TSTP Solution File: NUM721^4 by Satallax---3.5
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- Process Solution
%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : NUM721^4 : 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 : n027.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 13:55:41 EDT 2022
% Result : Theorem 172.68s 172.60s
% Output : Proof 172.68s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : NUM721^4 : TPTP v8.1.0. Released v7.1.0.
% 0.00/0.12 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.12/0.31 % Computer : n027.cluster.edu
% 0.12/0.31 % Model : x86_64 x86_64
% 0.12/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.31 % Memory : 8042.1875MB
% 0.12/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.31 % CPULimit : 300
% 0.12/0.32 % WCLimit : 600
% 0.12/0.32 % DateTime : Tue Jul 5 20:09:42 EDT 2022
% 0.12/0.32 % CPUTime :
% 172.68/172.60 % SZS status Theorem
% 172.68/172.60 % Mode: mode94:USE_SINE=true:SINE_TOLERANCE=2.0:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=2.:SINE_DEPTH=0
% 172.68/172.60 % Inferences: 10
% 172.68/172.60 % SZS output start Proof
% 172.68/172.60 thf(ty_univof, type, univof : ($i>$i)).
% 172.68/172.60 thf(ty_l_some, type, l_some : ($i>($i>$o)>$o)).
% 172.68/172.60 thf(ty_if, type, if : ($o>$i>$i>$i)).
% 172.68/172.60 thf(ty_eigen__2, type, eigen__2 : $i).
% 172.68/172.60 thf(ty_eps, type, eps : (($i>$o)>$i)).
% 172.68/172.60 thf(ty_d_Pi, type, d_Pi : ($i>($i>$i)>$i)).
% 172.68/172.60 thf(ty_d_ReplSep, type, d_ReplSep : ($i>($i>$o)>($i>$i)>$i)).
% 172.68/172.60 thf(ty_eigen__1, type, eigen__1 : $i).
% 172.68/172.60 thf(ty_eigen__0, type, eigen__0 : $i).
% 172.68/172.60 thf(ty_emptyset, type, emptyset : $i).
% 172.68/172.60 thf(ty_ind, type, ind : ($i>($i>$o)>$i)).
% 172.68/172.60 thf(ty_eigen__3, type, eigen__3 : $i).
% 172.68/172.60 thf(ty_plus, type, plus : ($i>$i)).
% 172.68/172.60 thf(ty_pair, type, pair : ($i>$i>$i)).
% 172.68/172.60 thf(ty_proj1, type, proj1 : ($i>$i)).
% 172.68/172.60 thf(ty_repl, type, repl : ($i>($i>$i)>$i)).
% 172.68/172.60 thf(ty_in, type, in : ($i>$i>$o)).
% 172.68/172.60 thf(ty_d_428_prop2, type, d_428_prop2 : ($i>$i>$o)).
% 172.68/172.60 thf(ty_nat_p, type, nat_p : ($i>$o)).
% 172.68/172.60 thf(sP1,plain,sP1 <=> (((in @ eigen__2) @ (((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)) => (((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:$i]:(~((X2 = ((pair @ X1) @ X3))))))))) @ proj1)))) => (((l_some @ (((if @ (~((![X1:$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__3 = (((d_ReplSep @ (plus @ eigen__2)) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ X1) @ X3))))))))) @ proj1)))) => ((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:$i]:((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X2:$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_428_prop2 @ eigen__1))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ eigen__3) @ X3))))))))) @ proj1) = (((d_ReplSep @ (plus @ (((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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__0))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ eigen__2) @ X3))))))))) @ proj1))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ X1) @ X3))))))))) @ proj1))))))),introduced(definition,[new_symbols(definition,[sP1])])).
% 172.68/172.60 thf(sP2,plain,sP2 <=> ((in @ eigen__1) @ (((if @ (~((![X1:$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:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))))))) @ emptyset)),introduced(definition,[new_symbols(definition,[sP2])])).
% 172.68/172.60 thf(sP3,plain,sP3 <=> (((in @ eigen__0) @ (((if @ (~((![X1:$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:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))))))) @ emptyset)) => (![X1:$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]:(eigen__1 = (((d_ReplSep @ (plus @ eigen__0)) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X3) @ X5))))))))) @ proj1)))) => (((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:$i]:(~((X4 = ((pair @ X3) @ X5))))))))) @ proj1)))) => ((l_some @ (((if @ (~((![X3:$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]:((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![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)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$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)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__1))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X1) @ X5))))))))) @ proj1) = (((d_ReplSep @ (plus @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X4:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$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)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__0))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X2) @ X5))))))))) @ proj1))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X3) @ X5))))))))) @ proj1))))))))))),introduced(definition,[new_symbols(definition,[sP3])])).
% 172.68/172.60 thf(sP4,plain,sP4 <=> ((in @ eigen__2) @ (((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])])).
% 172.68/172.60 thf(sP5,plain,sP5 <=> ((in @ eigen__3) @ (((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,[sP5])])).
% 172.68/172.60 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:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) => (![X2:$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)) => (![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$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)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) => (((l_some @ (((if @ (~((![X4:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$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)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:(eigen__1 = (((d_ReplSep @ (plus @ X1)) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ X4) @ X6))))))))) @ proj1)))) => (((l_some @ (((if @ (~((![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)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:(X2 = (((d_ReplSep @ (plus @ X3)) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ X4) @ X6))))))))) @ proj1)))) => ((l_some @ (((if @ (~((![X4:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$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)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((![X6:$i]:(((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))) @ ((repl @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__1))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ X2) @ X6))))))))) @ proj1) = (((d_ReplSep @ (plus @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$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)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((![X6:$i]:(((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))) @ ((repl @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ X1))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ X3) @ X6))))))))) @ proj1))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ X4) @ X6))))))))) @ proj1)))))))))))),introduced(definition,[new_symbols(definition,[sP6])])).
% 172.68/172.60 thf(sP7,plain,sP7 <=> ((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:$i]:((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X2:$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_428_prop2 @ eigen__1))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ eigen__3) @ X3))))))))) @ proj1) = (((d_ReplSep @ (plus @ (((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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__0))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ eigen__2) @ X3))))))))) @ proj1))) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ X1) @ X3))))))))) @ proj1)))),introduced(definition,[new_symbols(definition,[sP7])])).
% 172.68/172.60 thf(sP8,plain,sP8 <=> ((in @ eigen__0) @ (((if @ (~((![X1:$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:$i]:(~((((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))))))) @ emptyset)) => (X2 = emptyset)))))))))) @ emptyset)),introduced(definition,[new_symbols(definition,[sP8])])).
% 172.68/172.60 thf(sP9,plain,sP9 <=> (((l_some @ (((if @ (~((![X1:$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__3 = (((d_ReplSep @ (plus @ eigen__2)) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ X1) @ X3))))))))) @ proj1)))) => sP7),introduced(definition,[new_symbols(definition,[sP9])])).
% 172.68/172.60 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:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) => (![X2:$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)) => (![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$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)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) => (![X4:$i]:(((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)) => (((l_some @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)) @ (^[X5:$i]:(X1 = (((d_ReplSep @ (plus @ X2)) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((pair @ X5) @ X7))))))))) @ proj1)))) => (((l_some @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$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)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)) @ (^[X5:$i]:(X3 = (((d_ReplSep @ (plus @ X4)) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((pair @ X5) @ X7))))))))) @ proj1)))) => ((l_some @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)) @ (^[X5:$i]:((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X6:$i]:(((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))) @ ((repl @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((![X7:$i]:(((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))) @ ((repl @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (((if @ (~((![X9:$i]:(((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X9:$i]:(((if @ (nat_p @ X9)) @ X9) @ (eps @ (^[X10:$i]:(~((((in @ X10) @ (univof @ emptyset)) => (~((nat_p @ X10)))))))))))) @ emptyset)) => (X8 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ X1))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((pair @ X3) @ X7))))))))) @ proj1) = (((d_ReplSep @ (plus @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X6:$i]:(((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))) @ ((repl @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((![X7:$i]:(((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))) @ ((repl @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (((if @ (~((![X9:$i]:(((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X9:$i]:(((if @ (nat_p @ X9)) @ X9) @ (eps @ (^[X10:$i]:(~((((in @ X10) @ (univof @ emptyset)) => (~((nat_p @ X10)))))))))))) @ emptyset)) => (X8 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ X2))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((pair @ X4) @ X7))))))))) @ proj1))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((pair @ X5) @ X7))))))))) @ proj1)))))))))))))),introduced(definition,[new_symbols(definition,[sP10])])).
% 172.68/172.60 thf(sP11,plain,sP11 <=> ((l_some @ (((if @ (~((![X1:$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,[sP11])])).
% 172.68/172.60 thf(sP12,plain,sP12 <=> (![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:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) => (sP11 => (((l_some @ (((if @ (~((![X2:$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__3 = (((d_ReplSep @ (plus @ X1)) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1)))) => ((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:$i]:((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X3:$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]:(((if @ (~((![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)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__1))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ eigen__3) @ X4))))))))) @ proj1) = (((d_ReplSep @ (plus @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((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:$i]:(((if @ (~((![X4:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$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)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__0))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X1) @ X4))))))))) @ proj1))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1)))))))),introduced(definition,[new_symbols(definition,[sP12])])).
% 172.68/172.60 thf(sP13,plain,sP13 <=> (sP11 => sP9),introduced(definition,[new_symbols(definition,[sP13])])).
% 172.68/172.60 thf(sP14,plain,sP14 <=> (![X1:$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)) => (![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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)) => (sP11 => (((l_some @ (((if @ (~((![X3:$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)))) => ((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:$i]:((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X4:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$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)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__1))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X1) @ X5))))))))) @ proj1) = (((d_ReplSep @ (plus @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![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)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$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)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__0))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X2) @ X5))))))))) @ proj1))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X3) @ X5))))))))) @ proj1)))))))))),introduced(definition,[new_symbols(definition,[sP14])])).
% 172.68/172.60 thf(sP15,plain,sP15 <=> (sP2 => sP6),introduced(definition,[new_symbols(definition,[sP15])])).
% 172.68/172.60 thf(sP16,plain,sP16 <=> (sP5 => sP12),introduced(definition,[new_symbols(definition,[sP16])])).
% 172.68/172.60 thf(sP17,plain,sP17 <=> ((l_some @ (((if @ (~((![X1:$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__3 = (((d_ReplSep @ (plus @ eigen__2)) @ (^[X2:$i]:(~((![X3:$i]:(~((X2 = ((pair @ X1) @ X3))))))))) @ proj1)))),introduced(definition,[new_symbols(definition,[sP17])])).
% 172.68/172.60 thf(def_is_of,definition,(is_of = (^[X1:$i]:(^[X2:$i>$o]:(X2 @ X1))))).
% 172.68/172.60 thf(def_all_of,definition,(all_of = (^[X1:$i>$o]:(^[X2:$i>$o]:(![X3:$i]:(((is_of @ X3) @ X1) => (X2 @ X3))))))).
% 172.68/172.60 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))))).
% 172.68/172.60 thf(def_ordsucc,definition,(ordsucc = (^[X1:$i]:((binunion @ X1) @ (d_Sing @ X1))))).
% 172.68/172.60 thf(def_omega,definition,(omega = ((d_Sep @ (univof @ emptyset)) @ nat_p))).
% 172.68/172.60 thf(def_ap,definition,(ap = (^[X1:$i]:(^[X2:$i]:(((d_ReplSep @ X1) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1))))).
% 172.68/172.60 thf(def_d_not,definition,(d_not = (^[X1:$o]:((imp @ X1) @ $false)))).
% 172.68/172.60 thf(def_or3,definition,(or3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((l_or @ X1) @ ((l_or @ X2) @ X3))))))).
% 172.68/172.60 thf(def_ec3,definition,(ec3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:(((and3 @ ((l_ec @ X1) @ X2)) @ ((l_ec @ X2) @ X3)) @ ((l_ec @ X3) @ X1))))))).
% 172.68/172.60 thf(def_e_is,definition,(e_is = (^[X1:$i]:(=)))).
% 172.68/172.60 thf(def_e_in,definition,(e_in = (^[X1:$i]:(^[X2:$i>$o]:(^[X3:$i]:X3))))).
% 172.68/172.60 thf(def_nat,definition,(nat = ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))).
% 172.68/172.60 thf(def_n_is,definition,(n_is = (e_is @ nat))).
% 172.68/172.60 thf(def_nis,definition,(nis = (^[X1:$i]:(^[X2:$i]:(d_not @ ((n_is @ X1) @ X2)))))).
% 172.68/172.60 thf(def_n_some,definition,(n_some = (l_some @ nat))).
% 172.68/172.60 thf(def_n_1,definition,(n_1 = (ordsucc @ emptyset))).
% 172.68/172.60 thf(def_n_pl,definition,(n_pl = (^[X1:$i]:(ap @ (plus @ X1))))).
% 172.68/172.60 thf(def_diffprop,definition,(diffprop = (^[X1:$i]:(^[X2:$i]:(^[X3:$i]:((n_is @ X1) @ ((n_pl @ X2) @ X3))))))).
% 172.68/172.60 thf(def_d_29_ii,definition,(d_29_ii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X1) @ X2)))))).
% 172.68/172.60 thf(def_iii,definition,(iii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X2) @ X1)))))).
% 172.68/172.60 thf(def_moreis,definition,(moreis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((d_29_ii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))).
% 172.68/172.60 thf(def_lessis,definition,(lessis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((iii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))).
% 172.68/172.60 thf(def_times,definition,(times = (^[X1:$i]:((ind @ ((d_Pi @ nat) @ (^[X2:$i]:nat))) @ (d_428_prop2 @ X1))))).
% 172.68/172.60 thf(def_n_ts,definition,(n_ts = (^[X1:$i]:(ap @ (times @ X1))))).
% 172.68/172.60 thf(satz34a,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)) => (![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (~((~((X4 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X4:$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)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = emptyset)))))))))))))) @ emptyset)) => (![X4:$i]:(((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) => (((l_some @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ (^[X5:$i]:(X2 = (((d_ReplSep @ (plus @ X1)) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((pair @ X5) @ X7))))))))) @ proj1)))) => (((l_some @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X5:$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)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ (^[X5:$i]:(X4 = (((d_ReplSep @ (plus @ X3)) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((pair @ X5) @ X7))))))))) @ proj1)))) => ((l_some @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ (^[X5:$i]:((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X6:$i]:(((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (~((~((X6 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (~((~((X7 = emptyset)))))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((![X7:$i]:(((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (~((~((X7 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (((if @ (~((![X9:$i]:(((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X9:$i]:(((if @ (nat_p @ X9)) @ X9) @ (eps @ (^[X10:$i]:(~((((in @ X10) @ (univof @ emptyset)) => (~((nat_p @ X10)))))))))))) @ emptyset)) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)))) @ (d_428_prop2 @ X2))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((pair @ X4) @ X7))))))))) @ proj1) = (((d_ReplSep @ (plus @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X6:$i]:(((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (~((~((X6 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (~((~((X7 = emptyset)))))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((![X7:$i]:(((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (~((~((X7 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (((if @ (~((![X9:$i]:(((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X9:$i]:(((if @ (nat_p @ X9)) @ X9) @ (eps @ (^[X10:$i]:(~((((in @ X10) @ (univof @ emptyset)) => (~((nat_p @ X10)))))))))))) @ emptyset)) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)))) @ (d_428_prop2 @ X1))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((pair @ X3) @ X7))))))))) @ proj1))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((pair @ X5) @ X7))))))))) @ proj1))))))))))))))).
% 172.68/172.60 thf(h0,negated_conjecture,(~((![X1:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))))))) @ emptyset)) => (![X3:$i]:(((in @ X3) @ (((if @ (~((![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)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) => (![X4:$i]:(((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$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)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)) => (((l_some @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$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)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ 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(univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((![X7:$i]:(((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))) @ ((repl @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (((if @ (~((![X9:$i]:(((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X9:$i]:(((if @ (nat_p @ X9)) @ X9) @ (eps @ (^[X10:$i]:(~((((in @ X10) @ (univof @ emptyset)) => (~((nat_p @ X10)))))))))))) @ emptyset)) => (X8 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ X2))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((pair @ X4) @ X7))))))))) @ proj1) = (((d_ReplSep @ (plus @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X6:$i]:(((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))) @ ((repl @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((![X7:$i]:(((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))) @ ((repl @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (((if @ (~((![X9:$i]:(((in @ 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(univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((![X6:$i]:(((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))) @ ((repl @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ X1))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ X3) @ X6))))))))) @ proj1) = (((d_ReplSep @ (plus @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$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)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((![X6:$i]:(((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))) @ ((repl @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__0))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ X2) @ X6))))))))) @ proj1))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ X4) @ X6))))))))) @ proj1))))))))))))))),introduced(assumption,[])).
% 172.68/172.60 thf(h2,assumption,sP8,introduced(assumption,[])).
% 172.68/172.60 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:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) => (![X2:$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)) => (![X3:$i]:(((in @ X3) @ (((if @ (~((![X4:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$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)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) => (((l_some @ (((if @ (~((![X4:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$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)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:(X1 = (((d_ReplSep @ (plus @ eigen__0)) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ X4) @ X6))))))))) @ proj1)))) => (((l_some @ (((if @ (~((![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)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:(X3 = (((d_ReplSep @ (plus @ X2)) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ X4) @ X6))))))))) @ proj1)))) => ((l_some @ (((if @ (~((![X4:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$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)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((![X6:$i]:(((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))) @ ((repl @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ X1))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ X3) @ X6))))))))) @ proj1) = (((d_ReplSep @ (plus @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$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)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((![X6:$i]:(((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))) @ ((repl @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (X7 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__0))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ X2) @ X6))))))))) @ proj1))) @ (^[X5:$i]:(~((![X6:$i]:(~((X5 = ((pair @ X4) @ X6))))))))) @ proj1)))))))))))))),introduced(assumption,[])).
% 172.68/172.60 thf(h4,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:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) => (![X2:$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)) => (sP11 => (((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:$i]:(~((X4 = ((pair @ X3) @ X5))))))))) @ proj1)))) => ((l_some @ (((if @ (~((![X3:$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]:((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![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)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$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)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__1))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X2) @ X5))))))))) @ proj1) = (((d_ReplSep @ (plus @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X4:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$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)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__0))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X1) @ X5))))))))) @ proj1))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X3) @ X5))))))))) @ proj1))))))))))))),introduced(assumption,[])).
% 172.68/172.60 thf(h5,assumption,sP2,introduced(assumption,[])).
% 172.68/172.60 thf(h6,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:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) => (![X2:$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)) => (sP11 => (((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:$i]:(~((X4 = ((pair @ X3) @ X5))))))))) @ proj1)))) => ((l_some @ (((if @ (~((![X3:$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]:((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![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)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$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)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__1))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X2) @ X5))))))))) @ proj1) = (((d_ReplSep @ (plus @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X4:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$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)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (X6 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__0))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X1) @ X5))))))))) @ proj1))) @ (^[X4:$i]:(~((![X5:$i]:(~((X4 = ((pair @ X3) @ X5))))))))) @ proj1)))))))))))),introduced(assumption,[])).
% 172.68/172.61 thf(h7,assumption,(~((sP4 => (![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:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) => (X3 = emptyset)))))))))) @ emptyset)) => (sP11 => (((l_some @ (((if @ (~((![X2:$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__2)) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1)))) => ((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:$i]:((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X3:$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]:(((if @ (~((![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)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__1))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X1) @ X4))))))))) @ proj1) = (((d_ReplSep @ (plus @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((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:$i]:(((if @ (~((![X4:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$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)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__0))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ eigen__2) @ X4))))))))) @ proj1))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1))))))))))),introduced(assumption,[])).
% 172.68/172.61 thf(h8,assumption,sP4,introduced(assumption,[])).
% 172.68/172.61 thf(h9,assumption,(~((![X1:$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)) => (sP11 => (((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:$i]:(X1 = (((d_ReplSep @ (plus @ eigen__2)) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1)))) => ((l_some @ (((if @ (~((![X2:$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]:((((d_ReplSep @ ((ind @ ((d_Pi @ (((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:$i]:(((if @ (~((![X4:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (X4 = emptyset)))))) @ ((repl @ (((if @ (~((![X4:$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)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__1))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X1) @ X4))))))))) @ proj1) = (((d_ReplSep @ (plus @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X3:$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]:(((if @ (~((![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)))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (X5 = emptyset)))))))))) @ emptyset)))) @ (d_428_prop2 @ eigen__0))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ eigen__2) @ X4))))))))) @ proj1))) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1)))))))))),introduced(assumption,[])).
% 172.68/172.61 thf(h10,assumption,(~((sP5 => sP13))),introduced(assumption,[])).
% 172.68/172.61 thf(h11,assumption,sP5,introduced(assumption,[])).
% 172.68/172.61 thf(h12,assumption,(~(sP13)),introduced(assumption,[])).
% 172.68/172.61 thf(h13,assumption,sP11,introduced(assumption,[])).
% 172.68/172.61 thf(h14,assumption,(~(sP9)),introduced(assumption,[])).
% 172.68/172.61 thf(h15,assumption,sP17,introduced(assumption,[])).
% 172.68/172.61 thf(h16,assumption,(~(sP7)),introduced(assumption,[])).
% 172.68/172.61 thf(1,plain,(~(sP12) | sP1),inference(all_rule,[status(thm)],[])).
% 172.68/172.61 thf(2,plain,((~(sP1) | ~(sP4)) | sP13),inference(prop_rule,[status(thm)],[])).
% 172.68/172.61 thf(3,plain,((~(sP13) | ~(sP11)) | sP9),inference(prop_rule,[status(thm)],[])).
% 172.68/172.61 thf(4,plain,((~(sP9) | ~(sP17)) | sP7),inference(prop_rule,[status(thm)],[])).
% 172.68/172.61 thf(5,plain,(~(sP14) | sP16),inference(all_rule,[status(thm)],[])).
% 172.68/172.61 thf(6,plain,((~(sP16) | ~(sP5)) | sP12),inference(prop_rule,[status(thm)],[])).
% 172.68/172.61 thf(7,plain,(~(sP6) | sP3),inference(all_rule,[status(thm)],[])).
% 172.68/172.61 thf(8,plain,((~(sP3) | ~(sP8)) | sP14),inference(prop_rule,[status(thm)],[])).
% 172.68/172.61 thf(9,plain,(~(sP10) | sP15),inference(all_rule,[status(thm)],[])).
% 172.68/172.61 thf(10,plain,((~(sP15) | ~(sP2)) | sP6),inference(prop_rule,[status(thm)],[])).
% 172.68/172.61 thf(satz34,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ nat))) @ (^[X2:$i]:((all_of @ (^[X3:$i]:((in @ X3) @ nat))) @ (^[X3:$i]:((all_of @ (^[X4:$i]:((in @ X4) @ nat))) @ (^[X4:$i]:(((d_29_ii @ X1) @ X2) => (((d_29_ii @ X3) @ X4) => ((d_29_ii @ ((n_ts @ X1) @ X3)) @ ((n_ts @ X2) @ X4))))))))))))).
% 172.68/172.61 thf(11,plain,sP10,inference(preprocess,[status(thm)],[satz34]).
% 172.68/172.61 thf(12,plain,$false,inference(prop_unsat,[status(thm),assumptions([h15,h16,h13,h14,h11,h12,h10,h8,h9,h7,h5,h6,h4,h2,h3,h1,h0])],[1,2,3,4,5,6,7,8,9,10,h2,h5,h8,h11,h13,h15,h16,11])).
% 172.68/172.61 thf(13,plain,$false,inference(tab_negimp,[status(thm),assumptions([h13,h14,h11,h12,h10,h8,h9,h7,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h15,h16])],[h14,12,h15,h16])).
% 172.68/172.61 thf(14,plain,$false,inference(tab_negimp,[status(thm),assumptions([h11,h12,h10,h8,h9,h7,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h13,h14])],[h12,13,h13,h14])).
% 172.68/172.61 thf(15,plain,$false,inference(tab_negimp,[status(thm),assumptions([h10,h8,h9,h7,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h11,h12])],[h10,14,h11,h12])).
% 172.68/172.61 thf(16,plain,$false,inference(tab_negall,[status(thm),assumptions([h8,h9,h7,h5,h6,h4,h2,h3,h1,h0]),tab_negall(discharge,[h10]),tab_negall(eigenvar,eigen__3)],[h9,15,h10])).
% 172.68/172.61 thf(17,plain,$false,inference(tab_negimp,[status(thm),assumptions([h7,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h8,h9])],[h7,16,h8,h9])).
% 172.68/172.61 thf(18,plain,$false,inference(tab_negall,[status(thm),assumptions([h5,h6,h4,h2,h3,h1,h0]),tab_negall(discharge,[h7]),tab_negall(eigenvar,eigen__2)],[h6,17,h7])).
% 172.68/172.61 thf(19,plain,$false,inference(tab_negimp,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,18,h5,h6])).
% 172.68/172.61 thf(20,plain,$false,inference(tab_negall,[status(thm),assumptions([h2,h3,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,19,h4])).
% 172.68/172.61 thf(21,plain,$false,inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,20,h2,h3])).
% 172.68/172.61 thf(22,plain,$false,inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,21,h1])).
% 172.68/172.61 thf(0,theorem,(![X1:$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:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) => (~((~((X4 = emptyset)))))))))))))) @ emptyset)) => (![X3:$i]:(((in @ X3) @ (((if @ (~((![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)))))))))) @ ((repl @ (((if @ (~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(((if @ (nat_p @ X4)) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))))))) @ emptyset)) @ (^[X4:$i]:(((if @ (~((X4 = emptyset)))) @ X4) @ (eps @ (^[X5:$i]:(~((((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = emptyset)))))))))))))) @ emptyset)) => (![X4:$i]:(((in @ X4) @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X5:$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)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) => (((l_some @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X5:$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)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ (^[X5:$i]:(X2 = (((d_ReplSep @ (plus @ X1)) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((pair @ X5) @ X7))))))))) @ proj1)))) => (((l_some @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(((if @ (nat_p @ X5)) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))))))) @ emptyset)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ (^[X5:$i]:(X4 = (((d_ReplSep @ (plus @ X3)) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((pair @ X5) @ X7))))))))) @ proj1)))) => ((l_some @ (((if @ (~((![X5:$i]:(((in @ X5) @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) => (~((~((X5 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X5:$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)) @ (^[X5:$i]:(((if @ (~((X5 = emptyset)))) @ X5) @ (eps @ (^[X6:$i]:(~((((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (~((~((X6 = emptyset)))))))))))))) @ emptyset)) @ (^[X5:$i]:((((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X6:$i]:(((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (~((~((X6 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (~((~((X7 = emptyset)))))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((![X7:$i]:(((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (~((~((X7 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (((if @ (~((![X9:$i]:(((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X9:$i]:(((if @ (nat_p @ X9)) @ X9) @ (eps @ (^[X10:$i]:(~((((in @ X10) @ (univof @ emptyset)) => (~((nat_p @ X10)))))))))))) @ emptyset)) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)))) @ (d_428_prop2 @ X2))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((pair @ X4) @ X7))))))))) @ proj1) = (((d_ReplSep @ (plus @ (((d_ReplSep @ ((ind @ ((d_Pi @ (((if @ (~((![X6:$i]:(((in @ X6) @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) => (~((~((X6 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(((if @ (nat_p @ X6)) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((X6 = emptyset)))) @ X6) @ (eps @ (^[X7:$i]:(~((((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (~((~((X7 = emptyset)))))))))))))) @ emptyset)) @ (^[X6:$i]:(((if @ (~((![X7:$i]:(((in @ X7) @ (((if @ (~((![X8:$i]:(((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X8:$i]:(((if @ (nat_p @ X8)) @ X8) @ (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))))))) @ emptyset)) => (~((~((X7 = emptyset)))))))))) @ ((repl @ (((if @ (~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(((if @ (nat_p @ X7)) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8)))))))))))) @ emptyset)) @ (^[X7:$i]:(((if @ (~((X7 = emptyset)))) @ X7) @ (eps @ (^[X8:$i]:(~((((in @ X8) @ (((if @ (~((![X9:$i]:(((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9)))))))) @ ((repl @ (univof @ emptyset)) @ (^[X9:$i]:(((if @ (nat_p @ X9)) @ X9) @ (eps @ (^[X10:$i]:(~((((in @ X10) @ (univof @ emptyset)) => (~((nat_p @ X10)))))))))))) @ emptyset)) => (~((~((X8 = emptyset)))))))))))))) @ emptyset)))) @ (d_428_prop2 @ X1))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((pair @ X3) @ X7))))))))) @ proj1))) @ (^[X6:$i]:(~((![X7:$i]:(~((X6 = ((pair @ X5) @ X7))))))))) @ proj1)))))))))))))),inference(contra,[status(thm),contra(discharge,[h0])],[22,h0])).
% 172.68/172.61 % SZS output end Proof
%------------------------------------------------------------------------------