%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : NUM769^4 : 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 : n006.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 02:12:07 PM UTC 2025 % Result : Theorem 231.06s 230.31s % Output : Proof 231.06s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.12/0.12 % Problem : NUM769^4 : TPTP v9.2.0. Released v7.1.0. % 0.12/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.12/0.34 % Computer : n006.cluster.edu % 0.12/0.34 % Model : x86_64 x86_64 % 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.34 % Memory : 8042.1875MB % 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.34 % CPULimit : 300 % 0.12/0.34 % WCLimit : 300 % 0.12/0.34 % DateTime : Wed Oct 8 22:55:08 EDT 2025 % 0.12/0.34 % CPUTime : % 231.06/230.31 % SZS status Theorem % 231.06/230.31 % Mode: mode92:USE_SINE=true:SINE_TOLERANCE=5.0:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=2.:SINE_DEPTH=0 % 231.06/230.31 % Inferences: 5 % 231.06/230.31 % SZS output start Proof % 231.06/230.31 thf(ty_if, type, if : ($o>$i>$i>$i)). % 231.06/230.31 thf(ty_times, type, times : ($i>$i)). % 231.06/230.31 thf(ty_power, type, power : ($i>$i)). % 231.06/230.31 thf(ty_d_1to, type, d_1to : ($i>$i)). % 231.06/230.31 thf(ty_d_Inj0, type, d_Inj0 : ($i>$i)). % 231.06/230.31 thf(ty_union, type, union : ($i>$i)). % 231.06/230.31 thf(ty_eps, type, eps : (($i>$o)>$i)). % 231.06/230.31 thf(ty_non, type, non : ($i>($i>$o)>$i>$o)). % 231.06/230.31 thf(ty_d_ReplSep, type, d_ReplSep : ($i>($i>$o)>($i>$i)>$i)). % 231.06/230.31 thf(ty_eigen__1, type, eigen__1 : $i). % 231.06/230.31 thf(ty_eigen__0, type, eigen__0 : $i). % 231.06/230.31 thf(ty_famunion, type, famunion : ($i>($i>$i)>$i)). % 231.06/230.31 thf(ty_second1, type, second1 : ($i>$i>$i)). % 231.06/230.31 thf(ty_d_Unj, type, d_Unj : ($i>$i)). % 231.06/230.31 thf(ty_emptyset, type, emptyset : $i). % 231.06/230.31 thf(ty_first1, type, first1 : ($i>$i>$i)). % 231.06/230.31 thf(ty_ind, type, ind : ($i>($i>$o)>$i)). % 231.06/230.31 thf(ty_plus, type, plus : ($i>$i)). % 231.06/230.31 thf(ty_d_Inj1, type, d_Inj1 : ($i>$i)). % 231.06/230.31 thf(ty_omega, type, omega : $i). % 231.06/230.31 thf(ty_imp, type, imp : ($o>$o>$o)). % 231.06/230.31 thf(ty_pair1, type, pair1 : ($i>$i>$i>$i)). % 231.06/230.31 thf(ty_repl, type, repl : ($i>($i>$i)>$i)). % 231.06/230.31 thf(ty_in, type, in : ($i>$i>$o)). % 231.06/230.31 thf(ty_binunion, type, binunion : ($i>$i>$i)). % 231.06/230.31 thf(ty_n_2, type, n_2 : $i). % 231.06/230.31 thf(sP1,plain,sP1 <=> (![X1:$i]:(((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X2:$i]:(((in @ X2) @ omega) => (X2 = emptyset)))))) @ ((repl @ omega) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ omega) => (X3 = emptyset)))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X2:$i]:(((in @ X2) @ omega) => (X2 = emptyset)))))) @ ((repl @ omega) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ omega) => (X3 = emptyset)))))))))) @ emptyset)) @ (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X2:$i]:(((in @ X2) @ omega) => (X2 = emptyset)))))) @ ((repl @ omega) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ omega) => (X3 = emptyset)))))))))) @ emptyset)) @ eigen__1))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ ((ind @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (X8 = emptyset)))))))))) @ emptyset)) @ eigen__1)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))) = (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ eigen__1))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (X8 = emptyset)))))))))) @ emptyset)) @ eigen__0)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ X4) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))))))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ eigen__1)) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))))) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))))) @ (^[X2:$i]:(((d_ReplSep @ X2) @ (^[X3:$i]:(~((![X4:$i]:(~(((d_Inj1 @ X4) = X3)))))))) @ d_Unj))))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((binunion @ ((repl @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ ((ind @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (X8 = emptyset)))))))))) @ emptyset)) @ eigen__1)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))) = (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ eigen__1))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (X8 = emptyset)))))))))) @ emptyset)) @ eigen__0)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ X4) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))))))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ eigen__1)) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj))))))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ eigen__1)) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))))) @ (^[X2:$i]:(((d_ReplSep @ X2) @ (^[X3:$i]:(~((![X4:$i]:(~(((d_Inj1 @ X4) = X3)))))))) @ d_Unj)))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X2:$i]:(((in @ X2) @ omega) => (X2 = emptyset)))))) @ ((repl @ omega) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ omega) => (X3 = emptyset)))))))))) @ emptyset)) @ eigen__1))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ ((ind @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (X8 = emptyset)))))))))) @ emptyset)) @ eigen__1)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))) = (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ eigen__1))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (X8 = emptyset)))))))))) @ emptyset)) @ eigen__0)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ X4) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))))))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ eigen__1)) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))))) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))))) @ (^[X2:$i]:(((d_ReplSep @ X2) @ (^[X3:$i]:(~((![X4:$i]:(~(((d_Inj1 @ X4) = X3)))))))) @ d_Unj))))))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ eigen__0)) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))))) @ (^[X2:$i]:(((d_ReplSep @ X2) @ (^[X3:$i]:(~((![X4:$i]:(~(((d_Inj1 @ X4) = X3)))))))) @ d_Unj))) = X1) => (X1 = (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X2:$i]:(((in @ X2) @ omega) => (X2 = emptyset)))))) @ ((repl @ omega) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ omega) => (X3 = emptyset)))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X2:$i]:(((in @ X2) @ omega) => (X2 = emptyset)))))) @ ((repl @ omega) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ omega) => (X3 = emptyset)))))))))) @ emptyset)) @ (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X2:$i]:(((in @ X2) @ omega) => (X2 = emptyset)))))) @ ((repl @ omega) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ omega) => (X3 = emptyset)))))))))) @ emptyset)) @ eigen__1))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ ((ind @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ 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(^[X1:$i]:(~((![X2:$i]:(~((X1 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset)) @ eigen__0)) @ d_Inj0)) @ ((repl @ X2) @ d_Inj1)))))))))) @ (^[X1:$i]:(((d_ReplSep @ X1) @ (^[X2:$i]:(~((![X3:$i]:(~(((d_Inj1 @ X3) = X2)))))))) @ d_Unj)))),introduced(definition,[new_symbols(definition,[sP2])])). % 231.06/230.31 thf(sP3,plain,sP3 <=> ((imp @ (![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ omega) => (X2 = emptyset)))))) @ ((repl @ omega) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ omega) => (X3 = emptyset)))))))))) @ emptyset)) => (((non @ (((if @ (~((![X2:$i]:(((in @ X2) @ omega) => (X2 = emptyset)))))) @ ((repl @ omega) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ omega) => (X3 = emptyset)))))))))) @ emptyset)) @ (^[X2:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ eigen__1)) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj))) = (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset)) @ eigen__1))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ eigen__0)) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj))))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ X2) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj)))))) @ X1)))) @ $false),introduced(definition,[new_symbols(definition,[sP3])])). % 231.06/230.31 thf(sP4,plain,sP4 <=> (![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X3:$i]:((repl @ (union @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ 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(^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ ((ind @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ (^[X6:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (X8 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X7:$i]:(~((![X8:$i]:(~((X7 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X9:$i]:(((in @ X9) @ omega) => (X9 = emptyset)))))) @ ((repl @ omega) @ (^[X9:$i]:(((if @ (~((X9 = 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(^[X7:$i]:(~((![X8:$i]:(~((X7 = ((binunion @ ((repl @ X6) @ d_Inj0)) @ ((repl @ X8) @ d_Inj1)))))))))) @ (^[X7:$i]:(((d_ReplSep @ X7) @ (^[X8:$i]:(~((![X9:$i]:(~(((d_Inj1 @ X9) = X8)))))))) @ d_Unj))))))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (X8 = emptyset)))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (X9 = emptyset)))))))))) @ emptyset)) @ X1)) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj))))))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (X8 = emptyset)))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (X9 = emptyset)))))))))) @ emptyset)) @ X1)) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj)))) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ X1))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ ((ind @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ (^[X6:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (X8 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X7:$i]:(~((![X8:$i]:(~((X7 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X9:$i]:(((in @ X9) @ omega) => (X9 = emptyset)))))) @ ((repl @ omega) @ (^[X9:$i]:(((if @ (~((X9 = emptyset)))) @ X9) @ (eps @ (^[X10:$i]:(~((((in @ X10) @ omega) => (X10 = emptyset)))))))))) @ emptyset)) @ X1)) @ d_Inj0)) @ ((repl @ X8) @ d_Inj1)))))))))) @ (^[X7:$i]:(((d_ReplSep @ X7) @ (^[X8:$i]:(~((![X9:$i]:(~(((d_Inj1 @ X9) = X8)))))))) @ d_Unj))) = (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (X8 = emptyset)))))))))) @ emptyset)) @ X1))) @ (^[X7:$i]:(~((![X8:$i]:(~((X7 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X9:$i]:(((in @ X9) @ omega) => (X9 = emptyset)))))) @ ((repl @ omega) @ (^[X9:$i]:(((if @ (~((X9 = emptyset)))) @ X9) @ (eps @ (^[X10:$i]:(~((((in @ X10) @ omega) => (X10 = emptyset)))))))))) @ emptyset)) @ eigen__0)) @ d_Inj0)) @ ((repl @ X8) @ d_Inj1)))))))))) @ (^[X7:$i]:(((d_ReplSep @ X7) @ (^[X8:$i]:(~((![X9:$i]:(~(((d_Inj1 @ X9) = X8)))))))) @ d_Unj))))) @ (^[X7:$i]:(~((![X8:$i]:(~((X7 = ((binunion @ ((repl @ X6) @ d_Inj0)) @ ((repl @ X8) @ d_Inj1)))))))))) @ (^[X7:$i]:(((d_ReplSep @ X7) @ (^[X8:$i]:(~((![X9:$i]:(~(((d_Inj1 @ X9) = X8)))))))) @ d_Unj))))))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (X8 = emptyset)))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (X9 = emptyset)))))))))) @ emptyset)) @ X1)) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj)))))) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))))) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))))) @ (^[X2:$i]:(((d_ReplSep @ X2) @ (^[X3:$i]:(~((![X4:$i]:(~(((d_Inj1 @ X4) = X3)))))))) @ d_Unj))))))),introduced(definition,[new_symbols(definition,[sP4])])). % 231.06/230.31 thf(sP5,plain,sP5 <=> (((in @ eigen__0) @ (((if @ (~((![X1:$i]:(((in @ X1) @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X2:$i]:((repl @ (union @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset))) @ (^[X3:$i]:((binunion @ ((repl @ X2) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))) => (~((![X2:$i]:(((in @ X2) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X1) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ X2) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj)))) @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset))))))))))) @ ((repl @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X1:$i]:((repl @ (union @ (((if @ (~((![X2:$i]:(((in @ X2) @ omega) => (X2 = emptyset)))))) @ ((repl @ omega) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ omega) => (X3 = emptyset)))))))))) @ emptyset))) @ (^[X2:$i]:((binunion @ ((repl @ X1) @ d_Inj0)) @ ((repl @ X2) @ d_Inj1)))))))) @ (^[X1:$i]:(((if @ (![X2:$i]:(((in @ X2) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X1) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ X2) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj)))) @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset))))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X3:$i]:((repl @ (union @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset))) @ (^[X4:$i]:((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))) => (~((![X3:$i]:(((in @ X3) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X2) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))) @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset))))))))))))))) @ emptyset)) => sP4),introduced(definition,[new_symbols(definition,[sP5])])). % 231.06/230.31 thf(sP6,plain,sP6 <=> ((in @ eigen__0) @ (((if @ (~((![X1:$i]:(((in @ X1) @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X2:$i]:((repl @ (union @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset))) @ (^[X3:$i]:((binunion @ ((repl @ X2) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))) => (~((![X2:$i]:(((in @ X2) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X1) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ X2) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj)))) @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset))))))))))) @ ((repl @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X1:$i]:((repl @ (union @ (((if @ (~((![X2:$i]:(((in @ X2) @ omega) => (X2 = emptyset)))))) @ ((repl @ omega) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ omega) => (X3 = emptyset)))))))))) @ emptyset))) @ (^[X2:$i]:((binunion @ ((repl @ X1) @ d_Inj0)) @ ((repl @ X2) @ d_Inj1)))))))) @ (^[X1:$i]:(((if @ (![X2:$i]:(((in @ X2) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X1) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ X2) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj)))) @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset))))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X3:$i]:((repl @ (union @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset))) @ (^[X4:$i]:((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))) => (~((![X3:$i]:(((in @ X3) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X2) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))) @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset))))))))))))))) @ emptyset)),introduced(definition,[new_symbols(definition,[sP6])])). % 231.06/230.31 thf(sP7,plain,sP7 <=> ((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X1:$i]:(((in @ X1) @ omega) => (X1 = emptyset)))))) @ ((repl @ omega) @ (^[X1:$i]:(((if @ (~((X1 = emptyset)))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ omega) => (X2 = emptyset)))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X1:$i]:(((in @ X1) @ omega) => (X1 = emptyset)))))) @ ((repl @ omega) @ (^[X1:$i]:(((if @ (~((X1 = emptyset)))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ omega) => (X2 = emptyset)))))))))) @ emptyset)) @ (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X1:$i]:(((in @ X1) @ omega) => (X1 = emptyset)))))) @ ((repl @ omega) @ (^[X1:$i]:(((if @ (~((X1 = emptyset)))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ omega) => (X2 = emptyset)))))))))) @ emptyset)) @ eigen__1))) @ (^[X1:$i]:(~((![X2:$i]:(~((X1 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset)) @ ((ind @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset)) @ (^[X3:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ eigen__1)) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj))) = (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ eigen__1))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ eigen__0)) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj))))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl 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emptyset)))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (X9 = emptyset)))))))))) @ emptyset)) @ eigen__0)) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj))))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ X5) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj))))))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (X8 = emptyset)))))))))) @ emptyset)) @ eigen__1)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))))))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (X8 = emptyset)))))))))) @ emptyset)) @ eigen__1)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj)))) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj)))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset)) @ eigen__1))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ ((ind @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ (^[X5:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (X8 = emptyset)))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (X9 = emptyset)))))))))) @ emptyset)) @ eigen__1)) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj))) = (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ eigen__1))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (X8 = emptyset)))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (X9 = emptyset)))))))))) @ emptyset)) @ eigen__0)) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj))))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ X5) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj))))))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (X8 = emptyset)))))))))) @ emptyset)) @ eigen__1)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj)))))) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj)))))) @ d_Inj0)) @ ((repl @ X2) @ d_Inj1)))))))))) @ (^[X1:$i]:(((d_ReplSep @ X1) @ (^[X2:$i]:(~((![X3:$i]:(~(((d_Inj1 @ X3) = X2)))))))) @ d_Unj)))),introduced(definition,[new_symbols(definition,[sP7])])). % 231.06/230.31 thf(sP8,plain,sP8 <=> (sP7 => sP2),introduced(definition,[new_symbols(definition,[sP8])])). % 231.06/230.31 thf(sP9,plain,sP9 <=> (sP3 => sP7),introduced(definition,[new_symbols(definition,[sP9])])). % 231.06/230.31 thf(sP10,plain,sP10 <=> (![X1:$i]:(![X2:$i]:((X1 = X2) => (X2 = X1)))),introduced(definition,[new_symbols(definition,[sP10])])). % 231.06/230.31 thf(sP11,plain,sP11 <=> ((in @ eigen__1) @ (((if @ (~((![X1:$i]:(((in @ X1) @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X2:$i]:((repl @ (union @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset))) @ (^[X3:$i]:((binunion @ ((repl @ X2) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))) => (~((![X2:$i]:(((in @ X2) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X1) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ X2) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj)))) @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset))))))))))) @ ((repl @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X1:$i]:((repl @ (union @ (((if @ (~((![X2:$i]:(((in @ X2) @ omega) => (X2 = emptyset)))))) @ ((repl @ omega) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ omega) => (X3 = emptyset)))))))))) @ emptyset))) @ (^[X2:$i]:((binunion @ ((repl @ X1) @ d_Inj0)) @ ((repl @ X2) @ d_Inj1)))))))) @ (^[X1:$i]:(((if @ (![X2:$i]:(((in @ X2) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X1) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ X2) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj)))) @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset))))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X3:$i]:((repl @ (union @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset))) @ (^[X4:$i]:((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))) => (~((![X3:$i]:(((in @ X3) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X2) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))) @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset))))))))))))))) @ emptyset)),introduced(definition,[new_symbols(definition,[sP11])])). % 231.06/230.31 thf(sP12,plain,sP12 <=> (sP11 => sP9),introduced(definition,[new_symbols(definition,[sP12])])). % 231.06/230.31 thf(sP13,plain,sP13 <=> (![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X3:$i]:((repl @ (union @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset))) @ (^[X4:$i]:((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))) => (~((![X3:$i]:(((in @ X3) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X2) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))) @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset))))))))))) @ ((repl @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X2:$i]:((repl @ (union @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset))) @ (^[X3:$i]:((binunion @ ((repl @ X2) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))) @ (^[X2:$i]:(((if @ (![X3:$i]:(((in @ X3) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X2) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))) @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset))))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X4:$i]:((repl @ (union @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset))) @ (^[X5:$i]:((binunion @ ((repl @ X4) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))) => (~((![X4:$i]:(((in @ X4) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X3) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ X4) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj)))) @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset))))))))))))))) @ emptyset)) => (![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X4:$i]:((repl @ (union @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset))) 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d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj))) = (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset)) @ X1))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ 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(^[X1:$i]:(^[X2:$i>$o]:(X2 @ X1))))). % 231.06/230.32 thf(def_all_of,definition,(all_of = (^[X1:$i>$o]:(^[X2:$i>$o]:(![X3:$i]:(((is_of @ X3) @ X1) => (X2 @ X3))))))). % 231.06/230.32 thf(def_d_Subq,definition,(d_Subq = (^[X1:$i]:(^[X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))))))). % 231.06/230.32 thf(def_d_Sep,definition,(d_Sep = (^[X1:$i]:(^[X2:$i>$o]:(((if @ (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) @ ((repl @ X1) @ (^[X3:$i]:(((if @ (X2 @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ X1) => (~((X2 @ X4)))))))))))) @ emptyset))))). % 231.06/230.32 thf(def_ordsucc,definition,(ordsucc = (^[X1:$i]:((binunion @ X1) @ (d_Sing @ X1))))). % 231.06/230.32 thf(def_pair,definition,(pair = (^[X1:$i]:(^[X2:$i]:((binunion @ ((repl @ X1) @ d_Inj0)) @ ((repl @ X2) @ d_Inj1)))))). % 231.06/230.32 thf(def_proj0,definition,(proj0 = (^[X1:$i]:(((d_ReplSep @ X1) @ (^[X2:$i]:(~((![X3:$i]:(~(((d_Inj0 @ X3) = X2)))))))) @ d_Unj)))). % 231.06/230.32 thf(def_proj1,definition,(proj1 = (^[X1:$i]:(((d_ReplSep @ X1) @ (^[X2:$i]:(~((![X3:$i]:(~(((d_Inj1 @ X3) = X2)))))))) @ d_Unj)))). % 231.06/230.32 thf(def_d_Sigma,definition,(d_Sigma = (^[X1:$i]:(^[X2:$i>$i]:((famunion @ X1) @ (^[X3:$i]:((repl @ (X2 @ X3)) @ (pair @ X3)))))))). % 231.06/230.32 thf(def_setprod,definition,(setprod = (^[X1:$i]:(^[X2:$i]:((d_Sigma @ X1) @ (^[X3:$i]:X2)))))). % 231.06/230.32 thf(def_ap,definition,(ap = (^[X1:$i]:(^[X2:$i]:(((d_ReplSep @ X1) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1))))). % 231.06/230.32 thf(def_d_Pi,definition,(d_Pi = (^[X1:$i]:(^[X2:$i>$i]:((d_Sep @ (power @ ((d_Sigma @ X1) @ (^[X3:$i]:(union @ (X2 @ X3)))))) @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X1) => ((in @ ((ap @ X3) @ X4)) @ (X2 @ X4)))))))))). % 231.06/230.32 thf(def_d_not,definition,(d_not = (^[X1:$o]:((imp @ X1) @ $false)))). % 231.06/230.32 thf(def_d_and,definition,(d_and = (^[X1:$o]:(^[X2:$o]:(d_not @ ((l_ec @ X1) @ X2)))))). % 231.06/230.32 thf(def_l_some,definition,(l_some = (^[X1:$i]:(^[X2:$i>$o]:(d_not @ ((all_of @ (^[X3:$i]:((in @ X3) @ X1))) @ ((non @ X1) @ X2))))))). % 231.06/230.32 thf(def_or3,definition,(or3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((l_or @ X1) @ ((l_or @ X2) @ X3))))))). % 231.06/230.32 thf(def_ec3,definition,(ec3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:(((and3 @ ((l_ec @ X1) @ X2)) @ ((l_ec @ X2) @ X3)) @ ((l_ec @ X3) @ X1))))))). % 231.06/230.32 thf(def_orec3,definition,(orec3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((d_and @ (((or3 @ X1) @ X2) @ X3)) @ (((ec3 @ X1) @ X2) @ X3))))))). % 231.06/230.32 thf(def_e_is,definition,(e_is = (^[X1:$i]:(=)))). % 231.06/230.32 thf(def_amone,definition,(amone = (^[X1:$i]:(^[X2:$i>$o]:((all_of @ (^[X3:$i]:((in @ X3) @ X1))) @ (^[X3:$i]:((all_of @ (^[X4:$i]:((in @ X4) @ X1))) @ (^[X4:$i]:((X2 @ X3) => ((X2 @ X4) => (((e_is @ X1) @ X3) @ X4))))))))))). % 231.06/230.32 thf(def_one,definition,(one = (^[X1:$i]:(^[X2:$i>$o]:((d_and @ ((amone @ X1) @ X2)) @ ((l_some @ X1) @ X2)))))). % 231.06/230.32 thf(def_image,definition,(image = (^[X1:$i]:(^[X2:$i]:(^[X3:$i]:(^[X4:$i]:((l_some @ X1) @ (^[X5:$i]:(((e_is @ X2) @ X4) @ ((ap @ X3) @ X5)))))))))). % 231.06/230.32 thf(def_e_in,definition,(e_in = (^[X1:$i]:(^[X2:$i>$o]:(^[X3:$i]:X3))))). % 231.06/230.32 thf(def_d_pair,definition,(d_pair = (^[X1:$i]:(^[X2:$i]:pair)))). % 231.06/230.32 thf(def_first,definition,(first = (^[X1:$i]:(^[X2:$i]:proj0)))). % 231.06/230.32 thf(def_second,definition,(second = (^[X1:$i]:(^[X2:$i]:proj1)))). % 231.06/230.32 thf(def_esti,definition,(esti = (^[X1:$i]:in))). % 231.06/230.32 thf(def_incl,definition,(incl = (^[X1:$i]:(^[X2:$i]:(^[X3:$i]:((all @ X1) @ (^[X4:$i]:((imp @ (((esti @ X1) @ X4) @ X2)) @ (((esti @ X1) @ X4) @ X3))))))))). % 231.06/230.32 thf(def_nat,definition,(nat = ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))). % 231.06/230.32 thf(def_n_is,definition,(n_is = (e_is @ nat))). % 231.06/230.32 thf(def_nis,definition,(nis = (^[X1:$i]:(^[X2:$i]:(d_not @ ((n_is @ X1) @ X2)))))). % 231.06/230.32 thf(def_n_in,definition,(n_in = (esti @ nat))). % 231.06/230.32 thf(def_n_some,definition,(n_some = (l_some @ nat))). % 231.06/230.32 thf(def_n_all,definition,(n_all = (all @ nat))). % 231.06/230.32 thf(def_n_one,definition,(n_one = (one @ nat))). % 231.06/230.32 thf(def_n_1,definition,(n_1 = (ordsucc @ emptyset))). % 231.06/230.32 thf(def_cond1,definition,(cond1 = (n_in @ n_1))). % 231.06/230.32 thf(def_cond2,definition,(cond2 = (^[X1:$i]:(n_all @ (^[X2:$i]:((imp @ ((n_in @ X2) @ X1)) @ ((n_in @ (ordsucc @ X2)) @ X1))))))). % 231.06/230.32 thf(def_n_pl,definition,(n_pl = (^[X1:$i]:(ap @ (plus @ X1))))). % 231.06/230.32 thf(def_diffprop,definition,(diffprop = (^[X1:$i]:(^[X2:$i]:(^[X3:$i]:((n_is @ X1) @ ((n_pl @ X2) @ X3))))))). % 231.06/230.32 thf(def_d_29_ii,definition,(d_29_ii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X1) @ X2)))))). % 231.06/230.32 thf(def_iii,definition,(iii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X2) @ X1)))))). % 231.06/230.32 thf(def_moreis,definition,(moreis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((d_29_ii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))). % 231.06/230.32 thf(def_lessis,definition,(lessis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((iii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))). % 231.06/230.32 thf(def_n_ts,definition,(n_ts = (^[X1:$i]:(ap @ (times @ X1))))). % 231.06/230.32 thf(def_pair1type,definition,(pair1type = (^[X1:$i]:((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:X1))))). % 231.06/230.32 thf(def_frac,definition,(frac = (pair1type @ nat))). % 231.06/230.32 thf(def_n_fr,definition,(n_fr = (pair1 @ nat))). % 231.06/230.32 thf(def_num,definition,(num = (first1 @ nat))). % 231.06/230.32 thf(def_den,definition,(den = (second1 @ nat))). % 231.06/230.32 thf(def_n_eq,definition,(n_eq = (^[X1:$i]:(^[X2:$i]:((n_is @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1))))))). % 231.06/230.32 thf(def_moref,definition,(moref = (^[X1:$i]:(^[X2:$i]:((d_29_ii @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1))))))). % 231.06/230.32 thf(def_lessf,definition,(lessf = (^[X1:$i]:(^[X2:$i]:((iii @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1))))))). % 231.06/230.32 thf(def_moreq,definition,(moreq = (^[X1:$i]:(^[X2:$i]:((l_or @ ((moref @ X1) @ X2)) @ ((n_eq @ X1) @ X2)))))). % 231.06/230.32 thf(def_lesseq,definition,(lesseq = (^[X1:$i]:(^[X2:$i]:((l_or @ ((lessf @ X1) @ X2)) @ ((n_eq @ X1) @ X2)))))). % 231.06/230.32 thf(def_n_pf,definition,(n_pf = (^[X1:$i]:(^[X2:$i]:((n_fr @ ((n_pl @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1)))) @ ((n_ts @ (den @ X1)) @ (den @ X2))))))). % 231.06/230.32 thf(def_d_367_vo,definition,(d_367_vo = (^[X1:$i]:(^[X2:$i]:((ind @ nat) @ ((diffprop @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1)))))))). % 231.06/230.32 thf(def_d_367_w,definition,(d_367_w = (^[X1:$i]:(^[X2:$i]:((n_fr @ ((d_367_vo @ X1) @ X2)) @ ((n_ts @ (den @ X1)) @ (den @ X2))))))). % 231.06/230.32 thf(satz67d,conjecture,(![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X3:$i]:((repl @ (union @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (~((~((X4 = emptyset)))))))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))))))) @ emptyset))) @ (^[X4:$i]:((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))) => (~((![X3:$i]:(((in @ X3) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X2) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))) @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (~((~((X4 = emptyset)))))))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))))))) @ emptyset))))))))))) @ ((repl @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X2:$i]:((repl @ (union @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (~((~((X3 = emptyset)))))))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (~((~((X4 = emptyset)))))))))))))) @ emptyset))) @ (^[X3:$i]:((binunion @ ((repl @ X2) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))) @ (^[X2:$i]:(((if @ (![X3:$i]:(((in @ X3) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X2) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))) @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (~((~((X4 = emptyset)))))))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))))))) @ emptyset))))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X4:$i]:((repl @ (union @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset))) @ (^[X5:$i]:((binunion @ ((repl @ X4) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))) => (~((![X4:$i]:(((in @ X4) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X3) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ X4) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj)))) @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset))))))))))))))) @ emptyset)) => (![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X4:$i]:((repl @ (union @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset))) @ (^[X5:$i]:((binunion @ ((repl @ X4) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))) => (~((![X4:$i]:(((in @ X4) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X3) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ X4) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj)))) @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset))))))))))) @ ((repl @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X3:$i]:((repl @ (union @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (~((~((X4 = emptyset)))))))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))))))) @ emptyset))) @ (^[X4:$i]:((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))) @ (^[X3:$i]:(((if @ (![X4:$i]:(((in @ X4) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X3) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ X4) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj)))) @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset))))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X5:$i]:((repl @ (union @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))))))) @ emptyset))) @ (^[X6:$i]:((binunion @ ((repl @ X5) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))) => (~((![X5:$i]:(((in @ X5) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X4) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ X5) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj)))) @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))))))) @ emptyset))))))))))))))) @ emptyset)) => (((imp @ (![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (~((~((X4 = emptyset)))))))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))))))) @ emptyset)) => (((non @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (~((~((X4 = emptyset)))))))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))))))) @ emptyset)) @ (^[X4:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ X1))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)) @ X2)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))) = (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ X2))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)) @ X1)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ X4) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj)))))) @ X3)))) @ $false) => ((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (~((~((X3 = emptyset)))))))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (~((~((X4 = emptyset)))))))))))))) @ emptyset)) @ X1))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 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X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset))) @ (^[X3:$i]:((binunion @ ((repl @ X2) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))) @ (^[X2:$i]:(((if @ (![X3:$i]:(((in @ X3) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X2) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))) @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset))))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (power @ ((famunion @ (d_1to @ n_2)) @ (^[X4:$i]:((repl @ (union @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset))) @ (^[X5:$i]:((binunion @ ((repl @ X4) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))) => (~((![X4:$i]:(((in @ X4) @ (d_1to @ n_2)) => ((in @ (((d_ReplSep @ X3) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ X4) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj)))) @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset))))))))))))))) @ emptyset)) => (((imp @ (![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset)) => (((non @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (X3 = emptyset)))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (X4 = emptyset)))))))))) @ emptyset)) @ (^[X3:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ X1)) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj))) = (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ X1))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ eigen__0)) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj))))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ X3) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))))) @ X2)))) @ $false) => ((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X2:$i]:(((in @ X2) @ omega) => (X2 = emptyset)))))) @ ((repl @ omega) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ omega) => (X3 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ X1))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ ((ind @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ (^[X6:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (X8 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X7:$i]:(~((![X8:$i]:(~((X7 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X9:$i]:(((in @ X9) @ omega) => (X9 = emptyset)))))) @ ((repl @ omega) @ (^[X9:$i]:(((if @ (~((X9 = emptyset)))) @ X9) @ (eps @ (^[X10:$i]:(~((((in @ X10) @ omega) => (X10 = emptyset)))))))))) @ emptyset)) @ X1)) @ d_Inj0)) @ ((repl @ X8) @ d_Inj1)))))))))) @ (^[X7:$i]:(((d_ReplSep @ X7) @ (^[X8:$i]:(~((![X9:$i]:(~(((d_Inj1 @ X9) = X8)))))))) @ d_Unj))) = (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (X8 = emptyset)))))))))) @ emptyset)) @ X1))) @ (^[X7:$i]:(~((![X8:$i]:(~((X7 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X9:$i]:(((in @ X9) @ omega) => (X9 = emptyset)))))) @ ((repl @ omega) @ (^[X9:$i]:(((if @ (~((X9 = emptyset)))) @ X9) @ (eps @ (^[X10:$i]:(~((((in @ X10) @ omega) => (X10 = emptyset)))))))))) @ emptyset)) @ eigen__0)) @ d_Inj0)) @ ((repl @ X8) @ d_Inj1)))))))))) @ (^[X7:$i]:(((d_ReplSep @ X7) @ (^[X8:$i]:(~((![X9:$i]:(~(((d_Inj1 @ X9) = X8)))))))) @ d_Unj))))) @ (^[X7:$i]:(~((![X8:$i]:(~((X7 = ((binunion @ ((repl @ X6) @ d_Inj0)) @ ((repl @ X8) @ d_Inj1)))))))))) @ (^[X7:$i]:(((d_ReplSep @ X7) @ (^[X8:$i]:(~((![X9:$i]:(~(((d_Inj1 @ X9) = X8)))))))) @ d_Unj))))))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (X8 = emptyset)))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (X9 = emptyset)))))))))) @ emptyset)) @ X1)) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj)))))) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj))))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ ((ind @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ (^[X6:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ 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((repl @ omega) @ (^[X9:$i]:(((if @ (~((X9 = emptyset)))) @ X9) @ (eps @ (^[X10:$i]:(~((((in @ X10) @ omega) => (X10 = emptyset)))))))))) @ emptyset)) @ eigen__0)) @ d_Inj0)) @ ((repl @ X8) @ d_Inj1)))))))))) @ (^[X7:$i]:(((d_ReplSep @ X7) @ (^[X8:$i]:(~((![X9:$i]:(~(((d_Inj1 @ X9) = X8)))))))) @ d_Unj))))) @ (^[X7:$i]:(~((![X8:$i]:(~((X7 = ((binunion @ ((repl @ X6) @ d_Inj0)) @ ((repl @ X8) @ d_Inj1)))))))))) @ (^[X7:$i]:(((d_ReplSep @ X7) @ (^[X8:$i]:(~((![X9:$i]:(~(((d_Inj1 @ X9) = X8)))))))) @ d_Unj))))))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (X8 = emptyset)))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ 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(^[X4:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (X8 = emptyset)))))))))) @ emptyset)) @ X1)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))) = (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (X5 = emptyset)))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (X6 = emptyset)))))))))) @ emptyset)) @ X1))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (X7 = emptyset)))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (X8 = emptyset)))))))))) @ emptyset)) @ eigen__0)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ X4) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))))))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ ((second1 @ (((if @ 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(^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))))) @ (^[X2:$i]:(((d_ReplSep @ X2) @ (^[X3:$i]:(~((![X4:$i]:(~(((d_Inj1 @ X4) = X3)))))))) @ d_Unj)))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X2:$i]:(((in @ X2) @ omega) => (X2 = emptyset)))))) @ ((repl @ omega) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ omega) => (X3 = emptyset)))))))))) @ emptyset)) @ X1))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ ((ind @ (((if 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(^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))))))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ X1)) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))))) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))))) @ (^[X2:$i]:(((d_ReplSep @ X2) @ (^[X3:$i]:(~((![X4:$i]:(~(((d_Inj1 @ X4) = X3)))))))) @ d_Unj))))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((binunion @ 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d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ X4) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))))))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ eigen__0))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (X6 = emptyset)))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (X7 = emptyset)))))))))) @ emptyset)) @ X1)) @ d_Inj0)) @ ((repl @ X5) @ d_Inj1)))))))))) @ (^[X4:$i]:(((d_ReplSep @ X4) @ (^[X5:$i]:(~((![X6:$i]:(~(((d_Inj1 @ X6) = X5)))))))) @ d_Unj)))))) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))))) @ (^[X2:$i]:(((d_ReplSep @ X2) @ (^[X3:$i]:(~((![X4:$i]:(~(((d_Inj1 @ X4) = X3)))))))) @ d_Unj))))))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X4:$i]:(((in @ X4) @ omega) => (X4 = emptyset)))))) @ ((repl @ omega) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ omega) => (X5 = emptyset)))))))))) @ emptyset)) @ eigen__0)) @ d_Inj0)) @ ((repl @ X3) @ d_Inj1)))))))))) @ (^[X2:$i]:(((d_ReplSep @ X2) @ (^[X3:$i]:(~((![X4:$i]:(~(((d_Inj1 @ X4) = X3)))))))) @ d_Unj))))))))),introduced(assumption,[])). % 231.06/230.33 thf(h4,assumption,(~((sP11 => (sP3 => sP2)))),introduced(assumption,[])). % 231.06/230.33 thf(h5,assumption,sP11,introduced(assumption,[])). % 231.06/230.33 thf(h6,assumption,(~((sP3 => sP2))),introduced(assumption,[])). % 231.06/230.33 thf(h7,assumption,sP3,introduced(assumption,[])). % 231.06/230.33 thf(h8,assumption,(~(sP2)),introduced(assumption,[])). % 231.06/230.33 thf(1,plain,(~(sP4) | sP12),inference(all_rule,[status(thm)],[])). % 231.06/230.33 thf(2,plain,((~(sP12) | ~(sP11)) | sP9),inference(prop_rule,[status(thm)],[])). % 231.06/230.33 thf(3,plain,((~(sP9) | ~(sP3)) | sP7),inference(prop_rule,[status(thm)],[])). % 231.06/230.33 thf(4,plain,(~(sP13) | sP5),inference(all_rule,[status(thm)],[])). % 231.06/230.33 thf(5,plain,((~(sP5) | ~(sP6)) | sP4),inference(prop_rule,[status(thm)],[])). % 231.06/230.33 thf(6,plain,((~(sP8) | ~(sP7)) | sP2),inference(prop_rule,[status(thm)],[])). % 231.06/230.33 thf(7,plain,(~(sP1) | sP8),inference(all_rule,[status(thm)],[])). % 231.06/230.33 thf(8,plain,(~(sP10) | sP1),inference(all_rule,[status(thm)],[])). % 231.06/230.33 thf(9,plain,sP10,inference(eq_sym,[status(thm)],[])). % 231.06/230.33 thf(k_satz67c,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ frac))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ frac))) @ (^[X2:$i]:(((moref @ X1) @ X2) => ((n_eq @ ((n_pf @ X2) @ ((d_367_w @ X1) @ X2))) @ X1))))))). % 231.06/230.33 thf(10,plain,sP13,inference(preprocess,[status(thm)],[k_satz67c]). % 231.06/230.33 thf(11,plain,$false,inference(prop_unsat,[status(thm),assumptions([h7,h8,h5,h6,h4,h2,h3,h1,h0])],[1,2,3,4,5,6,7,8,9,h2,h5,h7,h8,10])). % 231.06/230.33 thf(12,plain,$false,inference(tab_negimp,[status(thm),assumptions([h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h7,h8])],[h6,11,h7,h8])). % 231.06/230.33 thf(13,plain,$false,inference(tab_negimp,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,12,h5,h6])). % 231.06/230.33 thf(14,plain,$false,inference(tab_negall,[status(thm),assumptions([h2,h3,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,13,h4])). % 231.06/230.33 thf(15,plain,$false,inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,14,h2,h3])). % 231.06/230.33 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(~((~((X7 = emptyset)))))))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)) @ X1))) @ (^[X7:$i]:(~((![X8:$i]:(~((X7 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X9:$i]:(((in @ X9) @ omega) => (~((~((X9 = emptyset)))))))))) @ ((repl @ omega) @ (^[X9:$i]:(((if @ (~((X9 = emptyset)))) @ X9) @ (eps @ (^[X10:$i]:(~((((in @ X10) @ omega) => (~((~((X10 = emptyset)))))))))))))) @ emptyset)) @ X2)) @ d_Inj0)) @ ((repl @ X8) @ d_Inj1)))))))))) @ (^[X7:$i]:(((d_ReplSep @ X7) @ (^[X8:$i]:(~((![X9:$i]:(~(((d_Inj1 @ X9) = X8)))))))) @ d_Unj)))))) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)) @ ((ind @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)) @ (^[X7:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (~((~((X9 = emptyset)))))))))))))) @ emptyset)) @ X1))) @ (^[X8:$i]:(~((![X9:$i]:(~((X8 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X10:$i]:(((in @ X10) @ omega) => (~((~((X10 = emptyset)))))))))) @ ((repl @ omega) @ (^[X10:$i]:(((if @ (~((X10 = emptyset)))) @ X10) @ (eps @ (^[X11:$i]:(~((((in @ X11) @ omega) => (~((~((X11 = emptyset)))))))))))))) @ emptyset)) @ X2)) @ d_Inj0)) @ ((repl @ X9) @ d_Inj1)))))))))) @ (^[X8:$i]:(((d_ReplSep @ X8) @ (^[X9:$i]:(~((![X10:$i]:(~(((d_Inj1 @ X10) = X9)))))))) @ d_Unj))) = (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (~((~((X9 = emptyset)))))))))))))) @ emptyset)) @ X2))) @ (^[X8:$i]:(~((![X9:$i]:(~((X8 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X10:$i]:(((in @ X10) @ omega) => (~((~((X10 = emptyset)))))))))) @ ((repl @ omega) @ (^[X10:$i]:(((if @ (~((X10 = emptyset)))) @ X10) @ (eps @ (^[X11:$i]:(~((((in @ X11) @ omega) => (~((~((X11 = emptyset)))))))))))))) @ emptyset)) @ X1)) @ d_Inj0)) @ ((repl @ X9) @ d_Inj1)))))))))) @ (^[X8:$i]:(((d_ReplSep @ X8) @ (^[X9:$i]:(~((![X10:$i]:(~(((d_Inj1 @ X10) = X9)))))))) @ d_Unj))))) @ (^[X8:$i]:(~((![X9:$i]:(~((X8 = ((binunion @ ((repl @ X7) @ d_Inj0)) @ ((repl @ X9) @ d_Inj1)))))))))) @ (^[X8:$i]:(((d_ReplSep @ X8) @ (^[X9:$i]:(~((![X10:$i]:(~(((d_Inj1 @ X10) = X9)))))))) @ d_Unj))))))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)) @ X1))) @ (^[X7:$i]:(~((![X8:$i]:(~((X7 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X9:$i]:(((in @ X9) @ omega) => (~((~((X9 = emptyset)))))))))) @ ((repl @ omega) @ (^[X9:$i]:(((if @ (~((X9 = emptyset)))) @ X9) @ (eps @ (^[X10:$i]:(~((((in @ X10) @ omega) => (~((~((X10 = emptyset)))))))))))))) @ emptyset)) @ X2)) @ d_Inj0)) @ ((repl @ X8) @ d_Inj1)))))))))) @ (^[X7:$i]:(((d_ReplSep @ X7) @ (^[X8:$i]:(~((![X9:$i]:(~(((d_Inj1 @ X9) = X8)))))))) @ d_Unj))))))) @ (^[X7:$i]:(~((![X8:$i]:(~((X7 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X9:$i]:(((in @ X9) @ omega) => (~((~((X9 = emptyset)))))))))) @ ((repl @ omega) @ (^[X9:$i]:(((if @ (~((X9 = emptyset)))) @ X9) @ (eps @ (^[X10:$i]:(~((((in @ X10) @ omega) => (~((~((X10 = emptyset)))))))))))))) @ emptyset)) @ X2)) @ d_Inj0)) @ ((repl @ X8) @ d_Inj1)))))))))) @ (^[X7:$i]:(((d_ReplSep @ X7) @ (^[X8:$i]:(~((![X9:$i]:(~(((d_Inj1 @ X9) = X8)))))))) @ d_Unj)))) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj)))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ X2))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)) @ ((ind @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)) @ (^[X7:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (~((~((X9 = emptyset)))))))))))))) @ emptyset)) @ X1))) @ (^[X8:$i]:(~((![X9:$i]:(~((X8 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X10:$i]:(((in @ X10) @ omega) => (~((~((X10 = emptyset)))))))))) @ ((repl @ omega) @ (^[X10:$i]:(((if @ (~((X10 = emptyset)))) @ X10) @ (eps @ (^[X11:$i]:(~((((in @ X11) @ omega) => (~((~((X11 = emptyset)))))))))))))) @ emptyset)) @ X2)) @ d_Inj0)) @ ((repl @ X9) @ d_Inj1)))))))))) @ (^[X8:$i]:(((d_ReplSep @ X8) @ (^[X9:$i]:(~((![X10:$i]:(~(((d_Inj1 @ X10) = X9)))))))) @ d_Unj))) = (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (~((~((X9 = emptyset)))))))))))))) @ emptyset)) @ X2))) @ (^[X8:$i]:(~((![X9:$i]:(~((X8 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X10:$i]:(((in @ X10) @ omega) => (~((~((X10 = 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(^[X9:$i]:(((if @ (~((X9 = emptyset)))) @ X9) @ (eps @ (^[X10:$i]:(~((((in @ X10) @ omega) => (~((~((X10 = emptyset)))))))))))))) @ emptyset)) @ X2)) @ d_Inj0)) @ ((repl @ X8) @ d_Inj1)))))))))) @ (^[X7:$i]:(((d_ReplSep @ X7) @ (^[X8:$i]:(~((![X9:$i]:(~(((d_Inj1 @ X9) = X8)))))))) @ d_Unj)))))) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj)))))) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj))) = (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (~((~((X3 = emptyset)))))))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (~((~((X4 = emptyset)))))))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (~((~((X3 = emptyset)))))))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (~((~((X4 = emptyset)))))))))))))) @ emptyset)) @ (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (~((~((X3 = emptyset)))))))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (~((~((X4 = emptyset)))))))))))))) @ emptyset)) @ X2))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ ((ind @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ (^[X5:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))))))) @ emptyset)) @ X1))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (~((~((X9 = emptyset)))))))))))))) @ emptyset)) @ X2)) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj))) = (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))))))) @ emptyset)) @ X2))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (~((~((X9 = emptyset)))))))))))))) @ emptyset)) @ X1)) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj))))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ X5) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj))))))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ X1))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)) @ X2)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj)))))) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj))))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ ((ind @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ (^[X5:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))))))) @ emptyset)) @ X1))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (~((~((X9 = emptyset)))))))))))))) @ emptyset)) @ X2)) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj))) = (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))))))) @ emptyset)) @ X2))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (~((~((X9 = emptyset)))))))))))))) @ emptyset)) @ X1)) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj))))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ X5) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj))))))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ X1))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)) @ X2)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj))))))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)) @ X2)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj)))) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj)))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X3:$i]:(((in @ X3) @ omega) => (~((~((X3 = emptyset)))))))))) @ ((repl @ omega) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ omega) => (~((~((X4 = emptyset)))))))))))))) @ emptyset)) @ X2))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ (((pair1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ ((ind @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ (^[X5:$i]:((((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))))))) @ emptyset)) @ X1))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (~((~((X9 = emptyset)))))))))))))) @ emptyset)) @ X2)) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj))) = (((d_ReplSep @ (plus @ (((d_ReplSep @ (times @ ((first1 @ (((if @ (~((![X6:$i]:(((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))) @ ((repl @ omega) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))))))) @ emptyset)) @ X2))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X8:$i]:(((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))) @ ((repl @ omega) @ (^[X8:$i]:(((if @ (~((X8 = emptyset)))) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ omega) => (~((~((X9 = emptyset)))))))))))))) @ emptyset)) @ X1)) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj))))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((binunion @ ((repl @ X5) @ d_Inj0)) @ ((repl @ X7) @ d_Inj1)))))))))) @ (^[X6:$i]:(((d_ReplSep @ X6) @ (^[X7:$i]:(~((![X8:$i]:(~(((d_Inj1 @ X8) = X7)))))))) @ d_Unj))))))) @ (((d_ReplSep @ (times @ ((second1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ X1))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X7:$i]:(((in @ X7) @ omega) => (~((~((X7 = emptyset)))))))))) @ ((repl @ omega) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ omega) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)) @ X2)) @ d_Inj0)) @ ((repl @ X6) @ d_Inj1)))))))))) @ (^[X5:$i]:(((d_ReplSep @ X5) @ (^[X6:$i]:(~((![X7:$i]:(~(((d_Inj1 @ X7) = X6)))))))) @ d_Unj)))))) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj))))))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((binunion @ ((repl @ ((second1 @ (((if @ (~((![X5:$i]:(((in @ X5) @ omega) => (~((~((X5 = emptyset)))))))))) @ ((repl @ omega) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ omega) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ X1)) @ d_Inj0)) @ ((repl @ X4) @ d_Inj1)))))))))) @ (^[X3:$i]:(((d_ReplSep @ X3) @ (^[X4:$i]:(~((![X5:$i]:(~(((d_Inj1 @ X5) = X4)))))))) @ d_Unj))))))))),inference(contra,[status(thm),contra(discharge,[h0])],[16,h0])). % 231.06/230.33 % SZS output end Proof %------------------------------------------------------------------------------