%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : NUM791^4 : 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 : n020.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 02:12:12 PM UTC 2025 % Result : Theorem 86.42s 86.38s % Output : Proof 86.42s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.05/0.13 % Problem : NUM791^4 : TPTP v9.2.0. Released v7.1.0. % 0.13/0.14 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.14/0.35 % Computer : n020.cluster.edu % 0.14/0.35 % Model : x86_64 x86_64 % 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.35 % Memory : 8042.1875MB % 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.35 % CPULimit : 300 % 0.14/0.35 % WCLimit : 300 % 0.14/0.35 % DateTime : Wed Oct 8 22:33:08 EDT 2025 % 0.14/0.35 % CPUTime : % 86.42/86.38 % SZS status Theorem % 86.42/86.38 % Mode: mode94:USE_SINE=true:SINE_TOLERANCE=2.0:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=2.:SINE_DEPTH=0 % 86.42/86.38 % Inferences: 6 % 86.42/86.38 % SZS output start Proof % 86.42/86.38 thf(ty_times, type, times : ($i>$i)). % 86.42/86.38 thf(ty_power, type, power : ($i>$i)). % 86.42/86.38 thf(ty_esti, type, esti : ($i>$i>$i>$o)). % 86.42/86.38 thf(ty_non, type, non : ($i>($i>$o)>$i>$o)). % 86.42/86.38 thf(ty_eigen__1, type, eigen__1 : $i). % 86.42/86.38 thf(ty_eigen__0, type, eigen__0 : $i). % 86.42/86.38 thf(ty_d_Sep, type, d_Sep : ($i>($i>$o)>$i)). % 86.42/86.38 thf(ty_second1, type, second1 : ($i>$i>$i)). % 86.42/86.38 thf(ty_emptyset, type, emptyset : $i). % 86.42/86.38 thf(ty_first1, type, first1 : ($i>$i>$i)). % 86.42/86.38 thf(ty_l_ec, type, l_ec : ($o>$o>$o)). % 86.42/86.38 thf(ty_ap, type, ap : ($i>$i>$i)). % 86.42/86.38 thf(ty_n_some, type, n_some : (($i>$o)>$o)). % 86.42/86.38 thf(ty_pair1type, type, pair1type : ($i>$i)). % 86.42/86.38 thf(ty_diffprop, type, diffprop : ($i>$i>$i>$o)). % 86.42/86.38 thf(ty_omega, type, omega : $i). % 86.42/86.38 thf(ty_ecect, type, ecect : ($i>($i>$i>$o)>$i>$i)). % 86.42/86.38 thf(ty_anec, type, anec : ($i>($i>$i>$o)>$i>$o)). % 86.42/86.38 thf(ty_in, type, in : ($i>$i>$o)). % 86.42/86.38 thf(ty_e_is, type, e_is : ($i>$i>$i>$o)). % 86.42/86.38 thf(sP1,plain,sP1 <=> ((![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]:(~(((l_ec @ (((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:$i]:(~((X7 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5)))))) @ eigen__0))) @ (~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5)))))) @ eigen__1))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ 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:$i]:(~((X3 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1)))))))) @ eigen__0) @ eigen__1)),introduced(definition,[new_symbols(definition,[sP1])])). % 86.42/86.38 thf(sP2,plain,sP2 <=> (![X1:$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:$i]:(~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5)))))) @ eigen__0))) @ (~(((l_ec @ (((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:$i]:(~((X7 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5)))))) @ eigen__1))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X2)))))))))))) @ X3))))))) @ X1))),introduced(definition,[new_symbols(definition,[sP2])])). % 86.42/86.38 thf(sP3,plain,sP3 <=> ((in @ eigen__0) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1)))))))),introduced(definition,[new_symbols(definition,[sP3])])). % 86.42/86.38 thf(sP4,plain,sP4 <=> (sP3 => (![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 @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ 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]:(~(((l_ec @ (((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:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6)))))) @ eigen__0))) @ (~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X5))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ 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:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2)))))))) @ eigen__0) @ X1)))) => (~((![X2:$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:$i]:(~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6)))))) @ eigen__0))) @ (~(((l_ec @ (((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:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X3))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X5)))))))))))) @ X4))))))) @ X2))))))))),introduced(definition,[new_symbols(definition,[sP4])])). % 86.42/86.38 thf(sP5,plain,sP5 <=> ((in @ eigen__1) @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X1:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1)))))))),introduced(definition,[new_symbols(definition,[sP5])])). % 86.42/86.38 thf(sP6,plain,sP6 <=> (sP5 => ((~(sP1)) => (~((![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]:(~(((l_ec @ (((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:$i]:(~((X7 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5)))))) @ eigen__0))) @ (~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5)))))) @ eigen__1))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X2))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4)))))))))))) @ X3))))))) @ X1))))))),introduced(definition,[new_symbols(definition,[sP6])])). % 86.42/86.38 thf(sP7,plain,sP7 <=> (((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:$i]:(~((X3 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1)))))))) @ eigen__0) @ eigen__1),introduced(definition,[new_symbols(definition,[sP7])])). % 86.42/86.38 thf(sP8,plain,sP8 <=> (![X1:$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:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2)))))))) => (![X2:$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:$i]:(~((X5 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X3)))))))) => ((~(((![X3:$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:$i]:(~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7)))))) @ X1))) @ (~(((l_ec @ (((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:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X4)))))))))))) @ X5))))))) @ X3))) => (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ 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]:(~(((l_ec @ (((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:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7)))))) @ X1))) @ (~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X4))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6)))))))))))) @ X5))))))) @ X3)))))))))),introduced(definition,[new_symbols(definition,[sP8])])). % 86.42/86.38 thf(sP9,plain,sP9 <=> (![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 @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ 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]:(~(((l_ec @ (((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:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6)))))) @ eigen__0))) @ (~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X5))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ 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:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2)))))))) @ eigen__0) @ X1)))) => (~((![X2:$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:$i]:(~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6)))))) @ eigen__0))) @ (~(((l_ec @ (((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:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X3))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X5)))))))))))) @ X4))))))) @ X2)))))))),introduced(definition,[new_symbols(definition,[sP9])])). % 86.42/86.38 thf(sP10,plain,sP10 <=> ((~(sP1)) => (~((![X1:$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:$i]:(~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5)))))) @ eigen__0))) @ (~(((l_ec @ (((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:$i]:(~((X7 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5)))))) @ eigen__1))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X2))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4)))))))))))) @ X3))))))) @ X1)))))),introduced(definition,[new_symbols(definition,[sP10])])). % 86.42/86.38 thf(sP11,plain,sP11 <=> (![X1:$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:$i]:(~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X5:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5)))))) @ eigen__0))) @ (~(((l_ec @ (((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:$i]:(~((X7 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X5)))))) @ eigen__1))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X2))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4)))))))))))) @ X3))))))) @ X1))),introduced(definition,[new_symbols(definition,[sP11])])). % 86.42/86.38 thf(def_is_of,definition,(is_of = (^[X1:$i]:(^[X2:$i>$o]:(X2 @ X1))))). % 86.42/86.38 thf(def_all_of,definition,(all_of = (^[X1:$i>$o]:(^[X2:$i>$o]:(![X3:$i]:(((is_of @ X3) @ X1) => (X2 @ X3))))))). % 86.42/86.38 thf(def_ordsucc,definition,(ordsucc = (^[X1:$i]:((binunion @ X1) @ (d_Sing @ X1))))). % 86.42/86.38 thf(def_imp,definition,(imp = (^[X1:$o]:(^[X2:$o]:(X1 => X2))))). % 86.42/86.38 thf(def_d_not,definition,(d_not = (^[X1:$o]:((imp @ X1) @ $false)))). % 86.42/86.38 thf(def_d_and,definition,(d_and = (^[X1:$o]:(^[X2:$o]:(d_not @ ((l_ec @ X1) @ X2)))))). % 86.42/86.38 thf(def_l_or,definition,(l_or = (^[X1:$o]:(imp @ (d_not @ X1))))). % 86.42/86.38 thf(def_l_some,definition,(l_some = (^[X1:$i]:(^[X2:$i>$o]:(d_not @ ((all_of @ (^[X3:$i]:((in @ X3) @ X1))) @ ((non @ X1) @ X2))))))). % 86.42/86.38 thf(def_or3,definition,(or3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((l_or @ X1) @ ((l_or @ X2) @ X3))))))). % 86.42/86.38 thf(def_and3,definition,(and3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((d_and @ X1) @ ((d_and @ X2) @ X3))))))). % 86.42/86.38 thf(def_ec3,definition,(ec3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:(((and3 @ ((l_ec @ X1) @ X2)) @ ((l_ec @ X2) @ X3)) @ ((l_ec @ X3) @ X1))))))). % 86.42/86.38 thf(def_orec3,definition,(orec3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((d_and @ (((or3 @ X1) @ X2) @ X3)) @ (((ec3 @ X1) @ X2) @ X3))))))). % 86.42/86.38 thf(def_ect,definition,(ect = (^[X1:$i]:(^[X2:$i>$i>$o]:((d_Sep @ (power @ X1)) @ ((anec @ X1) @ X2)))))). % 86.42/86.38 thf(def_nat,definition,(nat = ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))). % 86.42/86.38 thf(def_n_is,definition,(n_is = (e_is @ nat))). % 86.42/86.38 thf(def_n_1,definition,(n_1 = (ordsucc @ emptyset))). % 86.42/86.38 thf(def_n_pl,definition,(n_pl = (^[X1:$i]:(ap @ (plus @ X1))))). % 86.42/86.38 thf(def_d_29_ii,definition,(d_29_ii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X1) @ X2)))))). % 86.42/86.38 thf(def_iii,definition,(iii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X2) @ X1)))))). % 86.42/86.38 thf(def_moreis,definition,(moreis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((d_29_ii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))). % 86.42/86.38 thf(def_lessis,definition,(lessis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((iii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))). % 86.42/86.38 thf(def_n_ts,definition,(n_ts = (^[X1:$i]:(ap @ (times @ X1))))). % 86.42/86.38 thf(def_frac,definition,(frac = (pair1type @ nat))). % 86.42/86.38 thf(def_n_fr,definition,(n_fr = (pair1 @ nat))). % 86.42/86.38 thf(def_num,definition,(num = (first1 @ nat))). % 86.42/86.38 thf(def_den,definition,(den = (second1 @ nat))). % 86.42/86.38 thf(def_n_eq,definition,(n_eq = (^[X1:$i]:(^[X2:$i]:((n_is @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1))))))). % 86.42/86.38 thf(def_moref,definition,(moref = (^[X1:$i]:(^[X2:$i]:((d_29_ii @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1))))))). % 86.42/86.38 thf(def_lessf,definition,(lessf = (^[X1:$i]:(^[X2:$i]:((iii @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1))))))). % 86.42/86.38 thf(def_moreq,definition,(moreq = (^[X1:$i]:(^[X2:$i]:((l_or @ ((moref @ X1) @ X2)) @ ((n_eq @ X1) @ X2)))))). % 86.42/86.38 thf(def_lesseq,definition,(lesseq = (^[X1:$i]:(^[X2:$i]:((l_or @ ((lessf @ X1) @ X2)) @ ((n_eq @ X1) @ X2)))))). % 86.42/86.38 thf(def_n_pf,definition,(n_pf = (^[X1:$i]:(^[X2:$i]:((n_fr @ ((n_pl @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1)))) @ ((n_ts @ (den @ X1)) @ (den @ X2))))))). % 86.42/86.38 thf(def_d_367_w,definition,(d_367_w = (^[X1:$i]:(^[X2:$i]:((n_fr @ ((d_367_vo @ X1) @ X2)) @ ((n_ts @ (den @ X1)) @ (den @ X2))))))). % 86.42/86.38 thf(def_n_tf,definition,(n_tf = (^[X1:$i]:(^[X2:$i]:((n_fr @ ((n_ts @ (num @ X1)) @ (num @ X2))) @ ((n_ts @ (den @ X1)) @ (den @ X2))))))). % 86.42/86.38 thf(def_inf,definition,(inf = (esti @ frac))). % 86.42/86.38 thf(def_rat,definition,(rat = ((ect @ frac) @ n_eq))). % 86.42/86.38 thf(def_rt_is,definition,(rt_is = (e_is @ rat))). % 86.42/86.38 thf(def_class,definition,(class = ((ecect @ frac) @ n_eq))). % 86.42/86.38 thf(def_rt_more,definition,(rt_more = (^[X1:$i]:(^[X2:$i]:((l_some @ frac) @ (^[X3:$i]:((l_some @ frac) @ (^[X4:$i]:(((and3 @ ((inf @ X3) @ (class @ X1))) @ ((inf @ X4) @ (class @ X2))) @ ((moref @ X3) @ X4)))))))))). % 86.42/86.38 thf(def_rt_less,definition,(rt_less = (^[X1:$i]:(^[X2:$i]:((l_some @ frac) @ (^[X3:$i]:((l_some @ frac) @ (^[X4:$i]:(((and3 @ ((inf @ X3) @ (class @ X1))) @ ((inf @ X4) @ (class @ X2))) @ ((lessf @ X3) @ X4)))))))))). % 86.42/86.38 thf(def_rt_moreis,definition,(rt_moreis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((rt_more @ X1) @ X2)) @ ((rt_is @ X1) @ X2)))))). % 86.42/86.38 thf(def_rt_lessis,definition,(rt_lessis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((rt_less @ X1) @ X2)) @ ((rt_is @ X1) @ X2)))))). % 86.42/86.38 thf(satz81k,conjecture,(![X1:$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:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2)))))))) => (![X2:$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:$i]:(~((X5 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X3)))))))) => ((~(((~((~((![X3:$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:$i]:(~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7)))))) @ X1))) @ (~(((l_ec @ (((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:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X4))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6)))))))))))) @ X5))))))) @ X3))))))) => (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X3:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ 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]:(~(((l_ec @ (((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:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7)))))) @ X1))) @ (~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X4)))))))))))) @ X5))))))) @ X3))))))))))). % 86.42/86.38 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 @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ 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 @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ 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]:(~(((l_ec @ (((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:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7)))))) @ X1))) @ (~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X4))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ 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:$i]:(~((X5 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X3)))))))) @ X1) @ X2)))) => (~((![X3:$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:$i]:(~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7)))))) @ X1))) @ (~(((l_ec @ (((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:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X4)))))))))))) @ X5))))))) @ X3)))))))))))),inference(assume_negation,[status(cth)],[satz81k])). % 86.42/86.38 thf(h1,assumption,(~((sP3 => (![X1:$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:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2)))))))) => ((~(((![X2:$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:$i]:(~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6)))))) @ eigen__0))) @ (~(((l_ec @ (((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:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X3))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X5)))))))))))) @ X4))))))) @ X2))) => (((e_is @ ((d_Sep @ (power @ (pair1type @ ((d_Sep @ omega) @ (^[X2:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ 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]:(~(((l_ec @ (((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:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6)))))) @ eigen__0))) @ (~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X5))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X3)))))))))))) @ X4))))))) @ X2))))))))))),introduced(assumption,[])). % 86.42/86.38 thf(h2,assumption,sP3,introduced(assumption,[])). % 86.42/86.38 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 @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ 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]:(~(((l_ec @ (((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:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6)))))) @ eigen__0))) @ (~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X3))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ 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:$i]:(~((X4 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2)))))))) @ eigen__0) @ X1)))) => (~((![X2:$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:$i]:(~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X6:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6)))))) @ eigen__0))) @ (~(((l_ec @ (((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:$i]:(~((X8 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X8:$i]:(~((X8 = emptyset)))))) @ X6)))))) @ X1))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X5))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X5))) @ ((second1 @ ((d_Sep @ omega) @ (^[X6:$i]:(~((X6 = emptyset)))))) @ X3)))))))))))) @ X4))))))) @ X2)))))))))),introduced(assumption,[])). % 86.42/86.38 thf(h4,assumption,(~((sP5 => ((~((sP11 => sP7))) => (~(sP2)))))),introduced(assumption,[])). % 86.42/86.38 thf(h5,assumption,sP5,introduced(assumption,[])). % 86.42/86.38 thf(h6,assumption,(~(((~((sP11 => sP7))) => (~(sP2))))),introduced(assumption,[])). % 86.42/86.38 thf(h7,assumption,(~((sP11 => sP7))),introduced(assumption,[])). % 86.42/86.38 thf(h8,assumption,sP2,introduced(assumption,[])). % 86.42/86.38 thf(h9,assumption,sP11,introduced(assumption,[])). % 86.42/86.38 thf(h10,assumption,(~(sP7)),introduced(assumption,[])). % 86.42/86.38 thf(1,plain,(~(sP9) | sP6),inference(all_rule,[status(thm)],[])). % 86.42/86.38 thf(2,plain,((~(sP6) | ~(sP5)) | sP10),inference(prop_rule,[status(thm)],[])). % 86.42/86.38 thf(3,plain,((~(sP10) | sP1) | ~(sP11)),inference(prop_rule,[status(thm)],[])). % 86.42/86.38 thf(4,plain,((~(sP1) | ~(sP2)) | sP7),inference(prop_rule,[status(thm)],[])). % 86.42/86.38 thf(5,plain,(~(sP8) | sP4),inference(all_rule,[status(thm)],[])). % 86.42/86.38 thf(6,plain,((~(sP4) | ~(sP3)) | sP9),inference(prop_rule,[status(thm)],[])). % 86.42/86.38 thf(satz81j,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ rat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ rat))) @ (^[X2:$i]:((d_not @ ((rt_moreis @ X1) @ X2)) => ((rt_less @ X1) @ X2))))))). % 86.42/86.38 thf(7,plain,sP8,inference(preprocess,[status(thm)],[satz81j]). % 86.42/86.38 thf(8,plain,$false,inference(prop_unsat,[status(thm),assumptions([h9,h10,h7,h8,h5,h6,h4,h2,h3,h1,h0])],[1,2,3,4,5,6,h2,h5,h9,h10,h8,7])). % 86.42/86.38 thf(9,plain,$false,inference(tab_negimp,[status(thm),assumptions([h7,h8,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h9,h10])],[h7,8,h9,h10])). % 86.42/86.38 thf(10,plain,$false,inference(tab_negimp,[status(thm),assumptions([h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h7,h8])],[h6,9,h7,h8])). % 86.42/86.38 thf(11,plain,$false,inference(tab_negimp,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,10,h5,h6])). % 86.42/86.38 thf(12,plain,$false,inference(tab_negall,[status(thm),assumptions([h2,h3,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,11,h4])). % 86.42/86.38 thf(13,plain,$false,inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,12,h2,h3])). % 86.42/86.38 thf(14,plain,$false,inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,13,h1])). % 86.42/86.38 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 @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ 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 @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ 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]:(~(((l_ec @ (((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:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7)))))) @ X1))) @ (~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X4))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ 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:$i]:(~((X5 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))) @ X3)))))))) @ X1) @ X2)))) => (~((![X3:$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:$i]:(~(((l_ec @ (((esti @ (pair1type @ ((d_Sep @ omega) @ (^[X7:$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 @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7)))))) @ X1))) @ (~(((l_ec @ (((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:$i]:(~((X9 = emptyset)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X8))) @ ((second1 @ ((d_Sep @ omega) @ (^[X9:$i]:(~((X9 = emptyset)))))) @ X7)))))) @ X2))) @ (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X4))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X6))) @ ((second1 @ ((d_Sep @ omega) @ (^[X7:$i]:(~((X7 = emptyset)))))) @ X4)))))))))))) @ X5))))))) @ X3)))))))))),inference(contra,[status(thm),contra(discharge,[h0])],[14,h0])). % 86.42/86.38 % SZS output end Proof %------------------------------------------------------------------------------