%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : NUM656^4 : 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 : n003.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 02:11:39 PM UTC 2025 % Result : Theorem 230.05s 230.11s % Output : Proof 230.05s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.11/0.12 % Problem : NUM656^4 : TPTP v9.2.0. Released v7.1.0. % 0.11/0.12 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.12/0.33 % Computer : n003.cluster.edu % 0.12/0.33 % Model : x86_64 x86_64 % 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.33 % Memory : 8042.1875MB % 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.33 % CPULimit : 300 % 0.12/0.33 % WCLimit : 300 % 0.12/0.33 % DateTime : Wed Oct 8 22:57:08 EDT 2025 % 0.18/0.33 % CPUTime : % 230.05/230.11 % SZS status Theorem % 230.05/230.11 % Mode: mode92:USE_SINE=true:SINE_TOLERANCE=5.0:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=2.:SINE_DEPTH=0 % 230.05/230.11 % Inferences: 253 % 230.05/230.11 % SZS output start Proof % 230.05/230.11 thf(ty_univof, type, univof : ($i>$i)). % 230.05/230.11 thf(ty_l_some, type, l_some : ($i>($i>$o)>$o)). % 230.05/230.11 thf(ty_d_Inj0, type, d_Inj0 : ($i>$i)). % 230.05/230.11 thf(ty_eps, type, eps : (($i>$o)>$i)). % 230.05/230.11 thf(ty_d_ReplSep, type, d_ReplSep : ($i>($i>$o)>($i>$i)>$i)). % 230.05/230.11 thf(ty_eigen__1, type, eigen__1 : $i). % 230.05/230.11 thf(ty_eigen__0, type, eigen__0 : $i). % 230.05/230.11 thf(ty_d_Unj, type, d_Unj : ($i>$i)). % 230.05/230.11 thf(ty_emptyset, type, emptyset : $i). % 230.05/230.11 thf(ty_plus, type, plus : ($i>$i)). % 230.05/230.11 thf(ty_d_Inj1, type, d_Inj1 : ($i>$i)). % 230.05/230.11 thf(ty_repl, type, repl : ($i>($i>$i)>$i)). % 230.05/230.11 thf(ty_in, type, in : ($i>$i>$o)). % 230.05/230.11 thf(ty_nat_p, type, nat_p : ($i>$o)). % 230.05/230.11 thf(ty_binunion, type, binunion : ($i>$i>$i)). % 230.05/230.11 thf(sP1,plain,sP1 <=> ((~(((l_some @ (eps @ (^[X1:$i]:(((~((![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))) => (~((X3 = emptyset)))))))))) => (X2 = emptyset))))) => (~((X1 = ((repl @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3))))))) => (~((X2 = ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((nat_p @ X3) => (~((X4 = X3)))) => (~(((~((nat_p @ X3))) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))))))))))))))))))) => (~(((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3))))) => (~((X2 = emptyset)))))))))) @ (^[X2:$i]:(eps @ (^[X3:$i]:(((~((X2 = emptyset))) => (~((X3 = X2)))) => (~(((X2 = emptyset) => (~((X3 = (eps @ (^[X4:$i]:(~((((in @ X4) @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((nat_p @ X6) => (~((X7 = X6)))) => (~(((~((nat_p @ X6))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))))))))))))))))))))) => (~(((![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))) => (~((X3 = emptyset)))))))))) => (X2 = emptyset))) => (~((X1 = emptyset)))))))))) @ (^[X1:$i]:(eigen__1 = (((d_ReplSep @ (plus @ eigen__0)) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((binunion @ ((repl @ X1) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))))) @ (^[X2:$i]:(((d_ReplSep @ X2) @ (^[X3:$i]:(~((![X4:$i]:(~(((d_Inj1 @ X4) = X3)))))))) @ d_Unj)))))))) => (eigen__0 = eigen__1)),introduced(definition,[new_symbols(definition,[sP1])])). % 230.05/230.11 thf(sP2,plain,sP2 <=> (![X1:$i]:(((in @ X1) @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))))) => (~((X2 = ((repl @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))) => (~((X3 = emptyset)))))))))) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((~((X3 = emptyset))) => (~((X4 = X3)))) => (~(((X3 = emptyset) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (eps @ (^[X6:$i]:(((~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))) => (~((X6 = ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(eps @ (^[X8:$i]:(((nat_p @ X7) => (~((X8 = X7)))) => (~(((~((nat_p @ X7))) => (~((X8 = (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9))))))))))))))))))))))) => (~(((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))) => (~((X6 = emptyset)))))))))) => (X5 = emptyset))))))))))))))))))))) => (~(((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))) => (~((X2 = emptyset)))))))))) => (![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((nat_p @ X6) => (~((X7 = X6)))) => (~(((~((nat_p @ X6))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))))) => (~((X3 = ((repl @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))) => (~((X4 = emptyset)))))))))) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((~((X4 = emptyset))) => (~((X5 = X4)))) => (~(((X4 = emptyset) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (eps @ (^[X7:$i]:(((~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))))) => (~((X7 = ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(eps @ (^[X9:$i]:(((nat_p @ X8) => (~((X9 = X8)))) => (~(((~((nat_p @ X8))) => (~((X9 = (eps @ (^[X10:$i]:(~((((in @ X10) @ (univof @ emptyset)) => (~((nat_p @ X10))))))))))))))))))))))) => (~(((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))) => (~((X7 = emptyset)))))))))) => (X6 = emptyset))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((nat_p @ X6) => (~((X7 = X6)))) => (~(((~((nat_p @ X6))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))) => (~((X3 = emptyset)))))))))) => (((~(((l_some @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((nat_p @ X6) => (~((X7 = X6)))) => (~(((~((nat_p @ X6))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))))) => (~((X3 = ((repl @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))) => (~((X4 = emptyset)))))))))) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((~((X4 = emptyset))) => (~((X5 = X4)))) => (~(((X4 = emptyset) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (eps @ (^[X7:$i]:(((~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))))) => (~((X7 = ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(eps @ (^[X9:$i]:(((nat_p @ X8) => (~((X9 = X8)))) => (~(((~((nat_p @ X8))) => (~((X9 = (eps @ (^[X10:$i]:(~((((in @ X10) @ (univof @ emptyset)) => (~((nat_p @ X10))))))))))))))))))))))) => (~(((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))) => (~((X7 = emptyset)))))))))) => (X6 = emptyset))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((nat_p @ X6) => (~((X7 = X6)))) => (~(((~((nat_p @ X6))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))) => (~((X3 = emptyset)))))))))) @ (^[X3:$i]:(X2 = (((d_ReplSep @ (plus @ X1)) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))))))) => (X1 = X2)) => (~(((l_some @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((nat_p @ X6) => (~((X7 = X6)))) => (~(((~((nat_p @ X6))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))))) => (~((X3 = ((repl @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))) => (~((X4 = emptyset)))))))))) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((~((X4 = emptyset))) => (~((X5 = X4)))) => (~(((X4 = emptyset) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (eps @ (^[X7:$i]:(((~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))))) => (~((X7 = ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(eps @ (^[X9:$i]:(((nat_p @ X8) => (~((X9 = X8)))) => (~(((~((nat_p @ X8))) => (~((X9 = (eps @ (^[X10:$i]:(~((((in @ X10) @ (univof @ emptyset)) => (~((nat_p @ X10))))))))))))))))))))))) => (~(((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))) => (~((X7 = emptyset)))))))))) => (X6 = emptyset))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((nat_p @ X6) => (~((X7 = X6)))) => (~(((~((nat_p @ X6))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))) => (~((X3 = emptyset)))))))))) @ (^[X3:$i]:(X1 = (((d_ReplSep @ (plus @ X2)) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj))))))))))))),introduced(definition,[new_symbols(definition,[sP2])])). % 230.05/230.11 thf(sP3,plain,sP3 <=> ((l_some @ (eps @ (^[X1:$i]:(((~((![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))) => (~((X3 = emptyset)))))))))) => (X2 = emptyset))))) => (~((X1 = ((repl @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3))))))) => (~((X2 = ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((nat_p @ X3) => (~((X4 = X3)))) => (~(((~((nat_p @ X3))) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))))))))))))))))))) => (~(((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3))))) => (~((X2 = emptyset)))))))))) @ (^[X2:$i]:(eps @ (^[X3:$i]:(((~((X2 = emptyset))) => (~((X3 = X2)))) => (~(((X2 = emptyset) => (~((X3 = (eps @ (^[X4:$i]:(~((((in @ X4) @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((nat_p @ X6) => (~((X7 = X6)))) => (~(((~((nat_p @ X6))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))))))))))))))))))))) => (~(((![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))) => (~((X3 = emptyset)))))))))) => (X2 = emptyset))) => (~((X1 = emptyset)))))))))) @ (^[X1:$i]:(eigen__0 = (((d_ReplSep @ (plus @ eigen__1)) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((binunion @ ((repl @ X1) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))))) @ (^[X2:$i]:(((d_ReplSep @ X2) @ (^[X3:$i]:(~((![X4:$i]:(~(((d_Inj1 @ X4) = X3)))))))) @ d_Unj)))))),introduced(definition,[new_symbols(definition,[sP3])])). % 230.05/230.11 thf(sP4,plain,sP4 <=> ((in @ eigen__1) @ (eps @ (^[X1:$i]:(((~((![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ 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(~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((nat_p @ X6) => (~((X7 = X6)))) => (~(((~((nat_p @ X6))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))))))))))))))))))))) => (~(((![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))) => (~((X3 = emptyset)))))))))) => (X2 = emptyset))) => (~((X1 = emptyset)))))))))),introduced(definition,[new_symbols(definition,[sP4])])). % 230.05/230.11 thf(sP5,plain,sP5 <=> ((l_some @ (eps @ (^[X1:$i]:(((~((![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))) => (~((X3 = emptyset)))))))))) => (X2 = emptyset))))) => (~((X1 = ((repl @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3))))))) => (~((X2 = ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((nat_p @ X3) => (~((X4 = X3)))) => (~(((~((nat_p @ X3))) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))))))))))))))))))) => (~(((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3))))) => (~((X2 = emptyset)))))))))) @ (^[X2:$i]:(eps @ 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=> (~((X7 = X6)))) => (~(((~((nat_p @ X6))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))))))))))))))))))))) => (~(((![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))) => (~((X3 = emptyset)))))))))) => (X2 = emptyset))) => (~((X1 = emptyset)))))))))) => (![X1:$i]:(((in @ X1) @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl 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emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))) => (~((X3 = emptyset)))))))))) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((~((X3 = emptyset))) => (~((X4 = X3)))) => (~(((X3 = emptyset) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (eps @ (^[X6:$i]:(((~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))) => (~((X6 = ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(eps @ (^[X8:$i]:(((nat_p @ X7) => (~((X8 = X7)))) => (~(((~((nat_p @ X7))) => (~((X8 = (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9))))))))))))))))))))))) => (~(((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))) => (~((X6 = emptyset)))))))))) => (X5 = emptyset))))))))))))))))))))) => (~(((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))) => (~((X2 = emptyset)))))))))) @ (^[X2:$i]:(eigen__0 = (((d_ReplSep @ (plus @ X1)) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ X2) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj)))))))))))),introduced(definition,[new_symbols(definition,[sP8])])). % 230.05/230.11 thf(sP9,plain,sP9 <=> ((in @ eigen__0) @ (eps @ (^[X1:$i]:(((~((![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))) => (~((X3 = emptyset)))))))))) => (X2 = emptyset))))) => (~((X1 = ((repl @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3))))))) => (~((X2 = ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((nat_p @ X3) => (~((X4 = X3)))) => (~(((~((nat_p @ X3))) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))))))))))))))))))) => (~(((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3))))) => (~((X2 = emptyset)))))))))) @ (^[X2:$i]:(eps @ (^[X3:$i]:(((~((X2 = emptyset))) => (~((X3 = X2)))) => (~(((X2 = emptyset) => (~((X3 = (eps @ (^[X4:$i]:(~((((in @ X4) @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((nat_p @ X6) => (~((X7 = X6)))) => (~(((~((nat_p @ X6))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))))))))))))))))))))) => (~(((![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))) => (~((X3 = emptyset)))))))))) => (X2 = emptyset))) => (~((X1 = emptyset)))))))))),introduced(definition,[new_symbols(definition,[sP9])])). % 230.05/230.11 thf(sP10,plain,sP10 <=> (![X1:$i]:(((in @ X1) @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))))) => (~((X2 = ((repl @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))) => (~((X3 = emptyset)))))))))) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((~((X3 = emptyset))) => (~((X4 = X3)))) => (~(((X3 = emptyset) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (eps @ (^[X6:$i]:(((~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))) => (~((X6 = ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(eps @ (^[X8:$i]:(((nat_p @ X7) => (~((X8 = X7)))) => (~(((~((nat_p @ X7))) => (~((X8 = (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9))))))))))))))))))))))) => (~(((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))) => (~((X6 = emptyset)))))))))) => (X5 = emptyset))))))))))))))))))))) => (~(((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))) => (~((X2 = emptyset)))))))))) => (((~(((l_some @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))))) => (~((X2 = ((repl @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))) => (~((X3 = emptyset)))))))))) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((~((X3 = emptyset))) => (~((X4 = X3)))) => (~(((X3 = emptyset) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (eps @ (^[X6:$i]:(((~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))) => (~((X6 = ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(eps @ (^[X8:$i]:(((nat_p @ X7) => (~((X8 = X7)))) => (~(((~((nat_p @ X7))) => (~((X8 = (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9))))))))))))))))))))))) => (~(((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))) => (~((X6 = emptyset)))))))))) => (X5 = emptyset))))))))))))))))))))) => (~(((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))) => (~((X2 = emptyset)))))))))) @ (^[X2:$i]:(X1 = (((d_ReplSep @ (plus @ eigen__0)) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ X2) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj)))))))) => (eigen__0 = X1)) => (~(((l_some @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))))) => (~((X2 = ((repl @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))) => (~((X3 = emptyset)))))))))) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((~((X3 = emptyset))) => (~((X4 = X3)))) => (~(((X3 = emptyset) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (eps @ (^[X6:$i]:(((~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))) => (~((X6 = ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(eps @ (^[X8:$i]:(((nat_p @ X7) => (~((X8 = X7)))) => (~(((~((nat_p @ X7))) => (~((X8 = (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9))))))))))))))))))))))) => (~(((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))) => (~((X6 = emptyset)))))))))) => (X5 = emptyset))))))))))))))))))))) => (~(((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))) => (~((X2 = emptyset)))))))))) @ (^[X2:$i]:(eigen__0 = (((d_ReplSep @ (plus @ X1)) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ X2) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj))))))))))),introduced(definition,[new_symbols(definition,[sP10])])). % 230.05/230.11 thf(sP11,plain,sP11 <=> (eigen__0 = eigen__1),introduced(definition,[new_symbols(definition,[sP11])])). % 230.05/230.11 thf(def_is_of,definition,(is_of = (^[X1:$i]:(^[X2:$i>$o]:(X2 @ X1))))). % 230.05/230.11 thf(def_all_of,definition,(all_of = (^[X1:$i>$o]:(^[X2:$i>$o]:(![X3:$i]:(((is_of @ X3) @ X1) => (X2 @ X3))))))). % 230.05/230.11 thf(def_d_Subq,definition,(d_Subq = (^[X1:$i]:(^[X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))))))). % 230.05/230.11 thf(def_if,definition,(if = (^[X1:$o]:(^[X2:$i]:(^[X3:$i]:(eps @ (^[X4:$i]:((X1 => (~((X4 = X2)))) => (~(((~(X1)) => (~((X4 = X3)))))))))))))). % 230.05/230.11 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))))). % 230.05/230.11 thf(def_ordsucc,definition,(ordsucc = (^[X1:$i]:((binunion @ X1) @ (d_Sing @ X1))))). % 230.05/230.11 thf(def_omega,definition,(omega = ((d_Sep @ (univof @ emptyset)) @ nat_p))). % 230.05/230.11 thf(def_pair,definition,(pair = (^[X1:$i]:(^[X2:$i]:((binunion @ ((repl @ X1) @ d_Inj0)) @ ((repl @ X2) @ d_Inj1)))))). % 230.05/230.11 thf(def_proj0,definition,(proj0 = (^[X1:$i]:(((d_ReplSep @ X1) @ (^[X2:$i]:(~((![X3:$i]:(~(((d_Inj0 @ X3) = X2)))))))) @ d_Unj)))). % 230.05/230.11 thf(def_proj1,definition,(proj1 = (^[X1:$i]:(((d_ReplSep @ X1) @ (^[X2:$i]:(~((![X3:$i]:(~(((d_Inj1 @ X3) = X2)))))))) @ d_Unj)))). % 230.05/230.11 thf(def_d_Sigma,definition,(d_Sigma = (^[X1:$i]:(^[X2:$i>$i]:((famunion @ X1) @ (^[X3:$i]:((repl @ (X2 @ X3)) @ (pair @ X3)))))))). % 230.05/230.11 thf(def_setprod,definition,(setprod = (^[X1:$i]:(^[X2:$i]:((d_Sigma @ X1) @ (^[X3:$i]:X2)))))). % 230.05/230.11 thf(def_ap,definition,(ap = (^[X1:$i]:(^[X2:$i]:(((d_ReplSep @ X1) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1))))). % 230.05/230.11 thf(def_d_Pi,definition,(d_Pi = (^[X1:$i]:(^[X2:$i>$i]:((d_Sep @ (power @ ((d_Sigma @ X1) @ (^[X3:$i]:(union @ (X2 @ X3)))))) @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X1) => ((in @ ((ap @ X3) @ X4)) @ (X2 @ X4)))))))))). % 230.05/230.11 thf(def_imp,definition,(imp = (^[X1:$o]:(^[X2:$o]:(X1 => X2))))). % 230.05/230.11 thf(def_d_not,definition,(d_not = (^[X1:$o]:((imp @ X1) @ $false)))). % 230.05/230.11 thf(def_d_and,definition,(d_and = (^[X1:$o]:(^[X2:$o]:(d_not @ ((l_ec @ X1) @ X2)))))). % 230.05/230.11 thf(def_l_or,definition,(l_or = (^[X1:$o]:(imp @ (d_not @ X1))))). % 230.05/230.11 thf(def_or3,definition,(or3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((l_or @ X1) @ ((l_or @ X2) @ X3))))))). % 230.05/230.11 thf(def_ec3,definition,(ec3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:(((and3 @ ((l_ec @ X1) @ X2)) @ ((l_ec @ X2) @ X3)) @ ((l_ec @ X3) @ X1))))))). % 230.05/230.11 thf(def_orec3,definition,(orec3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((d_and @ (((or3 @ X1) @ X2) @ X3)) @ (((ec3 @ X1) @ X2) @ X3))))))). % 230.05/230.11 thf(def_e_is,definition,(e_is = (^[X1:$i]:(=)))). % 230.05/230.11 thf(def_amone,definition,(amone = (^[X1:$i]:(^[X2:$i>$o]:((all_of @ (^[X3:$i]:((in @ X3) @ X1))) @ (^[X3:$i]:((all_of @ (^[X4:$i]:((in @ X4) @ X1))) @ (^[X4:$i]:((X2 @ X3) => ((X2 @ X4) => (((e_is @ X1) @ X3) @ X4))))))))))). % 230.05/230.11 thf(def_one,definition,(one = (^[X1:$i]:(^[X2:$i>$o]:((d_and @ ((amone @ X1) @ X2)) @ ((l_some @ X1) @ X2)))))). % 230.05/230.11 thf(def_image,definition,(image = (^[X1:$i]:(^[X2:$i]:(^[X3:$i]:(^[X4:$i]:((l_some @ X1) @ (^[X5:$i]:(((e_is @ X2) @ X4) @ ((ap @ X3) @ X5)))))))))). % 230.05/230.11 thf(def_e_in,definition,(e_in = (^[X1:$i]:(^[X2:$i>$o]:(^[X3:$i]:X3))))). % 230.05/230.11 thf(def_d_pair,definition,(d_pair = (^[X1:$i]:(^[X2:$i]:pair)))). % 230.05/230.11 thf(def_first,definition,(first = (^[X1:$i]:(^[X2:$i]:proj0)))). % 230.05/230.11 thf(def_second,definition,(second = (^[X1:$i]:(^[X2:$i]:proj1)))). % 230.05/230.11 thf(def_esti,definition,(esti = (^[X1:$i]:in))). % 230.05/230.11 thf(def_incl,definition,(incl = (^[X1:$i]:(^[X2:$i]:(^[X3:$i]:((all @ X1) @ (^[X4:$i]:((imp @ (((esti @ X1) @ X4) @ X2)) @ (((esti @ X1) @ X4) @ X3))))))))). % 230.05/230.11 thf(def_nat,definition,(nat = ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))). % 230.05/230.11 thf(def_n_is,definition,(n_is = (e_is @ nat))). % 230.05/230.11 thf(def_nis,definition,(nis = (^[X1:$i]:(^[X2:$i]:(d_not @ ((n_is @ X1) @ X2)))))). % 230.05/230.11 thf(def_n_in,definition,(n_in = (esti @ nat))). % 230.05/230.11 thf(def_n_some,definition,(n_some = (l_some @ nat))). % 230.05/230.11 thf(def_n_all,definition,(n_all = (all @ nat))). % 230.05/230.11 thf(def_n_one,definition,(n_one = (one @ nat))). % 230.05/230.11 thf(def_n_1,definition,(n_1 = (ordsucc @ emptyset))). % 230.05/230.11 thf(def_cond1,definition,(cond1 = (n_in @ n_1))). % 230.05/230.11 thf(def_cond2,definition,(cond2 = (^[X1:$i]:(n_all @ (^[X2:$i]:((imp @ ((n_in @ X2) @ X1)) @ ((n_in @ (ordsucc @ X2)) @ X1))))))). % 230.05/230.11 thf(def_n_pl,definition,(n_pl = (^[X1:$i]:(ap @ (plus @ X1))))). % 230.05/230.11 thf(def_diffprop,definition,(diffprop = (^[X1:$i]:(^[X2:$i]:(^[X3:$i]:((n_is @ X1) @ ((n_pl @ X2) @ X3))))))). % 230.05/230.11 thf(def_d_29_ii,definition,(d_29_ii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X1) @ X2)))))). % 230.05/230.11 thf(def_iii,definition,(iii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X2) @ X1)))))). % 230.05/230.11 thf(def_moreis,definition,(moreis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((d_29_ii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))). % 230.05/230.11 thf(def_lessis,definition,(lessis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((iii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))). % 230.05/230.11 thf(satz10g,conjecture,(![X1:$i]:(((in @ X1) @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((~((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))))) => (~((X4 = emptyset)))))))))) => (~((~((X3 = emptyset))))))))) => (~((X2 = ((repl @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((~((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))))) => (~((X3 = emptyset)))))))))) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((~((X3 = emptyset))) => (~((X4 = X3)))) => (~(((~((~((X3 = emptyset))))) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (eps @ (^[X6:$i]:(((~((![X7:$i]:(((in @ X7) @ (univof @ 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=> (~((X6 = emptyset)))))))))) => (X5 = emptyset))))))))))))))))))))) => (~(((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))) => (~((X2 = emptyset)))))))))) => (((l_some @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => 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~(sP5)),inference(prop_rule,[status(thm)],[])). % 230.05/230.12 thf(2,plain,(~(sP10) | sP7),inference(all_rule,[status(thm)],[])). % 230.05/230.12 thf(3,plain,((~(sP7) | ~(sP4)) | sP6),inference(prop_rule,[status(thm)],[])). % 230.05/230.12 thf(4,plain,((~(sP6) | ~(sP1)) | ~(sP3)),inference(prop_rule,[status(thm)],[])). % 230.05/230.12 thf(5,plain,(~(sP2) | sP8),inference(all_rule,[status(thm)],[])). % 230.05/230.12 thf(6,plain,((~(sP8) | ~(sP9)) | sP10),inference(prop_rule,[status(thm)],[])). % 230.05/230.12 thf(satz10d,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ nat))) @ (^[X2:$i]:(((lessis @ X1) @ X2) => (d_not @ ((d_29_ii @ X1) @ X2)))))))). % 230.05/230.12 thf(7,plain,sP2,inference(preprocess,[status(thm)],[satz10d]). % 230.05/230.12 thf(8,plain,$false,inference(prop_unsat,[status(thm),assumptions([h9,h7,h8,h5,h6,h4,h2,h3,h1,h0])],[1,2,3,4,5,6,h2,h5,h7,h9,7])). % 230.05/230.12 thf(9,plain,(sP1 | ~(sP11)),inference(prop_rule,[status(thm)],[])). % 230.05/230.12 thf(10,plain,(~(sP10) | sP7),inference(all_rule,[status(thm)],[])). % 230.05/230.12 thf(11,plain,((~(sP7) | ~(sP4)) | sP6),inference(prop_rule,[status(thm)],[])). % 230.05/230.12 thf(12,plain,((~(sP6) | ~(sP1)) | ~(sP3)),inference(prop_rule,[status(thm)],[])). % 230.05/230.12 thf(13,plain,(~(sP2) | sP8),inference(all_rule,[status(thm)],[])). % 230.05/230.12 thf(14,plain,((~(sP8) | ~(sP9)) | sP10),inference(prop_rule,[status(thm)],[])). % 230.05/230.12 thf(15,plain,$false,inference(prop_unsat,[status(thm),assumptions([h10,h7,h8,h5,h6,h4,h2,h3,h1,h0])],[9,10,11,12,13,14,h2,h5,h7,h10,7])). % 230.05/230.12 thf(16,plain,$false,inference(tab_imp,[status(thm),assumptions([h7,h8,h5,h6,h4,h2,h3,h1,h0]),tab_imp(discharge,[h9]),tab_imp(discharge,[h10])],[h8,8,15,h9,h10])). % 230.05/230.12 thf(17,plain,$false,inference(tab_negimp,[status(thm),assumptions([h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h7,h8])],[h6,16,h7,h8])). % 230.05/230.12 thf(18,plain,$false,inference(tab_negimp,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,17,h5,h6])). % 230.05/230.12 thf(19,plain,$false,inference(tab_negall,[status(thm),assumptions([h2,h3,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,18,h4])). % 230.05/230.12 thf(20,plain,$false,inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,19,h2,h3])). % 230.05/230.12 thf(21,plain,$false,inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,20,h1])). % 230.05/230.12 thf(0,theorem,(![X1:$i]:(((in @ X1) @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((~((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))))) => (~((X4 = emptyset)))))))))) => (~((~((X3 = emptyset))))))))) => (~((X2 = ((repl @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((~((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))))) => (~((X3 = emptyset)))))))))) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((~((X3 = emptyset))) => (~((X4 = X3)))) => (~(((~((~((X3 = emptyset))))) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (eps @ (^[X6:$i]:(((~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))) => (~((X6 = ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(eps @ (^[X8:$i]:(((nat_p @ X7) => 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