TSTP Solution File: NUM757^4 by Satallax---3.5
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : NUM757^4 : 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 : n009.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 13:56:11 EDT 2022
% Result : Theorem 172.20s 172.07s
% Output : Proof 172.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : NUM757^4 : 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
% 0.12/0.33 % Computer : n009.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 600
% 0.12/0.33 % DateTime : Tue Jul 5 07:03:07 EDT 2022
% 0.12/0.33 % CPUTime :
% 172.20/172.07 % SZS status Theorem
% 172.20/172.07 % Mode: mode94:USE_SINE=true:SINE_TOLERANCE=2.0:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=2.:SINE_DEPTH=0
% 172.20/172.07 % Inferences: 8
% 172.20/172.07 % SZS output start Proof
% 172.20/172.07 thf(ty_eigen__2, type, eigen__2 : $i).
% 172.20/172.07 thf(ty_times, type, times : ($i>$i)).
% 172.20/172.07 thf(ty_d_1to, type, d_1to : ($i>$i)).
% 172.20/172.07 thf(ty_d_Pi, type, d_Pi : ($i>($i>$i)>$i)).
% 172.20/172.07 thf(ty_eigen__1, type, eigen__1 : $i).
% 172.20/172.07 thf(ty_eigen__0, type, eigen__0 : $i).
% 172.20/172.07 thf(ty_d_Sep, type, d_Sep : ($i>($i>$o)>$i)).
% 172.20/172.07 thf(ty_second1, type, second1 : ($i>$i>$i)).
% 172.20/172.07 thf(ty_emptyset, type, emptyset : $i).
% 172.20/172.07 thf(ty_first1, type, first1 : ($i>$i>$i)).
% 172.20/172.07 thf(ty_plus, type, plus : ($i>$i)).
% 172.20/172.07 thf(ty_ap, type, ap : ($i>$i>$i)).
% 172.20/172.07 thf(ty_n_some, type, n_some : (($i>$o)>$o)).
% 172.20/172.07 thf(ty_diffprop, type, diffprop : ($i>$i>$i>$o)).
% 172.20/172.07 thf(ty_omega, type, omega : $i).
% 172.20/172.07 thf(ty_pair1, type, pair1 : ($i>$i>$i>$i)).
% 172.20/172.07 thf(ty_in, type, in : ($i>$i>$o)).
% 172.20/172.07 thf(ty_n_2, type, n_2 : $i).
% 172.20/172.07 thf(sP1,plain,sP1 <=> (![X1:$i]:(((in @ X1) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (![X2:$i]:(((in @ X2) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X3:$i]:((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))))) => (![X3:$i]:(((in @ X3) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X4:$i]:((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))))) => ((n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))))))) => (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))))))))))),introduced(definition,[new_symbols(definition,[sP1])])).
% 172.20/172.07 thf(sP2,plain,sP2 <=> ((n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2))))))) => (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1))))),introduced(definition,[new_symbols(definition,[sP2])])).
% 172.20/172.07 thf(sP3,plain,sP3 <=> (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__2))))))),introduced(definition,[new_symbols(definition,[sP3])])).
% 172.20/172.07 thf(sP4,plain,sP4 <=> (((in @ eigen__1) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X1:$i]:((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) => (![X1:$i]:(((in @ X1) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (![X2:$i]:(((in @ X2) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X3:$i]:((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))))) => ((n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))))))) => (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__1)))))))))),introduced(definition,[new_symbols(definition,[sP4])])).
% 172.20/172.07 thf(sP5,plain,sP5 <=> (((in @ eigen__2) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X1:$i]:((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) => sP2),introduced(definition,[new_symbols(definition,[sP5])])).
% 172.20/172.07 thf(sP6,plain,sP6 <=> (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1)))),introduced(definition,[new_symbols(definition,[sP6])])).
% 172.20/172.07 thf(sP7,plain,sP7 <=> ((in @ eigen__0) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X1:$i]:((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))),introduced(definition,[new_symbols(definition,[sP7])])).
% 172.20/172.07 thf(sP8,plain,sP8 <=> (![X1:$i]:(((in @ X1) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => ((n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__0)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__0)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1))))))) => sP6))),introduced(definition,[new_symbols(definition,[sP8])])).
% 172.20/172.07 thf(sP9,plain,sP9 <=> ((in @ eigen__1) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X1:$i]:((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))),introduced(definition,[new_symbols(definition,[sP9])])).
% 172.20/172.07 thf(sP10,plain,sP10 <=> ((in @ eigen__2) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X1:$i]:((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))),introduced(definition,[new_symbols(definition,[sP10])])).
% 172.20/172.07 thf(sP11,plain,sP11 <=> (sP7 => sP8),introduced(definition,[new_symbols(definition,[sP11])])).
% 172.20/172.07 thf(sP12,plain,sP12 <=> (![X1:$i]:(((in @ X1) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (![X2:$i]:(((in @ X2) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X3:$i]:((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))))) => ((n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))))))) => (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__1))))))))),introduced(definition,[new_symbols(definition,[sP12])])).
% 172.20/172.07 thf(def_is_of,definition,(is_of = (^[X1:$i]:(^[X2:$i>$o]:(X2 @ X1))))).
% 172.20/172.07 thf(def_all_of,definition,(all_of = (^[X1:$i>$o]:(^[X2:$i>$o]:(![X3:$i]:(((is_of @ X3) @ X1) => (X2 @ X3))))))).
% 172.20/172.07 thf(def_ordsucc,definition,(ordsucc = (^[X1:$i]:((binunion @ X1) @ (d_Sing @ X1))))).
% 172.20/172.07 thf(def_d_not,definition,(d_not = (^[X1:$o]:((imp @ X1) @ $false)))).
% 172.20/172.07 thf(def_d_and,definition,(d_and = (^[X1:$o]:(^[X2:$o]:(d_not @ ((l_ec @ X1) @ X2)))))).
% 172.20/172.07 thf(def_l_some,definition,(l_some = (^[X1:$i]:(^[X2:$i>$o]:(d_not @ ((all_of @ (^[X3:$i]:((in @ X3) @ X1))) @ ((non @ X1) @ X2))))))).
% 172.20/172.07 thf(def_or3,definition,(or3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((l_or @ X1) @ ((l_or @ X2) @ X3))))))).
% 172.20/172.07 thf(def_ec3,definition,(ec3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:(((and3 @ ((l_ec @ X1) @ X2)) @ ((l_ec @ X2) @ X3)) @ ((l_ec @ X3) @ X1))))))).
% 172.20/172.07 thf(def_orec3,definition,(orec3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((d_and @ (((or3 @ X1) @ X2) @ X3)) @ (((ec3 @ X1) @ X2) @ X3))))))).
% 172.20/172.07 thf(def_nat,definition,(nat = ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))).
% 172.20/172.07 thf(def_n_is,definition,(n_is = (e_is @ nat))).
% 172.20/172.07 thf(def_n_1,definition,(n_1 = (ordsucc @ emptyset))).
% 172.20/172.07 thf(def_n_pl,definition,(n_pl = (^[X1:$i]:(ap @ (plus @ X1))))).
% 172.20/172.07 thf(def_d_29_ii,definition,(d_29_ii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X1) @ X2)))))).
% 172.20/172.07 thf(def_iii,definition,(iii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X2) @ X1)))))).
% 172.20/172.07 thf(def_moreis,definition,(moreis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((d_29_ii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))).
% 172.20/172.07 thf(def_lessis,definition,(lessis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((iii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))).
% 172.20/172.07 thf(def_n_ts,definition,(n_ts = (^[X1:$i]:(ap @ (times @ X1))))).
% 172.20/172.07 thf(def_pair1type,definition,(pair1type = (^[X1:$i]:((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:X1))))).
% 172.20/172.07 thf(def_frac,definition,(frac = (pair1type @ nat))).
% 172.20/172.07 thf(def_n_fr,definition,(n_fr = (pair1 @ nat))).
% 172.20/172.07 thf(def_num,definition,(num = (first1 @ nat))).
% 172.20/172.07 thf(def_den,definition,(den = (second1 @ nat))).
% 172.20/172.07 thf(def_n_eq,definition,(n_eq = (^[X1:$i]:(^[X2:$i]:((n_is @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1))))))).
% 172.20/172.07 thf(def_moref,definition,(moref = (^[X1:$i]:(^[X2:$i]:((d_29_ii @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1))))))).
% 172.20/172.07 thf(def_lessf,definition,(lessf = (^[X1:$i]:(^[X2:$i]:((iii @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1))))))).
% 172.20/172.07 thf(def_moreq,definition,(moreq = (^[X1:$i]:(^[X2:$i]:((l_or @ ((moref @ X1) @ X2)) @ ((n_eq @ X1) @ X2)))))).
% 172.20/172.07 thf(def_lesseq,definition,(lesseq = (^[X1:$i]:(^[X2:$i]:((l_or @ ((lessf @ X1) @ X2)) @ ((n_eq @ X1) @ X2)))))).
% 172.20/172.07 thf(def_n_pf,definition,(n_pf = (^[X1:$i]:(^[X2:$i]:((n_fr @ ((n_pl @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1)))) @ ((n_ts @ (den @ X1)) @ (den @ X2))))))).
% 172.20/172.07 thf(satz63c,conjecture,(![X1:$i]:(((in @ X1) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (![X2:$i]:(((in @ X2) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X3:$i]:((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))))) => (![X3:$i]:(((in @ X3) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X4:$i]:((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))))) => ((n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))))))) => (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2)))))))))))).
% 172.20/172.07 thf(h0,negated_conjecture,(~((![X1:$i]:(((in @ X1) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (![X2:$i]:(((in @ X2) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X3:$i]:((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))))) => (![X3:$i]:(((in @ X3) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X4:$i]:((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))))) => ((n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))))))) => (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))))))))))))),inference(assume_negation,[status(cth)],[satz63c])).
% 172.20/172.07 thf(h1,assumption,(~((sP7 => (![X1:$i]:(((in @ X1) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (![X2:$i]:(((in @ X2) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X3:$i]:((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))))) => ((n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))))))) => (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1)))))))))))),introduced(assumption,[])).
% 172.20/172.07 thf(h2,assumption,sP7,introduced(assumption,[])).
% 172.20/172.07 thf(h3,assumption,(~((![X1:$i]:(((in @ X1) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (![X2:$i]:(((in @ X2) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X3:$i]:((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))))) => ((n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))))))) => (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))))))))))),introduced(assumption,[])).
% 172.20/172.07 thf(h4,assumption,(~((sP9 => sP8))),introduced(assumption,[])).
% 172.20/172.07 thf(h5,assumption,sP9,introduced(assumption,[])).
% 172.20/172.07 thf(h6,assumption,(~(sP8)),introduced(assumption,[])).
% 172.20/172.07 thf(h7,assumption,(~(sP5)),introduced(assumption,[])).
% 172.20/172.07 thf(h8,assumption,sP10,introduced(assumption,[])).
% 172.20/172.07 thf(h9,assumption,(~(sP2)),introduced(assumption,[])).
% 172.20/172.07 thf(h10,assumption,sP3,introduced(assumption,[])).
% 172.20/172.07 thf(h11,assumption,(~(sP6)),introduced(assumption,[])).
% 172.20/172.07 thf(1,plain,(~(sP8) | sP5),inference(all_rule,[status(thm)],[])).
% 172.20/172.07 thf(2,plain,((~(sP5) | ~(sP10)) | sP2),inference(prop_rule,[status(thm)],[])).
% 172.20/172.07 thf(3,plain,((~(sP2) | ~(sP3)) | sP6),inference(prop_rule,[status(thm)],[])).
% 172.20/172.07 thf(4,plain,(~(sP12) | sP11),inference(all_rule,[status(thm)],[])).
% 172.20/172.07 thf(5,plain,((~(sP11) | ~(sP7)) | sP8),inference(prop_rule,[status(thm)],[])).
% 172.20/172.07 thf(6,plain,(~(sP1) | sP4),inference(all_rule,[status(thm)],[])).
% 172.20/172.07 thf(7,plain,((~(sP4) | ~(sP9)) | sP12),inference(prop_rule,[status(thm)],[])).
% 172.20/172.07 thf(satz63a,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ frac))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ frac))) @ (^[X2:$i]:((all_of @ (^[X3:$i]:((in @ X3) @ frac))) @ (^[X3:$i]:(((moref @ ((n_pf @ X1) @ X3)) @ ((n_pf @ X2) @ X3)) => ((moref @ X1) @ X2))))))))).
% 172.20/172.07 thf(8,plain,sP1,inference(preprocess,[status(thm)],[satz63a]).
% 172.20/172.07 thf(9,plain,$false,inference(prop_unsat,[status(thm),assumptions([h10,h11,h8,h9,h7,h5,h6,h4,h2,h3,h1,h0])],[1,2,3,4,5,6,7,h2,h5,h8,h10,h11,8])).
% 172.20/172.07 thf(10,plain,$false,inference(tab_negimp,[status(thm),assumptions([h8,h9,h7,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h10,h11])],[h9,9,h10,h11])).
% 172.20/172.07 thf(11,plain,$false,inference(tab_negimp,[status(thm),assumptions([h7,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h8,h9])],[h7,10,h8,h9])).
% 172.20/172.07 thf(12,plain,$false,inference(tab_negall,[status(thm),assumptions([h5,h6,h4,h2,h3,h1,h0]),tab_negall(discharge,[h7]),tab_negall(eigenvar,eigen__2)],[h6,11,h7])).
% 172.20/172.07 thf(13,plain,$false,inference(tab_negimp,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,12,h5,h6])).
% 172.20/172.07 thf(14,plain,$false,inference(tab_negall,[status(thm),assumptions([h2,h3,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,13,h4])).
% 172.20/172.07 thf(15,plain,$false,inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,14,h2,h3])).
% 172.20/172.07 thf(16,plain,$false,inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,15,h1])).
% 172.20/172.07 thf(0,theorem,(![X1:$i]:(((in @ X1) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (![X2:$i]:(((in @ X2) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X3:$i]:((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))))) => (![X3:$i]:(((in @ X3) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X4:$i]:((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))))) => ((n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X3))))))) => (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2))))))))))),inference(contra,[status(thm),contra(discharge,[h0])],[16,h0])).
% 172.20/172.07 % SZS output end Proof
%------------------------------------------------------------------------------