%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : NUM728^4 : TPTP v9.2.0. Released v7.1.0. % Transfm : none % Format : tptp:raw % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % Computer : n018.cluster.edu % Model : x86_64 x86_64 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz % Memory : 8042.1875MB % OS : Linux 3.10.0-693.el7.x86_64 % CPULimit : 300s % WCLimit : 300s % DateTime : Thu Oct 9 02:11:57 PM UTC 2025 % Result : Theorem 86.16s 86.41s % Output : Proof 86.16s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.11/0.12 % Problem : NUM728^4 : TPTP v9.2.0. Released v7.1.0. % 0.11/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.13/0.34 % Computer : n018.cluster.edu % 0.13/0.34 % Model : x86_64 x86_64 % 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.13/0.34 % Memory : 8042.1875MB % 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.13/0.34 % CPULimit : 300 % 0.13/0.34 % WCLimit : 300 % 0.13/0.34 % DateTime : Wed Oct 8 22:08:53 EDT 2025 % 0.13/0.34 % CPUTime : % 86.16/86.41 % SZS status Theorem % 86.16/86.41 % Mode: mode94:USE_SINE=true:SINE_TOLERANCE=2.0:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=2.:SINE_DEPTH=0 % 86.16/86.41 % Inferences: 4 % 86.16/86.41 % SZS output start Proof % 86.16/86.41 thf(ty_univof, type, univof : ($i>$i)). % 86.16/86.41 thf(ty_if, type, if : ($o>$i>$i>$i)). % 86.16/86.41 thf(ty_d_1to, type, d_1to : ($i>$i)). % 86.16/86.41 thf(ty_eps, type, eps : (($i>$o)>$i)). % 86.16/86.41 thf(ty_d_Pi, type, d_Pi : ($i>($i>$i)>$i)). % 86.16/86.41 thf(ty_d_ReplSep, type, d_ReplSep : ($i>($i>$o)>($i>$i)>$i)). % 86.16/86.41 thf(ty_eigen__1, type, eigen__1 : $i). % 86.16/86.41 thf(ty_ite, type, ite : ($o>$i>$i>$i>$i)). % 86.16/86.41 thf(ty_eigen__0, type, eigen__0 : $i). % 86.16/86.41 thf(ty_emptyset, type, emptyset : $i). % 86.16/86.41 thf(ty_n_1t, type, n_1t : $i). % 86.16/86.41 thf(ty_ind, type, ind : ($i>($i>$o)>$i)). % 86.16/86.41 thf(ty_pair, type, pair : ($i>$i>$i)). % 86.16/86.41 thf(ty_proj1, type, proj1 : ($i>$i)). % 86.16/86.41 thf(ty_n_2t, type, n_2t : $i). % 86.16/86.41 thf(ty_d_Sigma, type, d_Sigma : ($i>($i>$i)>$i)). % 86.16/86.41 thf(ty_repl, type, repl : ($i>($i>$i)>$i)). % 86.16/86.41 thf(ty_in, type, in : ($i>$i>$o)). % 86.16/86.41 thf(ty_d_428_prop2, type, d_428_prop2 : ($i>$i>$o)). % 86.16/86.41 thf(ty_nat_p, type, nat_p : ($i>$o)). % 86.16/86.41 thf(ty_n_2, type, n_2 : $i). % 86.16/86.41 thf(sP1,plain,sP1 <=> (![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) => ((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ eigen__0) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ n_1t) @ X3))))))))) @ proj1)))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ (((d_ReplSep @ ((d_Sigma @ (d_1to @ n_2)) @ (^[X4:$i]:((((ite @ (X4 = n_1t)) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)) @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((![X6:$i]:(((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))) @ ((repl @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ eigen__0) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ n_1t) @ X6))))))))) @ proj1)))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ X1) @ X6))))))))) @ proj1)) @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((![X6:$i]:(((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))) @ ((repl @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ eigen__0) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ n_2t) @ X6))))))))) @ proj1)))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ X1) @ X6))))))))) @ proj1))))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ n_2t) @ X5))))))))) @ proj1)) @ X3))))))))) @ proj1) = (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ ((d_Sigma @ (d_1to @ n_2)) @ (^[X2:$i]:((((ite @ (X2 = n_1t)) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)) @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((![X4:$i]:(((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ eigen__0) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ n_1t) @ X4))))))))) @ proj1)))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X1) @ X4))))))))) @ proj1)) @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))) @ ((repl @ (((if @ 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proj1)) @ X3))))))))) @ proj1)))),introduced(definition,[new_symbols(definition,[sP1])])). % 86.16/86.41 thf(sP2,plain,sP2 <=> (((in @ eigen__1) @ (((if @ (~((![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) => (X1 = emptyset)))))) @ ((repl @ (((if @ (~((![X1:$i]:(((in @ X1) @ (univof @ emptyset)) => (~((nat_p @ X1)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X1:$i]:(((if @ (nat_p @ X1)) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((X1 = emptyset)))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ 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emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ eigen__0) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ n_1t) @ X3))))))))) @ proj1)))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ eigen__1) @ X3))))))))) @ proj1)) @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ eigen__0) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ n_2t) @ X3))))))))) @ proj1)))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ eigen__1) @ X3))))))))) @ proj1))))) @ (^[X1:$i]:(~((![X2:$i]:(~((X1 = ((pair @ n_1t) @ X2))))))))) @ proj1)))) @ (^[X1:$i]:(~((![X2:$i]:(~((X1 = ((pair @ (((d_ReplSep @ eigen__0) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ n_2t) @ X4))))))))) @ proj1)) @ X2))))))))) @ proj1))),introduced(definition,[new_symbols(definition,[sP2])])). % 86.16/86.41 thf(sP3,plain,sP3 <=> (((in @ eigen__0) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X1:$i]:(((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)))) => sP1),introduced(definition,[new_symbols(definition,[sP3])])). % 86.16/86.41 thf(sP4,plain,sP4 <=> ((in @ eigen__1) @ (((if @ (~((![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) => (X1 = emptyset)))))) @ ((repl @ (((if @ (~((![X1:$i]:(((in @ X1) @ (univof @ emptyset)) => (~((nat_p @ X1)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X1:$i]:(((if @ (nat_p @ X1)) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((X1 = emptyset)))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))))))) @ emptyset)),introduced(definition,[new_symbols(definition,[sP4])])). % 86.16/86.41 thf(sP5,plain,sP5 <=> ((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) => (X1 = emptyset)))))) @ ((repl @ (((if @ (~((![X1:$i]:(((in @ X1) @ (univof @ emptyset)) => (~((nat_p @ X1)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X1:$i]:(((if @ (nat_p @ X1)) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((X1 = emptyset)))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ ((d_Sigma @ (d_1to @ n_2)) @ (^[X1:$i]:((((ite @ (X1 = n_1t)) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ eigen__0) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ n_1t) @ X3))))))))) @ proj1)))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ eigen__1) @ X3))))))))) @ proj1)) @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ eigen__0) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ n_2t) @ X3))))))))) @ proj1)))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ eigen__1) @ X3))))))))) @ proj1))))) @ (^[X1:$i]:(~((![X2:$i]:(~((X1 = ((pair @ n_1t) @ X2))))))))) @ proj1)))) @ (^[X1:$i]:(~((![X2:$i]:(~((X1 = ((pair @ (((d_ReplSep @ eigen__0) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ n_2t) @ X4))))))))) @ proj1)) @ X2))))))))) @ proj1) = (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) => (X1 = emptyset)))))) @ ((repl @ (((if @ (~((![X1:$i]:(((in @ X1) @ (univof @ emptyset)) => (~((nat_p @ X1)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X1:$i]:(((if @ (nat_p @ X1)) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((X1 = emptyset)))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ eigen__0) @ (^[X1:$i]:(~((![X2:$i]:(~((X1 = ((pair @ n_1t) @ X2))))))))) @ proj1)))) @ (^[X1:$i]:(~((![X2:$i]:(~((X1 = ((pair @ (((d_ReplSep @ ((d_Sigma @ (d_1to @ n_2)) @ (^[X3:$i]:((((ite @ (X3 = n_1t)) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X4:$i]:(((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ eigen__0) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ n_1t) @ X5))))))))) @ proj1)))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ eigen__1) @ X5))))))))) @ proj1)) @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X4:$i]:(((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ eigen__0) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ n_2t) @ X5))))))))) @ proj1)))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ eigen__1) @ X5))))))))) @ proj1))))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ n_2t) @ X4))))))))) @ proj1)) @ X2))))))))) @ proj1)),introduced(definition,[new_symbols(definition,[sP5])])). % 86.16/86.41 thf(sP6,plain,sP6 <=> (![X1:$i]:(![X2:$i]:((X1 = X2) => (X2 = X1)))),introduced(definition,[new_symbols(definition,[sP6])])). % 86.16/86.41 thf(sP7,plain,sP7 <=> (((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) => (X1 = emptyset)))))) @ ((repl @ (((if @ (~((![X1:$i]:(((in @ X1) @ (univof @ emptyset)) => (~((nat_p @ X1)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X1:$i]:(((if @ (nat_p @ X1)) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((X1 = emptyset)))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ eigen__0) @ (^[X1:$i]:(~((![X2:$i]:(~((X1 = ((pair @ n_1t) @ X2))))))))) @ proj1)))) @ (^[X1:$i]:(~((![X2:$i]:(~((X1 = ((pair @ (((d_ReplSep @ ((d_Sigma @ (d_1to @ n_2)) @ (^[X3:$i]:((((ite @ (X3 = n_1t)) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X4:$i]:(((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ eigen__0) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ n_1t) @ X5))))))))) @ proj1)))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ eigen__1) @ X5))))))))) @ proj1)) @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X4:$i]:(((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in 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(^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ eigen__1) @ X6))))))))) @ proj1))))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ n_2t) @ X5))))))))) @ proj1)) @ X3))))))))) @ proj1)))),introduced(definition,[new_symbols(definition,[sP11])])). % 86.16/86.42 thf(def_is_of,definition,(is_of = (^[X1:$i]:(^[X2:$i>$o]:(X2 @ X1))))). % 86.16/86.42 thf(def_all_of,definition,(all_of = (^[X1:$i>$o]:(^[X2:$i>$o]:(![X3:$i]:(((is_of @ X3) @ X1) => (X2 @ X3))))))). % 86.16/86.42 thf(def_d_Sep,definition,(d_Sep = (^[X1:$i]:(^[X2:$i>$o]:(((if @ (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) @ ((repl @ X1) @ (^[X3:$i]:(((if @ (X2 @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ X1) => (~((X2 @ X4)))))))))))) @ emptyset))))). % 86.16/86.42 thf(def_ordsucc,definition,(ordsucc = (^[X1:$i]:((binunion @ X1) @ (d_Sing @ X1))))). % 86.16/86.42 thf(def_omega,definition,(omega = ((d_Sep @ (univof @ emptyset)) @ nat_p))). % 86.16/86.42 thf(def_ap,definition,(ap = (^[X1:$i]:(^[X2:$i]:(((d_ReplSep @ X1) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1))))). % 86.16/86.42 thf(def_d_not,definition,(d_not = (^[X1:$o]:((imp @ X1) @ $false)))). % 86.16/86.42 thf(def_e_is,definition,(e_is = (^[X1:$i]:(=)))). % 86.16/86.42 thf(def_nat,definition,(nat = ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))). % 86.16/86.42 thf(def_n_is,definition,(n_is = (e_is @ nat))). % 86.16/86.42 thf(def_nis,definition,(nis = (^[X1:$i]:(^[X2:$i]:(d_not @ ((n_is @ X1) @ X2)))))). % 86.16/86.42 thf(def_n_some,definition,(n_some = (l_some @ nat))). % 86.16/86.42 thf(def_n_1,definition,(n_1 = (ordsucc @ emptyset))). % 86.16/86.42 thf(def_n_pl,definition,(n_pl = (^[X1:$i]:(ap @ (plus @ X1))))). % 86.16/86.42 thf(def_d_29_ii,definition,(d_29_ii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X1) @ X2)))))). % 86.16/86.42 thf(def_iii,definition,(iii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X2) @ X1)))))). % 86.16/86.42 thf(def_moreis,definition,(moreis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((d_29_ii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))). % 86.16/86.42 thf(def_lessis,definition,(lessis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((iii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))). % 86.16/86.42 thf(def_times,definition,(times = (^[X1:$i]:((ind @ ((d_Pi @ nat) @ (^[X2:$i]:nat))) @ (d_428_prop2 @ X1))))). % 86.16/86.42 thf(def_n_ts,definition,(n_ts = (^[X1:$i]:(ap @ (times @ X1))))). % 86.16/86.42 thf(def_pair1type,definition,(pair1type = (^[X1:$i]:((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:X1))))). % 86.16/86.42 thf(def_pair1,definition,(pair1 = (^[X1:$i]:(^[X2:$i]:(^[X3:$i]:((d_Sigma @ (d_1to @ n_2)) @ (^[X4:$i]:((((ite @ (((e_is @ (d_1to @ n_2)) @ X4) @ n_1t)) @ X1) @ X2) @ X3)))))))). % 86.16/86.42 thf(def_first1,definition,(first1 = (^[X1:$i]:(^[X2:$i]:((ap @ X2) @ n_1t))))). % 86.16/86.42 thf(def_second1,definition,(second1 = (^[X1:$i]:(^[X2:$i]:((ap @ X2) @ n_2t))))). % 86.16/86.42 thf(def_frac,definition,(frac = (pair1type @ nat))). % 86.16/86.42 thf(def_n_fr,definition,(n_fr = (pair1 @ nat))). % 86.16/86.42 thf(def_num,definition,(num = (first1 @ nat))). % 86.16/86.42 thf(def_den,definition,(den = (second1 @ nat))). % 86.16/86.42 thf(def_n_eq,definition,(n_eq = (^[X1:$i]:(^[X2:$i]:((n_is @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1))))))). % 86.16/86.42 thf(satz40a,conjecture,(![X1:$i]:(((in @ X1) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:(((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (~((~((X3 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (~((~((X4 = emptyset)))))))))))))) @ emptyset)))) => (![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (~((~((X3 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (~((~((X4 = emptyset)))))))))))))) @ emptyset)) => ((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (~((~((X3 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (~((~((X4 = emptyset)))))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((![X4:$i]:(((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (~((~((X4 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = emptyset)))))))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ ((d_Sigma @ (d_1to @ n_2)) @ (^[X3:$i]:((((ite @ (X3 = n_1t)) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (~((~((X4 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = emptyset)))))))))))))) @ emptyset)) @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X4:$i]:(((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (~((~((X4 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = emptyset)))))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ X1) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ n_1t) @ X5))))))))) @ proj1)))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X2) @ X5))))))))) @ proj1)) @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X4:$i]:(((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (~((~((X4 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = emptyset)))))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = 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@ (^[X5:$i]:(((if @ (~((![X6:$i]:(((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))) @ ((repl @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep 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X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ eigen__0) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ n_2t) @ X6))))))))) @ proj1)))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ X1) @ X6))))))))) @ proj1))))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ n_2t) @ X5))))))))) @ proj1)) @ X3))))))))) @ proj1)))))),introduced(assumption,[])). % 86.16/86.42 thf(h4,assumption,(~((sP4 => sP5))),introduced(assumption,[])). % 86.16/86.42 thf(h5,assumption,sP4,introduced(assumption,[])). % 86.16/86.42 thf(h6,assumption,(~(sP5)),introduced(assumption,[])). % 86.16/86.42 thf(1,plain,(~(sP1) | sP2),inference(all_rule,[status(thm)],[])). % 86.16/86.42 thf(2,plain,((~(sP2) | ~(sP4)) | sP10),inference(prop_rule,[status(thm)],[])). % 86.16/86.42 thf(3,plain,(~(sP8) | sP3),inference(all_rule,[status(thm)],[])). % 86.16/86.42 thf(4,plain,((~(sP3) | ~(sP9)) | sP1),inference(prop_rule,[status(thm)],[])). % 86.16/86.42 thf(5,plain,((~(sP7) | ~(sP10)) | sP5),inference(prop_rule,[status(thm)],[])). % 86.16/86.42 thf(6,plain,(~(sP11) | sP7),inference(all_rule,[status(thm)],[])). % 86.16/86.42 thf(7,plain,(~(sP6) | sP11),inference(all_rule,[status(thm)],[])). % 86.16/86.42 thf(8,plain,sP6,inference(eq_sym,[status(thm)],[])). % 86.16/86.42 thf(satz40,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ frac))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ nat))) @ (^[X2:$i]:((n_eq @ X1) @ ((n_fr @ ((n_ts @ (num @ X1)) @ X2)) @ ((n_ts @ (den @ X1)) @ X2)))))))). % 86.16/86.42 thf(9,plain,sP8,inference(preprocess,[status(thm)],[satz40]). % 86.16/86.42 thf(10,plain,$false,inference(prop_unsat,[status(thm),assumptions([h5,h6,h4,h2,h3,h1,h0])],[1,2,3,4,5,6,7,8,h2,h5,h6,9])). % 86.16/86.42 thf(11,plain,$false,inference(tab_negimp,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,10,h5,h6])). % 86.16/86.42 thf(12,plain,$false,inference(tab_negall,[status(thm),assumptions([h2,h3,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,11,h4])). % 86.16/86.42 thf(13,plain,$false,inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,12,h2,h3])). % 86.16/86.42 thf(14,plain,$false,inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,13,h1])). % 86.16/86.42 thf(0,theorem,(![X1:$i]:(((in @ X1) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:(((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (~((~((X3 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (~((~((X4 = emptyset)))))))))))))) @ emptyset)))) => (![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (~((~((X3 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (~((~((X4 = emptyset)))))))))))))) @ emptyset)) => ((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ 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=> (~((~((X4 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = emptyset)))))))))))))) @ emptyset)))) @ (d_428_prop2 @ (((d_ReplSep @ ((d_Sigma @ (d_1to @ n_2)) @ (^[X3:$i]:((((ite @ (X3 = n_1t)) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ 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(^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ n_2t) @ X6))))))))) @ proj1)) @ X4))))))))) @ proj1)))))),inference(contra,[status(thm),contra(discharge,[h0])],[14,h0])). % 86.16/86.43 % SZS output end Proof %------------------------------------------------------------------------------