TSTP Solution File: NUM659^4 by Satallax---3.5
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%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : NUM659^4 : TPTP v8.1.0. Released v7.1.0.
% Transfm : none
% Format : tptp:raw
% Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% Computer : n021.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 600s
% DateTime : Mon Jul 18 13:54:48 EDT 2022
% Result : Theorem 171.66s 171.90s
% Output : Proof 171.66s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : NUM659^4 : TPTP v8.1.0. Released v7.1.0.
% 0.11/0.12 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.12/0.33 % Computer : n021.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 : 300
% 0.12/0.33 % WCLimit : 600
% 0.12/0.33 % DateTime : Tue Jul 5 19:17:48 EDT 2022
% 0.12/0.33 % CPUTime :
% 171.66/171.90 % SZS status Theorem
% 171.66/171.90 % Mode: mode94:USE_SINE=true:SINE_TOLERANCE=2.0:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=2.:SINE_DEPTH=0
% 171.66/171.90 % Inferences: 6
% 171.66/171.90 % SZS output start Proof
% 171.66/171.90 thf(ty_univof, type, univof : ($i>$i)).
% 171.66/171.90 thf(ty_l_some, type, l_some : ($i>($i>$o)>$o)).
% 171.66/171.90 thf(ty_if, type, if : ($o>$i>$i>$i)).
% 171.66/171.90 thf(ty_eps, type, eps : (($i>$o)>$i)).
% 171.66/171.90 thf(ty_d_ReplSep, type, d_ReplSep : ($i>($i>$o)>($i>$i)>$i)).
% 171.66/171.90 thf(ty_eigen__1, type, eigen__1 : $i).
% 171.66/171.90 thf(ty_eigen__0, type, eigen__0 : $i).
% 171.66/171.90 thf(ty_emptyset, type, emptyset : $i).
% 171.66/171.90 thf(ty_plus, type, plus : ($i>$i)).
% 171.66/171.90 thf(ty_pair, type, pair : ($i>$i>$i)).
% 171.66/171.90 thf(ty_proj1, type, proj1 : ($i>$i)).
% 171.66/171.90 thf(ty_repl, type, repl : ($i>($i>$i)>$i)).
% 171.66/171.90 thf(ty_in, type, in : ($i>$i>$o)).
% 171.66/171.90 thf(ty_nat_p, type, nat_p : ($i>$o)).
% 171.66/171.90 thf(sP1,plain,sP1 <=> ((~(((l_some @ (((if @ (~((![X1:$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]:(eigen__1 = (((d_ReplSep @ (plus @ eigen__0)) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ X1) @ X3))))))))) @ proj1)))))) => (eigen__1 = eigen__0)),introduced(definition,[new_symbols(definition,[sP1])])).
% 171.66/171.90 thf(sP2,plain,sP2 <=> ((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,[sP2])])).
% 171.66/171.90 thf(sP3,plain,sP3 <=> (eigen__1 = eigen__0),introduced(definition,[new_symbols(definition,[sP3])])).
% 171.66/171.90 thf(sP4,plain,sP4 <=> ((in @ eigen__1) @ (((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,[sP4])])).
% 171.66/171.90 thf(sP5,plain,sP5 <=> (![X1:$i]:(![X2:$i]:((X1 = X2) => (X2 = X1)))),introduced(definition,[new_symbols(definition,[sP5])])).
% 171.66/171.90 thf(sP6,plain,sP6 <=> ((~(sP1)) => ((l_some @ (((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:$i]:(eigen__0 = (((d_ReplSep @ (plus @ eigen__1)) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ X1) @ X3))))))))) @ proj1))))),introduced(definition,[new_symbols(definition,[sP6])])).
% 171.66/171.90 thf(sP7,plain,sP7 <=> (![X1:$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)) => (![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)) => ((~(((~(((l_some @ (((if @ (~((![X3:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)) @ (^[X3:$i]:(X1 = (((d_ReplSep @ (plus @ X2)) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X3) @ X5))))))))) @ proj1)))))) => (X1 = X2)))) => ((l_some @ (((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)) @ (^[X3:$i]:(X2 = (((d_ReplSep @ (plus @ X1)) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X3) @ X5))))))))) @ proj1))))))))),introduced(definition,[new_symbols(definition,[sP7])])).
% 171.66/171.90 thf(sP8,plain,sP8 <=> (sP2 => sP6),introduced(definition,[new_symbols(definition,[sP8])])).
% 171.66/171.90 thf(sP9,plain,sP9 <=> ((l_some @ (((if @ (~((![X1:$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]:(eigen__0 = (((d_ReplSep @ (plus @ eigen__1)) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ X1) @ X3))))))))) @ proj1)))),introduced(definition,[new_symbols(definition,[sP9])])).
% 171.66/171.90 thf(sP10,plain,sP10 <=> ((l_some @ (((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:$i]:(eigen__1 = (((d_ReplSep @ (plus @ eigen__0)) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ X1) @ X3))))))))) @ proj1)))),introduced(definition,[new_symbols(definition,[sP10])])).
% 171.66/171.90 thf(sP11,plain,sP11 <=> (![X1:$i]:((eigen__1 = X1) => (X1 = eigen__1))),introduced(definition,[new_symbols(definition,[sP11])])).
% 171.66/171.90 thf(sP12,plain,sP12 <=> (sP3 => (eigen__0 = eigen__1)),introduced(definition,[new_symbols(definition,[sP12])])).
% 171.66/171.90 thf(sP13,plain,sP13 <=> (eigen__0 = eigen__1),introduced(definition,[new_symbols(definition,[sP13])])).
% 171.66/171.90 thf(sP14,plain,sP14 <=> (![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:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) => ((~(((~(((l_some @ (((if @ (~((![X2:$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]:(eigen__1 = (((d_ReplSep @ (plus @ X1)) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1)))))) => (eigen__1 = X1)))) => ((l_some @ (((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:$i]:(X1 = (((d_ReplSep @ (plus @ eigen__1)) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1))))))),introduced(definition,[new_symbols(definition,[sP14])])).
% 171.66/171.90 thf(sP15,plain,sP15 <=> (sP4 => sP14),introduced(definition,[new_symbols(definition,[sP15])])).
% 171.66/171.90 thf(def_is_of,definition,(is_of = (^[X1:$i]:(^[X2:$i>$o]:(X2 @ X1))))).
% 171.66/171.90 thf(def_all_of,definition,(all_of = (^[X1:$i>$o]:(^[X2:$i>$o]:(![X3:$i]:(((is_of @ X3) @ X1) => (X2 @ X3))))))).
% 171.66/171.90 thf(def_d_Sep,definition,(d_Sep = (^[X1:$i]:(^[X2:$i>$o]:(((if @ (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) @ ((repl @ X1) @ (^[X3:$i]:(((if @ (X2 @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ X1) => (~((X2 @ X4)))))))))))) @ emptyset))))).
% 171.66/171.90 thf(def_ordsucc,definition,(ordsucc = (^[X1:$i]:((binunion @ X1) @ (d_Sing @ X1))))).
% 171.66/171.90 thf(def_omega,definition,(omega = ((d_Sep @ (univof @ emptyset)) @ nat_p))).
% 171.66/171.90 thf(def_setprod,definition,(setprod = (^[X1:$i]:(^[X2:$i]:((d_Sigma @ X1) @ (^[X3:$i]:X2)))))).
% 171.66/171.90 thf(def_ap,definition,(ap = (^[X1:$i]:(^[X2:$i]:(((d_ReplSep @ X1) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1))))).
% 171.66/171.90 thf(def_d_Pi,definition,(d_Pi = (^[X1:$i]:(^[X2:$i>$i]:((d_Sep @ (power @ ((d_Sigma @ X1) @ (^[X3:$i]:(union @ (X2 @ X3)))))) @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X1) => ((in @ ((ap @ X3) @ X4)) @ (X2 @ X4)))))))))).
% 171.66/171.90 thf(def_imp,definition,(imp = (^[X1:$o]:(^[X2:$o]:(X1 => X2))))).
% 171.66/171.90 thf(def_d_not,definition,(d_not = (^[X1:$o]:((imp @ X1) @ $false)))).
% 171.66/171.90 thf(def_d_and,definition,(d_and = (^[X1:$o]:(^[X2:$o]:(d_not @ ((l_ec @ X1) @ X2)))))).
% 171.66/171.90 thf(def_l_or,definition,(l_or = (^[X1:$o]:(imp @ (d_not @ X1))))).
% 171.66/171.90 thf(def_or3,definition,(or3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((l_or @ X1) @ ((l_or @ X2) @ X3))))))).
% 171.66/171.90 thf(def_ec3,definition,(ec3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:(((and3 @ ((l_ec @ X1) @ X2)) @ ((l_ec @ X2) @ X3)) @ ((l_ec @ X3) @ X1))))))).
% 171.66/171.90 thf(def_orec3,definition,(orec3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((d_and @ (((or3 @ X1) @ X2) @ X3)) @ (((ec3 @ X1) @ X2) @ X3))))))).
% 171.66/171.90 thf(def_e_is,definition,(e_is = (^[X1:$i]:(=)))).
% 171.66/171.90 thf(def_one,definition,(one = (^[X1:$i]:(^[X2:$i>$o]:((d_and @ ((amone @ X1) @ X2)) @ ((l_some @ X1) @ X2)))))).
% 171.66/171.90 thf(def_e_in,definition,(e_in = (^[X1:$i]:(^[X2:$i>$o]:(^[X3:$i]:X3))))).
% 171.66/171.90 thf(def_d_pair,definition,(d_pair = (^[X1:$i]:(^[X2:$i]:pair)))).
% 171.66/171.90 thf(def_esti,definition,(esti = (^[X1:$i]:in))).
% 171.66/171.90 thf(def_nat,definition,(nat = ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))).
% 171.66/171.90 thf(def_n_is,definition,(n_is = (e_is @ nat))).
% 171.66/171.90 thf(def_nis,definition,(nis = (^[X1:$i]:(^[X2:$i]:(d_not @ ((n_is @ X1) @ X2)))))).
% 171.66/171.90 thf(def_n_some,definition,(n_some = (l_some @ nat))).
% 171.66/171.90 thf(def_n_all,definition,(n_all = (all @ nat))).
% 171.66/171.90 thf(def_n_one,definition,(n_one = (one @ nat))).
% 171.66/171.90 thf(def_n_1,definition,(n_1 = (ordsucc @ emptyset))).
% 171.66/171.90 thf(def_n_pl,definition,(n_pl = (^[X1:$i]:(ap @ (plus @ X1))))).
% 171.66/171.90 thf(def_diffprop,definition,(diffprop = (^[X1:$i]:(^[X2:$i]:(^[X3:$i]:((n_is @ X1) @ ((n_pl @ X2) @ X3))))))).
% 171.66/171.90 thf(def_d_29_ii,definition,(d_29_ii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X1) @ X2)))))).
% 171.66/171.90 thf(def_iii,definition,(iii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X2) @ X1)))))).
% 171.66/171.90 thf(def_moreis,definition,(moreis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((d_29_ii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))).
% 171.66/171.90 thf(def_lessis,definition,(lessis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((iii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))).
% 171.66/171.90 thf(satz10k,conjecture,(![X1:$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)) => (![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (~((~((X4 = emptyset)))))))))))))) @ emptyset)) => ((~(((~(((l_some @ (((if @ (~((![X3:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (~((~((X4 = emptyset)))))))))))))) @ emptyset)) @ (^[X3:$i]:(X2 = (((d_ReplSep @ (plus @ X1)) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X3) @ X5))))))))) @ proj1)))))) => (X1 = X2)))) => ((l_some @ (((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)) @ (^[X3:$i]:(X1 = (((d_ReplSep @ (plus @ X2)) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X3) @ X5))))))))) @ proj1)))))))))).
% 171.66/171.90 thf(h0,negated_conjecture,(~((![X1:$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)) => (![X2:$i]:(((in @ X2) @ (((if @ (~((![X3:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)) => ((~(((~(((l_some @ (((if @ (~((![X3:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)) @ (^[X3:$i]:(X2 = (((d_ReplSep @ (plus @ X1)) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X3) @ X5))))))))) @ proj1)))))) => (X1 = X2)))) => ((l_some @ (((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)) @ (^[X3:$i]:(X1 = (((d_ReplSep @ (plus @ X2)) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X3) @ X5))))))))) @ proj1))))))))))),inference(assume_negation,[status(cth)],[satz10k])).
% 171.66/171.90 thf(h1,assumption,(~((sP2 => (![X1:$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)) => ((~(((~(((l_some @ (((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:$i]:(X1 = (((d_ReplSep @ (plus @ eigen__0)) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1)))))) => (eigen__0 = X1)))) => ((l_some @ (((if @ (~((![X2:$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]:(eigen__0 = (((d_ReplSep @ (plus @ X1)) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1)))))))))),introduced(assumption,[])).
% 171.66/171.90 thf(h2,assumption,sP2,introduced(assumption,[])).
% 171.66/171.90 thf(h3,assumption,(~((![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:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) => ((~(((~(((l_some @ (((if @ (~((![X2:$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]:(X1 = (((d_ReplSep @ (plus @ eigen__0)) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1)))))) => (eigen__0 = X1)))) => ((l_some @ (((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:$i]:(eigen__0 = (((d_ReplSep @ (plus @ X1)) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1))))))))),introduced(assumption,[])).
% 171.66/171.90 thf(h4,assumption,(~((sP4 => ((~(((~(sP10)) => sP13))) => sP9)))),introduced(assumption,[])).
% 171.66/171.90 thf(h5,assumption,sP4,introduced(assumption,[])).
% 171.66/171.90 thf(h6,assumption,(~(((~(((~(sP10)) => sP13))) => sP9))),introduced(assumption,[])).
% 171.66/171.90 thf(h7,assumption,(~(((~(sP10)) => sP13))),introduced(assumption,[])).
% 171.66/171.90 thf(h8,assumption,(~(sP9)),introduced(assumption,[])).
% 171.66/171.90 thf(h9,assumption,(~(sP10)),introduced(assumption,[])).
% 171.66/171.90 thf(h10,assumption,(~(sP13)),introduced(assumption,[])).
% 171.66/171.90 thf(1,plain,(~(sP14) | sP8),inference(all_rule,[status(thm)],[])).
% 171.66/171.90 thf(2,plain,((~(sP8) | ~(sP2)) | sP6),inference(prop_rule,[status(thm)],[])).
% 171.66/171.90 thf(3,plain,((~(sP6) | sP1) | sP9),inference(prop_rule,[status(thm)],[])).
% 171.66/171.90 thf(4,plain,((~(sP1) | sP10) | sP3),inference(prop_rule,[status(thm)],[])).
% 171.66/171.90 thf(5,plain,(~(sP7) | sP15),inference(all_rule,[status(thm)],[])).
% 171.66/171.90 thf(6,plain,((~(sP15) | ~(sP4)) | sP14),inference(prop_rule,[status(thm)],[])).
% 171.66/171.90 thf(7,plain,((~(sP12) | ~(sP3)) | sP13),inference(prop_rule,[status(thm)],[])).
% 171.66/171.90 thf(8,plain,(~(sP11) | sP12),inference(all_rule,[status(thm)],[])).
% 171.66/171.90 thf(9,plain,(~(sP5) | sP11),inference(all_rule,[status(thm)],[])).
% 171.66/171.90 thf(10,plain,sP5,inference(eq_sym,[status(thm)],[])).
% 171.66/171.90 thf(satz10j,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ nat))) @ (^[X2:$i]:((d_not @ ((moreis @ X1) @ X2)) => ((iii @ X1) @ X2))))))).
% 171.66/171.90 thf(11,plain,sP7,inference(preprocess,[status(thm)],[satz10j]).
% 171.66/171.90 thf(12,plain,$false,inference(prop_unsat,[status(thm),assumptions([h9,h10,h7,h8,h5,h6,h4,h2,h3,h1,h0])],[1,2,3,4,5,6,7,8,9,10,h2,h5,h9,h10,h8,11])).
% 171.66/171.90 thf(13,plain,$false,inference(tab_negimp,[status(thm),assumptions([h7,h8,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h9,h10])],[h7,12,h9,h10])).
% 171.66/171.90 thf(14,plain,$false,inference(tab_negimp,[status(thm),assumptions([h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h7,h8])],[h6,13,h7,h8])).
% 171.66/171.90 thf(15,plain,$false,inference(tab_negimp,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,14,h5,h6])).
% 171.66/171.90 thf(16,plain,$false,inference(tab_negall,[status(thm),assumptions([h2,h3,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,15,h4])).
% 171.66/171.90 thf(17,plain,$false,inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,16,h2,h3])).
% 171.66/171.90 thf(18,plain,$false,inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,17,h1])).
% 171.66/171.90 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:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (~((~((X3 = emptyset)))))))))))))) @ emptyset)) => (![X2:$i]:(((in @ X2) @ (((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)) => ((~(((~(((l_some @ (((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)) @ (^[X3:$i]:(X2 = (((d_ReplSep @ (plus @ X1)) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X3) @ X5))))))))) @ proj1)))))) => (X1 = X2)))) => ((l_some @ (((if @ (~((![X3:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (~((~((X4 = emptyset)))))))))))))) @ emptyset)) @ (^[X3:$i]:(X1 = (((d_ReplSep @ (plus @ X2)) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X3) @ X5))))))))) @ proj1))))))))),inference(contra,[status(thm),contra(discharge,[h0])],[18,h0])).
% 171.66/171.90 % SZS output end Proof
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