct_0(B)) ) => ( r1_xreal_0(k3_algseq_1(B,C),A) => k2_normsp_1(B,C,A) = k1_rlvect_1(B) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d4_algseq_1,d3_algseq_1]), [file(algseq_1,t22_algseq_1),interesting(0.35)]). fof(t29_algseq_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l2_struct_0(A) ) => ( u1_struct_0(A) != k1_struct_0(A,k1_rlvect_1(A)) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ? [C] : ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,u1_struct_0(A)) & v1_algseq_1(C,A) & m2_relset_1(C,k5_numbers,u1_struct_0(A)) & k3_algseq_1(A,C) = B ) ) ) ) ), inference(mizar_proof,[status(thm)],[t41_zfmisc_1,s1_algseq_1,t24_algseq_1,t1_xreal_1]), [file(algseq_1,t29_algseq_1),interesting(0.30)]). fof(s1_algseq_1,theorem,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,u1_struct_0(f1_s1_algseq_1)) & v1_algseq_1(A,f1_s1_algseq_1) & m2_relset_1(A,k5_numbers,u1_struct_0(f1_s1_algseq_1)) & r1_xreal_0(k3_algseq_1(f1_s1_algseq_1,A),f2_s1_algseq_1) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(f2_s1_algseq_1,B) => k2_normsp_1(f1_s1_algseq_1,A,B) = f3_s1_algseq_1(B) ) ) ) ), inference(mizar_proof,[status(thm)],[s3_funct_2,d2_algseq_1,d3_algseq_1,d4_algseq_1]), [file(algseq_1,s1_algseq_1),interesting(0.29)]). fof(t24_algseq_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v3_struct_0(B) & l2_struct_0(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,u1_struct_0(B)) & v1_algseq_1(C,B) & m2_relset_1(C,k5_numbers,u1_struct_0(B)) ) => ( ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ~ ( ~ r1_xreal_0(A,D) & k2_normsp_1(B,C,D) = k1_rlvect_1(B) ) ) => r1_xreal_0(A,k3_algseq_1(B,C)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t22_algseq_1]), [file(algseq_1,t24_algseq_1),interesting(0.29)]). fof(t25_algseq_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v3_struct_0(B) & l2_struct_0(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,u1_struct_0(B)) & v1_algseq_1(C,B) & m2_relset_1(C,k5_numbers,u1_struct_0(B)) ) => ~ ( k3_algseq_1(B,C) = k1_nat_1(A,1) & k2_normsp_1(B,C,A) = k1_rlvect_1(B) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t31_xreal_1,d4_algseq_1,d3_algseq_1,t22_algseq_1,t38_nat_1,t1_xreal_1]), [file(algseq_1,t25_algseq_1),interesting(0.27)]). fof(t28_algseq_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l2_struct_0(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & v1_algseq_1(B,A) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,u1_struct_0(A)) & v1_algseq_1(C,A) & m2_relset_1(C,k5_numbers,u1_struct_0(A)) ) => ( ( k3_algseq_1(A,B) = k3_algseq_1(A,C) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(k3_algseq_1(A,B),D) => k2_normsp_1(A,B,D) = k2_normsp_1(A,C,D) ) ) ) => B = C ) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_funct_2,t22_algseq_1,t9_funct_1]), [file(algseq_1,t28_algseq_1),interesting(0.23)]). fof(t27_algseq_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v3_struct_0(B) & l2_struct_0(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,u1_struct_0(B)) & v1_algseq_1(C,B) & m2_relset_1(C,k5_numbers,u1_struct_0(B)) ) => ( A = k3_algseq_1(B,C) <=> k2_algseq_1(A) = k4_algseq_1(B,C) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t14_algseq_1]), [file(algseq_1,t27_algseq_1),interesting(0.21)]). fof(d2_algseq_1,definition,( ! [A] : ( ( ~ v3_struct_0(A) & l2_struct_0(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ( v1_algseq_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) => k2_normsp_1(A,B,D) = k1_rlvect_1(A) ) ) ) ) ) ) ), file(algseq_1,d2_algseq_1), [interesting(0.00)]). fof(d3_algseq_1,definition,( ! [A] : ( ( ~ v3_struct_0(A) & l2_struct_0(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & v1_algseq_1(B,A) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r1_algseq_1(A,B,C) <=> ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r1_xreal_0(C,D) => k2_normsp_1(A,B,D) = k1_rlvect_1(A) ) ) ) ) ) ) ), file(algseq_1,d3_algseq_1), [interesting(0.00)]). fof(s5_nat_1,theorem, ( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & p1_s5_nat_1(A) ) => ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & p1_s5_nat_1(A) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( p1_s5_nat_1(B) => r1_xreal_0(A,B) ) ) ) ), file(nat_1,s5_nat_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(l24_algseq_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( ( ~ v3_struct_0(C) & l2_struct_0(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k5_numbers,u1_struct_0(C)) & v1_algseq_1(D,C) & m2_relset_1(D,k5_numbers,u1_struct_0(C)) ) => ( ( r1_algseq_1(C,D,A) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( r1_algseq_1(C,D,E) => r1_xreal_0(A,E) ) ) & r1_algseq_1(C,D,B) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( r1_algseq_1(C,D,E) => r1_xreal_0(B,E) ) ) ) => A = B ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t1_xreal_1]), [file(algseq_1,l24_algseq_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(t10_algseq_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r2_hidden(A,k2_algseq_1(B)) <=> ~ r1_xreal_0(B,A) ) ) ) ), file(algseq_1,t10_algseq_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(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(t15_algseq_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r1_xreal_0(A,B) => ( k2_algseq_1(A) = k5_subset_1(k5_numbers,k2_algseq_1(A),k2_algseq_1(B)) & k2_algseq_1(A) = k5_subset_1(k5_numbers,k2_algseq_1(B),k2_algseq_1(A)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t13_algseq_1,t28_xboole_1]), [file(algseq_1,t15_algseq_1),interesting(0.00)]). fof(t16_algseq_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( k2_algseq_1(A) = k5_subset_1(k5_numbers,k2_algseq_1(A),k2_algseq_1(B)) | k2_algseq_1(A) = k5_subset_1(k5_numbers,k2_algseq_1(B),k2_algseq_1(A)) ) => r1_xreal_0(A,B) ) ) ) ), inference(mizar_proof,[status(thm)],[t15_algseq_1,t14_algseq_1]), [file(algseq_1,t16_algseq_1),interesting(0.00)]). fof(t9_xreal_1,theorem,( ! [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(k2_xcmplx_0(A,C),k2_xcmplx_0(B,D)) ) ) ) ) ) ), file(xreal_1,t9_xreal_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(t31_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(k2_xcmplx_0(B,A),B) ) ) ) ), file(xreal_1,t31_xreal_1), [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(t70_nat_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( ~ r1_xreal_0(k1_nat_1(B,1),A) & r1_xreal_0(B,A) & A != B ) ) ) ), file(nat_1,t70_nat_1), [interesting(0.00)]). fof(t18_algseq_1,theorem,( $true ), file(algseq_1,t18_algseq_1), [interesting(0.00)]). fof(t19_algseq_1,theorem,( $true ), file(algseq_1,t19_algseq_1), [interesting(0.00)]). fof(t1_algseq_1,theorem,( $true ), file(algseq_1,t1_algseq_1), [interesting(0.00)]). fof(t20_algseq_1,theorem,( $true ), file(algseq_1,t20_algseq_1), [interesting(0.00)]). fof(t21_algseq_1,theorem,( $true ), file(algseq_1,t21_algseq_1), [interesting(0.00)]). fof(t23_algseq_1,theorem,( $true ), file(algseq_1,t23_algseq_1), [interesting(0.00)]). fof(t26_algseq_1,theorem,( $true ), file(algseq_1,t26_algseq_1), [interesting(0.00)]). fof(t41_zfmisc_1,theorem,( ! [A,B] : ~ ( A != k1_tarski(B) & A != k1_xboole_0 & ! [C] : ~ ( r2_hidden(C,A) & C != B ) ) ), file(zfmisc_1,t41_zfmisc_1), [interesting(0.00)]). fof(s3_funct_2,theorem, ( ! [A] : ( m1_subset_1(A,f1_s3_funct_2) => ? [B] : ( m1_subset_1(B,f2_s3_funct_2) & p1_s3_funct_2(A,B) ) ) => ? [A] : ( v1_funct_1(A) & v1_funct_2(A,f1_s3_funct_2,f2_s3_funct_2) & m2_relset_1(A,f1_s3_funct_2,f2_s3_funct_2) & ! [B] : ( m1_subset_1(B,f1_s3_funct_2) => p1_s3_funct_2(B,k8_funct_2(f1_s3_funct_2,f2_s3_funct_2,A,B)) ) ) ), file(funct_2,s3_funct_2), [interesting(0.00)]). fof(d4_algseq_1,definition,( ! [A] : ( ( ~ v3_struct_0(A) & l2_struct_0(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & v1_algseq_1(B,A) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( C = k3_algseq_1(A,B) <=> ( r1_algseq_1(A,B,C) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r1_algseq_1(A,B,D) => r1_xreal_0(C,D) ) ) ) ) ) ) ) ), file(algseq_1,d4_algseq_1), [interesting(0.00)]). fof(t2_algseq_1,theorem,( $true ), file(algseq_1,t2_algseq_1), [interesting(0.00)]). fof(t30_algseq_1,theorem,( $true ), file(algseq_1,t30_algseq_1), [interesting(0.00)]). fof(d6_algseq_1,definition,( ! [A] : ( ( ~ v3_struct_0(A) & l2_struct_0(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,u1_struct_0(A)) & v1_algseq_1(C,A) & m2_relset_1(C,k5_numbers,u1_struct_0(A)) ) => ( C = k5_algseq_1(A,B) <=> ( r1_xreal_0(k3_algseq_1(A,C),1) & k2_normsp_1(A,C,0) = B ) ) ) ) ) ), file(algseq_1,d6_algseq_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(d5_real_1,definition,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) <=> ~ ( r1_xreal_0(B,A) & B != A ) ) ) ) ), file(real_1,d5_real_1), [interesting(0.00)]). fof(t39_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ( ~ r1_xreal_0(1,A) => A = 0 ) ) ), file(nat_1,t39_nat_1), [interesting(0.00)]). fof(l2_algseq_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( r2_hidden(B,k2_algseq_1(A)) => m2_subset_1(B,k1_numbers,k5_numbers) ) ) ), file(algseq_1,l2_algseq_1), [interesting(0.00)]). fof(t18_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => r1_xreal_0(0,A) ) ), file(nat_1,t18_nat_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(t71_nat_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( ~ r1_xreal_0(2,A) & A != 0 & A != 1 ) ) ), file(nat_1,t71_nat_1), [interesting(0.00)]). fof(d2_tarski,definition,( ! [A,B,C] : ( C = k2_tarski(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( D = A | D = B ) ) ) ), file(tarski,d2_tarski), [interesting(0.00)]). fof(d1_funct_2,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( ( ( B = k1_xboole_0 => A = k1_xboole_0 ) => ( v1_funct_2(C,A,B) <=> A = k4_relset_1(A,B,C) ) ) & ( B = k1_xboole_0 => ( A = k1_xboole_0 | ( v1_funct_2(C,A,B) <=> C = k1_xboole_0 ) ) ) ) ) ), file(funct_2,d1_funct_2), [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_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(t19_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ~ ( 0 != A & r1_xreal_0(A,0) ) ) ), file(nat_1,t19_nat_1), [interesting(0.00)]). fof(t3_algseq_1,theorem,( $true ), file(algseq_1,t3_algseq_1), [interesting(0.00)]). fof(t4_algseq_1,theorem,( $true ), file(algseq_1,t4_algseq_1), [interesting(0.00)]). fof(t5_algseq_1,theorem,( $true ), file(algseq_1,t5_algseq_1), [interesting(0.00)]). fof(t6_algseq_1,theorem,( $true ), file(algseq_1,t6_algseq_1), [interesting(0.00)]). fof(t7_algseq_1,theorem,( $true ), file(algseq_1,t7_algseq_1), [interesting(0.00)]). fof(t8_algseq_1,theorem,( $true ), file(algseq_1,t8_algseq_1), [interesting(0.00)]). fof(t9_algseq_1,theorem,( $true ), file(algseq_1,t9_algseq_1), [interesting(0.00)]).