0.06/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.06/0.12 % Command : satallax -s schedule_3_1 -E eprover -P picomus -M modes -p tstp -t %d %s 0.12/0.33 % Computer : n018.cluster.edu 0.12/0.33 % Model : x86_64 x86_64 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.12/0.33 % Memory : 8042.1875MB 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 0.12/0.33 % CPULimit : 180 0.12/0.33 % DateTime : Fri Sep 6 10:25:32 EDT 2019 0.12/0.33 % CPUTime : 55.53/55.59 % SZS status Theorem 55.53/55.59 % Mode: mode454 55.53/55.59 % Inferences: 70 55.53/55.59 % SZS output start Proof 55.53/55.59 thf(ty_$i, type, $i : $tType). 55.53/55.59 thf(ty_univof, type, univof : ($i>$i)). 55.53/55.59 thf(ty_power, type, power : ($i>$i)). 55.53/55.59 thf(ty_eps, type, eps : (($i>$o)>$i)). 55.53/55.59 thf(ty_union, type, union : ($i>$i)). 55.53/55.59 thf(ty_eigen__1, type, eigen__1 : $i). 55.53/55.59 thf(ty_eigen__0, type, eigen__0 : $i). 55.53/55.59 thf(ty_emptyset, type, emptyset : $i). 55.53/55.59 thf(ty_repl, type, repl : ($i>($i>$i)>$i)). 55.53/55.59 thf(ty_in, type, in : ($i>$i>$o)). 55.53/55.59 thf(h0, assumption, (![X1:$i>$o]:(![X2:$i]:((X1 @ X2) => (X1 @ (eps__0 @ X1))))),introduced(assumption,[])). 55.53/55.59 thf(eigendef_eigen__1, definition, (eigen__1 = (eps__0 @ (^[X1:$i]:(~((((in @ X1) @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))) => (~((X6 = X5)))) => (~(((~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))))) => (~((X2 = ((repl @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X4)))))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X4)))) => (~((X5 = X4)))) => (~(((~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X4)))))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X4)))))))) => (~((X3 = emptyset)))))))))) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((~((X3 = emptyset))) => (~((X4 = X3)))) => (~(((X3 = emptyset) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (eps @ (^[X6:$i]:(((~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))))) => (~((X6 = ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(eps @ (^[X8:$i]:(((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X7)))) => (~((X8 = X7)))) => (~(((~((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X7)))))) => (~((X8 = (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((![X10:$i>$o]:((X10 @ emptyset) => ((![X11:$i]:((X10 @ X11) => (X10 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X11)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = ((repl @ (power @ (power @ emptyset))) @ (^[X14:$i]:(eps @ (^[X15:$i]:((((in @ emptyset) @ X14) => (~((X15 = X11)))) => (~(((~(((in @ emptyset) @ X14))) => (~((X15 = X11)))))))))))))))))))))))))) => (X10 @ X9)))))))))))))))))))))))))) => (~(((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))) => (~((X6 = emptyset)))))))))) => (X5 = emptyset))))))))))))))))))))) => (~(((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))) => (~((X6 = X5)))) => (~(((~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))) => (~((X2 = emptyset)))))))))) => ((~((eigen__0 = X1))) => (~(((union @ ((repl @ (power @ (power @ emptyset))) @ (^[X2:$i]:(eps @ (^[X3:$i]:((((in @ emptyset) @ X2) => (~((X3 = eigen__0)))) => (~(((~(((in @ emptyset) @ X2))) => (~((X3 = ((repl @ (power @ (power @ emptyset))) @ (^[X4:$i]:(eps @ (^[X5:$i]:((((in @ emptyset) @ X4) => (~((X5 = eigen__0)))) => (~(((~(((in @ emptyset) @ X4))) => (~((X5 = eigen__0))))))))))))))))))))))) = (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X2:$i]:(eps @ (^[X3:$i]:((((in @ emptyset) @ X2) => (~((X3 = X1)))) => (~(((~(((in @ emptyset) @ X2))) => (~((X3 = ((repl @ (power @ (power @ emptyset))) @ (^[X4:$i]:(eps @ (^[X5:$i]:((((in @ emptyset) @ X4) => (~((X5 = X1)))) => (~(((~(((in @ emptyset) @ X4))) => (~((X5 = X1))))))))))))))))))))))))))))))))), introduced(definition,[new_symbols(definition,[eigen__1]))). 55.53/55.59 thf(eigendef_eigen__0, definition, (eigen__0 = (eps__0 @ (^[X1:$i]:(~((((in @ X1) @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))) => (~((X6 = X5)))) => (~(((~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))))) => (~((X2 = ((repl @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X4)))))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X4)))) => (~((X5 = X4)))) => (~(((~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X4)))))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X4)))))))) => (~((X3 = emptyset)))))))))) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((~((X3 = emptyset))) => (~((X4 = X3)))) => (~(((X3 = emptyset) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (eps @ (^[X6:$i]:(((~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))))) => (~((X6 = ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(eps @ (^[X8:$i]:(((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X7)))) => (~((X8 = X7)))) => (~(((~((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X7)))))) => (~((X8 = (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((![X10:$i>$o]:((X10 @ emptyset) => ((![X11:$i]:((X10 @ X11) => (X10 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X11)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = ((repl @ (power @ (power @ emptyset))) @ (^[X14:$i]:(eps @ (^[X15:$i]:((((in @ emptyset) @ X14) => (~((X15 = X11)))) => (~(((~(((in @ emptyset) @ X14))) => (~((X15 = X11)))))))))))))))))))))))))) => (X10 @ X9)))))))))))))))))))))))))) => (~(((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))) => (~((X6 = emptyset)))))))))) => (X5 = emptyset))))))))))))))))))))) => (~(((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))) => (~((X6 = X5)))) => (~(((~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))) => (~((X2 = emptyset)))))))))) => (![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X6)))) => (~((X7 = X6)))) => (~(((~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X6)))))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X8)))))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))))) => (~((X3 = ((repl @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))) => (~((X6 = X5)))) => (~(((~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))) => (~((X4 = emptyset)))))))))) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((~((X4 = emptyset))) => (~((X5 = X4)))) => (~(((X4 = emptyset) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (eps @ (^[X7:$i]:(((~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X8)))))))))) => (~((X7 = ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(eps @ (^[X9:$i]:(((![X10:$i>$o]:((X10 @ emptyset) => ((![X11:$i]:((X10 @ X11) => (X10 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X11)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = ((repl @ (power @ (power @ emptyset))) @ (^[X14:$i]:(eps @ (^[X15:$i]:((((in @ emptyset) @ X14) => (~((X15 = X11)))) => (~(((~(((in @ emptyset) @ X14))) => (~((X15 = X11)))))))))))))))))))))))))) => (X10 @ X8)))) => (~((X9 = X8)))) => (~(((~((![X10:$i>$o]:((X10 @ emptyset) => ((![X11:$i]:((X10 @ X11) => (X10 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X11)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = ((repl @ (power @ (power @ emptyset))) @ (^[X14:$i]:(eps @ (^[X15:$i]:((((in @ emptyset) @ X14) => (~((X15 = X11)))) => (~(((~(((in @ emptyset) @ X14))) => (~((X15 = X11)))))))))))))))))))))))))) => (X10 @ X8)))))) => (~((X9 = (eps @ (^[X10:$i]:(~((((in @ X10) @ (univof @ emptyset)) => (~((![X11:$i>$o]:((X11 @ emptyset) => ((![X12:$i]:((X11 @ X12) => (X11 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X12)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = ((repl @ (power @ (power @ emptyset))) @ (^[X15:$i]:(eps @ (^[X16:$i]:((((in @ emptyset) @ X15) => (~((X16 = X12)))) => (~(((~(((in @ emptyset) @ X15))) => (~((X16 = X12)))))))))))))))))))))))))) => (X11 @ X10)))))))))))))))))))))))))) => (~(((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X8)))))))) => (~((X7 = emptyset)))))))))) => (X6 = emptyset))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X6)))) => (~((X7 = X6)))) => (~(((~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X6)))))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X8)))))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))) => (~((X3 = emptyset)))))))))) => ((~((X1 = X2))) => (~(((union @ ((repl @ (power @ (power @ emptyset))) @ (^[X3:$i]:(eps @ (^[X4:$i]:((((in @ emptyset) @ X3) => (~((X4 = X1)))) => (~(((~(((in @ emptyset) @ X3))) => (~((X4 = ((repl @ (power @ (power @ emptyset))) @ (^[X5:$i]:(eps @ (^[X6:$i]:((((in @ emptyset) @ X5) => (~((X6 = X1)))) => (~(((~(((in @ emptyset) @ X5))) => (~((X6 = X1))))))))))))))))))))))) = (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X3:$i]:(eps @ (^[X4:$i]:((((in @ emptyset) @ X3) => (~((X4 = X2)))) => (~(((~(((in @ emptyset) @ X3))) => (~((X4 = ((repl @ (power @ (power @ emptyset))) @ (^[X5:$i]:(eps @ (^[X6:$i]:((((in @ emptyset) @ X5) => (~((X6 = X2)))) => (~(((~(((in @ emptyset) @ X5))) => (~((X6 = X2))))))))))))))))))))))))))))))))))), introduced(definition,[new_symbols(definition,[eigen__0]))). 55.53/55.59 thf(sP1,plain,(sP1 <=> ((~((eigen__0 = eigen__1))) => (~(((union @ ((repl @ (power @ (power @ emptyset))) @ (^[X1:$i]:(eps @ (^[X2:$i]:((((in @ emptyset) @ X1) => (~((X2 = eigen__0)))) => (~(((~(((in @ emptyset) @ X1))) => (~((X2 = ((repl @ (power @ (power @ emptyset))) @ (^[X3:$i]:(eps @ (^[X4:$i]:((((in @ emptyset) @ X3) => (~((X4 = eigen__0)))) => (~(((~(((in @ emptyset) @ X3))) => (~((X4 = eigen__0))))))))))))))))))))))) = (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X1:$i]:(eps @ (^[X2:$i]:((((in @ emptyset) @ X1) => (~((X2 = eigen__1)))) => (~(((~(((in @ emptyset) @ X1))) => (~((X2 = ((repl @ (power @ (power @ emptyset))) @ (^[X3:$i]:(eps @ (^[X4:$i]:((((in @ emptyset) @ X3) => (~((X4 = eigen__1)))) => (~(((~(((in @ emptyset) @ X3))) => (~((X4 = eigen__1))))))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP1])]))). 55.53/55.59 thf(sP2,plain,(sP2 <=> ((union @ ((repl @ (power @ (power @ emptyset))) @ (^[X1:$i]:(eps @ (^[X2:$i]:((((in @ emptyset) @ X1) => (~((X2 = eigen__0)))) => (~(((~(((in @ emptyset) @ X1))) => (~((X2 = ((repl @ (power @ (power @ emptyset))) @ (^[X3:$i]:(eps @ (^[X4:$i]:((((in @ emptyset) @ X3) => (~((X4 = eigen__0)))) => (~(((~(((in @ emptyset) @ X3))) => (~((X4 = eigen__0))))))))))))))))))))))) = (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X1:$i]:(eps @ (^[X2:$i]:((((in @ emptyset) @ X1) => (~((X2 = eigen__1)))) => (~(((~(((in @ emptyset) @ X1))) => (~((X2 = ((repl @ (power @ (power @ emptyset))) @ (^[X3:$i]:(eps @ (^[X4:$i]:((((in @ emptyset) @ X3) => (~((X4 = eigen__1)))) => (~(((~(((in @ emptyset) @ X3))) => (~((X4 = eigen__1)))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP2])]))). 55.53/55.59 thf(sP3,plain,(sP3 <=> (![X1:$i]:(((in @ X1) @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))) => (~((X6 = X5)))) => (~(((~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))))) => (~((X2 = ((repl @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X4)))))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X4)))) => (~((X5 = X4)))) => (~(((~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X4)))))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X4)))))))) => (~((X3 = emptyset)))))))))) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((~((X3 = emptyset))) => (~((X4 = X3)))) => (~(((X3 = emptyset) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (eps @ (^[X6:$i]:(((~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))))) => (~((X6 = ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(eps @ (^[X8:$i]:(((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X7)))) => (~((X8 = X7)))) => (~(((~((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X7)))))) => (~((X8 = (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((![X10:$i>$o]:((X10 @ emptyset) => ((![X11:$i]:((X10 @ X11) => (X10 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X11)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = ((repl @ (power @ (power @ emptyset))) @ (^[X14:$i]:(eps @ (^[X15:$i]:((((in @ emptyset) @ X14) => (~((X15 = X11)))) => (~(((~(((in @ emptyset) @ X14))) => (~((X15 = X11)))))))))))))))))))))))))) => (X10 @ X9)))))))))))))))))))))))))) => (~(((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))) => (~((X6 = emptyset)))))))))) => (X5 = emptyset))))))))))))))))))))) => (~(((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))) => (~((X6 = X5)))) => (~(((~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))) => (~((X2 = emptyset)))))))))) => ((~((eigen__0 = X1))) => (~(((union @ ((repl @ (power @ (power @ emptyset))) @ (^[X2:$i]:(eps @ (^[X3:$i]:((((in @ emptyset) @ X2) => (~((X3 = eigen__0)))) => (~(((~(((in @ emptyset) @ X2))) => (~((X3 = ((repl @ (power @ (power @ emptyset))) @ (^[X4:$i]:(eps @ (^[X5:$i]:((((in @ emptyset) @ X4) => (~((X5 = eigen__0)))) => (~(((~(((in @ emptyset) @ X4))) => (~((X5 = eigen__0))))))))))))))))))))))) = (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X2:$i]:(eps @ (^[X3:$i]:((((in @ emptyset) @ X2) => (~((X3 = X1)))) => (~(((~(((in @ emptyset) @ X2))) => (~((X3 = ((repl @ (power @ (power @ emptyset))) @ (^[X4:$i]:(eps @ (^[X5:$i]:((((in @ emptyset) @ X4) => (~((X5 = X1)))) => (~(((~(((in @ emptyset) @ X4))) => (~((X5 = X1))))))))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP3])]))). 55.53/55.59 thf(sP4,plain,(sP4 <=> (((in @ eigen__0) @ (eps @ (^[X1:$i]:(((~((![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X4)))))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X4)))) => (~((X5 = X4)))) => (~(((~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X4)))))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X4)))))))) => (~((X3 = emptyset)))))))))) => (X2 = emptyset))))) => (~((X1 = ((repl @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((![X4:$i>$o]:((X4 @ emptyset) => ((![X5:$i]:((X4 @ X5) => (X4 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X6:$i]:(eps @ (^[X7:$i]:((((in @ emptyset) @ X6) => (~((X7 = X5)))) => (~(((~(((in @ emptyset) @ X6))) => (~((X7 = ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X5)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = X5)))))))))))))))))))))))))) => (X4 @ X3)))))))))) => (~((X2 = ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X3)))) => (~((X4 = X3)))) => (~(((~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X3)))))) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))))))))))))))))))))) => (~(((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((![X4:$i>$o]:((X4 @ emptyset) => ((![X5:$i]:((X4 @ X5) => (X4 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X6:$i]:(eps @ (^[X7:$i]:((((in @ emptyset) @ X6) => (~((X7 = X5)))) => (~(((~(((in @ emptyset) @ X6))) => (~((X7 = ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X5)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = X5)))))))))))))))))))))))))) => (X4 @ X3)))))))) => (~((X2 = emptyset)))))))))) @ (^[X2:$i]:(eps @ (^[X3:$i]:(((~((X2 = emptyset))) => (~((X3 = X2)))) => (~(((X2 = emptyset) => (~((X3 = (eps @ (^[X4:$i]:(~((((in @ X4) @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X6)))) => (~((X7 = X6)))) => (~(((~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X6)))))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X8)))))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))))))))))))))))))))) => (~(((![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X4)))))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X4)))) => (~((X5 = X4)))) => (~(((~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X4)))))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X4)))))))) => (~((X3 = emptyset)))))))))) => (X2 = emptyset))) => (~((X1 = emptyset)))))))))) => sP3),introduced(definition,[new_symbols(definition,[sP4])]))). 55.53/55.59 thf(sP5,plain,(sP5 <=> (sP2 => (eigen__0 = eigen__1)),introduced(definition,[new_symbols(definition,[sP5])]))). 55.53/55.59 thf(sP6,plain,(sP6 <=> (![X1:$i]:(((union @ ((repl @ (power @ (power @ emptyset))) @ (^[X2:$i]:(eps @ (^[X3:$i]:((((in @ emptyset) @ X2) => (~((X3 = eigen__0)))) => (~(((~(((in @ emptyset) @ X2))) => (~((X3 = ((repl @ (power @ (power @ emptyset))) @ (^[X4:$i]:(eps @ (^[X5:$i]:((((in @ emptyset) @ X4) => (~((X5 = eigen__0)))) => (~(((~(((in @ emptyset) @ X4))) => (~((X5 = eigen__0))))))))))))))))))))))) = (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X2:$i]:(eps @ (^[X3:$i]:((((in @ emptyset) @ X2) => (~((X3 = X1)))) => (~(((~(((in @ emptyset) @ X2))) => (~((X3 = ((repl @ (power @ (power @ emptyset))) @ (^[X4:$i]:(eps @ (^[X5:$i]:((((in @ emptyset) @ X4) => (~((X5 = X1)))) => (~(((~(((in @ emptyset) @ X4))) => (~((X5 = X1)))))))))))))))))))))))) => (eigen__0 = X1))),introduced(definition,[new_symbols(definition,[sP6])]))). 55.53/55.59 thf(sP7,plain,(sP7 <=> (eigen__0 = eigen__1),introduced(definition,[new_symbols(definition,[sP7])]))). 55.53/55.59 thf(sP8,plain,(sP8 <=> (![X1:$i]:(((in @ X1) @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))) => (~((X6 = X5)))) => (~(((~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))))) => (~((X2 = ((repl @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X4)))))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X4)))) => (~((X5 = X4)))) => (~(((~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X4)))))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X4)))))))) => (~((X3 = emptyset)))))))))) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((~((X3 = emptyset))) => (~((X4 = X3)))) => (~(((X3 = emptyset) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (eps @ (^[X6:$i]:(((~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))))) => (~((X6 = ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(eps @ (^[X8:$i]:(((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X7)))) => (~((X8 = X7)))) => (~(((~((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X7)))))) => (~((X8 = (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((![X10:$i>$o]:((X10 @ emptyset) => ((![X11:$i]:((X10 @ X11) => (X10 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X11)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = ((repl @ (power @ (power @ emptyset))) @ (^[X14:$i]:(eps @ (^[X15:$i]:((((in @ emptyset) @ X14) => (~((X15 = X11)))) => (~(((~(((in @ emptyset) @ X14))) => (~((X15 = X11)))))))))))))))))))))))))) => (X10 @ X9)))))))))))))))))))))))))) => (~(((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))) => (~((X6 = emptyset)))))))))) => (X5 = emptyset))))))))))))))))))))) => (~(((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))) => (~((X6 = X5)))) => (~(((~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))) => (~((X4 = emptyset)))))))))) => (X3 = emptyset))) => (~((X2 = emptyset)))))))))) => (![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X6)))) => (~((X7 = X6)))) => (~(((~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X6)))))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X8)))))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))))) => (~((X3 = ((repl @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))) => (~((X6 = X5)))) => (~(((~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X5)))))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X7)))))))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))) => (~((X4 = emptyset)))))))))) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((~((X4 = emptyset))) => (~((X5 = X4)))) => (~(((X4 = emptyset) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (eps @ (^[X7:$i]:(((~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X8)))))))))) => (~((X7 = ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(eps @ (^[X9:$i]:(((![X10:$i>$o]:((X10 @ emptyset) => ((![X11:$i]:((X10 @ X11) => (X10 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X11)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = ((repl @ (power @ (power @ emptyset))) @ (^[X14:$i]:(eps @ (^[X15:$i]:((((in @ emptyset) @ X14) => (~((X15 = X11)))) => (~(((~(((in @ emptyset) @ X14))) => (~((X15 = X11)))))))))))))))))))))))))) => (X10 @ X8)))) => (~((X9 = X8)))) => (~(((~((![X10:$i>$o]:((X10 @ emptyset) => ((![X11:$i]:((X10 @ X11) => (X10 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X11)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = ((repl @ (power @ (power @ emptyset))) @ (^[X14:$i]:(eps @ (^[X15:$i]:((((in @ emptyset) @ X14) => (~((X15 = X11)))) => (~(((~(((in @ emptyset) @ X14))) => (~((X15 = X11)))))))))))))))))))))))))) => (X10 @ X8)))))) => (~((X9 = (eps @ (^[X10:$i]:(~((((in @ X10) @ (univof @ emptyset)) => (~((![X11:$i>$o]:((X11 @ emptyset) => ((![X12:$i]:((X11 @ X12) => (X11 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X12)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = ((repl @ (power @ (power @ emptyset))) @ (^[X15:$i]:(eps @ (^[X16:$i]:((((in @ emptyset) @ X15) => (~((X16 = X12)))) => (~(((~(((in @ emptyset) @ X15))) => (~((X16 = X12)))))))))))))))))))))))))) => (X11 @ X10)))))))))))))))))))))))))) => (~(((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X8)))))))) => (~((X7 = emptyset)))))))))) => (X6 = emptyset))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X6)))) => (~((X7 = X6)))) => (~(((~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X6)))))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X8)))))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))) => (~((X3 = emptyset)))))))))) => ((~((X1 = X2))) => (~(((union @ ((repl @ (power @ (power @ emptyset))) @ (^[X3:$i]:(eps @ (^[X4:$i]:((((in @ emptyset) @ X3) => (~((X4 = X1)))) => (~(((~(((in @ emptyset) @ X3))) => (~((X4 = ((repl @ (power @ (power @ emptyset))) @ (^[X5:$i]:(eps @ (^[X6:$i]:((((in @ emptyset) @ X5) => (~((X6 = X1)))) => (~(((~(((in @ emptyset) @ X5))) => (~((X6 = X1))))))))))))))))))))))) = (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X3:$i]:(eps @ (^[X4:$i]:((((in @ emptyset) @ X3) => (~((X4 = X2)))) => (~(((~(((in @ emptyset) @ X3))) => (~((X4 = ((repl @ (power @ (power @ emptyset))) @ (^[X5:$i]:(eps @ (^[X6:$i]:((((in @ emptyset) @ X5) => (~((X6 = X2)))) => (~(((~(((in @ emptyset) @ X5))) => (~((X6 = X2))))))))))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP8])]))). 55.53/55.59 thf(sP9,plain,(sP9 <=> (((in @ eigen__1) @ (eps @ (^[X1:$i]:(((~((![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X4)))))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X4)))) => (~((X5 = X4)))) => (~(((~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X4)))))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X4)))))))) => (~((X3 = emptyset)))))))))) => (X2 = emptyset))))) => (~((X1 = ((repl @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((![X4:$i>$o]:((X4 @ emptyset) => ((![X5:$i]:((X4 @ X5) => (X4 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X6:$i]:(eps @ (^[X7:$i]:((((in @ emptyset) @ X6) => (~((X7 = X5)))) => (~(((~(((in @ emptyset) @ X6))) => (~((X7 = ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X5)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = X5)))))))))))))))))))))))))) => (X4 @ X3)))))))))) => (~((X2 = ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X3)))) => (~((X4 = X3)))) => (~(((~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X3)))))) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X5)))))))))))))))))))))))))) => (~(((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((![X4:$i>$o]:((X4 @ emptyset) => ((![X5:$i]:((X4 @ X5) => (X4 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X6:$i]:(eps @ (^[X7:$i]:((((in @ emptyset) @ X6) => (~((X7 = X5)))) => (~(((~(((in @ emptyset) @ X6))) => (~((X7 = ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X5)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = X5)))))))))))))))))))))))))) => (X4 @ X3)))))))) => (~((X2 = emptyset)))))))))) @ (^[X2:$i]:(eps @ (^[X3:$i]:(((~((X2 = emptyset))) => (~((X3 = X2)))) => (~(((X2 = emptyset) => (~((X3 = (eps @ (^[X4:$i]:(~((((in @ X4) @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X6)))) => (~((X7 = X6)))) => (~(((~((![X8:$i>$o]:((X8 @ emptyset) => ((![X9:$i]:((X8 @ X9) => (X8 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X9)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = ((repl @ (power @ (power @ emptyset))) @ (^[X12:$i]:(eps @ (^[X13:$i]:((((in @ emptyset) @ X12) => (~((X13 = X9)))) => (~(((~(((in @ emptyset) @ X12))) => (~((X13 = X9)))))))))))))))))))))))))) => (X8 @ X6)))))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((![X9:$i>$o]:((X9 @ emptyset) => ((![X10:$i]:((X9 @ X10) => (X9 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X10)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = ((repl @ (power @ (power @ emptyset))) @ (^[X13:$i]:(eps @ (^[X14:$i]:((((in @ emptyset) @ X13) => (~((X14 = X10)))) => (~(((~(((in @ emptyset) @ X13))) => (~((X14 = X10)))))))))))))))))))))))))) => (X9 @ X8)))))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))) => (~((X5 = emptyset)))))))))) => (X4 = emptyset))))))))))))))))))))) => (~(((![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X4)))))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X4)))) => (~((X5 = X4)))) => (~(((~((![X6:$i>$o]:((X6 @ emptyset) => ((![X7:$i]:((X6 @ X7) => (X6 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X8:$i]:(eps @ (^[X9:$i]:((((in @ emptyset) @ X8) => (~((X9 = X7)))) => (~(((~(((in @ emptyset) @ X8))) => (~((X9 = ((repl @ (power @ (power @ emptyset))) @ (^[X10:$i]:(eps @ (^[X11:$i]:((((in @ emptyset) @ X10) => (~((X11 = X7)))) => (~(((~(((in @ emptyset) @ X10))) => (~((X11 = X7)))))))))))))))))))))))))) => (X6 @ X4)))))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((![X7:$i>$o]:((X7 @ emptyset) => ((![X8:$i]:((X7 @ X8) => (X7 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X8)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = ((repl @ (power @ (power @ emptyset))) @ (^[X11:$i]:(eps @ (^[X12:$i]:((((in @ emptyset) @ X11) => (~((X12 = X8)))) => (~(((~(((in @ emptyset) @ X11))) => (~((X12 = X8)))))))))))))))))))))))))) => (X7 @ X6)))))))))))))))))))))))))) => (~(((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((![X5:$i>$o]:((X5 @ emptyset) => ((![X6:$i]:((X5 @ X6) => (X5 @ (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X7:$i]:(eps @ (^[X8:$i]:((((in @ emptyset) @ X7) => (~((X8 = X6)))) => (~(((~(((in @ emptyset) @ X7))) => (~((X8 = ((repl @ (power @ (power @ emptyset))) @ (^[X9:$i]:(eps @ (^[X10:$i]:((((in @ emptyset) @ X9) => (~((X10 = X6)))) => (~(((~(((in @ emptyset) @ X9))) => (~((X10 = X6)))))))))))))))))))))))))) => (X5 @ X4)))))))) => (~((X3 = emptyset)))))))))) => (X2 = emptyset))) => (~((X1 = emptyset)))))))))) => sP1),introduced(definition,[new_symbols(definition,[sP9])]))). 55.53/55.59 thf(sP10,plain,(sP10 <=> (![X1:$i]:(![X2:$i]:(((union @ ((repl @ (power @ (power @ emptyset))) @ (^[X3:$i]:(eps @ (^[X4:$i]:((((in @ emptyset) @ X3) => (~((X4 = X1)))) => (~(((~(((in @ emptyset) @ X3))) => (~((X4 = ((repl @ (power @ (power @ emptyset))) @ (^[X5:$i]:(eps @ (^[X6:$i]:((((in @ emptyset) @ X5) => (~((X6 = X1)))) => (~(((~(((in @ emptyset) @ X5))) => (~((X6 = X1))))))))))))))))))))))) = (union @ ((repl @ (power @ (power @ emptyset))) @ (^[X3:$i]:(eps @ (^[X4:$i]:((((in @ emptyset) @ X3) => (~((X4 = X2)))) => (~(((~(((in @ emptyset) @ X3))) => (~((X4 = ((repl @ (power @ (power @ emptyset))) @ (^[X5:$i]:(eps @ (^[X6:$i]:((((in @ emptyset) @ X5) => (~((X6 = X2)))) => (~(((~(((in @ emptyset) @ X5))) => (~((X6 = X2)))))))))))))))))))))))) => (X1 = X2)))),introduced(definition,[new_symbols(definition,[sP10])]))). 55.53/55.59 thf(def_is_of,definition,(is_of = (^[X1:$i]:(^[X2:$i>$o]:(X2 @ X1))))). 55.53/55.59 thf(def_all_of,definition,(all_of = (^[X1:$i>$o]:(^[X2:$i>$o]:(![X3:$i]:(((is_of @ X3) @ X1) => (X2 @ X3))))))). 55.53/55.59 thf(def_d_Subq,definition,(d_Subq = (^[X1:$i]:(^[X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))))))). 55.53/55.59 thf(def_if,definition,(if = (^[X1:$o]:(^[X2:$i]:(^[X3:$i]:(eps @ (^[X4:$i]:((X1 => (~((X4 = X2)))) => (~(((~(X1)) => (~((X4 = X3)))))))))))))). 55.53/55.59 thf(def_d_UPair,definition,(d_UPair = (^[X1:$i]:(^[X2:$i]:((repl @ (power @ (power @ emptyset))) @ (^[X3:$i]:(((if @ ((in @ emptyset) @ X3)) @ X1) @ X2))))))). 55.53/55.59 thf(def_d_Sing,definition,(d_Sing = (^[X1:$i]:((d_UPair @ X1) @ X1)))). 55.53/55.59 thf(def_binunion,definition,(binunion = (^[X1:$i]:(^[X2:$i]:(union @ ((d_UPair @ X1) @ X2)))))). 55.53/55.59 thf(def_famunion,definition,(famunion = (^[X1:$i]:(^[X2:$i>$i]:(union @ ((repl @ X1) @ X2)))))). 55.53/55.59 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))))). 55.53/55.59 thf(def_d_ReplSep,definition,(d_ReplSep = (^[X1:$i]:(^[X2:$i>$o]:(repl @ ((d_Sep @ X1) @ X2)))))). 55.53/55.59 thf(def_ordsucc,definition,(ordsucc = (^[X1:$i]:((binunion @ X1) @ (d_Sing @ X1))))). 55.53/55.59 thf(def_nat_p,definition,(nat_p = (^[X1:$i]:(![X2:$i>$o]:((X2 @ emptyset) => ((![X3:$i]:((X2 @ X3) => (X2 @ (ordsucc @ X3)))) => (X2 @ X1))))))). 55.53/55.59 thf(def_omega,definition,(omega = ((d_Sep @ (univof @ emptyset)) @ nat_p))). 55.53/55.59 thf(def_d_Inj1,definition,(d_Inj1 = (d_In_rec @ (^[X1:$i]:(^[X2:$i>$i]:((binunion @ (d_Sing @ emptyset)) @ ((repl @ X1) @ X2))))))). 55.53/55.59 thf(def_d_Inj0,definition,(d_Inj0 = (^[X1:$i]:((repl @ X1) @ d_Inj1)))). 55.53/55.59 thf(def_d_Unj,definition,(d_Unj = (d_In_rec @ (^[X1:$i]:(repl @ ((setminus @ X1) @ (d_Sing @ emptyset))))))). 55.53/55.59 thf(def_pair,definition,(pair = (^[X1:$i]:(^[X2:$i]:((binunion @ ((repl @ X1) @ d_Inj0)) @ ((repl @ X2) @ d_Inj1)))))). 55.53/55.59 thf(def_proj0,definition,(proj0 = (^[X1:$i]:(((d_ReplSep @ X1) @ (^[X2:$i]:(~((![X3:$i]:(~(((d_Inj0 @ X3) = X2)))))))) @ d_Unj)))). 55.53/55.59 thf(def_proj1,definition,(proj1 = (^[X1:$i]:(((d_ReplSep @ X1) @ (^[X2:$i]:(~((![X3:$i]:(~(((d_Inj1 @ X3) = X2)))))))) @ d_Unj)))). 55.53/55.59 thf(def_d_Sigma,definition,(d_Sigma = (^[X1:$i]:(^[X2:$i>$i]:((famunion @ X1) @ (^[X3:$i]:((repl @ (X2 @ X3)) @ (pair @ X3)))))))). 55.53/55.59 thf(def_setprod,definition,(setprod = (^[X1:$i]:(^[X2:$i]:((d_Sigma @ X1) @ (^[X3:$i]:X2)))))). 55.53/55.59 thf(def_ap,definition,(ap = (^[X1:$i]:(^[X2:$i]:(((d_ReplSep @ X1) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1))))). 55.53/55.59 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)))))))))). 55.53/55.59 thf(def_imp,definition,(imp = (^[X1:$o]:(^[X2:$o]:(X1 => X2))))). 55.53/55.59 thf(def_d_not,definition,(d_not = (^[X1:$o]:((imp @ X1) @ $false)))). 55.53/55.59 thf(def_e_is,definition,(e_is = (^[X1:$i]:(=)))). 55.53/55.59 thf(def_image,definition,(image = (^[X1:$i]:(^[X2:$i]:(^[X3:$i]:(^[X4:$i]:((l_some @ X1) @ (^[X5:$i]:(((e_is @ X2) @ X4) @ ((ap @ X3) @ X5)))))))))). 55.53/55.59 thf(def_e_in,definition,(e_in = (^[X1:$i]:(^[X2:$i>$o]:(^[X3:$i]:X3))))). 55.53/55.59 thf(def_d_pair,definition,(d_pair = (^[X1:$i]:(^[X2:$i]:pair)))). 55.53/55.59 thf(def_first,definition,(first = (^[X1:$i]:(^[X2:$i]:proj0)))). 55.53/55.59 thf(def_second,definition,(second = (^[X1:$i]:(^[X2:$i]:proj1)))). 55.53/55.59 thf(def_esti,definition,(esti = (^[X1:$i]:in))). 55.53/55.59 thf(def_incl,definition,(incl = (^[X1:$i]:(^[X2:$i]:(^[X3:$i]:((all @ X1) @ (^[X4:$i]:((imp @ (((esti @ X1) @ X4) @ X2)) @ (((esti @ X1) @ X4) @ X3))))))))). 55.53/55.59 thf(def_nat,definition,(nat = ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))). 55.53/55.59 thf(def_n_is,definition,(n_is = (e_is @ nat))). 55.53/55.59 thf(def_nis,definition,(nis = (^[X1:$i]:(^[X2:$i]:(d_not @ ((n_is @ X1) @ X2)))))). 55.53/55.59 thf(def_n_in,definition,(n_in = (esti @ nat))). 55.53/55.59 thf(def_n_all,definition,(n_all = (all @ nat))). 55.53/55.59 thf(def_n_1,definition,(n_1 = (ordsucc @ emptyset))). 55.53/55.59 thf(def_cond1,definition,(cond1 = (n_in @ n_1))). 55.53/55.59 thf(def_cond2,definition,(cond2 = (^[X1:$i]:(n_all @ (^[X2:$i]:((imp @ ((n_in @ X2) @ X1)) @ ((n_in @ (ordsucc @ X2)) @ X1))))))). 55.53/55.59 thf(satz1,conjecture,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ nat))) @ (^[X2:$i]:(((nis @ X1) @ X2) => ((nis @ (ordsucc @ X1)) @ (ordsucc @ X2)))))))). 55.53/55.59 thf(h1,negated_conjecture,(~(sP8)),inference(assume_negation,[status(cth)],[satz1])). 55.53/55.59 thf(ordsucc_inj,axiom,(![X1:$i]:(![X2:$i]:(((ordsucc @ X1) = (ordsucc @ X2)) => (X1 = X2))))). 55.53/55.59 thf(1,plain,sP10,inference(preprocess,[status(thm)],[ordsucc_inj]). 55.53/55.59 thf(2,plain,(sP8 | ~(sP4)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__0])). 55.53/55.59 thf(3,plain,(sP4 | ~(sP3)),inference(prop_rule,[status(thm)],[])). 55.53/55.59 thf(4,plain,(sP3 | ~(sP9)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1])). 55.53/55.59 thf(5,plain,(sP9 | ~(sP1)),inference(prop_rule,[status(thm)],[])). 55.53/55.59 thf(6,plain,(sP1 | ~(sP7)),inference(prop_rule,[status(thm)],[])). 55.53/55.59 thf(7,plain,(sP1 | sP2),inference(prop_rule,[status(thm)],[])). 55.53/55.59 thf(8,plain,((~(sP5) | ~(sP2)) | sP7),inference(prop_rule,[status(thm)],[])). 55.53/55.59 thf(9,plain,(~(sP6) | sP5),inference(all_rule,[status(thm)],[])). 55.53/55.59 thf(10,plain,(~(sP10) | sP6),inference(all_rule,[status(thm)],[])). 55.53/55.59 thf(11,plain,$false,inference(prop_unsat,[status(thm),assumptions([h1,h0])],[h1,1,2,3,4,5,6,7,8,9,10])). 55.53/55.59 thf(12,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[11,h0])). 55.53/55.59 thf(0,theorem,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ nat))) @ (^[X2:$i]:(((nis @ X1) @ X2) => ((nis @ (ordsucc @ X1)) @ (ordsucc @ X2))))))),inference(contra,[status(thm),contra(discharge,[h1])],[11,h1])). 55.53/55.59 % SZS output end Proof 55.53/55.59 EOF