%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : NUM781^4 : TPTP v8.1.2. Released v7.1.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n031.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 : 300s
% DateTime : Thu Aug 31 11:41:04 EDT 2023
% Result : Theorem 0.20s 0.44s
% Output : Proof 0.20s
% Verified :
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)
% Comments :
%------------------------------------------------------------------------------
thf(ty_d_Pi,type,
d_Pi: $i > ( $i > $i ) > $i ).
thf(ty_ap,type,
ap: $i > $i > $i ).
thf(ty_omega,type,
omega: $i ).
thf(ty_second1,type,
second1: $i > $i > $i ).
thf(ty_n_2,type,
n_2: $i ).
thf(ty_first1,type,
first1: $i > $i > $i ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_emptyset,type,
emptyset: $i ).
thf(ty_d_1to,type,
d_1to: $i > $i ).
thf(ty_l_some,type,
l_some: $i > ( $i > $o ) > $o ).
thf(ty_repl,type,
repl: $i > ( $i > $i ) > $i ).
thf(ty_eps,type,
eps: ( $i > $o ) > $i ).
thf(ty_times,type,
times: $i > $i ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_power,type,
power: $i > $i ).
thf(ty_ecp,type,
ecp: $i > ( $i > $i > $o ) > $i > $i > $o ).
thf(ty_if,type,
if: $o > $i > $i > $i ).
thf(ty_in,type,
in: $i > $i > $o ).
thf(sP1,plain,
( sP1
<=> ( eigen__0 = eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( eigen__1 = eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(def_is_of,definition,
( is_of
= ( ^ [X1: $i,X2: $i > $o] : ( X2 @ X1 ) ) ) ).
thf(def_all_of,definition,
( all_of
= ( ^ [X1: $i > $o,X2: $i > $o] :
! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( is_of @ X3 @ X1 )
@ ( X2 @ X3 ) ) ) ) ).
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 ) ) ) ).
thf(def_e_is,definition,
( e_is
= ( ^ [X1: $i,X2: $i,X3: $i] : ( X2 = X3 ) ) ) ).
thf(def_anec,definition,
( anec
= ( ^ [X1: $i,X2: $i > $i > $o,X3: $i] : ( l_some @ X1 @ ( ecp @ X1 @ X2 @ X3 ) ) ) ) ).
thf(def_ect,definition,
( ect
= ( ^ [X1: $i,X2: $i > $i > $o] : ( d_Sep @ ( power @ X1 ) @ ( anec @ X1 @ X2 ) ) ) ) ).
thf(def_nat,definition,
( nat
= ( d_Sep @ omega
@ ^ [X1: $i] : ( (~) @ ( X1 = emptyset ) ) ) ) ).
thf(def_n_is,definition,
( n_is
= ( e_is @ nat ) ) ).
thf(def_n_ts,definition,
( n_ts
= ( ^ [X1: $i] : ( ap @ ( times @ X1 ) ) ) ) ).
thf(def_pair1type,definition,
( pair1type
= ( ^ [X1: $i] :
( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] : X1 ) ) ) ).
thf(def_frac,definition,
( frac
= ( pair1type @ nat ) ) ).
thf(def_num,definition,
( num
= ( first1 @ nat ) ) ).
thf(def_den,definition,
( den
= ( second1 @ nat ) ) ).
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 ) ) ) ) ) ).
thf(def_rat,definition,
( rat
= ( ect @ frac @ n_eq ) ) ).
thf(def_rt_is,definition,
( rt_is
= ( e_is @ rat ) ) ).
thf(satz79,conjecture,
! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X3: $i,X4: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) ) )
@ X2 ) ) )
@ ( repl
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ ^ [X2: $i] :
( if
@ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X3: $i,X4: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) ) )
@ X2 ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X4: $i,X5: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) ) )
@ X3 ) ) ) ) ) )
@ emptyset ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X4: $i,X5: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) ) )
@ X3 ) ) )
@ ( repl
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ ^ [X3: $i] :
( if
@ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X4: $i,X5: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) ) )
@ X3 ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X5: $i] :
( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X5: $i] :
( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X5: $i] :
( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X5: $i,X6: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ omega )
=> ( X7 = emptyset ) )
@ ( repl @ omega
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ omega )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
@ ( second1
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ omega )
=> ( X7 = emptyset ) )
@ ( repl @ omega
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ omega )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset )
@ X6 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ omega )
=> ( X7 = emptyset ) )
@ ( repl @ omega
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ omega )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset )
@ X6 ) )
@ ( second1
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ omega )
=> ( X7 = emptyset ) )
@ ( repl @ omega
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ omega )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) ) )
@ X4 ) ) ) ) ) )
@ emptyset ) )
=> ( ( X1 = X2 )
=> ( X2 = X1 ) ) ) ) ).
thf(h0,negated_conjecture,
~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X3: $i,X4: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) ) )
@ X2 ) ) )
@ ( repl
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ ^ [X2: $i] :
( if
@ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X3: $i,X4: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) ) )
@ X2 ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X4: $i,X5: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) ) )
@ X3 ) ) ) ) ) )
@ emptyset ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X4: $i,X5: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) ) )
@ X3 ) ) )
@ ( repl
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ ^ [X3: $i] :
( if
@ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X4: $i,X5: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) ) )
@ X3 ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X5: $i] :
( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X5: $i] :
( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X5: $i] :
( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X5: $i,X6: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ omega )
=> ( X7 = emptyset ) )
@ ( repl @ omega
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ omega )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
@ ( second1
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ omega )
=> ( X7 = emptyset ) )
@ ( repl @ omega
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ omega )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset )
@ X6 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ omega )
=> ( X7 = emptyset ) )
@ ( repl @ omega
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ omega )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset )
@ X6 ) )
@ ( second1
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ omega )
=> ( X7 = emptyset ) )
@ ( repl @ omega
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ omega )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) ) )
@ X4 ) ) ) ) ) )
@ emptyset ) )
=> ( ( X1 = X2 )
=> ( X2 = X1 ) ) ) ),
inference(assume_negation,[status(cth)],[satz79]) ).
thf(h1,assumption,
~ ( ( in @ eigen__0
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X2: $i,X3: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X2 ) )
@ ( second1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X2 ) ) )
@ X1 ) ) )
@ ( repl
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X1: $i] :
( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( X2 = emptyset ) )
@ ( repl @ omega
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ omega )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ ^ [X1: $i] :
( if
@ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X2: $i,X3: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X2 ) )
@ ( second1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X2 ) ) )
@ X1 ) )
@ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X3: $i,X4: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) ) )
@ X2 ) ) ) ) ) )
@ emptyset ) )
=> ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X3: $i,X4: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) ) )
@ X2 ) ) )
@ ( repl
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ ^ [X2: $i] :
( if
@ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X3: $i,X4: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) ) )
@ X2 ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X4: $i,X5: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) ) )
@ X3 ) ) ) ) ) )
@ emptyset ) )
=> ( ( eigen__0 = X1 )
=> ( X1 = eigen__0 ) ) ) ),
introduced(assumption,[]) ).
thf(h2,assumption,
( in @ eigen__0
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X2: $i,X3: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X2 ) )
@ ( second1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X2 ) ) )
@ X1 ) ) )
@ ( repl
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X1: $i] :
( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( X2 = emptyset ) )
@ ( repl @ omega
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ omega )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ ^ [X1: $i] :
( if
@ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X2: $i,X3: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X2 ) )
@ ( second1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X2 ) ) )
@ X1 ) )
@ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X3: $i,X4: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) ) )
@ X2 ) ) ) ) ) )
@ emptyset ) ),
introduced(assumption,[]) ).
thf(h3,assumption,
~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X3: $i,X4: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) ) )
@ X2 ) ) )
@ ( repl
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ ^ [X2: $i] :
( if
@ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X3: $i,X4: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) ) )
@ X2 ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X4: $i,X5: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) ) )
@ X3 ) ) ) ) ) )
@ emptyset ) )
=> ( ( eigen__0 = X1 )
=> ( X1 = eigen__0 ) ) ),
introduced(assumption,[]) ).
thf(h4,assumption,
~ ( ( in @ eigen__1
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X2: $i,X3: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X2 ) )
@ ( second1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X2 ) ) )
@ X1 ) ) )
@ ( repl
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X1: $i] :
( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( X2 = emptyset ) )
@ ( repl @ omega
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ omega )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ ^ [X1: $i] :
( if
@ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X2: $i,X3: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X2 ) )
@ ( second1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X2 ) ) )
@ X1 ) )
@ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X3: $i,X4: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) ) )
@ X2 ) ) ) ) ) )
@ emptyset ) )
=> ( sP1
=> sP2 ) ),
introduced(assumption,[]) ).
thf(h5,assumption,
( in @ eigen__1
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X2: $i,X3: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X2 ) )
@ ( second1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X2 ) ) )
@ X1 ) ) )
@ ( repl
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X1: $i] :
( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( X2 = emptyset ) )
@ ( repl @ omega
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ omega )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ ^ [X1: $i] :
( if
@ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X2: $i,X3: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X2 ) )
@ ( second1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ X2 ) ) )
@ X1 ) )
@ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X3: $i,X4: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) ) )
@ X2 ) ) ) ) ) )
@ emptyset ) ),
introduced(assumption,[]) ).
thf(h6,assumption,
~ ( sP1
=> sP2 ),
introduced(assumption,[]) ).
thf(h7,assumption,
sP1,
introduced(assumption,[]) ).
thf(h8,assumption,
~ sP2,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP1
| sP2 ),
inference(symeq,[status(thm)],]) ).
thf(2,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h7,h8,h5,h6,h4,h2,h3,h1,h0])],[1,h7,h8]) ).
thf(3,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h7,h8])],[h6,2,h7,h8]) ).
thf(4,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,3,h5,h6]) ).
thf(5,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h2,h3,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,4,h4]) ).
thf(6,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,5,h2,h3]) ).
thf(7,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,6,h1]) ).
thf(0,theorem,
! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X3: $i,X4: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) ) )
@ X2 ) ) )
@ ( repl
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ omega )
=> ( X3 = emptyset ) )
@ ( repl @ omega
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ omega )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ ^ [X2: $i] :
( if
@ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X3: $i,X4: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ X3 ) ) )
@ X2 ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X4: $i,X5: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) ) )
@ X3 ) ) ) ) ) )
@ emptyset ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X4: $i,X5: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) ) )
@ X3 ) ) )
@ ( repl
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( X4 = emptyset ) )
@ ( repl @ omega
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ omega )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ ^ [X3: $i] :
( if
@ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ omega )
=> ( X5 = emptyset ) )
@ ( repl @ omega
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ omega )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X4: $i,X5: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
@ ( second1
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ X4 ) ) )
@ X3 ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( power
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X5: $i] :
( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
=> ~ ( l_some
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X5: $i] :
( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( ecp
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [X5: $i] :
( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ omega )
=> ( X6 = emptyset ) )
@ ( repl @ omega
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ omega )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
@ ^ [X5: $i,X6: $i] :
( ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ omega )
=> ( X7 = emptyset ) )
@ ( repl @ omega
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ omega )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) )
@ ( second1
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ omega )
=> ( X7 = emptyset ) )
@ ( repl @ omega
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ omega )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset )
@ X6 ) )
= ( ap
@ ( times
@ ( first1
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ omega )
=> ( X7 = emptyset ) )
@ ( repl @ omega
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ omega )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset )
@ X6 ) )
@ ( second1
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ omega )
=> ( X7 = emptyset ) )
@ ( repl @ omega
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ omega )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset )
@ X5 ) ) )
@ X4 ) ) ) ) ) )
@ emptyset ) )
=> ( ( X1 = X2 )
=> ( X2 = X1 ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h0])],[7,h0]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : NUM781^4 : TPTP v8.1.2. Released v7.1.0.
% 0.12/0.13 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.14/0.34 % Computer : n031.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Fri Aug 25 18:23:07 EDT 2023
% 0.14/0.34 % CPUTime :
% 0.20/0.44 % SZS status Theorem
% 0.20/0.44 % Mode: cade22sinegrackle2x6978
% 0.20/0.44 % Steps: 1
% 0.20/0.44 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------