0.07/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.07/0.12	% Command    : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
0.12/0.33	% Computer   : n019.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   : Thu Aug 29 16:18:22 EDT 2019
0.12/0.34	% CPUTime    : 
160.10/159.48	% SZS status Theorem
160.10/159.48	% Mode: mode474:USE_SINE=true:SINE_TOLERANCE=1.0:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=2.:SINE_DEPTH=0
160.10/159.48	% Inferences: 8850
160.10/159.48	% SZS output start Proof
160.10/159.48	thf(ty_univof, type, univof : ($i>$i)).
160.10/159.48	thf(ty_if, type, if : ($o>$i>$i>$i)).
160.10/159.48	thf(ty_eps, type, eps : (($i>$o)>$i)).
160.10/159.48	thf(ty_union, type, union : ($i>$i)).
160.10/159.48	thf(ty_d_Pi, type, d_Pi : ($i>($i>$i)>$i)).
160.10/159.48	thf(ty_d_ReplSep, type, d_ReplSep : ($i>($i>$o)>($i>$i)>$i)).
160.10/159.48	thf(ty_eigen__0, type, eigen__0 : $i).
160.10/159.48	thf(ty_emptyset, type, emptyset : $i).
160.10/159.48	thf(ty_ind, type, ind : ($i>($i>$o)>$i)).
160.10/159.48	thf(ty_pair, type, pair : ($i>$i>$i)).
160.10/159.48	thf(ty_proj1, type, proj1 : ($i>$i)).
160.10/159.48	thf(ty_d_24_prop2, type, d_24_prop2 : ($i>$i>$o)).
160.10/159.48	thf(ty_repl, type, repl : ($i>($i>$i)>$i)).
160.10/159.48	thf(ty_in, type, in : ($i>$i>$o)).
160.10/159.48	thf(ty_d_UPair, type, d_UPair : ($i>$i>$i)).
160.10/159.48	thf(ty_nat_p, type, nat_p : ($i>$o)).
160.10/159.48	thf(h0, assumption, (![X1:$i>$o]:(![X2:$i]:((X1 @ X2) => (X1 @ (eps__0 @ X1))))),introduced(assumption,[])).
160.10/159.48	thf(eigendef_eigen__0, definition, eigen__0 = (eps__0 @ (^[X1:$i]:(~((((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) => ((union @ ((d_UPair @ X1) @ ((d_UPair @ X1) @ X1))) = (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)))) @ (d_24_prop2 @ (union @ ((d_UPair @ emptyset) @ ((d_UPair @ emptyset) @ emptyset)))))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ X1) @ X3))))))))) @ proj1))))))), introduced(definition,[new_symbols(definition,[eigen__0])])).
160.10/159.48	thf(sP1,plain,sP1 <=> (((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) => (X1 = emptyset)))))) @ ((repl @ (((if @ (~((![X1:$i]:(((in @ X1) @ (univof @ emptyset)) => (~((nat_p @ X1)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X1:$i]:(((if @ (nat_p @ X1)) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((X1 = emptyset)))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)))) @ (d_24_prop2 @ (union @ ((d_UPair @ emptyset) @ ((d_UPair @ emptyset) @ emptyset)))))) @ (^[X1:$i]:(~((![X2:$i]:(~((X1 = ((pair @ eigen__0) @ X2))))))))) @ proj1) = (union @ ((d_UPair @ eigen__0) @ ((d_UPair @ eigen__0) @ eigen__0)))) => ((union @ ((d_UPair @ eigen__0) @ ((d_UPair @ eigen__0) @ eigen__0))) = (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) => (X1 = emptyset)))))) @ ((repl @ (((if @ (~((![X1:$i]:(((in @ X1) @ (univof @ emptyset)) => (~((nat_p @ X1)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X1:$i]:(((if @ (nat_p @ X1)) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((X1 = emptyset)))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)))) @ (d_24_prop2 @ (union @ ((d_UPair @ emptyset) @ ((d_UPair @ emptyset) @ emptyset)))))) @ (^[X1:$i]:(~((![X2:$i]:(~((X1 = ((pair @ eigen__0) @ X2))))))))) @ proj1))),introduced(definition,[new_symbols(definition,[sP1])])).
160.10/159.48	thf(sP2,plain,sP2 <=> (![X1:$i]:(((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)))) @ (d_24_prop2 @ (union @ ((d_UPair @ emptyset) @ ((d_UPair @ emptyset) @ emptyset)))))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ eigen__0) @ X3))))))))) @ proj1) = X1) => (X1 = (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)))) @ (d_24_prop2 @ (union @ ((d_UPair @ emptyset) @ ((d_UPair @ emptyset) @ emptyset)))))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ eigen__0) @ X3))))))))) @ proj1)))),introduced(definition,[new_symbols(definition,[sP2])])).
160.10/159.48	thf(sP3,plain,sP3 <=> (![X1:$i]:(![X2:$i]:((X1 = X2) => (X2 = X1)))),introduced(definition,[new_symbols(definition,[sP3])])).
160.10/159.48	thf(sP4,plain,sP4 <=> ((union @ ((d_UPair @ eigen__0) @ ((d_UPair @ eigen__0) @ eigen__0))) = (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) => (X1 = emptyset)))))) @ ((repl @ (((if @ (~((![X1:$i]:(((in @ X1) @ (univof @ emptyset)) => (~((nat_p @ X1)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X1:$i]:(((if @ (nat_p @ X1)) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((X1 = emptyset)))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)))) @ (d_24_prop2 @ (union @ ((d_UPair @ emptyset) @ ((d_UPair @ emptyset) @ emptyset)))))) @ (^[X1:$i]:(~((![X2:$i]:(~((X1 = ((pair @ eigen__0) @ X2))))))))) @ proj1)),introduced(definition,[new_symbols(definition,[sP4])])).
160.10/159.48	thf(sP5,plain,sP5 <=> ((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) => (X1 = emptyset)))))) @ ((repl @ (((if @ (~((![X1:$i]:(((in @ X1) @ (univof @ emptyset)) => (~((nat_p @ X1)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X1:$i]:(((if @ (nat_p @ X1)) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((X1 = emptyset)))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)))) @ (d_24_prop2 @ (union @ ((d_UPair @ emptyset) @ ((d_UPair @ emptyset) @ emptyset)))))) @ (^[X1:$i]:(~((![X2:$i]:(~((X1 = ((pair @ eigen__0) @ X2))))))))) @ proj1) = (union @ ((d_UPair @ eigen__0) @ ((d_UPair @ eigen__0) @ eigen__0)))),introduced(definition,[new_symbols(definition,[sP5])])).
160.10/159.48	thf(sP6,plain,sP6 <=> (![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) => ((union @ ((d_UPair @ X1) @ ((d_UPair @ X1) @ X1))) = (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)))) @ (d_24_prop2 @ (union @ ((d_UPair @ emptyset) @ ((d_UPair @ emptyset) @ emptyset)))))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ X1) @ X3))))))))) @ proj1)))),introduced(definition,[new_symbols(definition,[sP6])])).
160.10/159.48	thf(sP7,plain,sP7 <=> (((in @ eigen__0) @ (((if @ (~((![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) => (X1 = emptyset)))))) @ ((repl @ (((if @ (~((![X1:$i]:(((in @ X1) @ (univof @ emptyset)) => (~((nat_p @ X1)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X1:$i]:(((if @ (nat_p @ X1)) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((X1 = emptyset)))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))))))) @ emptyset)) => sP4),introduced(definition,[new_symbols(definition,[sP7])])).
160.10/159.48	thf(sP8,plain,sP8 <=> ((in @ eigen__0) @ (((if @ (~((![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) => (X1 = emptyset)))))) @ ((repl @ (((if @ (~((![X1:$i]:(((in @ X1) @ (univof @ emptyset)) => (~((nat_p @ X1)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X1:$i]:(((if @ (nat_p @ X1)) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))))))) @ emptyset)) @ (^[X1:$i]:(((if @ (~((X1 = emptyset)))) @ X1) @ (eps @ (^[X2:$i]:(~((((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))))))) @ emptyset)),introduced(definition,[new_symbols(definition,[sP8])])).
160.10/159.48	thf(sP9,plain,sP9 <=> (sP8 => sP5),introduced(definition,[new_symbols(definition,[sP9])])).
160.10/159.48	thf(sP10,plain,sP10 <=> (![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) => ((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)))) @ (d_24_prop2 @ (union @ ((d_UPair @ emptyset) @ ((d_UPair @ emptyset) @ emptyset)))))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ X1) @ X3))))))))) @ proj1) = (union @ ((d_UPair @ X1) @ ((d_UPair @ X1) @ X1)))))),introduced(definition,[new_symbols(definition,[sP10])])).
160.10/159.48	thf(def_is_of,definition,(is_of = (^[X1:$i]:(^[X2:$i>$o]:(X2 @ X1))))).
160.10/159.48	thf(def_all_of,definition,(all_of = (^[X1:$i>$o]:(^[X2:$i>$o]:(![X3:$i]:(((is_of @ X3) @ X1) => (X2 @ X3))))))).
160.10/159.48	thf(def_d_Sing,definition,(d_Sing = (^[X1:$i]:((d_UPair @ X1) @ X1)))).
160.10/159.48	thf(def_binunion,definition,(binunion = (^[X1:$i]:(^[X2:$i]:(union @ ((d_UPair @ X1) @ X2)))))).
160.10/159.48	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))))).
160.10/159.48	thf(def_ordsucc,definition,(ordsucc = (^[X1:$i]:((binunion @ X1) @ (d_Sing @ X1))))).
160.10/159.48	thf(def_omega,definition,(omega = ((d_Sep @ (univof @ emptyset)) @ nat_p))).
160.10/159.48	thf(def_ap,definition,(ap = (^[X1:$i]:(^[X2:$i]:(((d_ReplSep @ X1) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1))))).
160.10/159.48	thf(def_e_is,definition,(e_is = (^[X1:$i]:(=)))).
160.10/159.48	thf(def_nat,definition,(nat = ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))).
160.10/159.48	thf(def_n_is,definition,(n_is = (e_is @ nat))).
160.10/159.48	thf(def_n_1,definition,(n_1 = (ordsucc @ emptyset))).
160.10/159.48	thf(def_plus,definition,(plus = (^[X1:$i]:((ind @ ((d_Pi @ nat) @ (^[X2:$i]:nat))) @ (d_24_prop2 @ X1))))).
160.10/159.48	thf(def_n_pl,definition,(n_pl = (^[X1:$i]:(ap @ (plus @ X1))))).
160.10/159.48	thf(satz4g,conjecture,(![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (~((~((X2 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (~((~((X3 = emptyset)))))))))))))) @ emptyset)) => ((union @ ((d_UPair @ X1) @ ((d_UPair @ X1) @ X1))) = (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (~((~((X2 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (~((~((X3 = emptyset)))))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (~((~((X3 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (~((~((X4 = emptyset)))))))))))))) @ emptyset)))) @ (d_24_prop2 @ (union @ ((d_UPair @ emptyset) @ ((d_UPair @ emptyset) @ emptyset)))))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ X1) @ X3))))))))) @ proj1))))).
160.10/159.48	thf(h1,negated_conjecture,(~(sP6)),inference(assume_negation,[status(cth)],[satz4g])).
160.10/159.48	thf(1,plain,((~(sP9) | ~(sP8)) | sP5),inference(prop_rule,[status(thm)],[])).
160.10/159.48	thf(2,plain,(~(sP10) | sP9),inference(all_rule,[status(thm)],[])).
160.10/159.48	thf(3,plain,((~(sP1) | ~(sP5)) | sP4),inference(prop_rule,[status(thm)],[])).
160.10/159.48	thf(4,plain,(~(sP2) | sP1),inference(all_rule,[status(thm)],[])).
160.10/159.48	thf(5,plain,(~(sP3) | sP2),inference(all_rule,[status(thm)],[])).
160.10/159.48	thf(6,plain,sP3,inference(@eq_sym,[status(thm)],[])).
160.10/159.48	thf(7,plain,(sP7 | ~(sP4)),inference(prop_rule,[status(thm)],[])).
160.10/159.48	thf(8,plain,(sP7 | sP8),inference(prop_rule,[status(thm)],[])).
160.10/159.48	thf(9,plain,(sP6 | ~(sP7)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__0])).
160.10/159.48	thf(satz4c,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((n_is @ ((n_pl @ n_1) @ X1)) @ (ordsucc @ X1))))).
160.10/159.48	thf(10,plain,sP10,inference(preprocess,[status(thm)],[satz4c]).
160.10/159.48	thf(11,plain,$false,inference(prop_unsat,[status(thm),assumptions([h1,h0])],[1,2,3,4,5,6,7,8,9,10,h1])).
160.10/159.48	thf(12,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[11,h0])).
160.10/159.48	thf(0,theorem,(![X1:$i]:(((in @ X1) @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (~((~((X2 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (~((~((X3 = emptyset)))))))))))))) @ emptyset)) => ((union @ ((d_UPair @ X1) @ ((d_UPair @ X1) @ X1))) = (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (~((~((X2 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X2:$i]:(((in @ X2) @ (univof @ emptyset)) => (~((nat_p @ X2)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X2:$i]:(((if @ (nat_p @ X2)) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((X2 = emptyset)))) @ X2) @ (eps @ (^[X3:$i]:(~((((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (~((~((X3 = emptyset)))))))))))))) @ emptyset)) @ (^[X2:$i]:(((if @ (~((![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (~((~((X3 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X3:$i]:(((in @ X3) @ (univof @ emptyset)) => (~((nat_p @ X3)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X3:$i]:(((if @ (nat_p @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) @ (^[X3:$i]:(((if @ (~((X3 = emptyset)))) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (~((~((X4 = emptyset)))))))))))))) @ emptyset)))) @ (d_24_prop2 @ (union @ ((d_UPair @ emptyset) @ ((d_UPair @ emptyset) @ emptyset)))))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ X1) @ X3))))))))) @ proj1)))),inference(contra,[status(thm),contra(discharge,[h1])],[11,h1])).
160.10/159.48	% SZS output end Proof
160.10/159.48	EOF