%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : NUM788^4 : 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 : n011.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 02:12:10 PM UTC 2025 % Result : Theorem 275.49s 275.40s % Output : Proof 275.49s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.04/0.12 % Problem : NUM788^4 : TPTP v9.2.0. Released v7.1.0. % 0.04/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.13/0.34 % Computer : n011.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 22:15:08 EDT 2025 % 0.13/0.34 % CPUTime : % 275.49/275.40 % SZS status Theorem % 275.49/275.40 % Mode: mode498:USE_SINE=true:SINE_TOLERANCE=1.5:SINE_GENERALITY_THRESHOLD=16:SINE_RANK_LIMIT=2.:SINE_DEPTH=2 % 275.49/275.40 % Inferences: 624 % 275.49/275.40 % SZS output start Proof % 275.49/275.40 thf(ty_eigen__2, type, eigen__2 : $i). % 275.49/275.40 thf(ty_times, type, times : ($i>$i)). % 275.49/275.40 thf(ty_power, type, power : ($i>$i)). % 275.49/275.40 thf(ty_esti, type, esti : ($i>$i>$i>$o)). % 275.49/275.40 thf(ty_num, type, num : ($i>$i)). % 275.49/275.40 thf(ty_non, type, non : ($i>($i>$o)>$i>$o)). % 275.49/275.40 thf(ty_eigen__1, type, eigen__1 : $i). % 275.49/275.40 thf(ty_eigen__0, type, eigen__0 : $i). % 275.49/275.40 thf(ty_d_Sep, type, d_Sep : ($i>($i>$o)>$i)). % 275.49/275.40 thf(ty_emptyset, type, emptyset : $i). % 275.49/275.40 thf(ty_d_and, type, d_and : ($o>$o>$o)). % 275.49/275.40 thf(ty_eigen__3, type, eigen__3 : $i). % 275.49/275.40 thf(ty_den, type, den : ($i>$i)). % 275.49/275.40 thf(ty_ap, type, ap : ($i>$i>$i)). % 275.49/275.40 thf(ty_n_some, type, n_some : (($i>$o)>$o)). % 275.49/275.40 thf(ty_pair1type, type, pair1type : ($i>$i)). % 275.49/275.40 thf(ty_diffprop, type, diffprop : ($i>$i>$i>$o)). % 275.49/275.40 thf(ty_omega, type, omega : $i). % 275.49/275.40 thf(ty_ecect, type, ecect : ($i>($i>$i>$o)>$i>$i)). % 275.49/275.40 thf(ty_anec, type, anec : ($i>($i>$i>$o)>$i>$o)). % 275.49/275.40 thf(ty_in, type, in : ($i>$i>$o)). % 275.49/275.40 thf(ty_e_is, type, e_is : ($i>$i>$i>$o)). % 275.49/275.40 thf(sP1,plain,sP1 <=> (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))) @ (^[X1:$i]:(~((![X2:$i]:(((in @ X2) @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ X1) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:(^[X5:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ ((ap @ (times @ (num @ X4))) @ (den @ X5))) @ ((ap @ (times @ (num @ X5))) @ (den @ X4)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ X3) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:(^[X5:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ ((ap @ (times @ (num @ X4))) @ (den @ X5))) @ ((ap @ (times @ (num @ X5))) @ (den @ X4)))))) @ eigen__1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X3))) @ (den @ X1))) @ ((ap @ (times @ (num @ X1))) @ (den @ X3)))))))) @ X2))))))) @ eigen__3),introduced(definition,[new_symbols(definition,[sP1])])). % 275.49/275.40 thf(sP2,plain,sP2 <=> ((![X1:$i]:(((in @ X1) @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(~((![X3:$i]:(((in @ X3) @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ X2) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:(^[X6:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ ((ap @ (times @ (num @ X5))) @ (den @ X6))) @ ((ap @ (times @ (num @ X6))) @ (den @ X5)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ X4) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:(^[X6:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ ((ap @ (times @ (num @ X5))) @ (den @ X6))) @ ((ap @ (times @ (num @ X6))) @ (den @ X5)))))) @ eigen__1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X2))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X2)))))))) @ X3))))))) @ X1))) => (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))) @ (^[X1:$i]:(^[X2:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (times @ (num @ X1))) @ (den @ X2))) @ ((ap @ (times @ (num @ X2))) @ (den @ X1)))))))) @ eigen__0) @ eigen__1)),introduced(definition,[new_symbols(definition,[sP2])])). % 275.49/275.40 thf(sP3,plain,sP3 <=> (![X1:$i]:(((in @ X1) @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(~((![X3:$i]:(((in @ X3) @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ X2) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:(^[X6:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ ((ap @ (times @ (num @ X5))) @ (den @ X6))) @ ((ap @ (times @ (num @ X6))) @ (den @ X5)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ X4) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:(^[X6:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ ((ap @ (times @ (num @ X5))) @ (den @ X6))) @ ((ap @ (times @ (num @ X6))) @ (den @ X5)))))) @ eigen__1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X2))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X2)))))))) @ X3))))))) @ X1))),introduced(definition,[new_symbols(definition,[sP3])])). % 275.49/275.40 thf(sP4,plain,sP4 <=> (((in @ eigen__0) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))) @ (^[X1:$i]:(^[X2:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (times @ (num @ X1))) @ (den @ X2))) @ ((ap @ (times @ (num @ X2))) @ (den @ X1)))))))) => (![X1:$i]:(((in @ X1) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) => (((![X2:$i]:(((in @ X2) @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(~((![X4:$i]:(((in @ X4) @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X3) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X5) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X5))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X5)))))))) @ X4))))))) @ X2))) => (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) @ eigen__0) @ X1)) => (![X2:$i]:(((in @ X2) @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(~((![X4:$i]:(((in @ X4) @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X3) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X5) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X3))) @ (den @ X5))) @ ((ap @ (times @ (num @ X5))) @ (den @ X3)))))))) @ X4))))))) @ X2))))))),introduced(definition,[new_symbols(definition,[sP4])])). % 275.49/275.40 thf(sP5,plain,sP5 <=> (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))) @ (^[X1:$i]:(~((![X2:$i]:(((in @ X2) @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ X1) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:(^[X5:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ ((ap @ (times @ (num @ X4))) @ (den @ X5))) @ ((ap @ (times @ (num @ X5))) @ (den @ X4)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ X3) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:(^[X5:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ ((ap @ (times @ (num @ X4))) @ (den @ X5))) @ ((ap @ (times @ (num @ X5))) @ (den @ X4)))))) @ eigen__1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X1))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X1)))))))) @ X2))))))) @ eigen__2),introduced(definition,[new_symbols(definition,[sP5])])). % 275.49/275.40 thf(sP6,plain,sP6 <=> (((in @ eigen__0) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))) @ (^[X1:$i]:(^[X2:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (times @ (num @ X1))) @ (den @ X2))) @ ((ap @ (times @ (num @ X2))) @ (den @ X1)))))))) => (![X1:$i]:(((in @ X1) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) => (((![X2:$i]:(((in @ X2) @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(~((![X4:$i]:(((in @ X4) @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X3) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X5) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X3))) @ (den @ X5))) @ ((ap @ (times @ (num @ X5))) @ (den @ X3)))))))) @ X4))))))) @ X2))) => (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) @ eigen__0) @ X1)) => (![X2:$i]:(((in @ X2) @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(~((![X4:$i]:(((in @ X4) @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X3) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X5) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X5))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X5)))))))) @ X4))))))) @ X2))))))),introduced(definition,[new_symbols(definition,[sP6])])). % 275.49/275.40 thf(sP7,plain,sP7 <=> (![X1:$i]:(((in @ X1) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) => (![X2:$i]:(((in @ X2) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(^[X4:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ ((ap @ (times @ (num @ X3))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X3)))))))) => (((![X3:$i]:(((in @ X3) @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:(~((![X5:$i]:(((in @ X5) @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X4) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X1))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X6) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X4))) @ (den @ X6))) @ ((ap @ (times @ (num @ X6))) @ (den @ X4)))))))) @ X5))))))) @ X3))) => (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(^[X4:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ ((ap @ (times @ (num @ X3))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X3)))))))) @ X1) @ X2)) => (![X3:$i]:(((in @ X3) @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:(~((![X5:$i]:(((in @ X5) @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X4) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X1))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X6) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X6))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X6)))))))) @ X5))))))) @ X3)))))))),introduced(definition,[new_symbols(definition,[sP7])])). % 275.49/275.40 thf(sP8,plain,sP8 <=> (![X1:$i]:(((in @ X1) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) => (((![X2:$i]:(((in @ X2) @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(~((![X4:$i]:(((in @ X4) @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X3) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X5) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X5))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X5)))))))) @ X4))))))) @ X2))) => (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) @ eigen__0) @ X1)) => (![X2:$i]:(((in @ X2) @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(~((![X4:$i]:(((in @ X4) @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X3) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X5) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X3))) @ (den @ X5))) @ ((ap @ (times @ (num @ X5))) @ (den @ X3)))))))) @ X4))))))) @ X2)))))),introduced(definition,[new_symbols(definition,[sP8])])). % 275.49/275.40 thf(sP9,plain,sP9 <=> ((in @ eigen__2) @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))),introduced(definition,[new_symbols(definition,[sP9])])). % 275.49/275.40 thf(sP10,plain,sP10 <=> (((in @ eigen__3) @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))) => sP1),introduced(definition,[new_symbols(definition,[sP10])])). % 275.49/275.40 thf(sP11,plain,sP11 <=> (![X1:$i]:(((in @ X1) @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(~((![X3:$i]:(((in @ X3) @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ X2) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:(^[X6:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ ((ap @ (times @ (num @ X5))) @ (den @ X6))) @ ((ap @ (times @ (num @ X6))) @ (den @ X5)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ X4) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:(^[X6:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ ((ap @ (times @ (num @ X5))) @ (den @ X6))) @ ((ap @ (times @ (num @ X6))) @ (den @ X5)))))) @ eigen__1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X4))) @ (den @ X2))) @ ((ap @ (times @ (num @ X2))) @ (den @ X4)))))))) @ X3))))))) @ X1))),introduced(definition,[new_symbols(definition,[sP11])])). % 275.49/275.40 thf(sP12,plain,sP12 <=> ((in @ eigen__0) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))) @ (^[X1:$i]:(^[X2:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (times @ (num @ X1))) @ (den @ X2))) @ ((ap @ (times @ (num @ X2))) @ (den @ X1)))))))),introduced(definition,[new_symbols(definition,[sP12])])). % 275.49/275.40 thf(sP13,plain,sP13 <=> ((sP11 => (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))) @ (^[X1:$i]:(^[X2:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (times @ (num @ X1))) @ (den @ X2))) @ ((ap @ (times @ (num @ X2))) @ (den @ X1)))))))) @ eigen__0) @ eigen__1)) => sP3),introduced(definition,[new_symbols(definition,[sP13])])). % 275.49/275.40 thf(sP14,plain,sP14 <=> (![X1:$i]:(((in @ X1) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) => (((![X2:$i]:(((in @ X2) @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(~((![X4:$i]:(((in @ X4) @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X3) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X5) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X3))) @ (den @ X5))) @ ((ap @ (times @ (num @ X5))) @ (den @ X3)))))))) @ X4))))))) @ X2))) => (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) @ eigen__0) @ X1)) => (![X2:$i]:(((in @ X2) @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(~((![X4:$i]:(((in @ X4) @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X3) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X5) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X5))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X5)))))))) @ X4))))))) @ X2)))))),introduced(definition,[new_symbols(definition,[sP14])])). % 275.49/275.40 thf(sP15,plain,sP15 <=> (sP9 => sP5),introduced(definition,[new_symbols(definition,[sP15])])). % 275.49/275.40 thf(sP16,plain,sP16 <=> (sP2 => sP11),introduced(definition,[new_symbols(definition,[sP16])])). % 275.49/275.40 thf(sP17,plain,sP17 <=> ((in @ eigen__1) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))) @ (^[X1:$i]:(^[X2:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (times @ (num @ X1))) @ (den @ X2))) @ ((ap @ (times @ (num @ X2))) @ (den @ X1)))))))),introduced(definition,[new_symbols(definition,[sP17])])). % 275.49/275.40 thf(sP18,plain,sP18 <=> (![X1:$i]:(((in @ X1) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) => (![X2:$i]:(((in @ X2) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(^[X4:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ ((ap @ (times @ (num @ X3))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X3)))))))) => (((![X3:$i]:(((in @ X3) @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:(~((![X5:$i]:(((in @ X5) @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X4) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X1))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X6) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X6))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X6)))))))) @ X5))))))) @ X3))) => (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(^[X4:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ ((ap @ (times @ (num @ X3))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X3)))))))) @ X1) @ X2)) => (![X3:$i]:(((in @ X3) @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:(~((![X5:$i]:(((in @ X5) @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X4) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X1))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X6) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X4))) @ (den @ X6))) @ ((ap @ (times @ (num @ X6))) @ (den @ X4)))))))) @ X5))))))) @ X3)))))))),introduced(definition,[new_symbols(definition,[sP18])])). % 275.49/275.40 thf(sP19,plain,sP19 <=> (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))) @ (^[X1:$i]:(^[X2:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (times @ (num @ X1))) @ (den @ X2))) @ ((ap @ (times @ (num @ X2))) @ (den @ X1)))))))) @ eigen__0) @ eigen__1),introduced(definition,[new_symbols(definition,[sP19])])). % 275.49/275.40 thf(sP20,plain,sP20 <=> (sP11 => sP19),introduced(definition,[new_symbols(definition,[sP20])])). % 275.49/275.40 thf(sP21,plain,sP21 <=> (sP17 => sP13),introduced(definition,[new_symbols(definition,[sP21])])). % 275.49/275.40 thf(sP22,plain,sP22 <=> (sP17 => sP16),introduced(definition,[new_symbols(definition,[sP22])])). % 275.49/275.40 thf(sP23,plain,sP23 <=> ((in @ eigen__3) @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))),introduced(definition,[new_symbols(definition,[sP23])])). % 275.49/275.40 thf(def_is_of,definition,(is_of = (^[X1:$i]:(^[X2:$i>$o]:(X2 @ X1))))). % 275.49/275.40 thf(def_all_of,definition,(all_of = (^[X1:$i>$o]:(^[X2:$i>$o]:(![X3:$i]:(((is_of @ X3) @ X1) => (X2 @ X3))))))). % 275.49/275.40 thf(def_ordsucc,definition,(ordsucc = (^[X1:$i]:((binunion @ X1) @ (d_Sing @ X1))))). % 275.49/275.40 thf(def_imp,definition,(imp = (^[X1:$o]:(^[X2:$o]:(X1 => X2))))). % 275.49/275.40 thf(def_d_not,definition,(d_not = (^[X1:$o]:((imp @ X1) @ $false)))). % 275.49/275.40 thf(def_l_or,definition,(l_or = (^[X1:$o]:(imp @ (d_not @ X1))))). % 275.49/275.40 thf(def_l_some,definition,(l_some = (^[X1:$i]:(^[X2:$i>$o]:(d_not @ ((all_of @ (^[X3:$i]:((in @ X3) @ X1))) @ ((non @ X1) @ X2))))))). % 275.49/275.40 thf(def_or3,definition,(or3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((l_or @ X1) @ ((l_or @ X2) @ X3))))))). % 275.49/275.40 thf(def_and3,definition,(and3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((d_and @ X1) @ ((d_and @ X2) @ X3))))))). % 275.49/275.40 thf(def_ec3,definition,(ec3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:(((and3 @ ((l_ec @ X1) @ X2)) @ ((l_ec @ X2) @ X3)) @ ((l_ec @ X3) @ X1))))))). % 275.49/275.40 thf(def_orec3,definition,(orec3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((d_and @ (((or3 @ X1) @ X2) @ X3)) @ (((ec3 @ X1) @ X2) @ X3))))))). % 275.49/275.40 thf(def_ect,definition,(ect = (^[X1:$i]:(^[X2:$i>$i>$o]:((d_Sep @ (power @ X1)) @ ((anec @ X1) @ X2)))))). % 275.49/275.40 thf(def_nat,definition,(nat = ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))). % 275.49/275.40 thf(def_n_is,definition,(n_is = (e_is @ nat))). % 275.49/275.40 thf(def_n_1,definition,(n_1 = (ordsucc @ emptyset))). % 275.49/275.40 thf(def_n_pl,definition,(n_pl = (^[X1:$i]:(ap @ (plus @ X1))))). % 275.49/275.40 thf(def_d_29_ii,definition,(d_29_ii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X1) @ X2)))))). % 275.49/275.40 thf(def_iii,definition,(iii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X2) @ X1)))))). % 275.49/275.40 thf(def_moreis,definition,(moreis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((d_29_ii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))). % 275.49/275.40 thf(def_lessis,definition,(lessis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((iii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))). % 275.49/275.40 thf(def_n_ts,definition,(n_ts = (^[X1:$i]:(ap @ (times @ X1))))). % 275.49/275.40 thf(def_frac,definition,(frac = (pair1type @ nat))). % 275.49/275.40 thf(def_n_eq,definition,(n_eq = (^[X1:$i]:(^[X2:$i]:((n_is @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1))))))). % 275.49/275.40 thf(def_moref,definition,(moref = (^[X1:$i]:(^[X2:$i]:((d_29_ii @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1))))))). % 275.49/275.40 thf(def_lessf,definition,(lessf = (^[X1:$i]:(^[X2:$i]:((iii @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1))))))). % 275.49/275.40 thf(def_moreq,definition,(moreq = (^[X1:$i]:(^[X2:$i]:((l_or @ ((moref @ X1) @ X2)) @ ((n_eq @ X1) @ X2)))))). % 275.49/275.40 thf(def_lesseq,definition,(lesseq = (^[X1:$i]:(^[X2:$i]:((l_or @ ((lessf @ X1) @ X2)) @ ((n_eq @ X1) @ X2)))))). % 275.49/275.40 thf(def_n_pf,definition,(n_pf = (^[X1:$i]:(^[X2:$i]:((n_fr @ ((n_pl @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1)))) @ ((n_ts @ (den @ X1)) @ (den @ X2))))))). % 275.49/275.40 thf(def_n_tf,definition,(n_tf = (^[X1:$i]:(^[X2:$i]:((n_fr @ ((n_ts @ (num @ X1)) @ (num @ X2))) @ ((n_ts @ (den @ X1)) @ (den @ X2))))))). % 275.49/275.40 thf(def_inf,definition,(inf = (esti @ frac))). % 275.49/275.40 thf(def_rat,definition,(rat = ((ect @ frac) @ n_eq))). % 275.49/275.40 thf(def_rt_is,definition,(rt_is = (e_is @ rat))). % 275.49/275.40 thf(def_class,definition,(class = ((ecect @ frac) @ n_eq))). % 275.49/275.40 thf(def_rt_more,definition,(rt_more = (^[X1:$i]:(^[X2:$i]:((l_some @ frac) @ (^[X3:$i]:((l_some @ frac) @ (^[X4:$i]:(((and3 @ ((inf @ X3) @ (class @ X1))) @ ((inf @ X4) @ (class @ X2))) @ ((moref @ X3) @ X4)))))))))). % 275.49/275.40 thf(def_rt_less,definition,(rt_less = (^[X1:$i]:(^[X2:$i]:((l_some @ frac) @ (^[X3:$i]:((l_some @ frac) @ (^[X4:$i]:(((and3 @ ((inf @ X3) @ (class @ X1))) @ ((inf @ X4) @ (class @ X2))) @ ((lessf @ X3) @ X4)))))))))). % 275.49/275.40 thf(def_rt_moreis,definition,(rt_moreis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((rt_more @ X1) @ X2)) @ ((rt_is @ X1) @ X2)))))). % 275.49/275.40 thf(def_rt_lessis,definition,(rt_lessis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((rt_less @ X1) @ X2)) @ ((rt_is @ X1) @ X2)))))). % 275.49/275.40 thf(satz81g,conjecture,(![X1:$i]:(((in @ X1) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) => (![X2:$i]:(((in @ X2) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(^[X4:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ ((ap @ (times @ (num @ X3))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X3)))))))) => ((~((![X3:$i]:(((in @ X3) @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:(~((![X5:$i]:(((in @ X5) @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X4) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X1))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X6) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X4))) @ (den @ X6))) @ ((ap @ (times @ (num @ X6))) @ (den @ X4)))))))) @ X5))))))) @ X3))))) => (~(((~((~((![X3:$i]:(((in @ X3) @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:(~((![X5:$i]:(((in @ X5) @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X4) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X1))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X6) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X6))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X6)))))))) @ X5))))))) @ X3))))))) => (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(^[X4:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ ((ap @ (times @ (num @ X3))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X3)))))))) @ X1) @ X2)))))))))). % 275.49/275.40 thf(h0,negated_conjecture,(~((![X1:$i]:(((in @ X1) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) => (![X2:$i]:(((in @ X2) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(^[X4:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ ((ap @ (times @ (num @ X3))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X3)))))))) => ((~((![X3:$i]:(((in @ X3) @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:(~((![X5:$i]:(((in @ X5) @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X4) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X1))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X6) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X4))) @ (den @ X6))) @ ((ap @ (times @ (num @ X6))) @ (den @ X4)))))))) @ X5))))))) @ X3))))) => (~(((![X3:$i]:(((in @ X3) @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:(~((![X5:$i]:(((in @ X5) @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X4) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X1))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X6) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X6))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X6)))))))) @ X5))))))) @ X3))) => (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(^[X4:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ ((ap @ (times @ (num @ X3))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X3)))))))) @ X1) @ X2))))))))))),inference(assume_negation,[status(cth)],[satz81g])). % 275.49/275.40 thf(h1,assumption,(~((sP12 => (![X1:$i]:(((in @ X1) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) => ((~((![X2:$i]:(((in @ X2) @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(~((![X4:$i]:(((in @ X4) @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X3) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X5) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X3))) @ (den @ X5))) @ ((ap @ (times @ (num @ X5))) @ (den @ X3)))))))) @ X4))))))) @ X2))))) => (~(((![X2:$i]:(((in @ X2) @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(~((![X4:$i]:(((in @ X4) @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X3) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X5) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X5))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X5)))))))) @ X4))))))) @ X2))) => (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) @ eigen__0) @ X1)))))))))),introduced(assumption,[])). % 275.49/275.40 thf(h2,assumption,sP12,introduced(assumption,[])). % 275.49/275.40 thf(h3,assumption,(~((![X1:$i]:(((in @ X1) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) => ((~((![X2:$i]:(((in @ X2) @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(~((![X4:$i]:(((in @ X4) @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X3) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X5) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X3))) @ (den @ X5))) @ ((ap @ (times @ (num @ X5))) @ (den @ X3)))))))) @ X4))))))) @ X2))))) => (~(((![X2:$i]:(((in @ X2) @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(~((![X4:$i]:(((in @ X4) @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset))))))) @ (^[X5:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X3) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ eigen__0))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ X5) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:(^[X7:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ (num @ X6))) @ (den @ X7))) @ ((ap @ (times @ (num @ X7))) @ (den @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X5))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X5)))))))) @ X4))))))) @ X2))) => (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) @ eigen__0) @ X1))))))))),introduced(assumption,[])). % 275.49/275.40 thf(h4,assumption,(~((sP17 => ((~(sP3)) => (~(sP20)))))),introduced(assumption,[])). % 275.49/275.40 thf(h5,assumption,sP17,introduced(assumption,[])). % 275.49/275.40 thf(h6,assumption,(~(((~(sP3)) => (~(sP20))))),introduced(assumption,[])). % 275.49/275.40 thf(h7,assumption,(~(sP3)),introduced(assumption,[])). % 275.49/275.40 thf(h8,assumption,sP20,introduced(assumption,[])). % 275.49/275.40 thf(h9,assumption,(~(sP15)),introduced(assumption,[])). % 275.49/275.40 thf(h10,assumption,sP9,introduced(assumption,[])). % 275.49/275.40 thf(h11,assumption,(~(sP5)),introduced(assumption,[])). % 275.49/275.40 thf(h12,assumption,(~(sP11)),introduced(assumption,[])). % 275.49/275.40 thf(h13,assumption,sP19,introduced(assumption,[])). % 275.49/275.40 thf(h14,assumption,(~(sP10)),introduced(assumption,[])). % 275.49/275.40 thf(h15,assumption,sP23,introduced(assumption,[])). % 275.49/275.40 thf(h16,assumption,(~(sP1)),introduced(assumption,[])). % 275.49/275.40 thf(1,plain,(sP2 | sP3),inference(prop_rule,[status(thm)],[])). % 275.49/275.40 thf(2,plain,(~(sP7) | sP6),inference(all_rule,[status(thm)],[])). % 275.49/275.40 thf(3,plain,((~(sP6) | ~(sP12)) | sP14),inference(prop_rule,[status(thm)],[])). % 275.49/275.40 thf(4,plain,(~(sP14) | sP22),inference(all_rule,[status(thm)],[])). % 275.49/275.40 thf(5,plain,((~(sP22) | ~(sP17)) | sP16),inference(prop_rule,[status(thm)],[])). % 275.49/275.40 thf(6,plain,((~(sP16) | ~(sP2)) | sP11),inference(prop_rule,[status(thm)],[])). % 275.49/275.40 thf(7,plain,(~(sP11) | sP10),inference(all_rule,[status(thm)],[])). % 275.49/275.40 thf(8,plain,((~(sP10) | ~(sP23)) | sP1),inference(prop_rule,[status(thm)],[])). % 275.49/275.40 thf(9,plain,(~(sP3) | sP15),inference(all_rule,[status(thm)],[])). % 275.49/275.40 thf(10,plain,((~(sP15) | ~(sP9)) | sP5),inference(prop_rule,[status(thm)],[])). % 275.49/275.40 thf(satz81c,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ rat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ rat))) @ (^[X2:$i]:(((rt_moreis @ X1) @ X2) => (d_not @ ((rt_less @ X1) @ X2)))))))). % 275.49/275.40 thf(11,plain,sP7,inference(preprocess,[status(thm)],[satz81c]). % 275.49/275.40 thf(12,plain,$false,inference(prop_unsat,[status(thm),assumptions([h15,h16,h14,h12,h10,h11,h9,h7,h8,h5,h6,h4,h2,h3,h1,h0])],[1,2,3,4,5,6,7,8,9,10,h2,h5,h10,h11,h15,h16,11])). % 275.49/275.40 thf(13,plain,$false,inference(tab_negimp,[status(thm),assumptions([h14,h12,h10,h11,h9,h7,h8,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h15,h16])],[h14,12,h15,h16])). % 275.49/275.40 thf(14,plain,$false,inference(tab_negall,[status(thm),assumptions([h12,h10,h11,h9,h7,h8,h5,h6,h4,h2,h3,h1,h0]),tab_negall(discharge,[h14]),tab_negall(eigenvar,eigen__3)],[h12,13,h14])). % 275.49/275.40 thf(15,plain,(sP20 | ~(sP19)),inference(prop_rule,[status(thm)],[])). % 275.49/275.40 thf(16,plain,(~(sP18) | sP4),inference(all_rule,[status(thm)],[])). % 275.49/275.40 thf(17,plain,((~(sP4) | ~(sP12)) | sP8),inference(prop_rule,[status(thm)],[])). % 275.49/275.40 thf(18,plain,(~(sP8) | sP21),inference(all_rule,[status(thm)],[])). % 275.49/275.40 thf(19,plain,((~(sP21) | ~(sP17)) | sP13),inference(prop_rule,[status(thm)],[])). % 275.49/275.40 thf(20,plain,((~(sP13) | ~(sP20)) | sP3),inference(prop_rule,[status(thm)],[])). % 275.49/275.40 thf(21,plain,(~(sP3) | sP15),inference(all_rule,[status(thm)],[])). % 275.49/275.40 thf(22,plain,((~(sP15) | ~(sP9)) | sP5),inference(prop_rule,[status(thm)],[])). % 275.49/275.40 thf(satz81d,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ rat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ rat))) @ (^[X2:$i]:(((rt_lessis @ X1) @ X2) => (d_not @ ((rt_more @ X1) @ X2)))))))). % 275.49/275.40 thf(23,plain,sP18,inference(preprocess,[status(thm)],[satz81d]). % 275.49/275.40 thf(24,plain,$false,inference(prop_unsat,[status(thm),assumptions([h13,h10,h11,h9,h7,h8,h5,h6,h4,h2,h3,h1,h0])],[15,16,17,18,19,20,21,22,h2,h5,h10,h11,h13,23])). % 275.49/275.40 thf(25,plain,$false,inference(tab_imp,[status(thm),assumptions([h10,h11,h9,h7,h8,h5,h6,h4,h2,h3,h1,h0]),tab_imp(discharge,[h12]),tab_imp(discharge,[h13])],[h8,14,24,h12,h13])). % 275.49/275.40 thf(26,plain,$false,inference(tab_negimp,[status(thm),assumptions([h9,h7,h8,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h10,h11])],[h9,25,h10,h11])). % 275.49/275.40 thf(27,plain,$false,inference(tab_negall,[status(thm),assumptions([h7,h8,h5,h6,h4,h2,h3,h1,h0]),tab_negall(discharge,[h9]),tab_negall(eigenvar,eigen__2)],[h7,26,h9])). % 275.49/275.40 thf(28,plain,$false,inference(tab_negimp,[status(thm),assumptions([h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h7,h8])],[h6,27,h7,h8])). % 275.49/275.40 thf(29,plain,$false,inference(tab_negimp,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,28,h5,h6])). % 275.49/275.40 thf(30,plain,$false,inference(tab_negall,[status(thm),assumptions([h2,h3,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,29,h4])). % 275.49/275.40 thf(31,plain,$false,inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,30,h2,h3])). % 275.49/275.40 thf(32,plain,$false,inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,31,h1])). % 275.49/275.40 thf(0,theorem,(![X1:$i]:(((in @ X1) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))) @ (^[X2:$i]:(^[X3:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ (num @ X2))) @ (den @ X3))) @ ((ap @ (times @ (num @ X3))) @ (den @ X2)))))))) => (![X2:$i]:(((in @ X2) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(^[X4:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ ((ap @ (times @ (num @ X3))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X3)))))))) => ((~((![X3:$i]:(((in @ X3) @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:(~((![X5:$i]:(((in @ X5) @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X4) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X1))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X6) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X4))) @ (den @ X6))) @ ((ap @ (times @ (num @ X6))) @ (den @ X4)))))))) @ X5))))))) @ X3))))) => (~(((~((~((![X3:$i]:(((in @ X3) @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset))))))) @ (^[X4:$i]:(~((![X5:$i]:(((in @ X5) @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) => (((non @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset))))))) @ (^[X6:$i]:((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X4) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X1))) @ ((d_and @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ X6) @ (((ecect @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset))))))) @ (^[X7:$i]:(^[X8:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ (num @ X7))) @ (den @ X8))) @ ((ap @ (times @ (num @ X8))) @ (den @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ (num @ X6))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X6)))))))) @ X5))))))) @ X3))))))) => (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) @ ((anec @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset))))))) @ (^[X3:$i]:(^[X4:$i]:(((e_is @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ ((ap @ (times @ (num @ X3))) @ (den @ X4))) @ ((ap @ (times @ (num @ X4))) @ (den @ X3)))))))) @ X1) @ X2))))))))),inference(contra,[status(thm),contra(discharge,[h0])],[32,h0])). % 275.49/275.40 % SZS output end Proof %------------------------------------------------------------------------------