TPTP Problem File: SLH0341^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Query_Optimization/0009_Dtree/prob_01195_052539__15097212_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1495 ( 646 unt; 206 typ; 0 def)
% Number of atoms : 3861 (1412 equ; 0 cnn)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 12541 ( 641 ~; 180 |; 260 &;10157 @)
% ( 0 <=>;1303 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 6 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 1089 (1089 >; 0 *; 0 +; 0 <<)
% Number of symbols : 189 ( 186 usr; 21 con; 0-4 aty)
% Number of variables : 4049 ( 697 ^;3304 !; 48 ?;4049 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:06:19.902
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_Mtf__b_J,type,
produc6708371838016462714_b_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J_J,type,
set_se3183138701204633190_a_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mt__Dtree__Odtree_Itf__a_Mtf__b_J_J,type,
produc5177672665255943253ee_a_b: $tType ).
thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J,type,
fset_P5281107635120001194_a_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J,type,
set_Pr3012420139608375472_a_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_M_062_Itf__b_Mtf__a_J_J,type,
produc1083523234014712191_b_b_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Dtree__Odtree_Itf__a_Mtf__b_J_J,type,
produc3469756349985706280ee_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J,type,
produc4558475209616630778_a_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Dtree__Odtree_Itf__a_Mtf__b_J_J,type,
set_dtree_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
product_prod_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
set_set_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Dtree__Odtree_Itf__a_Mtf__b_J,type,
dtree_a_b: $tType ).
thf(ty_n_t__FSet__Ofset_It__Nat__Onat_J,type,
fset_nat: $tType ).
thf(ty_n_t__FSet__Ofset_Itf__b_J,type,
fset_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (186)
thf(sy_c_Basic__BNFs_Ofsts_001t__Dtree__Odtree_Itf__a_Mtf__b_J_001tf__b,type,
basic_7578771248400840636_a_b_b: produc4558475209616630778_a_b_b > set_dtree_a_b ).
thf(sy_c_Dtree_Odhead_001tf__a_001tf__b,type,
dhead_a_b: dtree_a_b > ( b > a ) > b > a ).
thf(sy_c_Dtree_Odhead__rel_001tf__a_001tf__b,type,
dhead_rel_a_b: produc1083523234014712191_b_b_a > produc1083523234014712191_b_b_a > $o ).
thf(sy_c_Dtree_Odtail_001tf__a_001tf__b,type,
dtail_a_b: dtree_a_b > ( b > a ) > b > a ).
thf(sy_c_Dtree_Odtail__rel_001tf__a_001tf__b,type,
dtail_rel_a_b: produc1083523234014712191_b_b_a > produc1083523234014712191_b_b_a > $o ).
thf(sy_c_Dtree_Odtree_ONode_001tf__a_001tf__b,type,
node_a_b: a > fset_P5281107635120001194_a_b_b > dtree_a_b ).
thf(sy_c_Dtree_Odtree_Ocase__dtree_001tf__a_001tf__b_001tf__a,type,
case_dtree_a_b_a: ( a > fset_P5281107635120001194_a_b_b > a ) > dtree_a_b > a ).
thf(sy_c_Dtree_Odtree_Odarcs_001tf__a_001tf__b,type,
darcs_a_b: dtree_a_b > set_b ).
thf(sy_c_Dtree_Odtree_Odverts_001tf__a_001tf__b,type,
dverts_a_b: dtree_a_b > set_a ).
thf(sy_c_Dtree_Odtree_Oroot_001tf__a_001tf__b,type,
root_a_b: dtree_a_b > a ).
thf(sy_c_Dtree_Odtree_Osucs_001tf__a_001tf__b,type,
sucs_a_b: dtree_a_b > fset_P5281107635120001194_a_b_b ).
thf(sy_c_Dtree_Owf__darcs_001tf__a_001tf__b,type,
wf_darcs_a_b: dtree_a_b > $o ).
thf(sy_c_Dtree_Owf__darcs_H_001tf__a_001tf__b,type,
wf_darcs_a_b2: dtree_a_b > $o ).
thf(sy_c_Dtree_Owf__darcs_H__rel_001tf__a_001tf__b,type,
wf_darcs_rel_a_b: dtree_a_b > dtree_a_b > $o ).
thf(sy_c_Dtree_Owf__dverts_001tf__a_001tf__b,type,
wf_dverts_a_b: dtree_a_b > $o ).
thf(sy_c_Dtree_Owf__dverts_H_001tf__a_001tf__b,type,
wf_dverts_a_b2: dtree_a_b > $o ).
thf(sy_c_Dtree_Owf__dverts_H__rel_001tf__a_001tf__b,type,
wf_dverts_rel_a_b: dtree_a_b > dtree_a_b > $o ).
thf(sy_c_FSet_Offold_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_001_062_Itf__b_Mtf__a_J,type,
ffold_8367945289176929151_b_b_a: ( produc4558475209616630778_a_b_b > ( b > a ) > b > a ) > ( b > a ) > fset_P5281107635120001194_a_b_b > b > a ).
thf(sy_c_FSet_Offold_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_001tf__a,type,
ffold_2783168711033344739_b_b_a: ( produc4558475209616630778_a_b_b > a > a ) > a > fset_P5281107635120001194_a_b_b > a ).
thf(sy_c_FSet_Ofimage_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J,type,
fimage7457256623133068659_a_b_b: ( produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b ) > fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b ).
thf(sy_c_FSet_Ofimage_001tf__b_001tf__b,type,
fimage_b_b: ( b > b ) > fset_b > fset_b ).
thf(sy_c_FSet_Ofinsert_001t__Nat__Onat,type,
finsert_nat: nat > fset_nat > fset_nat ).
thf(sy_c_FSet_Ofinsert_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J,type,
finser8437519239679886002_a_b_b: produc4558475209616630778_a_b_b > fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b ).
thf(sy_c_FSet_Ofinsert_001tf__b,type,
finsert_b: b > fset_b > fset_b ).
thf(sy_c_FSet_Ofmember_001t__Nat__Onat,type,
fmember_nat: nat > fset_nat > $o ).
thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J,type,
fmembe3173364709796808819_a_b_b: produc4558475209616630778_a_b_b > fset_P5281107635120001194_a_b_b > $o ).
thf(sy_c_FSet_Ofmember_001tf__b,type,
fmember_b: b > fset_b > $o ).
thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J,type,
fset_P783253628892185035_a_b_b: fset_P5281107635120001194_a_b_b > set_Pr3012420139608375472_a_b_b ).
thf(sy_c_FSet_Ofset_Ofset_001tf__b,type,
fset_b2: fset_b > set_b ).
thf(sy_c_FSet_Ofthe__elem_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J,type,
fthe_e7442499522476018237_a_b_b: fset_P5281107635120001194_a_b_b > produc4558475209616630778_a_b_b ).
thf(sy_c_Finite__Set_Ocomp__fun__commute_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_001_062_Itf__b_Mtf__a_J,type,
finite7715548283558590705_b_b_a: ( produc4558475209616630778_a_b_b > ( b > a ) > b > a ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__commute_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_001tf__a,type,
finite414203908571218417_b_b_a: ( produc4558475209616630778_a_b_b > a > a ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__commute_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_Mtf__b_J_001t__Set__Oset_Itf__b_J,type,
finite4381541246406268242_set_b: ( produc6708371838016462714_b_b_b > set_b > set_b ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__commute_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Set__Oset_Itf__b_J,type,
finite7340995349656252681_set_b: ( product_prod_b_b > set_b > set_b ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__commute_001tf__b_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
finite3301421349078847953_set_b: ( b > set_set_b > set_set_b ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__commute_001tf__b_001t__Set__Oset_Itf__b_J,type,
finite4863250414163961073_set_b: ( b > set_b > set_b ) > $o ).
thf(sy_c_Fun_Ocomp_001_062_Itf__b_Mtf__a_J_001_062_Itf__b_Mtf__a_J_001_062_Itf__b_Mtf__a_J,type,
comp_b_a_b_a_b_a: ( ( b > a ) > b > a ) > ( ( b > a ) > b > a ) > ( b > a ) > b > a ).
thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_001t__Dtree__Odtree_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__b_Mt__Dtree__Odtree_Itf__a_Mtf__b_J_J,type,
comp_P1139271906589003649ee_a_b: ( produc4558475209616630778_a_b_b > dtree_a_b ) > ( produc3469756349985706280ee_a_b > produc4558475209616630778_a_b_b ) > produc3469756349985706280ee_a_b > dtree_a_b ).
thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_001tf__b_001t__Product____Type__Oprod_Itf__b_Mt__Dtree__Odtree_Itf__a_Mtf__b_J_J,type,
comp_P6702227762116406538ee_a_b: ( produc4558475209616630778_a_b_b > b ) > ( produc3469756349985706280ee_a_b > produc4558475209616630778_a_b_b ) > produc3469756349985706280ee_a_b > b ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__a,type,
comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_M_Eo_J,type,
minus_6397467918800550972_b_b_o: ( produc4558475209616630778_a_b_b > $o ) > ( produc4558475209616630778_a_b_b > $o ) > produc4558475209616630778_a_b_b > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__b_M_Eo_J,type,
minus_minus_b_o: ( b > $o ) > ( b > $o ) > b > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__FSet__Ofset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J,type,
minus_1250967532242559235_a_b_b: fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J,type,
minus_1392386589478415753_a_b_b: set_Pr3012420139608375472_a_b_b > set_Pr3012420139608375472_a_b_b > set_Pr3012420139608375472_a_b_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
minus_minus_set_b: set_b > set_b > set_b ).
thf(sy_c_If_001_062_Itf__b_Mtf__a_J,type,
if_b_a: $o > ( b > a ) > ( b > a ) > b > a ).
thf(sy_c_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J,type,
if_fse8812573537926886756_a_b_b: $o > fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b ).
thf(sy_c_If_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J,type,
if_Pro6329973184163622324_a_b_b: $o > produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b ).
thf(sy_c_If_001t__Set__Oset_Itf__b_J,type,
if_set_b: $o > set_b > set_b > set_b ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Dtree__Odtree_Itf__a_Mtf__b_J_M_062_Itf__b_M_Eo_J_J,type,
inf_in8207984165653407081_b_b_o: ( dtree_a_b > b > $o ) > ( dtree_a_b > b > $o ) > dtree_a_b > b > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_M_Eo_J,type,
inf_in55627642082981827_b_b_o: ( produc4558475209616630778_a_b_b > $o ) > ( produc4558475209616630778_a_b_b > $o ) > produc4558475209616630778_a_b_b > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__b_M_Eo_J,type,
inf_inf_b_o: ( b > $o ) > ( b > $o ) > b > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__FSet__Ofset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J,type,
inf_in7138637532943773244_a_b_b: fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J,type,
inf_in6138156342456174402_a_b_b: set_Pr3012420139608375472_a_b_b > set_Pr3012420139608375472_a_b_b > set_Pr3012420139608375472_a_b_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__b_J,type,
inf_inf_set_b: set_b > set_b > set_b ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_I_062_Itf__a_Mtf__a_J_M_Eo_J,type,
sup_sup_a_a_o: ( ( a > a ) > $o ) > ( ( a > a ) > $o ) > ( a > a ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Dtree__Odtree_Itf__a_Mtf__b_J_M_062_Itf__b_M_Eo_J_J,type,
sup_su6709851091347060739_b_b_o: ( dtree_a_b > b > $o ) > ( dtree_a_b > b > $o ) > dtree_a_b > b > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Dtree__Odtree_Itf__a_Mtf__b_J_M_Eo_J,type,
sup_sup_dtree_a_b_o: ( dtree_a_b > $o ) > ( dtree_a_b > $o ) > dtree_a_b > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_M_Eo_J,type,
sup_su4209747780764569001_b_b_o: ( produc4558475209616630778_a_b_b > $o ) > ( produc4558475209616630778_a_b_b > $o ) > produc4558475209616630778_a_b_b > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J_M_Eo_J,type,
sup_su5450082668191030131_b_b_o: ( set_Pr3012420139608375472_a_b_b > $o ) > ( set_Pr3012420139608375472_a_b_b > $o ) > set_Pr3012420139608375472_a_b_b > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_Itf__b_J_M_Eo_J,type,
sup_sup_set_b_o: ( set_b > $o ) > ( set_b > $o ) > set_b > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__b_M_Eo_J,type,
sup_sup_b_o: ( b > $o ) > ( b > $o ) > b > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
sup_sup_o: $o > $o > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__FSet__Ofset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J,type,
sup_su860928060825958358_a_b_b: fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__FSet__Ofset_Itf__b_J,type,
sup_sup_fset_b: fset_b > fset_b > fset_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
sup_sup_set_a_a: set_a_a > set_a_a > set_a_a ).
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image_6081965176830705659_a_b_b: ( produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b ) > set_Pr3012420139608375472_a_b_b > set_Pr3012420139608375472_a_b_b ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_001tf__b,type,
image_3908709015779211070_b_b_b: ( produc4558475209616630778_a_b_b > b ) > set_Pr3012420139608375472_a_b_b > set_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J,type,
image_4903599603319290215_a_b_b: ( set_Pr3012420139608375472_a_b_b > set_Pr3012420139608375472_a_b_b ) > set_se3183138701204633190_a_b_b > set_se3183138701204633190_a_b_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__b_J,type,
image_set_b_set_b: ( set_b > set_b ) > set_set_b > set_set_b ).
thf(sy_c_Set_Oimage_001tf__b_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J,type,
image_7642607452437185460_a_b_b: ( b > produc4558475209616630778_a_b_b ) > set_b > set_Pr3012420139608375472_a_b_b ).
thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
image_b_b: ( b > b ) > set_b > set_b ).
thf(sy_c_Set_Oinsert_001_062_Itf__a_Mtf__a_J,type,
insert_a_a: ( a > a ) > set_a_a > set_a_a ).
thf(sy_c_Set_Oinsert_001t__Dtree__Odtree_Itf__a_Mtf__b_J,type,
insert_dtree_a_b: dtree_a_b > set_dtree_a_b > set_dtree_a_b ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J,type,
insert1613891728210272810_a_b_b: produc4558475209616630778_a_b_b > set_Pr3012420139608375472_a_b_b > set_Pr3012420139608375472_a_b_b ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J,type,
insert8355695866607091424_a_b_b: set_Pr3012420139608375472_a_b_b > set_se3183138701204633190_a_b_b > set_se3183138701204633190_a_b_b ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__b_J,type,
insert_set_b: set_b > set_set_b > set_set_b ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J,type,
is_sin1118336051388392454_a_b_b: set_Pr3012420139608375472_a_b_b > $o ).
thf(sy_c_Set_Ois__singleton_001tf__b,type,
is_singleton_b: set_b > $o ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J,type,
the_el4127461656392778949_a_b_b: set_Pr3012420139608375472_a_b_b > produc4558475209616630778_a_b_b ).
thf(sy_c_Set_Othe__elem_001tf__b,type,
the_elem_b: set_b > b ).
thf(sy_c_Wellfounded_Oaccp_001t__Dtree__Odtree_Itf__a_Mtf__b_J,type,
accp_dtree_a_b: ( dtree_a_b > dtree_a_b > $o ) > dtree_a_b > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_M_062_Itf__b_Mtf__a_J_J,type,
accp_P1416650344722773512_b_b_a: ( produc1083523234014712191_b_b_a > produc1083523234014712191_b_b_a > $o ) > produc1083523234014712191_b_b_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__Dtree__Odtree_Itf__a_Mtf__b_J,type,
member_dtree_a_b: dtree_a_b > set_dtree_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J,type,
member4380921116106875537_a_b_b: produc4558475209616630778_a_b_b > set_Pr3012420139608375472_a_b_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J,type,
member7431159781899395911_a_b_b: set_Pr3012420139608375472_a_b_b > set_se3183138701204633190_a_b_b > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
member_set_b: set_b > set_set_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_def,type,
def: b > a ).
thf(sy_v_e,type,
e: b ).
thf(sy_v_e3____,type,
e3: b ).
thf(sy_v_r,type,
r: a ).
thf(sy_v_t,type,
t: dtree_a_b ).
thf(sy_v_x_H____,type,
x: produc4558475209616630778_a_b_b ).
thf(sy_v_x____,type,
x2: dtree_a_b ).
thf(sy_v_xs,type,
xs: fset_P5281107635120001194_a_b_b ).
thf(sy_v_xsa____,type,
xsa: fset_P5281107635120001194_a_b_b ).
% Relevant facts (1277)
thf(fact_0_False,axiom,
e3 != e ).
% False
thf(fact_1__C1_C,axiom,
wf_darcs_a_b @ ( node_a_b @ r @ xsa ) ).
% "1"
thf(fact_2__C0_C,axiom,
member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ t @ e ) @ ( fset_P783253628892185035_a_b_b @ xsa ) ).
% "0"
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_Ae3_O_Ax_H_A_061_A_Ix_M_Ae3_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X: dtree_a_b,E3: b] :
( x
!= ( produc331601717337510060_a_b_b @ X @ E3 ) ) ).
% \<open>\<And>thesis. (\<And>x e3. x' = (x, e3) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_insert_Oprems_I2_J,axiom,
wf_darcs_a_b @ ( node_a_b @ r @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) ).
% insert.prems(2)
thf(fact_5_assms_I2_J,axiom,
wf_darcs_a_b @ ( node_a_b @ r @ xs ) ).
% assms(2)
thf(fact_6_dtree_Oinject,axiom,
! [X1: a,X2: fset_P5281107635120001194_a_b_b,Y1: a,Y2: fset_P5281107635120001194_a_b_b] :
( ( ( node_a_b @ X1 @ X2 )
= ( node_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% dtree.inject
thf(fact_7_assms_I1_J,axiom,
member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ t @ e ) @ ( fset_P783253628892185035_a_b_b @ xs ) ).
% assms(1)
thf(fact_8_insert_Oprems_I1_J,axiom,
member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ t @ e ) @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) ).
% insert.prems(1)
thf(fact_9_insert_OIH,axiom,
( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ t @ e ) @ ( fset_P783253628892185035_a_b_b @ xsa ) )
=> ( ( wf_darcs_a_b @ ( node_a_b @ r @ xsa ) )
=> ( ( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ xsa ) )
| ~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ r @ xsa ) ) )
@ B
@ ( if_a @ ( e = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ def @ e ) ) ) )
@ ( def @ e )
@ xsa )
= ( root_a_b @ t ) ) ) ) ).
% insert.IH
thf(fact_10__092_060open_062ffold_A_I_092_060lambda_062_Ix_M_Ae2_J_Ab_O_Aif_A_Ix_M_Ae2_J_A_092_060notin_062_Afset_A_Ifinsert_Ax_H_Axs_J_A_092_060or_062_Ae_A_092_060notin_062_Adarcs_Ax_A_092_060union_062_A_123e2_125_A_092_060or_062_A_092_060not_062_Awf__darcs_A_INode_Ar_A_Ifinsert_Ax_H_Axs_J_J_Athen_Ab_Aelse_Aif_Ae_A_061_Ae2_Athen_Adtree_Oroot_Ax_Aelse_Adhead_Ax_Adef_Ae_J_A_Idef_Ae_J_Axs_A_061_Affold_A_I_092_060lambda_062_Ix_M_Ae2_J_Ab_O_Aif_A_Ix_M_Ae2_J_A_092_060notin_062_Afset_Axs_A_092_060or_062_Ae_A_092_060notin_062_Adarcs_Ax_A_092_060union_062_A_123e2_125_A_092_060or_062_A_092_060not_062_Awf__darcs_A_INode_Ar_Axs_J_Athen_Ab_Aelse_Aif_Ae_A_061_Ae2_Athen_Adtree_Oroot_Ax_Aelse_Adhead_Ax_Adef_Ae_J_A_Idef_Ae_J_Axs_092_060close_062,axiom,
( ( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) )
| ~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ r @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) ) )
@ B
@ ( if_a @ ( e = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ def @ e ) ) ) )
@ ( def @ e )
@ xsa )
= ( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ xsa ) )
| ~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ r @ xsa ) ) )
@ B
@ ( if_a @ ( e = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ def @ e ) ) ) )
@ ( def @ e )
@ xsa ) ) ).
% \<open>ffold (\<lambda>(x, e2) b. if (x, e2) \<notin> fset (finsert x' xs) \<or> e \<notin> darcs x \<union> {e2} \<or> \<not> wf_darcs (Node r (finsert x' xs)) then b else if e = e2 then dtree.root x else dhead x def e) (def e) xs = ffold (\<lambda>(x, e2) b. if (x, e2) \<notin> fset xs \<or> e \<notin> darcs x \<union> {e2} \<or> \<not> wf_darcs (Node r xs) then b else if e = e2 then dtree.root x else dhead x def e) (def e) xs\<close>
thf(fact_11__092_060open_062ffold_A_I_092_060lambda_062_Ix_M_Ae2_J_Ab_O_Aif_A_Ix_M_Ae2_J_A_092_060notin_062_Afset_A_Ifinsert_Ax_H_Axs_J_A_092_060or_062_Ae_A_092_060notin_062_Adarcs_Ax_A_092_060union_062_A_123e2_125_A_092_060or_062_A_092_060not_062_Awf__darcs_A_INode_Ar_A_Ifinsert_Ax_H_Axs_J_J_Athen_Ab_Aelse_Aif_Ae_A_061_Ae2_Athen_Adtree_Oroot_Ax_Aelse_Adhead_Ax_Adef_Ae_J_A_Idef_Ae_J_A_Ifinsert_Ax_H_Axs_J_A_061_A_Icase_Ax_H_Aof_A_Ix_M_Ae2_J_A_092_060Rightarrow_062_A_092_060lambda_062b_O_Aif_A_Ix_M_Ae2_J_A_092_060notin_062_Afset_A_Ifinsert_Ax_H_Axs_J_A_092_060or_062_Ae_A_092_060notin_062_Adarcs_Ax_A_092_060union_062_A_123e2_125_A_092_060or_062_A_092_060not_062_Awf__darcs_A_INode_Ar_A_Ifinsert_Ax_H_Axs_J_J_Athen_Ab_Aelse_Aif_Ae_A_061_Ae2_Athen_Adtree_Oroot_Ax_Aelse_Adhead_Ax_Adef_Ae_J_A_Iffold_A_I_092_060lambda_062_Ix_M_Ae2_J_Ab_O_Aif_A_Ix_M_Ae2_J_A_092_060notin_062_Afset_A_Ifinsert_Ax_H_Axs_J_A_092_060or_062_Ae_A_092_060notin_062_Adarcs_Ax_A_092_060union_062_A_123e2_125_A_092_060or_062_A_092_060not_062_Awf__darcs_A_INode_Ar_A_Ifinsert_Ax_H_Axs_J_J_Athen_Ab_Aelse_Aif_Ae_A_061_Ae2_Athen_Adtree_Oroot_Ax_Aelse_Adhead_Ax_Adef_Ae_J_A_Idef_Ae_J_Axs_J_092_060close_062,axiom,
( ( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) )
| ~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ r @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) ) )
@ B
@ ( if_a @ ( e = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ def @ e ) ) ) )
@ ( def @ e )
@ ( finser8437519239679886002_a_b_b @ x @ xsa ) )
= ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) )
| ~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ r @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) ) )
@ B
@ ( if_a @ ( e = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ def @ e ) ) )
@ x
@ ( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) )
| ~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ r @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) ) )
@ B
@ ( if_a @ ( e = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ def @ e ) ) ) )
@ ( def @ e )
@ xsa ) ) ) ).
% \<open>ffold (\<lambda>(x, e2) b. if (x, e2) \<notin> fset (finsert x' xs) \<or> e \<notin> darcs x \<union> {e2} \<or> \<not> wf_darcs (Node r (finsert x' xs)) then b else if e = e2 then dtree.root x else dhead x def e) (def e) (finsert x' xs) = (case x' of (x, e2) \<Rightarrow> \<lambda>b. if (x, e2) \<notin> fset (finsert x' xs) \<or> e \<notin> darcs x \<union> {e2} \<or> \<not> wf_darcs (Node r (finsert x' xs)) then b else if e = e2 then dtree.root x else dhead x def e) (ffold (\<lambda>(x, e2) b. if (x, e2) \<notin> fset (finsert x' xs) \<or> e \<notin> darcs x \<union> {e2} \<or> \<not> wf_darcs (Node r (finsert x' xs)) then b else if e = e2 then dtree.root x else dhead x def e) (def e) xs)\<close>
thf(fact_12_calculation,axiom,
( ( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) )
| ~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ r @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) ) )
@ B
@ ( if_a @ ( e = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ def @ e ) ) ) )
@ ( def @ e )
@ ( finser8437519239679886002_a_b_b @ x @ xsa ) )
= ( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ xsa ) )
| ~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ r @ xsa ) ) )
@ B
@ ( if_a @ ( e = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ def @ e ) ) ) )
@ ( def @ e )
@ xsa ) ) ).
% calculation
thf(fact_13_dtree_Osel_I1_J,axiom,
! [X1: a,X2: fset_P5281107635120001194_a_b_b] :
( ( root_a_b @ ( node_a_b @ X1 @ X2 ) )
= X1 ) ).
% dtree.sel(1)
thf(fact_14_dhead_Oelims,axiom,
! [X4: dtree_a_b,Xa: b > a,Y: b > a] :
( ( ( dhead_a_b @ X4 @ Xa )
= Y )
=> ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( ( X4
= ( node_a_b @ R @ Xs ) )
=> ( Y
!= ( ^ [E: b] :
( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs ) )
| ~ ( member_b @ E @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R @ Xs ) ) )
@ B
@ ( if_a @ ( E = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ Xa @ E ) ) ) )
@ ( Xa @ E )
@ Xs ) ) ) ) ) ).
% dhead.elims
thf(fact_15_dhead_Osimps,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,Def: b > a] :
( ( dhead_a_b @ ( node_a_b @ R2 @ Xs2 ) @ Def )
= ( ^ [E: b] :
( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( if_a @ ( E = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ Def @ E ) ) ) )
@ ( Def @ E )
@ Xs2 ) ) ) ).
% dhead.simps
thf(fact_16_dtree_Oexhaust,axiom,
! [Y: dtree_a_b] :
~ ! [X12: a,X22: fset_P5281107635120001194_a_b_b] :
( Y
!= ( node_a_b @ X12 @ X22 ) ) ).
% dtree.exhaust
thf(fact_17_dverts__mset_Ocases,axiom,
! [X4: dtree_a_b] :
~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( X4
!= ( node_a_b @ R @ Xs ) ) ).
% dverts_mset.cases
thf(fact_18_dhead__ffold__f__alt,axiom,
! [P: fset_P5281107635120001194_a_b_b > $o,R2: a,Q: produc4558475209616630778_a_b_b > a > $o,E4: b,R3: produc4558475209616630778_a_b_b > a > a,Def: b > a,Xs2: fset_P5281107635120001194_a_b_b] :
( ( P
= ( ^ [Xs3: fset_P5281107635120001194_a_b_b] : ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs3 ) ) ) )
=> ( ( Q
= ( produc6139810021161713496_b_a_o
@ ^ [X3: dtree_a_b,E2: b,Uu: a] : ( member_b @ E4 @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) ) ) )
=> ( ( R3
= ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,Uu: a] : ( if_a @ ( E4 = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ Def @ E4 ) ) ) )
=> ( ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E4 @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( if_a @ ( E4 = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ Def @ E4 ) ) ) )
= ( ^ [A: produc4558475209616630778_a_b_b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ A @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( Q @ A @ B )
| ~ ( P @ Xs2 ) )
@ B
@ ( R3 @ A @ B ) ) ) ) ) ) ) ).
% dhead_ffold_f_alt
thf(fact_19_dhead__notelem__eq__def,axiom,
! [E4: b,T: dtree_a_b,Def: b > a] :
( ~ ( member_b @ E4 @ ( darcs_a_b @ T ) )
=> ( ( dhead_a_b @ T @ Def @ E4 )
= ( Def @ E4 ) ) ) ).
% dhead_notelem_eq_def
thf(fact_20_dhead__in__child__eq__child,axiom,
! [T: dtree_a_b,E1: b,Xs2: fset_P5281107635120001194_a_b_b,E4: b,R2: a,Def: b > a] :
( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( ( member_b @ E4 @ ( darcs_a_b @ T ) )
=> ( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ( ( dhead_a_b @ ( node_a_b @ R2 @ Xs2 ) @ Def @ E4 )
= ( dhead_a_b @ T @ Def @ E4 ) ) ) ) ) ).
% dhead_in_child_eq_child
thf(fact_21_disjoint__darcs__if__wf__aux1,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T1: dtree_a_b,E1: b] :
( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T1 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ~ ( member_b @ E1 @ ( darcs_a_b @ T1 ) ) ) ) ).
% disjoint_darcs_if_wf_aux1
thf(fact_22_disjoint__darcs__if__wf__aux3,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T1: dtree_a_b,E1: b,T2: dtree_a_b,E22: b] :
( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T1 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T2 @ E22 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ~ ( member_b @ E22 @ ( darcs_a_b @ T1 ) ) ) ) ) ).
% disjoint_darcs_if_wf_aux3
thf(fact_23_disjoint__darcs__if__wf__aux4,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T1: dtree_a_b,E1: b,T2: dtree_a_b,E22: b] :
( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T1 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T2 @ E22 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( ( ( produc331601717337510060_a_b_b @ T1 @ E1 )
!= ( produc331601717337510060_a_b_b @ T2 @ E22 ) )
=> ( E1 != E22 ) ) ) ) ) ).
% disjoint_darcs_if_wf_aux4
thf(fact_24_dhead__in__child__eq__child__ffold,axiom,
! [T: dtree_a_b,E1: b,Xs2: fset_P5281107635120001194_a_b_b,E4: b,R2: a,Def: b > a] :
( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( ( member_b @ E4 @ ( darcs_a_b @ T ) )
=> ( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ( ( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E4 @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( if_a @ ( E4 = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ Def @ E4 ) ) ) )
@ ( Def @ E4 )
@ Xs2 )
= ( dhead_a_b @ T @ Def @ E4 ) ) ) ) ) ).
% dhead_in_child_eq_child_ffold
thf(fact_25_case__prod__conv,axiom,
! [F: dtree_a_b > b > a > $o,A2: dtree_a_b,B2: b] :
( ( produc6139810021161713496_b_a_o @ F @ ( produc331601717337510060_a_b_b @ A2 @ B2 ) )
= ( F @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_26_case__prod__conv,axiom,
! [F: dtree_a_b > b > b,A2: dtree_a_b,B2: b] :
( ( produc3664522937540588134_b_b_b @ F @ ( produc331601717337510060_a_b_b @ A2 @ B2 ) )
= ( F @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_27_case__prod__conv,axiom,
! [F: dtree_a_b > b > a,A2: dtree_a_b,B2: b] :
( ( produc3664522937540588133_b_b_a @ F @ ( produc331601717337510060_a_b_b @ A2 @ B2 ) )
= ( F @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_28_case__prod__conv,axiom,
! [F: b > dtree_a_b > produc4558475209616630778_a_b_b,A2: b,B2: dtree_a_b] :
( ( produc1296939142185513033_a_b_b @ F @ ( produc3542686128043086370ee_a_b @ A2 @ B2 ) )
= ( F @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_29_case__prod__conv,axiom,
! [F: b > b > set_b > set_b,A2: b,B2: b] :
( ( produc831963642587629969_set_b @ F @ ( product_Pair_b_b @ A2 @ B2 ) )
= ( F @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_30_case__prod__conv,axiom,
! [F: dtree_a_b > b > a > a,A2: dtree_a_b,B2: b] :
( ( produc2242037354397874494_b_a_a @ F @ ( produc331601717337510060_a_b_b @ A2 @ B2 ) )
= ( F @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_31_singleton__conv,axiom,
! [A2: dtree_a_b] :
( ( collect_dtree_a_b
@ ^ [X3: dtree_a_b] : ( X3 = A2 ) )
= ( insert_dtree_a_b @ A2 @ bot_bo8730652382759064772ee_a_b ) ) ).
% singleton_conv
thf(fact_32_singleton__conv,axiom,
! [A2: produc4558475209616630778_a_b_b] :
( ( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] : ( X3 = A2 ) )
= ( insert1613891728210272810_a_b_b @ A2 @ bot_bo3721250822024684356_a_b_b ) ) ).
% singleton_conv
thf(fact_33_singleton__conv,axiom,
! [A2: a] :
( ( collect_a
@ ^ [X3: a] : ( X3 = A2 ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_34_singleton__conv,axiom,
! [A2: b] :
( ( collect_b
@ ^ [X3: b] : ( X3 = A2 ) )
= ( insert_b @ A2 @ bot_bot_set_b ) ) ).
% singleton_conv
thf(fact_35_singleton__conv2,axiom,
! [A2: dtree_a_b] :
( ( collect_dtree_a_b
@ ( ^ [Y3: dtree_a_b,Z: dtree_a_b] : ( Y3 = Z )
@ A2 ) )
= ( insert_dtree_a_b @ A2 @ bot_bo8730652382759064772ee_a_b ) ) ).
% singleton_conv2
thf(fact_36_singleton__conv2,axiom,
! [A2: produc4558475209616630778_a_b_b] :
( ( collec1368399972772960719_a_b_b
@ ( ^ [Y3: produc4558475209616630778_a_b_b,Z: produc4558475209616630778_a_b_b] : ( Y3 = Z )
@ A2 ) )
= ( insert1613891728210272810_a_b_b @ A2 @ bot_bo3721250822024684356_a_b_b ) ) ).
% singleton_conv2
thf(fact_37_singleton__conv2,axiom,
! [A2: a] :
( ( collect_a
@ ( ^ [Y3: a,Z: a] : ( Y3 = Z )
@ A2 ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_38_singleton__conv2,axiom,
! [A2: b] :
( ( collect_b
@ ( ^ [Y3: b,Z: b] : ( Y3 = Z )
@ A2 ) )
= ( insert_b @ A2 @ bot_bot_set_b ) ) ).
% singleton_conv2
thf(fact_39__092_060open_062_Icase_A_Ix_M_Ae3_J_Aof_A_Ix_M_Ae2_J_A_092_060Rightarrow_062_A_092_060lambda_062b_O_Aif_A_Ix_M_Ae2_J_A_092_060notin_062_Afset_A_Ifinsert_Ax_H_Axs_J_A_092_060or_062_Ae_A_092_060notin_062_Adarcs_Ax_A_092_060union_062_A_123e2_125_A_092_060or_062_A_092_060not_062_Awf__darcs_A_INode_Ar_A_Ifinsert_Ax_H_Axs_J_J_Athen_Ab_Aelse_Aif_Ae_A_061_Ae2_Athen_Adtree_Oroot_Ax_Aelse_Adhead_Ax_Adef_Ae_J_A_Iffold_A_I_092_060lambda_062_Ix_M_Ae2_J_Ab_O_Aif_A_Ix_M_Ae2_J_A_092_060notin_062_Afset_A_Ifinsert_Ax_H_Axs_J_A_092_060or_062_Ae_A_092_060notin_062_Adarcs_Ax_A_092_060union_062_A_123e2_125_A_092_060or_062_A_092_060not_062_Awf__darcs_A_INode_Ar_A_Ifinsert_Ax_H_Axs_J_J_Athen_Ab_Aelse_Aif_Ae_A_061_Ae2_Athen_Adtree_Oroot_Ax_Aelse_Adhead_Ax_Adef_Ae_J_A_Idef_Ae_J_Axs_J_A_061_Affold_A_I_092_060lambda_062_Ix_M_Ae2_J_Ab_O_Aif_A_Ix_M_Ae2_J_A_092_060notin_062_Afset_A_Ifinsert_Ax_H_Axs_J_A_092_060or_062_Ae_A_092_060notin_062_Adarcs_Ax_A_092_060union_062_A_123e2_125_A_092_060or_062_A_092_060not_062_Awf__darcs_A_INode_Ar_A_Ifinsert_Ax_H_Axs_J_J_Athen_Ab_Aelse_Aif_Ae_A_061_Ae2_Athen_Adtree_Oroot_Ax_Aelse_Adhead_Ax_Adef_Ae_J_A_Idef_Ae_J_Axs_092_060close_062,axiom,
( ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) )
| ~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ r @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) ) )
@ B
@ ( if_a @ ( e = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ def @ e ) ) )
@ ( produc331601717337510060_a_b_b @ x2 @ e3 )
@ ( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) )
| ~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ r @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) ) )
@ B
@ ( if_a @ ( e = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ def @ e ) ) ) )
@ ( def @ e )
@ xsa ) )
= ( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) )
| ~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ r @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) ) )
@ B
@ ( if_a @ ( e = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ def @ e ) ) ) )
@ ( def @ e )
@ xsa ) ) ).
% \<open>(case (x, e3) of (x, e2) \<Rightarrow> \<lambda>b. if (x, e2) \<notin> fset (finsert x' xs) \<or> e \<notin> darcs x \<union> {e2} \<or> \<not> wf_darcs (Node r (finsert x' xs)) then b else if e = e2 then dtree.root x else dhead x def e) (ffold (\<lambda>(x, e2) b. if (x, e2) \<notin> fset (finsert x' xs) \<or> e \<notin> darcs x \<union> {e2} \<or> \<not> wf_darcs (Node r (finsert x' xs)) then b else if e = e2 then dtree.root x else dhead x def e) (def e) xs) = ffold (\<lambda>(x, e2) b. if (x, e2) \<notin> fset (finsert x' xs) \<or> e \<notin> darcs x \<union> {e2} \<or> \<not> wf_darcs (Node r (finsert x' xs)) then b else if e = e2 then dtree.root x else dhead x def e) (def e) xs\<close>
thf(fact_40__092_060open_062_Icase_Ax_H_Aof_A_Ix_M_Ae2_J_A_092_060Rightarrow_062_A_092_060lambda_062b_O_Aif_A_Ix_M_Ae2_J_A_092_060notin_062_Afset_A_Ifinsert_Ax_H_Axs_J_A_092_060or_062_Ae_A_092_060notin_062_Adarcs_Ax_A_092_060union_062_A_123e2_125_A_092_060or_062_A_092_060not_062_Awf__darcs_A_INode_Ar_A_Ifinsert_Ax_H_Axs_J_J_Athen_Ab_Aelse_Aif_Ae_A_061_Ae2_Athen_Adtree_Oroot_Ax_Aelse_Adhead_Ax_Adef_Ae_J_A_Iffold_A_I_092_060lambda_062_Ix_M_Ae2_J_Ab_O_Aif_A_Ix_M_Ae2_J_A_092_060notin_062_Afset_A_Ifinsert_Ax_H_Axs_J_A_092_060or_062_Ae_A_092_060notin_062_Adarcs_Ax_A_092_060union_062_A_123e2_125_A_092_060or_062_A_092_060not_062_Awf__darcs_A_INode_Ar_A_Ifinsert_Ax_H_Axs_J_J_Athen_Ab_Aelse_Aif_Ae_A_061_Ae2_Athen_Adtree_Oroot_Ax_Aelse_Adhead_Ax_Adef_Ae_J_A_Idef_Ae_J_Axs_J_A_061_A_Icase_A_Ix_M_Ae3_J_Aof_A_Ix_M_Ae2_J_A_092_060Rightarrow_062_A_092_060lambda_062b_O_Aif_A_Ix_M_Ae2_J_A_092_060notin_062_Afset_A_Ifinsert_Ax_H_Axs_J_A_092_060or_062_Ae_A_092_060notin_062_Adarcs_Ax_A_092_060union_062_A_123e2_125_A_092_060or_062_A_092_060not_062_Awf__darcs_A_INode_Ar_A_Ifinsert_Ax_H_Axs_J_J_Athen_Ab_Aelse_Aif_Ae_A_061_Ae2_Athen_Adtree_Oroot_Ax_Aelse_Adhead_Ax_Adef_Ae_J_A_Iffold_A_I_092_060lambda_062_Ix_M_Ae2_J_Ab_O_Aif_A_Ix_M_Ae2_J_A_092_060notin_062_Afset_A_Ifinsert_Ax_H_Axs_J_A_092_060or_062_Ae_A_092_060notin_062_Adarcs_Ax_A_092_060union_062_A_123e2_125_A_092_060or_062_A_092_060not_062_Awf__darcs_A_INode_Ar_A_Ifinsert_Ax_H_Axs_J_J_Athen_Ab_Aelse_Aif_Ae_A_061_Ae2_Athen_Adtree_Oroot_Ax_Aelse_Adhead_Ax_Adef_Ae_J_A_Idef_Ae_J_Axs_J_092_060close_062,axiom,
( ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) )
| ~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ r @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) ) )
@ B
@ ( if_a @ ( e = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ def @ e ) ) )
@ x
@ ( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) )
| ~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ r @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) ) )
@ B
@ ( if_a @ ( e = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ def @ e ) ) ) )
@ ( def @ e )
@ xsa ) )
= ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) )
| ~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ r @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) ) )
@ B
@ ( if_a @ ( e = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ def @ e ) ) )
@ ( produc331601717337510060_a_b_b @ x2 @ e3 )
@ ( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) )
| ~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ r @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ) ) )
@ B
@ ( if_a @ ( e = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ def @ e ) ) ) )
@ ( def @ e )
@ xsa ) ) ) ).
% \<open>(case x' of (x, e2) \<Rightarrow> \<lambda>b. if (x, e2) \<notin> fset (finsert x' xs) \<or> e \<notin> darcs x \<union> {e2} \<or> \<not> wf_darcs (Node r (finsert x' xs)) then b else if e = e2 then dtree.root x else dhead x def e) (ffold (\<lambda>(x, e2) b. if (x, e2) \<notin> fset (finsert x' xs) \<or> e \<notin> darcs x \<union> {e2} \<or> \<not> wf_darcs (Node r (finsert x' xs)) then b else if e = e2 then dtree.root x else dhead x def e) (def e) xs) = (case (x, e3) of (x, e2) \<Rightarrow> \<lambda>b. if (x, e2) \<notin> fset (finsert x' xs) \<or> e \<notin> darcs x \<union> {e2} \<or> \<not> wf_darcs (Node r (finsert x' xs)) then b else if e = e2 then dtree.root x else dhead x def e) (ffold (\<lambda>(x, e2) b. if (x, e2) \<notin> fset (finsert x' xs) \<or> e \<notin> darcs x \<union> {e2} \<or> \<not> wf_darcs (Node r (finsert x' xs)) then b else if e = e2 then dtree.root x else dhead x def e) (def e) xs)\<close>
thf(fact_41_Un__insert__left,axiom,
! [A2: produc4558475209616630778_a_b_b,B3: set_Pr3012420139608375472_a_b_b,C: set_Pr3012420139608375472_a_b_b] :
( ( sup_su2887895092731772380_a_b_b @ ( insert1613891728210272810_a_b_b @ A2 @ B3 ) @ C )
= ( insert1613891728210272810_a_b_b @ A2 @ ( sup_su2887895092731772380_a_b_b @ B3 @ C ) ) ) ).
% Un_insert_left
thf(fact_42_Un__insert__left,axiom,
! [A2: dtree_a_b,B3: set_dtree_a_b,C: set_dtree_a_b] :
( ( sup_su8994539500306794332ee_a_b @ ( insert_dtree_a_b @ A2 @ B3 ) @ C )
= ( insert_dtree_a_b @ A2 @ ( sup_su8994539500306794332ee_a_b @ B3 @ C ) ) ) ).
% Un_insert_left
thf(fact_43_Un__insert__left,axiom,
! [A2: set_b,B3: set_set_b,C: set_set_b] :
( ( sup_sup_set_set_b @ ( insert_set_b @ A2 @ B3 ) @ C )
= ( insert_set_b @ A2 @ ( sup_sup_set_set_b @ B3 @ C ) ) ) ).
% Un_insert_left
thf(fact_44_Un__insert__left,axiom,
! [A2: b,B3: set_b,C: set_b] :
( ( sup_sup_set_b @ ( insert_b @ A2 @ B3 ) @ C )
= ( insert_b @ A2 @ ( sup_sup_set_b @ B3 @ C ) ) ) ).
% Un_insert_left
thf(fact_45_Un__insert__right,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,A2: produc4558475209616630778_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( sup_su2887895092731772380_a_b_b @ A3 @ ( insert1613891728210272810_a_b_b @ A2 @ B3 ) )
= ( insert1613891728210272810_a_b_b @ A2 @ ( sup_su2887895092731772380_a_b_b @ A3 @ B3 ) ) ) ).
% Un_insert_right
thf(fact_46_Un__insert__right,axiom,
! [A3: set_dtree_a_b,A2: dtree_a_b,B3: set_dtree_a_b] :
( ( sup_su8994539500306794332ee_a_b @ A3 @ ( insert_dtree_a_b @ A2 @ B3 ) )
= ( insert_dtree_a_b @ A2 @ ( sup_su8994539500306794332ee_a_b @ A3 @ B3 ) ) ) ).
% Un_insert_right
thf(fact_47_Un__insert__right,axiom,
! [A3: set_set_b,A2: set_b,B3: set_set_b] :
( ( sup_sup_set_set_b @ A3 @ ( insert_set_b @ A2 @ B3 ) )
= ( insert_set_b @ A2 @ ( sup_sup_set_set_b @ A3 @ B3 ) ) ) ).
% Un_insert_right
thf(fact_48_Un__insert__right,axiom,
! [A3: set_b,A2: b,B3: set_b] :
( ( sup_sup_set_b @ A3 @ ( insert_b @ A2 @ B3 ) )
= ( insert_b @ A2 @ ( sup_sup_set_b @ A3 @ B3 ) ) ) ).
% Un_insert_right
thf(fact_49_Un__empty,axiom,
! [A3: set_set_b,B3: set_set_b] :
( ( ( sup_sup_set_set_b @ A3 @ B3 )
= bot_bot_set_set_b )
= ( ( A3 = bot_bot_set_set_b )
& ( B3 = bot_bot_set_set_b ) ) ) ).
% Un_empty
thf(fact_50_Un__empty,axiom,
! [A3: set_a,B3: set_a] :
( ( ( sup_sup_set_a @ A3 @ B3 )
= bot_bot_set_a )
= ( ( A3 = bot_bot_set_a )
& ( B3 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_51_Un__empty,axiom,
! [A3: set_b,B3: set_b] :
( ( ( sup_sup_set_b @ A3 @ B3 )
= bot_bot_set_b )
= ( ( A3 = bot_bot_set_b )
& ( B3 = bot_bot_set_b ) ) ) ).
% Un_empty
thf(fact_52_singletonI,axiom,
! [A2: a > a] : ( member_a_a @ A2 @ ( insert_a_a @ A2 @ bot_bot_set_a_a ) ) ).
% singletonI
thf(fact_53_singletonI,axiom,
! [A2: dtree_a_b] : ( member_dtree_a_b @ A2 @ ( insert_dtree_a_b @ A2 @ bot_bo8730652382759064772ee_a_b ) ) ).
% singletonI
thf(fact_54_singletonI,axiom,
! [A2: set_b] : ( member_set_b @ A2 @ ( insert_set_b @ A2 @ bot_bot_set_set_b ) ) ).
% singletonI
thf(fact_55_singletonI,axiom,
! [A2: set_Pr3012420139608375472_a_b_b] : ( member7431159781899395911_a_b_b @ A2 @ ( insert8355695866607091424_a_b_b @ A2 @ bot_bo2537099559385417978_a_b_b ) ) ).
% singletonI
thf(fact_56_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_57_singletonI,axiom,
! [A2: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ A2 @ ( insert1613891728210272810_a_b_b @ A2 @ bot_bo3721250822024684356_a_b_b ) ) ).
% singletonI
thf(fact_58_singletonI,axiom,
! [A2: b] : ( member_b @ A2 @ ( insert_b @ A2 @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_59_sup__bot__left,axiom,
! [X4: set_set_b] :
( ( sup_sup_set_set_b @ bot_bot_set_set_b @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_60_sup__bot__left,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_61_sup__bot__left,axiom,
! [X4: produc4558475209616630778_a_b_b > $o] :
( ( sup_su4209747780764569001_b_b_o @ bot_bo7321339186913516097_b_b_o @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_62_sup__bot__left,axiom,
! [X4: dtree_a_b > b > $o] :
( ( sup_su6709851091347060739_b_b_o @ bot_bo471016548657204587_b_b_o @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_63_sup__bot__left,axiom,
! [X4: b > $o] :
( ( sup_sup_b_o @ bot_bot_b_o @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_64_sup__bot__left,axiom,
! [X4: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_65_sup__bot__right,axiom,
! [X4: set_set_b] :
( ( sup_sup_set_set_b @ X4 @ bot_bot_set_set_b )
= X4 ) ).
% sup_bot_right
thf(fact_66_sup__bot__right,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ X4 @ bot_bot_set_a )
= X4 ) ).
% sup_bot_right
thf(fact_67_sup__bot__right,axiom,
! [X4: produc4558475209616630778_a_b_b > $o] :
( ( sup_su4209747780764569001_b_b_o @ X4 @ bot_bo7321339186913516097_b_b_o )
= X4 ) ).
% sup_bot_right
thf(fact_68_sup__bot__right,axiom,
! [X4: dtree_a_b > b > $o] :
( ( sup_su6709851091347060739_b_b_o @ X4 @ bot_bo471016548657204587_b_b_o )
= X4 ) ).
% sup_bot_right
thf(fact_69_sup__bot__right,axiom,
! [X4: b > $o] :
( ( sup_sup_b_o @ X4 @ bot_bot_b_o )
= X4 ) ).
% sup_bot_right
thf(fact_70_sup__bot__right,axiom,
! [X4: set_b] :
( ( sup_sup_set_b @ X4 @ bot_bot_set_b )
= X4 ) ).
% sup_bot_right
thf(fact_71_bot__eq__sup__iff,axiom,
! [X4: set_set_b,Y: set_set_b] :
( ( bot_bot_set_set_b
= ( sup_sup_set_set_b @ X4 @ Y ) )
= ( ( X4 = bot_bot_set_set_b )
& ( Y = bot_bot_set_set_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_72_bot__eq__sup__iff,axiom,
! [X4: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X4 @ Y ) )
= ( ( X4 = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_73_bot__eq__sup__iff,axiom,
! [X4: produc4558475209616630778_a_b_b > $o,Y: produc4558475209616630778_a_b_b > $o] :
( ( bot_bo7321339186913516097_b_b_o
= ( sup_su4209747780764569001_b_b_o @ X4 @ Y ) )
= ( ( X4 = bot_bo7321339186913516097_b_b_o )
& ( Y = bot_bo7321339186913516097_b_b_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_74_bot__eq__sup__iff,axiom,
! [X4: dtree_a_b > b > $o,Y: dtree_a_b > b > $o] :
( ( bot_bo471016548657204587_b_b_o
= ( sup_su6709851091347060739_b_b_o @ X4 @ Y ) )
= ( ( X4 = bot_bo471016548657204587_b_b_o )
& ( Y = bot_bo471016548657204587_b_b_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_75_bot__eq__sup__iff,axiom,
! [X4: b > $o,Y: b > $o] :
( ( bot_bot_b_o
= ( sup_sup_b_o @ X4 @ Y ) )
= ( ( X4 = bot_bot_b_o )
& ( Y = bot_bot_b_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_76_bot__eq__sup__iff,axiom,
! [X4: set_b,Y: set_b] :
( ( bot_bot_set_b
= ( sup_sup_set_b @ X4 @ Y ) )
= ( ( X4 = bot_bot_set_b )
& ( Y = bot_bot_set_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_77_x__def,axiom,
( x
= ( produc331601717337510060_a_b_b @ x2 @ e3 ) ) ).
% x_def
thf(fact_78_old_Oprod_Oinject,axiom,
! [A2: dtree_a_b,B2: b > a,A4: dtree_a_b,B4: b > a] :
( ( ( produc1993688775741047735_b_b_a @ A2 @ B2 )
= ( produc1993688775741047735_b_b_a @ A4 @ B4 ) )
= ( ( A2 = A4 )
& ( B2 = B4 ) ) ) ).
% old.prod.inject
thf(fact_79_old_Oprod_Oinject,axiom,
! [A2: dtree_a_b,B2: dtree_a_b,A4: dtree_a_b,B4: dtree_a_b] :
( ( ( produc7805419539522982029ee_a_b @ A2 @ B2 )
= ( produc7805419539522982029ee_a_b @ A4 @ B4 ) )
= ( ( A2 = A4 )
& ( B2 = B4 ) ) ) ).
% old.prod.inject
thf(fact_80_old_Oprod_Oinject,axiom,
! [A2: dtree_a_b,B2: b,A4: dtree_a_b,B4: b] :
( ( ( produc331601717337510060_a_b_b @ A2 @ B2 )
= ( produc331601717337510060_a_b_b @ A4 @ B4 ) )
= ( ( A2 = A4 )
& ( B2 = B4 ) ) ) ).
% old.prod.inject
thf(fact_81_prod_Oinject,axiom,
! [X1: dtree_a_b,X2: b > a,Y1: dtree_a_b,Y2: b > a] :
( ( ( produc1993688775741047735_b_b_a @ X1 @ X2 )
= ( produc1993688775741047735_b_b_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_82_prod_Oinject,axiom,
! [X1: dtree_a_b,X2: dtree_a_b,Y1: dtree_a_b,Y2: dtree_a_b] :
( ( ( produc7805419539522982029ee_a_b @ X1 @ X2 )
= ( produc7805419539522982029ee_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_83_prod_Oinject,axiom,
! [X1: dtree_a_b,X2: b,Y1: dtree_a_b,Y2: b] :
( ( ( produc331601717337510060_a_b_b @ X1 @ X2 )
= ( produc331601717337510060_a_b_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_84_empty__Collect__eq,axiom,
! [P: produc4558475209616630778_a_b_b > $o] :
( ( bot_bo3721250822024684356_a_b_b
= ( collec1368399972772960719_a_b_b @ P ) )
= ( ! [X3: produc4558475209616630778_a_b_b] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_85_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_86_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X3: b] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_87_Collect__empty__eq,axiom,
! [P: produc4558475209616630778_a_b_b > $o] :
( ( ( collec1368399972772960719_a_b_b @ P )
= bot_bo3721250822024684356_a_b_b )
= ( ! [X3: produc4558475209616630778_a_b_b] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_88_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_89_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X3: b] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_90_mem__Collect__eq,axiom,
! [A2: a > a,P: ( a > a ) > $o] :
( ( member_a_a @ A2 @ ( collect_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_91_mem__Collect__eq,axiom,
! [A2: dtree_a_b,P: dtree_a_b > $o] :
( ( member_dtree_a_b @ A2 @ ( collect_dtree_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_92_mem__Collect__eq,axiom,
! [A2: set_b,P: set_b > $o] :
( ( member_set_b @ A2 @ ( collect_set_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_93_mem__Collect__eq,axiom,
! [A2: set_Pr3012420139608375472_a_b_b,P: set_Pr3012420139608375472_a_b_b > $o] :
( ( member7431159781899395911_a_b_b @ A2 @ ( collec5997417077270831749_a_b_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_94_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_95_mem__Collect__eq,axiom,
! [A2: produc4558475209616630778_a_b_b,P: produc4558475209616630778_a_b_b > $o] :
( ( member4380921116106875537_a_b_b @ A2 @ ( collec1368399972772960719_a_b_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_96_mem__Collect__eq,axiom,
! [A2: b,P: b > $o] :
( ( member_b @ A2 @ ( collect_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_97_Collect__mem__eq,axiom,
! [A3: set_a_a] :
( ( collect_a_a
@ ^ [X3: a > a] : ( member_a_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_98_Collect__mem__eq,axiom,
! [A3: set_dtree_a_b] :
( ( collect_dtree_a_b
@ ^ [X3: dtree_a_b] : ( member_dtree_a_b @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_99_Collect__mem__eq,axiom,
! [A3: set_set_b] :
( ( collect_set_b
@ ^ [X3: set_b] : ( member_set_b @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_100_Collect__mem__eq,axiom,
! [A3: set_se3183138701204633190_a_b_b] :
( ( collec5997417077270831749_a_b_b
@ ^ [X3: set_Pr3012420139608375472_a_b_b] : ( member7431159781899395911_a_b_b @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_101_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_102_Collect__mem__eq,axiom,
! [A3: set_Pr3012420139608375472_a_b_b] :
( ( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_103_Collect__mem__eq,axiom,
! [A3: set_b] :
( ( collect_b
@ ^ [X3: b] : ( member_b @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_104_Collect__cong,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X: b] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_b @ P )
= ( collect_b @ Q ) ) ) ).
% Collect_cong
thf(fact_105_Collect__cong,axiom,
! [P: produc4558475209616630778_a_b_b > $o,Q: produc4558475209616630778_a_b_b > $o] :
( ! [X: produc4558475209616630778_a_b_b] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec1368399972772960719_a_b_b @ P )
= ( collec1368399972772960719_a_b_b @ Q ) ) ) ).
% Collect_cong
thf(fact_106_all__not__in__conv,axiom,
! [A3: set_a_a] :
( ( ! [X3: a > a] :
~ ( member_a_a @ X3 @ A3 ) )
= ( A3 = bot_bot_set_a_a ) ) ).
% all_not_in_conv
thf(fact_107_all__not__in__conv,axiom,
! [A3: set_dtree_a_b] :
( ( ! [X3: dtree_a_b] :
~ ( member_dtree_a_b @ X3 @ A3 ) )
= ( A3 = bot_bo8730652382759064772ee_a_b ) ) ).
% all_not_in_conv
thf(fact_108_all__not__in__conv,axiom,
! [A3: set_set_b] :
( ( ! [X3: set_b] :
~ ( member_set_b @ X3 @ A3 ) )
= ( A3 = bot_bot_set_set_b ) ) ).
% all_not_in_conv
thf(fact_109_all__not__in__conv,axiom,
! [A3: set_se3183138701204633190_a_b_b] :
( ( ! [X3: set_Pr3012420139608375472_a_b_b] :
~ ( member7431159781899395911_a_b_b @ X3 @ A3 ) )
= ( A3 = bot_bo2537099559385417978_a_b_b ) ) ).
% all_not_in_conv
thf(fact_110_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_111_all__not__in__conv,axiom,
! [A3: set_Pr3012420139608375472_a_b_b] :
( ( ! [X3: produc4558475209616630778_a_b_b] :
~ ( member4380921116106875537_a_b_b @ X3 @ A3 ) )
= ( A3 = bot_bo3721250822024684356_a_b_b ) ) ).
% all_not_in_conv
thf(fact_112_all__not__in__conv,axiom,
! [A3: set_b] :
( ( ! [X3: b] :
~ ( member_b @ X3 @ A3 ) )
= ( A3 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_113_empty__iff,axiom,
! [C2: a > a] :
~ ( member_a_a @ C2 @ bot_bot_set_a_a ) ).
% empty_iff
thf(fact_114_empty__iff,axiom,
! [C2: dtree_a_b] :
~ ( member_dtree_a_b @ C2 @ bot_bo8730652382759064772ee_a_b ) ).
% empty_iff
thf(fact_115_empty__iff,axiom,
! [C2: set_b] :
~ ( member_set_b @ C2 @ bot_bot_set_set_b ) ).
% empty_iff
thf(fact_116_empty__iff,axiom,
! [C2: set_Pr3012420139608375472_a_b_b] :
~ ( member7431159781899395911_a_b_b @ C2 @ bot_bo2537099559385417978_a_b_b ) ).
% empty_iff
thf(fact_117_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_118_empty__iff,axiom,
! [C2: produc4558475209616630778_a_b_b] :
~ ( member4380921116106875537_a_b_b @ C2 @ bot_bo3721250822024684356_a_b_b ) ).
% empty_iff
thf(fact_119_empty__iff,axiom,
! [C2: b] :
~ ( member_b @ C2 @ bot_bot_set_b ) ).
% empty_iff
thf(fact_120_insert__absorb2,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ( insert1613891728210272810_a_b_b @ X4 @ ( insert1613891728210272810_a_b_b @ X4 @ A3 ) )
= ( insert1613891728210272810_a_b_b @ X4 @ A3 ) ) ).
% insert_absorb2
thf(fact_121_insert__absorb2,axiom,
! [X4: dtree_a_b,A3: set_dtree_a_b] :
( ( insert_dtree_a_b @ X4 @ ( insert_dtree_a_b @ X4 @ A3 ) )
= ( insert_dtree_a_b @ X4 @ A3 ) ) ).
% insert_absorb2
thf(fact_122_insert__absorb2,axiom,
! [X4: b,A3: set_b] :
( ( insert_b @ X4 @ ( insert_b @ X4 @ A3 ) )
= ( insert_b @ X4 @ A3 ) ) ).
% insert_absorb2
thf(fact_123_insert__iff,axiom,
! [A2: a > a,B2: a > a,A3: set_a_a] :
( ( member_a_a @ A2 @ ( insert_a_a @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member_a_a @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_124_insert__iff,axiom,
! [A2: dtree_a_b,B2: dtree_a_b,A3: set_dtree_a_b] :
( ( member_dtree_a_b @ A2 @ ( insert_dtree_a_b @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member_dtree_a_b @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_125_insert__iff,axiom,
! [A2: set_b,B2: set_b,A3: set_set_b] :
( ( member_set_b @ A2 @ ( insert_set_b @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member_set_b @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_126_insert__iff,axiom,
! [A2: set_Pr3012420139608375472_a_b_b,B2: set_Pr3012420139608375472_a_b_b,A3: set_se3183138701204633190_a_b_b] :
( ( member7431159781899395911_a_b_b @ A2 @ ( insert8355695866607091424_a_b_b @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member7431159781899395911_a_b_b @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_127_insert__iff,axiom,
! [A2: a,B2: a,A3: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member_a @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_128_insert__iff,axiom,
! [A2: produc4558475209616630778_a_b_b,B2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ A2 @ ( insert1613891728210272810_a_b_b @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member4380921116106875537_a_b_b @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_129_insert__iff,axiom,
! [A2: b,B2: b,A3: set_b] :
( ( member_b @ A2 @ ( insert_b @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member_b @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_130_insertCI,axiom,
! [A2: a > a,B3: set_a_a,B2: a > a] :
( ( ~ ( member_a_a @ A2 @ B3 )
=> ( A2 = B2 ) )
=> ( member_a_a @ A2 @ ( insert_a_a @ B2 @ B3 ) ) ) ).
% insertCI
thf(fact_131_insertCI,axiom,
! [A2: dtree_a_b,B3: set_dtree_a_b,B2: dtree_a_b] :
( ( ~ ( member_dtree_a_b @ A2 @ B3 )
=> ( A2 = B2 ) )
=> ( member_dtree_a_b @ A2 @ ( insert_dtree_a_b @ B2 @ B3 ) ) ) ).
% insertCI
thf(fact_132_insertCI,axiom,
! [A2: set_b,B3: set_set_b,B2: set_b] :
( ( ~ ( member_set_b @ A2 @ B3 )
=> ( A2 = B2 ) )
=> ( member_set_b @ A2 @ ( insert_set_b @ B2 @ B3 ) ) ) ).
% insertCI
thf(fact_133_insertCI,axiom,
! [A2: set_Pr3012420139608375472_a_b_b,B3: set_se3183138701204633190_a_b_b,B2: set_Pr3012420139608375472_a_b_b] :
( ( ~ ( member7431159781899395911_a_b_b @ A2 @ B3 )
=> ( A2 = B2 ) )
=> ( member7431159781899395911_a_b_b @ A2 @ ( insert8355695866607091424_a_b_b @ B2 @ B3 ) ) ) ).
% insertCI
thf(fact_134_insertCI,axiom,
! [A2: a,B3: set_a,B2: a] :
( ( ~ ( member_a @ A2 @ B3 )
=> ( A2 = B2 ) )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B3 ) ) ) ).
% insertCI
thf(fact_135_insertCI,axiom,
! [A2: produc4558475209616630778_a_b_b,B3: set_Pr3012420139608375472_a_b_b,B2: produc4558475209616630778_a_b_b] :
( ( ~ ( member4380921116106875537_a_b_b @ A2 @ B3 )
=> ( A2 = B2 ) )
=> ( member4380921116106875537_a_b_b @ A2 @ ( insert1613891728210272810_a_b_b @ B2 @ B3 ) ) ) ).
% insertCI
thf(fact_136_insertCI,axiom,
! [A2: b,B3: set_b,B2: b] :
( ( ~ ( member_b @ A2 @ B3 )
=> ( A2 = B2 ) )
=> ( member_b @ A2 @ ( insert_b @ B2 @ B3 ) ) ) ).
% insertCI
thf(fact_137_sup__apply,axiom,
( sup_su4209747780764569001_b_b_o
= ( ^ [F2: produc4558475209616630778_a_b_b > $o,G: produc4558475209616630778_a_b_b > $o,X3: produc4558475209616630778_a_b_b] : ( sup_sup_o @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% sup_apply
thf(fact_138_sup__apply,axiom,
( sup_su6709851091347060739_b_b_o
= ( ^ [F2: dtree_a_b > b > $o,G: dtree_a_b > b > $o,X3: dtree_a_b] : ( sup_sup_b_o @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% sup_apply
thf(fact_139_sup__apply,axiom,
( sup_sup_b_o
= ( ^ [F2: b > $o,G: b > $o,X3: b] : ( sup_sup_o @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% sup_apply
thf(fact_140_sup_Oright__idem,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( sup_sup_set_set_b @ ( sup_sup_set_set_b @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_set_b @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_141_sup_Oright__idem,axiom,
! [A2: nat,B2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_142_sup_Oright__idem,axiom,
! [A2: produc4558475209616630778_a_b_b > $o,B2: produc4558475209616630778_a_b_b > $o] :
( ( sup_su4209747780764569001_b_b_o @ ( sup_su4209747780764569001_b_b_o @ A2 @ B2 ) @ B2 )
= ( sup_su4209747780764569001_b_b_o @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_143_sup_Oright__idem,axiom,
! [A2: dtree_a_b > b > $o,B2: dtree_a_b > b > $o] :
( ( sup_su6709851091347060739_b_b_o @ ( sup_su6709851091347060739_b_b_o @ A2 @ B2 ) @ B2 )
= ( sup_su6709851091347060739_b_b_o @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_144_sup_Oright__idem,axiom,
! [A2: b > $o,B2: b > $o] :
( ( sup_sup_b_o @ ( sup_sup_b_o @ A2 @ B2 ) @ B2 )
= ( sup_sup_b_o @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_145_sup_Oright__idem,axiom,
! [A2: set_b,B2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_b @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_146_sup__left__idem,axiom,
! [X4: set_set_b,Y: set_set_b] :
( ( sup_sup_set_set_b @ X4 @ ( sup_sup_set_set_b @ X4 @ Y ) )
= ( sup_sup_set_set_b @ X4 @ Y ) ) ).
% sup_left_idem
thf(fact_147_sup__left__idem,axiom,
! [X4: nat,Y: nat] :
( ( sup_sup_nat @ X4 @ ( sup_sup_nat @ X4 @ Y ) )
= ( sup_sup_nat @ X4 @ Y ) ) ).
% sup_left_idem
thf(fact_148_sup__left__idem,axiom,
! [X4: produc4558475209616630778_a_b_b > $o,Y: produc4558475209616630778_a_b_b > $o] :
( ( sup_su4209747780764569001_b_b_o @ X4 @ ( sup_su4209747780764569001_b_b_o @ X4 @ Y ) )
= ( sup_su4209747780764569001_b_b_o @ X4 @ Y ) ) ).
% sup_left_idem
thf(fact_149_sup__left__idem,axiom,
! [X4: dtree_a_b > b > $o,Y: dtree_a_b > b > $o] :
( ( sup_su6709851091347060739_b_b_o @ X4 @ ( sup_su6709851091347060739_b_b_o @ X4 @ Y ) )
= ( sup_su6709851091347060739_b_b_o @ X4 @ Y ) ) ).
% sup_left_idem
thf(fact_150_sup__left__idem,axiom,
! [X4: b > $o,Y: b > $o] :
( ( sup_sup_b_o @ X4 @ ( sup_sup_b_o @ X4 @ Y ) )
= ( sup_sup_b_o @ X4 @ Y ) ) ).
% sup_left_idem
thf(fact_151_sup__left__idem,axiom,
! [X4: set_b,Y: set_b] :
( ( sup_sup_set_b @ X4 @ ( sup_sup_set_b @ X4 @ Y ) )
= ( sup_sup_set_b @ X4 @ Y ) ) ).
% sup_left_idem
thf(fact_152_sup_Oleft__idem,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( sup_sup_set_set_b @ A2 @ ( sup_sup_set_set_b @ A2 @ B2 ) )
= ( sup_sup_set_set_b @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_153_sup_Oleft__idem,axiom,
! [A2: nat,B2: nat] :
( ( sup_sup_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) )
= ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_154_sup_Oleft__idem,axiom,
! [A2: produc4558475209616630778_a_b_b > $o,B2: produc4558475209616630778_a_b_b > $o] :
( ( sup_su4209747780764569001_b_b_o @ A2 @ ( sup_su4209747780764569001_b_b_o @ A2 @ B2 ) )
= ( sup_su4209747780764569001_b_b_o @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_155_sup_Oleft__idem,axiom,
! [A2: dtree_a_b > b > $o,B2: dtree_a_b > b > $o] :
( ( sup_su6709851091347060739_b_b_o @ A2 @ ( sup_su6709851091347060739_b_b_o @ A2 @ B2 ) )
= ( sup_su6709851091347060739_b_b_o @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_156_sup_Oleft__idem,axiom,
! [A2: b > $o,B2: b > $o] :
( ( sup_sup_b_o @ A2 @ ( sup_sup_b_o @ A2 @ B2 ) )
= ( sup_sup_b_o @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_157_sup_Oleft__idem,axiom,
! [A2: set_b,B2: set_b] :
( ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ A2 @ B2 ) )
= ( sup_sup_set_b @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_158_sup__idem,axiom,
! [X4: set_set_b] :
( ( sup_sup_set_set_b @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_159_sup__idem,axiom,
! [X4: nat] :
( ( sup_sup_nat @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_160_sup__idem,axiom,
! [X4: produc4558475209616630778_a_b_b > $o] :
( ( sup_su4209747780764569001_b_b_o @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_161_sup__idem,axiom,
! [X4: dtree_a_b > b > $o] :
( ( sup_su6709851091347060739_b_b_o @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_162_sup__idem,axiom,
! [X4: b > $o] :
( ( sup_sup_b_o @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_163_sup__idem,axiom,
! [X4: set_b] :
( ( sup_sup_set_b @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_164_sup_Oidem,axiom,
! [A2: set_set_b] :
( ( sup_sup_set_set_b @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_165_sup_Oidem,axiom,
! [A2: nat] :
( ( sup_sup_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_166_sup_Oidem,axiom,
! [A2: produc4558475209616630778_a_b_b > $o] :
( ( sup_su4209747780764569001_b_b_o @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_167_sup_Oidem,axiom,
! [A2: dtree_a_b > b > $o] :
( ( sup_su6709851091347060739_b_b_o @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_168_sup_Oidem,axiom,
! [A2: b > $o] :
( ( sup_sup_b_o @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_169_sup_Oidem,axiom,
! [A2: set_b] :
( ( sup_sup_set_b @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_170_Un__iff,axiom,
! [C2: a > a,A3: set_a_a,B3: set_a_a] :
( ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A3 @ B3 ) )
= ( ( member_a_a @ C2 @ A3 )
| ( member_a_a @ C2 @ B3 ) ) ) ).
% Un_iff
thf(fact_171_Un__iff,axiom,
! [C2: dtree_a_b,A3: set_dtree_a_b,B3: set_dtree_a_b] :
( ( member_dtree_a_b @ C2 @ ( sup_su8994539500306794332ee_a_b @ A3 @ B3 ) )
= ( ( member_dtree_a_b @ C2 @ A3 )
| ( member_dtree_a_b @ C2 @ B3 ) ) ) ).
% Un_iff
thf(fact_172_Un__iff,axiom,
! [C2: set_Pr3012420139608375472_a_b_b,A3: set_se3183138701204633190_a_b_b,B3: set_se3183138701204633190_a_b_b] :
( ( member7431159781899395911_a_b_b @ C2 @ ( sup_su5350426443513267090_a_b_b @ A3 @ B3 ) )
= ( ( member7431159781899395911_a_b_b @ C2 @ A3 )
| ( member7431159781899395911_a_b_b @ C2 @ B3 ) ) ) ).
% Un_iff
thf(fact_173_Un__iff,axiom,
! [C2: a,A3: set_a,B3: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) )
= ( ( member_a @ C2 @ A3 )
| ( member_a @ C2 @ B3 ) ) ) ).
% Un_iff
thf(fact_174_Un__iff,axiom,
! [C2: set_b,A3: set_set_b,B3: set_set_b] :
( ( member_set_b @ C2 @ ( sup_sup_set_set_b @ A3 @ B3 ) )
= ( ( member_set_b @ C2 @ A3 )
| ( member_set_b @ C2 @ B3 ) ) ) ).
% Un_iff
thf(fact_175_Un__iff,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ C2 @ ( sup_su2887895092731772380_a_b_b @ A3 @ B3 ) )
= ( ( member4380921116106875537_a_b_b @ C2 @ A3 )
| ( member4380921116106875537_a_b_b @ C2 @ B3 ) ) ) ).
% Un_iff
thf(fact_176_Un__iff,axiom,
! [C2: b,A3: set_b,B3: set_b] :
( ( member_b @ C2 @ ( sup_sup_set_b @ A3 @ B3 ) )
= ( ( member_b @ C2 @ A3 )
| ( member_b @ C2 @ B3 ) ) ) ).
% Un_iff
thf(fact_177_UnCI,axiom,
! [C2: a > a,B3: set_a_a,A3: set_a_a] :
( ( ~ ( member_a_a @ C2 @ B3 )
=> ( member_a_a @ C2 @ A3 ) )
=> ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_178_UnCI,axiom,
! [C2: dtree_a_b,B3: set_dtree_a_b,A3: set_dtree_a_b] :
( ( ~ ( member_dtree_a_b @ C2 @ B3 )
=> ( member_dtree_a_b @ C2 @ A3 ) )
=> ( member_dtree_a_b @ C2 @ ( sup_su8994539500306794332ee_a_b @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_179_UnCI,axiom,
! [C2: set_Pr3012420139608375472_a_b_b,B3: set_se3183138701204633190_a_b_b,A3: set_se3183138701204633190_a_b_b] :
( ( ~ ( member7431159781899395911_a_b_b @ C2 @ B3 )
=> ( member7431159781899395911_a_b_b @ C2 @ A3 ) )
=> ( member7431159781899395911_a_b_b @ C2 @ ( sup_su5350426443513267090_a_b_b @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_180_UnCI,axiom,
! [C2: a,B3: set_a,A3: set_a] :
( ( ~ ( member_a @ C2 @ B3 )
=> ( member_a @ C2 @ A3 ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_181_UnCI,axiom,
! [C2: set_b,B3: set_set_b,A3: set_set_b] :
( ( ~ ( member_set_b @ C2 @ B3 )
=> ( member_set_b @ C2 @ A3 ) )
=> ( member_set_b @ C2 @ ( sup_sup_set_set_b @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_182_UnCI,axiom,
! [C2: produc4558475209616630778_a_b_b,B3: set_Pr3012420139608375472_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ( ~ ( member4380921116106875537_a_b_b @ C2 @ B3 )
=> ( member4380921116106875537_a_b_b @ C2 @ A3 ) )
=> ( member4380921116106875537_a_b_b @ C2 @ ( sup_su2887895092731772380_a_b_b @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_183_UnCI,axiom,
! [C2: b,B3: set_b,A3: set_b] :
( ( ~ ( member_b @ C2 @ B3 )
=> ( member_b @ C2 @ A3 ) )
=> ( member_b @ C2 @ ( sup_sup_set_b @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_184_case__prodI2,axiom,
! [P2: produc1083523234014712191_b_b_a,C2: dtree_a_b > ( b > a ) > $o] :
( ! [A5: dtree_a_b,B5: b > a] :
( ( P2
= ( produc1993688775741047735_b_b_a @ A5 @ B5 ) )
=> ( C2 @ A5 @ B5 ) )
=> ( produc1457303364454389452_b_a_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_185_case__prodI2,axiom,
! [P2: produc5177672665255943253ee_a_b,C2: dtree_a_b > dtree_a_b > $o] :
( ! [A5: dtree_a_b,B5: dtree_a_b] :
( ( P2
= ( produc7805419539522982029ee_a_b @ A5 @ B5 ) )
=> ( C2 @ A5 @ B5 ) )
=> ( produc3512921791960644726_a_b_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_186_case__prodI2,axiom,
! [P2: produc4558475209616630778_a_b_b,C2: dtree_a_b > b > $o] :
( ! [A5: dtree_a_b,B5: b] :
( ( P2
= ( produc331601717337510060_a_b_b @ A5 @ B5 ) )
=> ( C2 @ A5 @ B5 ) )
=> ( produc1325217093046185599_b_b_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_187_case__prodI,axiom,
! [F: dtree_a_b > ( b > a ) > $o,A2: dtree_a_b,B2: b > a] :
( ( F @ A2 @ B2 )
=> ( produc1457303364454389452_b_a_o @ F @ ( produc1993688775741047735_b_b_a @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_188_case__prodI,axiom,
! [F: dtree_a_b > dtree_a_b > $o,A2: dtree_a_b,B2: dtree_a_b] :
( ( F @ A2 @ B2 )
=> ( produc3512921791960644726_a_b_o @ F @ ( produc7805419539522982029ee_a_b @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_189_case__prodI,axiom,
! [F: dtree_a_b > b > $o,A2: dtree_a_b,B2: b] :
( ( F @ A2 @ B2 )
=> ( produc1325217093046185599_b_b_o @ F @ ( produc331601717337510060_a_b_b @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_190_mem__case__prodI2,axiom,
! [P2: produc4558475209616630778_a_b_b,Z2: b,C2: dtree_a_b > b > set_b] :
( ! [A5: dtree_a_b,B5: b] :
( ( P2
= ( produc331601717337510060_a_b_b @ A5 @ B5 ) )
=> ( member_b @ Z2 @ ( C2 @ A5 @ B5 ) ) )
=> ( member_b @ Z2 @ ( produc5617419908695543622_set_b @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_191_mem__case__prodI2,axiom,
! [P2: produc4558475209616630778_a_b_b,Z2: produc4558475209616630778_a_b_b,C2: dtree_a_b > b > set_Pr3012420139608375472_a_b_b] :
( ! [A5: dtree_a_b,B5: b] :
( ( P2
= ( produc331601717337510060_a_b_b @ A5 @ B5 ) )
=> ( member4380921116106875537_a_b_b @ Z2 @ ( C2 @ A5 @ B5 ) ) )
=> ( member4380921116106875537_a_b_b @ Z2 @ ( produc2559767781221151625_a_b_b @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_192_mem__case__prodI2,axiom,
! [P2: produc4558475209616630778_a_b_b,Z2: a,C2: dtree_a_b > b > set_a] :
( ! [A5: dtree_a_b,B5: b] :
( ( P2
= ( produc331601717337510060_a_b_b @ A5 @ B5 ) )
=> ( member_a @ Z2 @ ( C2 @ A5 @ B5 ) ) )
=> ( member_a @ Z2 @ ( produc5617419904392314821_set_a @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_193_mem__case__prodI2,axiom,
! [P2: produc4558475209616630778_a_b_b,Z2: set_b,C2: dtree_a_b > b > set_set_b] :
( ! [A5: dtree_a_b,B5: b] :
( ( P2
= ( produc331601717337510060_a_b_b @ A5 @ B5 ) )
=> ( member_set_b @ Z2 @ ( C2 @ A5 @ B5 ) ) )
=> ( member_set_b @ Z2 @ ( produc3243052509580634150_set_b @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_194_mem__case__prodI2,axiom,
! [P2: produc4558475209616630778_a_b_b,Z2: a > a,C2: dtree_a_b > b > set_a_a] :
( ! [A5: dtree_a_b,B5: b] :
( ( P2
= ( produc331601717337510060_a_b_b @ A5 @ B5 ) )
=> ( member_a_a @ Z2 @ ( C2 @ A5 @ B5 ) ) )
=> ( member_a_a @ Z2 @ ( produc3604202691690672244et_a_a @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_195_mem__case__prodI2,axiom,
! [P2: produc4558475209616630778_a_b_b,Z2: dtree_a_b,C2: dtree_a_b > b > set_dtree_a_b] :
( ! [A5: dtree_a_b,B5: b] :
( ( P2
= ( produc331601717337510060_a_b_b @ A5 @ B5 ) )
=> ( member_dtree_a_b @ Z2 @ ( C2 @ A5 @ B5 ) ) )
=> ( member_dtree_a_b @ Z2 @ ( produc2854446210517299721ee_a_b @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_196_mem__case__prodI2,axiom,
! [P2: produc1083523234014712191_b_b_a,Z2: b,C2: dtree_a_b > ( b > a ) > set_b] :
( ! [A5: dtree_a_b,B5: b > a] :
( ( P2
= ( produc1993688775741047735_b_b_a @ A5 @ B5 ) )
=> ( member_b @ Z2 @ ( C2 @ A5 @ B5 ) ) )
=> ( member_b @ Z2 @ ( produc8900335307428211987_set_b @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_197_mem__case__prodI2,axiom,
! [P2: produc1083523234014712191_b_b_a,Z2: a,C2: dtree_a_b > ( b > a ) > set_a] :
( ! [A5: dtree_a_b,B5: b > a] :
( ( P2
= ( produc1993688775741047735_b_b_a @ A5 @ B5 ) )
=> ( member_a @ Z2 @ ( C2 @ A5 @ B5 ) ) )
=> ( member_a @ Z2 @ ( produc8900335303124983186_set_a @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_198_mem__case__prodI2,axiom,
! [P2: produc5177672665255943253ee_a_b,Z2: b,C2: dtree_a_b > dtree_a_b > set_b] :
( ! [A5: dtree_a_b,B5: dtree_a_b] :
( ( P2
= ( produc7805419539522982029ee_a_b @ A5 @ B5 ) )
=> ( member_b @ Z2 @ ( C2 @ A5 @ B5 ) ) )
=> ( member_b @ Z2 @ ( produc5055721716539340221_set_b @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_199_mem__case__prodI2,axiom,
! [P2: produc5177672665255943253ee_a_b,Z2: a,C2: dtree_a_b > dtree_a_b > set_a] :
( ! [A5: dtree_a_b,B5: dtree_a_b] :
( ( P2
= ( produc7805419539522982029ee_a_b @ A5 @ B5 ) )
=> ( member_a @ Z2 @ ( C2 @ A5 @ B5 ) ) )
=> ( member_a @ Z2 @ ( produc5055721712236111420_set_a @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_200_mem__case__prodI,axiom,
! [Z2: b,C2: dtree_a_b > b > set_b,A2: dtree_a_b,B2: b] :
( ( member_b @ Z2 @ ( C2 @ A2 @ B2 ) )
=> ( member_b @ Z2 @ ( produc5617419908695543622_set_b @ C2 @ ( produc331601717337510060_a_b_b @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_201_mem__case__prodI,axiom,
! [Z2: produc4558475209616630778_a_b_b,C2: dtree_a_b > b > set_Pr3012420139608375472_a_b_b,A2: dtree_a_b,B2: b] :
( ( member4380921116106875537_a_b_b @ Z2 @ ( C2 @ A2 @ B2 ) )
=> ( member4380921116106875537_a_b_b @ Z2 @ ( produc2559767781221151625_a_b_b @ C2 @ ( produc331601717337510060_a_b_b @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_202_mem__case__prodI,axiom,
! [Z2: a,C2: dtree_a_b > b > set_a,A2: dtree_a_b,B2: b] :
( ( member_a @ Z2 @ ( C2 @ A2 @ B2 ) )
=> ( member_a @ Z2 @ ( produc5617419904392314821_set_a @ C2 @ ( produc331601717337510060_a_b_b @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_203_mem__case__prodI,axiom,
! [Z2: set_b,C2: dtree_a_b > b > set_set_b,A2: dtree_a_b,B2: b] :
( ( member_set_b @ Z2 @ ( C2 @ A2 @ B2 ) )
=> ( member_set_b @ Z2 @ ( produc3243052509580634150_set_b @ C2 @ ( produc331601717337510060_a_b_b @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_204_mem__case__prodI,axiom,
! [Z2: a > a,C2: dtree_a_b > b > set_a_a,A2: dtree_a_b,B2: b] :
( ( member_a_a @ Z2 @ ( C2 @ A2 @ B2 ) )
=> ( member_a_a @ Z2 @ ( produc3604202691690672244et_a_a @ C2 @ ( produc331601717337510060_a_b_b @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_205_mem__case__prodI,axiom,
! [Z2: dtree_a_b,C2: dtree_a_b > b > set_dtree_a_b,A2: dtree_a_b,B2: b] :
( ( member_dtree_a_b @ Z2 @ ( C2 @ A2 @ B2 ) )
=> ( member_dtree_a_b @ Z2 @ ( produc2854446210517299721ee_a_b @ C2 @ ( produc331601717337510060_a_b_b @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_206_mem__case__prodI,axiom,
! [Z2: b,C2: dtree_a_b > ( b > a ) > set_b,A2: dtree_a_b,B2: b > a] :
( ( member_b @ Z2 @ ( C2 @ A2 @ B2 ) )
=> ( member_b @ Z2 @ ( produc8900335307428211987_set_b @ C2 @ ( produc1993688775741047735_b_b_a @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_207_mem__case__prodI,axiom,
! [Z2: a,C2: dtree_a_b > ( b > a ) > set_a,A2: dtree_a_b,B2: b > a] :
( ( member_a @ Z2 @ ( C2 @ A2 @ B2 ) )
=> ( member_a @ Z2 @ ( produc8900335303124983186_set_a @ C2 @ ( produc1993688775741047735_b_b_a @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_208_mem__case__prodI,axiom,
! [Z2: b,C2: dtree_a_b > dtree_a_b > set_b,A2: dtree_a_b,B2: dtree_a_b] :
( ( member_b @ Z2 @ ( C2 @ A2 @ B2 ) )
=> ( member_b @ Z2 @ ( produc5055721716539340221_set_b @ C2 @ ( produc7805419539522982029ee_a_b @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_209_mem__case__prodI,axiom,
! [Z2: a,C2: dtree_a_b > dtree_a_b > set_a,A2: dtree_a_b,B2: dtree_a_b] :
( ( member_a @ Z2 @ ( C2 @ A2 @ B2 ) )
=> ( member_a @ Z2 @ ( produc5055721712236111420_set_a @ C2 @ ( produc7805419539522982029ee_a_b @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_210_sup__bot_Oright__neutral,axiom,
! [A2: set_set_b] :
( ( sup_sup_set_set_b @ A2 @ bot_bot_set_set_b )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_211_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_212_sup__bot_Oright__neutral,axiom,
! [A2: produc4558475209616630778_a_b_b > $o] :
( ( sup_su4209747780764569001_b_b_o @ A2 @ bot_bo7321339186913516097_b_b_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_213_sup__bot_Oright__neutral,axiom,
! [A2: dtree_a_b > b > $o] :
( ( sup_su6709851091347060739_b_b_o @ A2 @ bot_bo471016548657204587_b_b_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_214_sup__bot_Oright__neutral,axiom,
! [A2: b > $o] :
( ( sup_sup_b_o @ A2 @ bot_bot_b_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_215_sup__bot_Oright__neutral,axiom,
! [A2: set_b] :
( ( sup_sup_set_b @ A2 @ bot_bot_set_b )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_216_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( bot_bot_set_set_b
= ( sup_sup_set_set_b @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_set_b )
& ( B2 = bot_bot_set_set_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_217_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_218_sup__bot_Oneutr__eq__iff,axiom,
! [A2: produc4558475209616630778_a_b_b > $o,B2: produc4558475209616630778_a_b_b > $o] :
( ( bot_bo7321339186913516097_b_b_o
= ( sup_su4209747780764569001_b_b_o @ A2 @ B2 ) )
= ( ( A2 = bot_bo7321339186913516097_b_b_o )
& ( B2 = bot_bo7321339186913516097_b_b_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_219_sup__bot_Oneutr__eq__iff,axiom,
! [A2: dtree_a_b > b > $o,B2: dtree_a_b > b > $o] :
( ( bot_bo471016548657204587_b_b_o
= ( sup_su6709851091347060739_b_b_o @ A2 @ B2 ) )
= ( ( A2 = bot_bo471016548657204587_b_b_o )
& ( B2 = bot_bo471016548657204587_b_b_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_220_sup__bot_Oneutr__eq__iff,axiom,
! [A2: b > $o,B2: b > $o] :
( ( bot_bot_b_o
= ( sup_sup_b_o @ A2 @ B2 ) )
= ( ( A2 = bot_bot_b_o )
& ( B2 = bot_bot_b_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_221_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_b,B2: set_b] :
( ( bot_bot_set_b
= ( sup_sup_set_b @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_b )
& ( B2 = bot_bot_set_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_222_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_b] :
( ( sup_sup_set_set_b @ bot_bot_set_set_b @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_223_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_224_sup__bot_Oleft__neutral,axiom,
! [A2: produc4558475209616630778_a_b_b > $o] :
( ( sup_su4209747780764569001_b_b_o @ bot_bo7321339186913516097_b_b_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_225_sup__bot_Oleft__neutral,axiom,
! [A2: dtree_a_b > b > $o] :
( ( sup_su6709851091347060739_b_b_o @ bot_bo471016548657204587_b_b_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_226_sup__bot_Oleft__neutral,axiom,
! [A2: b > $o] :
( ( sup_sup_b_o @ bot_bot_b_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_227_sup__bot_Oleft__neutral,axiom,
! [A2: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_228_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( ( sup_sup_set_set_b @ A2 @ B2 )
= bot_bot_set_set_b )
= ( ( A2 = bot_bot_set_set_b )
& ( B2 = bot_bot_set_set_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_229_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_230_sup__bot_Oeq__neutr__iff,axiom,
! [A2: produc4558475209616630778_a_b_b > $o,B2: produc4558475209616630778_a_b_b > $o] :
( ( ( sup_su4209747780764569001_b_b_o @ A2 @ B2 )
= bot_bo7321339186913516097_b_b_o )
= ( ( A2 = bot_bo7321339186913516097_b_b_o )
& ( B2 = bot_bo7321339186913516097_b_b_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_231_sup__bot_Oeq__neutr__iff,axiom,
! [A2: dtree_a_b > b > $o,B2: dtree_a_b > b > $o] :
( ( ( sup_su6709851091347060739_b_b_o @ A2 @ B2 )
= bot_bo471016548657204587_b_b_o )
= ( ( A2 = bot_bo471016548657204587_b_b_o )
& ( B2 = bot_bo471016548657204587_b_b_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_232_sup__bot_Oeq__neutr__iff,axiom,
! [A2: b > $o,B2: b > $o] :
( ( ( sup_sup_b_o @ A2 @ B2 )
= bot_bot_b_o )
= ( ( A2 = bot_bot_b_o )
& ( B2 = bot_bot_b_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_233_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_b,B2: set_b] :
( ( ( sup_sup_set_b @ A2 @ B2 )
= bot_bot_set_b )
= ( ( A2 = bot_bot_set_b )
& ( B2 = bot_bot_set_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_234_sup__eq__bot__iff,axiom,
! [X4: set_set_b,Y: set_set_b] :
( ( ( sup_sup_set_set_b @ X4 @ Y )
= bot_bot_set_set_b )
= ( ( X4 = bot_bot_set_set_b )
& ( Y = bot_bot_set_set_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_235_sup__eq__bot__iff,axiom,
! [X4: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X4 @ Y )
= bot_bot_set_a )
= ( ( X4 = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_236_sup__eq__bot__iff,axiom,
! [X4: produc4558475209616630778_a_b_b > $o,Y: produc4558475209616630778_a_b_b > $o] :
( ( ( sup_su4209747780764569001_b_b_o @ X4 @ Y )
= bot_bo7321339186913516097_b_b_o )
= ( ( X4 = bot_bo7321339186913516097_b_b_o )
& ( Y = bot_bo7321339186913516097_b_b_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_237_sup__eq__bot__iff,axiom,
! [X4: dtree_a_b > b > $o,Y: dtree_a_b > b > $o] :
( ( ( sup_su6709851091347060739_b_b_o @ X4 @ Y )
= bot_bo471016548657204587_b_b_o )
= ( ( X4 = bot_bo471016548657204587_b_b_o )
& ( Y = bot_bo471016548657204587_b_b_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_238_sup__eq__bot__iff,axiom,
! [X4: b > $o,Y: b > $o] :
( ( ( sup_sup_b_o @ X4 @ Y )
= bot_bot_b_o )
= ( ( X4 = bot_bot_b_o )
& ( Y = bot_bot_b_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_239_sup__eq__bot__iff,axiom,
! [X4: set_b,Y: set_b] :
( ( ( sup_sup_set_b @ X4 @ Y )
= bot_bot_set_b )
= ( ( X4 = bot_bot_set_b )
& ( Y = bot_bot_set_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_240_case__prodI2_H,axiom,
! [P2: produc4558475209616630778_a_b_b,C2: dtree_a_b > b > a > $o,X4: a] :
( ! [A5: dtree_a_b,B5: b] :
( ( ( produc331601717337510060_a_b_b @ A5 @ B5 )
= P2 )
=> ( C2 @ A5 @ B5 @ X4 ) )
=> ( produc6139810021161713496_b_a_o @ C2 @ P2 @ X4 ) ) ).
% case_prodI2'
thf(fact_241__C4_C,axiom,
~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ x2 ) @ ( insert_b @ e3 @ bot_bot_set_b ) ) ) ).
% "4"
thf(fact_242_insert_Ohyps,axiom,
~ ( fmembe3173364709796808819_a_b_b @ x @ xsa ) ).
% insert.hyps
thf(fact_243_bot__set__def,axiom,
( bot_bo3721250822024684356_a_b_b
= ( collec1368399972772960719_a_b_b @ bot_bo7321339186913516097_b_b_o ) ) ).
% bot_set_def
thf(fact_244_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_245_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_246_mem__case__prodE,axiom,
! [Z2: b,C2: dtree_a_b > b > set_b,P2: produc4558475209616630778_a_b_b] :
( ( member_b @ Z2 @ ( produc5617419908695543622_set_b @ C2 @ P2 ) )
=> ~ ! [X: dtree_a_b,Y4: b] :
( ( P2
= ( produc331601717337510060_a_b_b @ X @ Y4 ) )
=> ~ ( member_b @ Z2 @ ( C2 @ X @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_247_mem__case__prodE,axiom,
! [Z2: produc4558475209616630778_a_b_b,C2: dtree_a_b > b > set_Pr3012420139608375472_a_b_b,P2: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z2 @ ( produc2559767781221151625_a_b_b @ C2 @ P2 ) )
=> ~ ! [X: dtree_a_b,Y4: b] :
( ( P2
= ( produc331601717337510060_a_b_b @ X @ Y4 ) )
=> ~ ( member4380921116106875537_a_b_b @ Z2 @ ( C2 @ X @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_248_mem__case__prodE,axiom,
! [Z2: a,C2: dtree_a_b > b > set_a,P2: produc4558475209616630778_a_b_b] :
( ( member_a @ Z2 @ ( produc5617419904392314821_set_a @ C2 @ P2 ) )
=> ~ ! [X: dtree_a_b,Y4: b] :
( ( P2
= ( produc331601717337510060_a_b_b @ X @ Y4 ) )
=> ~ ( member_a @ Z2 @ ( C2 @ X @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_249_mem__case__prodE,axiom,
! [Z2: set_b,C2: dtree_a_b > b > set_set_b,P2: produc4558475209616630778_a_b_b] :
( ( member_set_b @ Z2 @ ( produc3243052509580634150_set_b @ C2 @ P2 ) )
=> ~ ! [X: dtree_a_b,Y4: b] :
( ( P2
= ( produc331601717337510060_a_b_b @ X @ Y4 ) )
=> ~ ( member_set_b @ Z2 @ ( C2 @ X @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_250_mem__case__prodE,axiom,
! [Z2: a > a,C2: dtree_a_b > b > set_a_a,P2: produc4558475209616630778_a_b_b] :
( ( member_a_a @ Z2 @ ( produc3604202691690672244et_a_a @ C2 @ P2 ) )
=> ~ ! [X: dtree_a_b,Y4: b] :
( ( P2
= ( produc331601717337510060_a_b_b @ X @ Y4 ) )
=> ~ ( member_a_a @ Z2 @ ( C2 @ X @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_251_mem__case__prodE,axiom,
! [Z2: dtree_a_b,C2: dtree_a_b > b > set_dtree_a_b,P2: produc4558475209616630778_a_b_b] :
( ( member_dtree_a_b @ Z2 @ ( produc2854446210517299721ee_a_b @ C2 @ P2 ) )
=> ~ ! [X: dtree_a_b,Y4: b] :
( ( P2
= ( produc331601717337510060_a_b_b @ X @ Y4 ) )
=> ~ ( member_dtree_a_b @ Z2 @ ( C2 @ X @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_252_mem__case__prodE,axiom,
! [Z2: b,C2: dtree_a_b > ( b > a ) > set_b,P2: produc1083523234014712191_b_b_a] :
( ( member_b @ Z2 @ ( produc8900335307428211987_set_b @ C2 @ P2 ) )
=> ~ ! [X: dtree_a_b,Y4: b > a] :
( ( P2
= ( produc1993688775741047735_b_b_a @ X @ Y4 ) )
=> ~ ( member_b @ Z2 @ ( C2 @ X @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_253_mem__case__prodE,axiom,
! [Z2: a,C2: dtree_a_b > ( b > a ) > set_a,P2: produc1083523234014712191_b_b_a] :
( ( member_a @ Z2 @ ( produc8900335303124983186_set_a @ C2 @ P2 ) )
=> ~ ! [X: dtree_a_b,Y4: b > a] :
( ( P2
= ( produc1993688775741047735_b_b_a @ X @ Y4 ) )
=> ~ ( member_a @ Z2 @ ( C2 @ X @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_254_mem__case__prodE,axiom,
! [Z2: b,C2: dtree_a_b > dtree_a_b > set_b,P2: produc5177672665255943253ee_a_b] :
( ( member_b @ Z2 @ ( produc5055721716539340221_set_b @ C2 @ P2 ) )
=> ~ ! [X: dtree_a_b,Y4: dtree_a_b] :
( ( P2
= ( produc7805419539522982029ee_a_b @ X @ Y4 ) )
=> ~ ( member_b @ Z2 @ ( C2 @ X @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_255_mem__case__prodE,axiom,
! [Z2: a,C2: dtree_a_b > dtree_a_b > set_a,P2: produc5177672665255943253ee_a_b] :
( ( member_a @ Z2 @ ( produc5055721712236111420_set_a @ C2 @ P2 ) )
=> ~ ! [X: dtree_a_b,Y4: dtree_a_b] :
( ( P2
= ( produc7805419539522982029ee_a_b @ X @ Y4 ) )
=> ~ ( member_a @ Z2 @ ( C2 @ X @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_256_case__prodE,axiom,
! [C2: dtree_a_b > ( b > a ) > $o,P2: produc1083523234014712191_b_b_a] :
( ( produc1457303364454389452_b_a_o @ C2 @ P2 )
=> ~ ! [X: dtree_a_b,Y4: b > a] :
( ( P2
= ( produc1993688775741047735_b_b_a @ X @ Y4 ) )
=> ~ ( C2 @ X @ Y4 ) ) ) ).
% case_prodE
thf(fact_257_case__prodE,axiom,
! [C2: dtree_a_b > dtree_a_b > $o,P2: produc5177672665255943253ee_a_b] :
( ( produc3512921791960644726_a_b_o @ C2 @ P2 )
=> ~ ! [X: dtree_a_b,Y4: dtree_a_b] :
( ( P2
= ( produc7805419539522982029ee_a_b @ X @ Y4 ) )
=> ~ ( C2 @ X @ Y4 ) ) ) ).
% case_prodE
thf(fact_258_case__prodE,axiom,
! [C2: dtree_a_b > b > $o,P2: produc4558475209616630778_a_b_b] :
( ( produc1325217093046185599_b_b_o @ C2 @ P2 )
=> ~ ! [X: dtree_a_b,Y4: b] :
( ( P2
= ( produc331601717337510060_a_b_b @ X @ Y4 ) )
=> ~ ( C2 @ X @ Y4 ) ) ) ).
% case_prodE
thf(fact_259_case__prodD,axiom,
! [F: dtree_a_b > ( b > a ) > $o,A2: dtree_a_b,B2: b > a] :
( ( produc1457303364454389452_b_a_o @ F @ ( produc1993688775741047735_b_b_a @ A2 @ B2 ) )
=> ( F @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_260_case__prodD,axiom,
! [F: dtree_a_b > dtree_a_b > $o,A2: dtree_a_b,B2: dtree_a_b] :
( ( produc3512921791960644726_a_b_o @ F @ ( produc7805419539522982029ee_a_b @ A2 @ B2 ) )
=> ( F @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_261_case__prodD,axiom,
! [F: dtree_a_b > b > $o,A2: dtree_a_b,B2: b] :
( ( produc1325217093046185599_b_b_o @ F @ ( produc331601717337510060_a_b_b @ A2 @ B2 ) )
=> ( F @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_262_sup__set__def,axiom,
( sup_sup_set_a_a
= ( ^ [A6: set_a_a,B6: set_a_a] :
( collect_a_a
@ ( sup_sup_a_a_o
@ ^ [X3: a > a] : ( member_a_a @ X3 @ A6 )
@ ^ [X3: a > a] : ( member_a_a @ X3 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_263_sup__set__def,axiom,
( sup_su8994539500306794332ee_a_b
= ( ^ [A6: set_dtree_a_b,B6: set_dtree_a_b] :
( collect_dtree_a_b
@ ( sup_sup_dtree_a_b_o
@ ^ [X3: dtree_a_b] : ( member_dtree_a_b @ X3 @ A6 )
@ ^ [X3: dtree_a_b] : ( member_dtree_a_b @ X3 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_264_sup__set__def,axiom,
( sup_su5350426443513267090_a_b_b
= ( ^ [A6: set_se3183138701204633190_a_b_b,B6: set_se3183138701204633190_a_b_b] :
( collec5997417077270831749_a_b_b
@ ( sup_su5450082668191030131_b_b_o
@ ^ [X3: set_Pr3012420139608375472_a_b_b] : ( member7431159781899395911_a_b_b @ X3 @ A6 )
@ ^ [X3: set_Pr3012420139608375472_a_b_b] : ( member7431159781899395911_a_b_b @ X3 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_265_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_266_sup__set__def,axiom,
( sup_sup_set_set_b
= ( ^ [A6: set_set_b,B6: set_set_b] :
( collect_set_b
@ ( sup_sup_set_b_o
@ ^ [X3: set_b] : ( member_set_b @ X3 @ A6 )
@ ^ [X3: set_b] : ( member_set_b @ X3 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_267_sup__set__def,axiom,
( sup_su2887895092731772380_a_b_b
= ( ^ [A6: set_Pr3012420139608375472_a_b_b,B6: set_Pr3012420139608375472_a_b_b] :
( collec1368399972772960719_a_b_b
@ ( sup_su4209747780764569001_b_b_o
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ A6 )
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_268_sup__set__def,axiom,
( sup_sup_set_b
= ( ^ [A6: set_b,B6: set_b] :
( collect_b
@ ( sup_sup_b_o
@ ^ [X3: b] : ( member_b @ X3 @ A6 )
@ ^ [X3: b] : ( member_b @ X3 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_269_case__prodE_H,axiom,
! [C2: dtree_a_b > b > a > $o,P2: produc4558475209616630778_a_b_b,Z2: a] :
( ( produc6139810021161713496_b_a_o @ C2 @ P2 @ Z2 )
=> ~ ! [X: dtree_a_b,Y4: b] :
( ( P2
= ( produc331601717337510060_a_b_b @ X @ Y4 ) )
=> ~ ( C2 @ X @ Y4 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_270_case__prodD_H,axiom,
! [R3: dtree_a_b > b > a > $o,A2: dtree_a_b,B2: b,C2: a] :
( ( produc6139810021161713496_b_a_o @ R3 @ ( produc331601717337510060_a_b_b @ A2 @ B2 ) @ C2 )
=> ( R3 @ A2 @ B2 @ C2 ) ) ).
% case_prodD'
thf(fact_271_dtail_Ocases,axiom,
! [X4: produc1083523234014712191_b_b_a] :
~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b,Def2: b > a] :
( X4
!= ( produc1993688775741047735_b_b_a @ ( node_a_b @ R @ Xs ) @ Def2 ) ) ).
% dtail.cases
thf(fact_272_is__subtree_Ocases,axiom,
! [X4: produc5177672665255943253ee_a_b] :
~ ! [X: dtree_a_b,R: a,Xs: fset_P5281107635120001194_a_b_b] :
( X4
!= ( produc7805419539522982029ee_a_b @ X @ ( node_a_b @ R @ Xs ) ) ) ).
% is_subtree.cases
thf(fact_273_singleton__uneq,axiom,
! [R2: a,T: dtree_a_b,E4: b] :
( ( node_a_b @ R2 @ ( finser8437519239679886002_a_b_b @ ( produc331601717337510060_a_b_b @ T @ E4 ) @ bot_bo2895716411488905534_a_b_b ) )
!= T ) ).
% singleton_uneq
thf(fact_274_Pair__inject,axiom,
! [A2: dtree_a_b,B2: b > a,A4: dtree_a_b,B4: b > a] :
( ( ( produc1993688775741047735_b_b_a @ A2 @ B2 )
= ( produc1993688775741047735_b_b_a @ A4 @ B4 ) )
=> ~ ( ( A2 = A4 )
=> ( B2 != B4 ) ) ) ).
% Pair_inject
thf(fact_275_Pair__inject,axiom,
! [A2: dtree_a_b,B2: dtree_a_b,A4: dtree_a_b,B4: dtree_a_b] :
( ( ( produc7805419539522982029ee_a_b @ A2 @ B2 )
= ( produc7805419539522982029ee_a_b @ A4 @ B4 ) )
=> ~ ( ( A2 = A4 )
=> ( B2 != B4 ) ) ) ).
% Pair_inject
thf(fact_276_Pair__inject,axiom,
! [A2: dtree_a_b,B2: b,A4: dtree_a_b,B4: b] :
( ( ( produc331601717337510060_a_b_b @ A2 @ B2 )
= ( produc331601717337510060_a_b_b @ A4 @ B4 ) )
=> ~ ( ( A2 = A4 )
=> ( B2 != B4 ) ) ) ).
% Pair_inject
thf(fact_277_prod__cases,axiom,
! [P: produc1083523234014712191_b_b_a > $o,P2: produc1083523234014712191_b_b_a] :
( ! [A5: dtree_a_b,B5: b > a] : ( P @ ( produc1993688775741047735_b_b_a @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_278_prod__cases,axiom,
! [P: produc5177672665255943253ee_a_b > $o,P2: produc5177672665255943253ee_a_b] :
( ! [A5: dtree_a_b,B5: dtree_a_b] : ( P @ ( produc7805419539522982029ee_a_b @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_279_prod__cases,axiom,
! [P: produc4558475209616630778_a_b_b > $o,P2: produc4558475209616630778_a_b_b] :
( ! [A5: dtree_a_b,B5: b] : ( P @ ( produc331601717337510060_a_b_b @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_280_surj__pair,axiom,
! [P2: produc1083523234014712191_b_b_a] :
? [X: dtree_a_b,Y4: b > a] :
( P2
= ( produc1993688775741047735_b_b_a @ X @ Y4 ) ) ).
% surj_pair
thf(fact_281_surj__pair,axiom,
! [P2: produc5177672665255943253ee_a_b] :
? [X: dtree_a_b,Y4: dtree_a_b] :
( P2
= ( produc7805419539522982029ee_a_b @ X @ Y4 ) ) ).
% surj_pair
thf(fact_282_surj__pair,axiom,
! [P2: produc4558475209616630778_a_b_b] :
? [X: dtree_a_b,Y4: b] :
( P2
= ( produc331601717337510060_a_b_b @ X @ Y4 ) ) ).
% surj_pair
thf(fact_283_old_Oprod_Oexhaust,axiom,
! [Y: produc1083523234014712191_b_b_a] :
~ ! [A5: dtree_a_b,B5: b > a] :
( Y
!= ( produc1993688775741047735_b_b_a @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_284_old_Oprod_Oexhaust,axiom,
! [Y: produc5177672665255943253ee_a_b] :
~ ! [A5: dtree_a_b,B5: dtree_a_b] :
( Y
!= ( produc7805419539522982029ee_a_b @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_285_old_Oprod_Oexhaust,axiom,
! [Y: produc4558475209616630778_a_b_b] :
~ ! [A5: dtree_a_b,B5: b] :
( Y
!= ( produc331601717337510060_a_b_b @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_286_ex__in__conv,axiom,
! [A3: set_a_a] :
( ( ? [X3: a > a] : ( member_a_a @ X3 @ A3 ) )
= ( A3 != bot_bot_set_a_a ) ) ).
% ex_in_conv
thf(fact_287_ex__in__conv,axiom,
! [A3: set_dtree_a_b] :
( ( ? [X3: dtree_a_b] : ( member_dtree_a_b @ X3 @ A3 ) )
= ( A3 != bot_bo8730652382759064772ee_a_b ) ) ).
% ex_in_conv
thf(fact_288_ex__in__conv,axiom,
! [A3: set_set_b] :
( ( ? [X3: set_b] : ( member_set_b @ X3 @ A3 ) )
= ( A3 != bot_bot_set_set_b ) ) ).
% ex_in_conv
thf(fact_289_ex__in__conv,axiom,
! [A3: set_se3183138701204633190_a_b_b] :
( ( ? [X3: set_Pr3012420139608375472_a_b_b] : ( member7431159781899395911_a_b_b @ X3 @ A3 ) )
= ( A3 != bot_bo2537099559385417978_a_b_b ) ) ).
% ex_in_conv
thf(fact_290_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_291_ex__in__conv,axiom,
! [A3: set_Pr3012420139608375472_a_b_b] :
( ( ? [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ A3 ) )
= ( A3 != bot_bo3721250822024684356_a_b_b ) ) ).
% ex_in_conv
thf(fact_292_ex__in__conv,axiom,
! [A3: set_b] :
( ( ? [X3: b] : ( member_b @ X3 @ A3 ) )
= ( A3 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_293_equals0I,axiom,
! [A3: set_a_a] :
( ! [Y4: a > a] :
~ ( member_a_a @ Y4 @ A3 )
=> ( A3 = bot_bot_set_a_a ) ) ).
% equals0I
thf(fact_294_equals0I,axiom,
! [A3: set_dtree_a_b] :
( ! [Y4: dtree_a_b] :
~ ( member_dtree_a_b @ Y4 @ A3 )
=> ( A3 = bot_bo8730652382759064772ee_a_b ) ) ).
% equals0I
thf(fact_295_equals0I,axiom,
! [A3: set_set_b] :
( ! [Y4: set_b] :
~ ( member_set_b @ Y4 @ A3 )
=> ( A3 = bot_bot_set_set_b ) ) ).
% equals0I
thf(fact_296_equals0I,axiom,
! [A3: set_se3183138701204633190_a_b_b] :
( ! [Y4: set_Pr3012420139608375472_a_b_b] :
~ ( member7431159781899395911_a_b_b @ Y4 @ A3 )
=> ( A3 = bot_bo2537099559385417978_a_b_b ) ) ).
% equals0I
thf(fact_297_equals0I,axiom,
! [A3: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_298_equals0I,axiom,
! [A3: set_Pr3012420139608375472_a_b_b] :
( ! [Y4: produc4558475209616630778_a_b_b] :
~ ( member4380921116106875537_a_b_b @ Y4 @ A3 )
=> ( A3 = bot_bo3721250822024684356_a_b_b ) ) ).
% equals0I
thf(fact_299_equals0I,axiom,
! [A3: set_b] :
( ! [Y4: b] :
~ ( member_b @ Y4 @ A3 )
=> ( A3 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_300_equals0D,axiom,
! [A3: set_a_a,A2: a > a] :
( ( A3 = bot_bot_set_a_a )
=> ~ ( member_a_a @ A2 @ A3 ) ) ).
% equals0D
thf(fact_301_equals0D,axiom,
! [A3: set_dtree_a_b,A2: dtree_a_b] :
( ( A3 = bot_bo8730652382759064772ee_a_b )
=> ~ ( member_dtree_a_b @ A2 @ A3 ) ) ).
% equals0D
thf(fact_302_equals0D,axiom,
! [A3: set_set_b,A2: set_b] :
( ( A3 = bot_bot_set_set_b )
=> ~ ( member_set_b @ A2 @ A3 ) ) ).
% equals0D
thf(fact_303_equals0D,axiom,
! [A3: set_se3183138701204633190_a_b_b,A2: set_Pr3012420139608375472_a_b_b] :
( ( A3 = bot_bo2537099559385417978_a_b_b )
=> ~ ( member7431159781899395911_a_b_b @ A2 @ A3 ) ) ).
% equals0D
thf(fact_304_equals0D,axiom,
! [A3: set_a,A2: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A3 ) ) ).
% equals0D
thf(fact_305_equals0D,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,A2: produc4558475209616630778_a_b_b] :
( ( A3 = bot_bo3721250822024684356_a_b_b )
=> ~ ( member4380921116106875537_a_b_b @ A2 @ A3 ) ) ).
% equals0D
thf(fact_306_equals0D,axiom,
! [A3: set_b,A2: b] :
( ( A3 = bot_bot_set_b )
=> ~ ( member_b @ A2 @ A3 ) ) ).
% equals0D
thf(fact_307_emptyE,axiom,
! [A2: a > a] :
~ ( member_a_a @ A2 @ bot_bot_set_a_a ) ).
% emptyE
thf(fact_308_emptyE,axiom,
! [A2: dtree_a_b] :
~ ( member_dtree_a_b @ A2 @ bot_bo8730652382759064772ee_a_b ) ).
% emptyE
thf(fact_309_emptyE,axiom,
! [A2: set_b] :
~ ( member_set_b @ A2 @ bot_bot_set_set_b ) ).
% emptyE
thf(fact_310_emptyE,axiom,
! [A2: set_Pr3012420139608375472_a_b_b] :
~ ( member7431159781899395911_a_b_b @ A2 @ bot_bo2537099559385417978_a_b_b ) ).
% emptyE
thf(fact_311_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_312_emptyE,axiom,
! [A2: produc4558475209616630778_a_b_b] :
~ ( member4380921116106875537_a_b_b @ A2 @ bot_bo3721250822024684356_a_b_b ) ).
% emptyE
thf(fact_313_emptyE,axiom,
! [A2: b] :
~ ( member_b @ A2 @ bot_bot_set_b ) ).
% emptyE
thf(fact_314_mk__disjoint__insert,axiom,
! [A2: a > a,A3: set_a_a] :
( ( member_a_a @ A2 @ A3 )
=> ? [B7: set_a_a] :
( ( A3
= ( insert_a_a @ A2 @ B7 ) )
& ~ ( member_a_a @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_315_mk__disjoint__insert,axiom,
! [A2: dtree_a_b,A3: set_dtree_a_b] :
( ( member_dtree_a_b @ A2 @ A3 )
=> ? [B7: set_dtree_a_b] :
( ( A3
= ( insert_dtree_a_b @ A2 @ B7 ) )
& ~ ( member_dtree_a_b @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_316_mk__disjoint__insert,axiom,
! [A2: set_b,A3: set_set_b] :
( ( member_set_b @ A2 @ A3 )
=> ? [B7: set_set_b] :
( ( A3
= ( insert_set_b @ A2 @ B7 ) )
& ~ ( member_set_b @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_317_mk__disjoint__insert,axiom,
! [A2: set_Pr3012420139608375472_a_b_b,A3: set_se3183138701204633190_a_b_b] :
( ( member7431159781899395911_a_b_b @ A2 @ A3 )
=> ? [B7: set_se3183138701204633190_a_b_b] :
( ( A3
= ( insert8355695866607091424_a_b_b @ A2 @ B7 ) )
& ~ ( member7431159781899395911_a_b_b @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_318_mk__disjoint__insert,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ? [B7: set_a] :
( ( A3
= ( insert_a @ A2 @ B7 ) )
& ~ ( member_a @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_319_mk__disjoint__insert,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ A2 @ A3 )
=> ? [B7: set_Pr3012420139608375472_a_b_b] :
( ( A3
= ( insert1613891728210272810_a_b_b @ A2 @ B7 ) )
& ~ ( member4380921116106875537_a_b_b @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_320_mk__disjoint__insert,axiom,
! [A2: b,A3: set_b] :
( ( member_b @ A2 @ A3 )
=> ? [B7: set_b] :
( ( A3
= ( insert_b @ A2 @ B7 ) )
& ~ ( member_b @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_321_insert__commute,axiom,
! [X4: produc4558475209616630778_a_b_b,Y: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ( insert1613891728210272810_a_b_b @ X4 @ ( insert1613891728210272810_a_b_b @ Y @ A3 ) )
= ( insert1613891728210272810_a_b_b @ Y @ ( insert1613891728210272810_a_b_b @ X4 @ A3 ) ) ) ).
% insert_commute
thf(fact_322_insert__commute,axiom,
! [X4: dtree_a_b,Y: dtree_a_b,A3: set_dtree_a_b] :
( ( insert_dtree_a_b @ X4 @ ( insert_dtree_a_b @ Y @ A3 ) )
= ( insert_dtree_a_b @ Y @ ( insert_dtree_a_b @ X4 @ A3 ) ) ) ).
% insert_commute
thf(fact_323_insert__commute,axiom,
! [X4: b,Y: b,A3: set_b] :
( ( insert_b @ X4 @ ( insert_b @ Y @ A3 ) )
= ( insert_b @ Y @ ( insert_b @ X4 @ A3 ) ) ) ).
% insert_commute
thf(fact_324_insert__eq__iff,axiom,
! [A2: a > a,A3: set_a_a,B2: a > a,B3: set_a_a] :
( ~ ( member_a_a @ A2 @ A3 )
=> ( ~ ( member_a_a @ B2 @ B3 )
=> ( ( ( insert_a_a @ A2 @ A3 )
= ( insert_a_a @ B2 @ B3 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B3 ) )
& ( ( A2 != B2 )
=> ? [C3: set_a_a] :
( ( A3
= ( insert_a_a @ B2 @ C3 ) )
& ~ ( member_a_a @ B2 @ C3 )
& ( B3
= ( insert_a_a @ A2 @ C3 ) )
& ~ ( member_a_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_325_insert__eq__iff,axiom,
! [A2: dtree_a_b,A3: set_dtree_a_b,B2: dtree_a_b,B3: set_dtree_a_b] :
( ~ ( member_dtree_a_b @ A2 @ A3 )
=> ( ~ ( member_dtree_a_b @ B2 @ B3 )
=> ( ( ( insert_dtree_a_b @ A2 @ A3 )
= ( insert_dtree_a_b @ B2 @ B3 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B3 ) )
& ( ( A2 != B2 )
=> ? [C3: set_dtree_a_b] :
( ( A3
= ( insert_dtree_a_b @ B2 @ C3 ) )
& ~ ( member_dtree_a_b @ B2 @ C3 )
& ( B3
= ( insert_dtree_a_b @ A2 @ C3 ) )
& ~ ( member_dtree_a_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_326_insert__eq__iff,axiom,
! [A2: set_b,A3: set_set_b,B2: set_b,B3: set_set_b] :
( ~ ( member_set_b @ A2 @ A3 )
=> ( ~ ( member_set_b @ B2 @ B3 )
=> ( ( ( insert_set_b @ A2 @ A3 )
= ( insert_set_b @ B2 @ B3 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B3 ) )
& ( ( A2 != B2 )
=> ? [C3: set_set_b] :
( ( A3
= ( insert_set_b @ B2 @ C3 ) )
& ~ ( member_set_b @ B2 @ C3 )
& ( B3
= ( insert_set_b @ A2 @ C3 ) )
& ~ ( member_set_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_327_insert__eq__iff,axiom,
! [A2: set_Pr3012420139608375472_a_b_b,A3: set_se3183138701204633190_a_b_b,B2: set_Pr3012420139608375472_a_b_b,B3: set_se3183138701204633190_a_b_b] :
( ~ ( member7431159781899395911_a_b_b @ A2 @ A3 )
=> ( ~ ( member7431159781899395911_a_b_b @ B2 @ B3 )
=> ( ( ( insert8355695866607091424_a_b_b @ A2 @ A3 )
= ( insert8355695866607091424_a_b_b @ B2 @ B3 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B3 ) )
& ( ( A2 != B2 )
=> ? [C3: set_se3183138701204633190_a_b_b] :
( ( A3
= ( insert8355695866607091424_a_b_b @ B2 @ C3 ) )
& ~ ( member7431159781899395911_a_b_b @ B2 @ C3 )
& ( B3
= ( insert8355695866607091424_a_b_b @ A2 @ C3 ) )
& ~ ( member7431159781899395911_a_b_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_328_insert__eq__iff,axiom,
! [A2: a,A3: set_a,B2: a,B3: set_a] :
( ~ ( member_a @ A2 @ A3 )
=> ( ~ ( member_a @ B2 @ B3 )
=> ( ( ( insert_a @ A2 @ A3 )
= ( insert_a @ B2 @ B3 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B3 ) )
& ( ( A2 != B2 )
=> ? [C3: set_a] :
( ( A3
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B3
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_329_insert__eq__iff,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B2: produc4558475209616630778_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ~ ( member4380921116106875537_a_b_b @ A2 @ A3 )
=> ( ~ ( member4380921116106875537_a_b_b @ B2 @ B3 )
=> ( ( ( insert1613891728210272810_a_b_b @ A2 @ A3 )
= ( insert1613891728210272810_a_b_b @ B2 @ B3 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B3 ) )
& ( ( A2 != B2 )
=> ? [C3: set_Pr3012420139608375472_a_b_b] :
( ( A3
= ( insert1613891728210272810_a_b_b @ B2 @ C3 ) )
& ~ ( member4380921116106875537_a_b_b @ B2 @ C3 )
& ( B3
= ( insert1613891728210272810_a_b_b @ A2 @ C3 ) )
& ~ ( member4380921116106875537_a_b_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_330_insert__eq__iff,axiom,
! [A2: b,A3: set_b,B2: b,B3: set_b] :
( ~ ( member_b @ A2 @ A3 )
=> ( ~ ( member_b @ B2 @ B3 )
=> ( ( ( insert_b @ A2 @ A3 )
= ( insert_b @ B2 @ B3 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B3 ) )
& ( ( A2 != B2 )
=> ? [C3: set_b] :
( ( A3
= ( insert_b @ B2 @ C3 ) )
& ~ ( member_b @ B2 @ C3 )
& ( B3
= ( insert_b @ A2 @ C3 ) )
& ~ ( member_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_331_insert__absorb,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ( ( insert_a @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_332_insert__absorb,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ A2 @ A3 )
=> ( ( insert1613891728210272810_a_b_b @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_333_insert__absorb,axiom,
! [A2: b,A3: set_b] :
( ( member_b @ A2 @ A3 )
=> ( ( insert_b @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_334_insert__ident,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ~ ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( ~ ( member4380921116106875537_a_b_b @ X4 @ B3 )
=> ( ( ( insert1613891728210272810_a_b_b @ X4 @ A3 )
= ( insert1613891728210272810_a_b_b @ X4 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_335_insert__ident,axiom,
! [X4: b,A3: set_b,B3: set_b] :
( ~ ( member_b @ X4 @ A3 )
=> ( ~ ( member_b @ X4 @ B3 )
=> ( ( ( insert_b @ X4 @ A3 )
= ( insert_b @ X4 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_336_Set_Oset__insert,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ~ ! [B7: set_Pr3012420139608375472_a_b_b] :
( ( A3
= ( insert1613891728210272810_a_b_b @ X4 @ B7 ) )
=> ( member4380921116106875537_a_b_b @ X4 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_337_Set_Oset__insert,axiom,
! [X4: b,A3: set_b] :
( ( member_b @ X4 @ A3 )
=> ~ ! [B7: set_b] :
( ( A3
= ( insert_b @ X4 @ B7 ) )
=> ( member_b @ X4 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_338_insertI2,axiom,
! [A2: produc4558475209616630778_a_b_b,B3: set_Pr3012420139608375472_a_b_b,B2: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ A2 @ B3 )
=> ( member4380921116106875537_a_b_b @ A2 @ ( insert1613891728210272810_a_b_b @ B2 @ B3 ) ) ) ).
% insertI2
thf(fact_339_insertI2,axiom,
! [A2: b,B3: set_b,B2: b] :
( ( member_b @ A2 @ B3 )
=> ( member_b @ A2 @ ( insert_b @ B2 @ B3 ) ) ) ).
% insertI2
thf(fact_340_insertI1,axiom,
! [A2: produc4558475209616630778_a_b_b,B3: set_Pr3012420139608375472_a_b_b] : ( member4380921116106875537_a_b_b @ A2 @ ( insert1613891728210272810_a_b_b @ A2 @ B3 ) ) ).
% insertI1
thf(fact_341_insertI1,axiom,
! [A2: b,B3: set_b] : ( member_b @ A2 @ ( insert_b @ A2 @ B3 ) ) ).
% insertI1
thf(fact_342_insertE,axiom,
! [A2: produc4558475209616630778_a_b_b,B2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ A2 @ ( insert1613891728210272810_a_b_b @ B2 @ A3 ) )
=> ( ( A2 != B2 )
=> ( member4380921116106875537_a_b_b @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_343_insertE,axiom,
! [A2: b,B2: b,A3: set_b] :
( ( member_b @ A2 @ ( insert_b @ B2 @ A3 ) )
=> ( ( A2 != B2 )
=> ( member_b @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_344_sup__left__commute,axiom,
! [X4: set_b,Y: set_b,Z2: set_b] :
( ( sup_sup_set_b @ X4 @ ( sup_sup_set_b @ Y @ Z2 ) )
= ( sup_sup_set_b @ Y @ ( sup_sup_set_b @ X4 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_345_sup_Oleft__commute,axiom,
! [B2: set_b,A2: set_b,C2: set_b] :
( ( sup_sup_set_b @ B2 @ ( sup_sup_set_b @ A2 @ C2 ) )
= ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_346_sup__commute,axiom,
( sup_sup_set_b
= ( ^ [X3: set_b,Y5: set_b] : ( sup_sup_set_b @ Y5 @ X3 ) ) ) ).
% sup_commute
thf(fact_347_sup_Ocommute,axiom,
( sup_sup_set_b
= ( ^ [A: set_b,B: set_b] : ( sup_sup_set_b @ B @ A ) ) ) ).
% sup.commute
thf(fact_348_sup__assoc,axiom,
! [X4: set_b,Y: set_b,Z2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ X4 @ Y ) @ Z2 )
= ( sup_sup_set_b @ X4 @ ( sup_sup_set_b @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_349_sup_Oassoc,axiom,
! [A2: set_b,B2: set_b,C2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_350_inf__sup__aci_I5_J,axiom,
( sup_sup_set_b
= ( ^ [X3: set_b,Y5: set_b] : ( sup_sup_set_b @ Y5 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_351_inf__sup__aci_I6_J,axiom,
! [X4: set_b,Y: set_b,Z2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ X4 @ Y ) @ Z2 )
= ( sup_sup_set_b @ X4 @ ( sup_sup_set_b @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_352_inf__sup__aci_I7_J,axiom,
! [X4: set_b,Y: set_b,Z2: set_b] :
( ( sup_sup_set_b @ X4 @ ( sup_sup_set_b @ Y @ Z2 ) )
= ( sup_sup_set_b @ Y @ ( sup_sup_set_b @ X4 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_353_inf__sup__aci_I8_J,axiom,
! [X4: set_b,Y: set_b] :
( ( sup_sup_set_b @ X4 @ ( sup_sup_set_b @ X4 @ Y ) )
= ( sup_sup_set_b @ X4 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_354_Un__left__commute,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( sup_sup_set_b @ A3 @ ( sup_sup_set_b @ B3 @ C ) )
= ( sup_sup_set_b @ B3 @ ( sup_sup_set_b @ A3 @ C ) ) ) ).
% Un_left_commute
thf(fact_355_Un__left__absorb,axiom,
! [A3: set_b,B3: set_b] :
( ( sup_sup_set_b @ A3 @ ( sup_sup_set_b @ A3 @ B3 ) )
= ( sup_sup_set_b @ A3 @ B3 ) ) ).
% Un_left_absorb
thf(fact_356_Un__commute,axiom,
( sup_sup_set_b
= ( ^ [A6: set_b,B6: set_b] : ( sup_sup_set_b @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_357_Un__absorb,axiom,
! [A3: set_b] :
( ( sup_sup_set_b @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_358_Un__assoc,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ A3 @ B3 ) @ C )
= ( sup_sup_set_b @ A3 @ ( sup_sup_set_b @ B3 @ C ) ) ) ).
% Un_assoc
thf(fact_359_ball__Un,axiom,
! [A3: set_b,B3: set_b,P: b > $o] :
( ( ! [X3: b] :
( ( member_b @ X3 @ ( sup_sup_set_b @ A3 @ B3 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: b] :
( ( member_b @ X3 @ A3 )
=> ( P @ X3 ) )
& ! [X3: b] :
( ( member_b @ X3 @ B3 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_360_bex__Un,axiom,
! [A3: set_b,B3: set_b,P: b > $o] :
( ( ? [X3: b] :
( ( member_b @ X3 @ ( sup_sup_set_b @ A3 @ B3 ) )
& ( P @ X3 ) ) )
= ( ? [X3: b] :
( ( member_b @ X3 @ A3 )
& ( P @ X3 ) )
| ? [X3: b] :
( ( member_b @ X3 @ B3 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_361_UnI2,axiom,
! [C2: produc4558475209616630778_a_b_b,B3: set_Pr3012420139608375472_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ C2 @ B3 )
=> ( member4380921116106875537_a_b_b @ C2 @ ( sup_su2887895092731772380_a_b_b @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_362_UnI2,axiom,
! [C2: b,B3: set_b,A3: set_b] :
( ( member_b @ C2 @ B3 )
=> ( member_b @ C2 @ ( sup_sup_set_b @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_363_UnI1,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ C2 @ A3 )
=> ( member4380921116106875537_a_b_b @ C2 @ ( sup_su2887895092731772380_a_b_b @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_364_UnI1,axiom,
! [C2: b,A3: set_b,B3: set_b] :
( ( member_b @ C2 @ A3 )
=> ( member_b @ C2 @ ( sup_sup_set_b @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_365_UnE,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ C2 @ ( sup_su2887895092731772380_a_b_b @ A3 @ B3 ) )
=> ( ~ ( member4380921116106875537_a_b_b @ C2 @ A3 )
=> ( member4380921116106875537_a_b_b @ C2 @ B3 ) ) ) ).
% UnE
thf(fact_366_UnE,axiom,
! [C2: b,A3: set_b,B3: set_b] :
( ( member_b @ C2 @ ( sup_sup_set_b @ A3 @ B3 ) )
=> ( ~ ( member_b @ C2 @ A3 )
=> ( member_b @ C2 @ B3 ) ) ) ).
% UnE
thf(fact_367_empty__def,axiom,
( bot_bot_set_b
= ( collect_b
@ ^ [X3: b] : $false ) ) ).
% empty_def
thf(fact_368_insert__Collect,axiom,
! [A2: b,P: b > $o] :
( ( insert_b @ A2 @ ( collect_b @ P ) )
= ( collect_b
@ ^ [U: b] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_369_insert__compr,axiom,
( insert1613891728210272810_a_b_b
= ( ^ [A: produc4558475209616630778_a_b_b,B6: set_Pr3012420139608375472_a_b_b] :
( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] :
( ( X3 = A )
| ( member4380921116106875537_a_b_b @ X3 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_370_insert__compr,axiom,
( insert_b
= ( ^ [A: b,B6: set_b] :
( collect_b
@ ^ [X3: b] :
( ( X3 = A )
| ( member_b @ X3 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_371_prod_Ocase__distrib,axiom,
! [H: ( a > a ) > a > a,F: dtree_a_b > b > a > a,Prod: produc4558475209616630778_a_b_b] :
( ( H @ ( produc2242037354397874494_b_a_a @ F @ Prod ) )
= ( produc2242037354397874494_b_a_a
@ ^ [X13: dtree_a_b,X23: b] : ( H @ ( F @ X13 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_372_Collect__disj__eq,axiom,
! [P: b > $o,Q: b > $o] :
( ( collect_b
@ ^ [X3: b] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_373_Un__def,axiom,
( sup_su2887895092731772380_a_b_b
= ( ^ [A6: set_Pr3012420139608375472_a_b_b,B6: set_Pr3012420139608375472_a_b_b] :
( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ A6 )
| ( member4380921116106875537_a_b_b @ X3 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_374_Un__def,axiom,
( sup_sup_set_b
= ( ^ [A6: set_b,B6: set_b] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A6 )
| ( member_b @ X3 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_375_singleton__inject,axiom,
! [A2: b,B2: b] :
( ( ( insert_b @ A2 @ bot_bot_set_b )
= ( insert_b @ B2 @ bot_bot_set_b ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_376_insert__not__empty,axiom,
! [A2: b,A3: set_b] :
( ( insert_b @ A2 @ A3 )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_377_doubleton__eq__iff,axiom,
! [A2: b,B2: b,C2: b,D: b] :
( ( ( insert_b @ A2 @ ( insert_b @ B2 @ bot_bot_set_b ) )
= ( insert_b @ C2 @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A2 = C2 )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_378_singleton__iff,axiom,
! [B2: produc4558475209616630778_a_b_b,A2: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ B2 @ ( insert1613891728210272810_a_b_b @ A2 @ bot_bo3721250822024684356_a_b_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_379_singleton__iff,axiom,
! [B2: b,A2: b] :
( ( member_b @ B2 @ ( insert_b @ A2 @ bot_bot_set_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_380_singletonD,axiom,
! [B2: produc4558475209616630778_a_b_b,A2: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ B2 @ ( insert1613891728210272810_a_b_b @ A2 @ bot_bo3721250822024684356_a_b_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_381_singletonD,axiom,
! [B2: b,A2: b] :
( ( member_b @ B2 @ ( insert_b @ A2 @ bot_bot_set_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_382_old_Oprod_Ocase,axiom,
! [F: dtree_a_b > b > a > a,X1: dtree_a_b,X2: b] :
( ( produc2242037354397874494_b_a_a @ F @ ( produc331601717337510060_a_b_b @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_383_Un__empty__right,axiom,
! [A3: set_b] :
( ( sup_sup_set_b @ A3 @ bot_bot_set_b )
= A3 ) ).
% Un_empty_right
thf(fact_384_Un__empty__left,axiom,
! [B3: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ B3 )
= B3 ) ).
% Un_empty_left
thf(fact_385_Collect__conv__if2,axiom,
! [P: b > $o,A2: b] :
( ( ( P @ A2 )
=> ( ( collect_b
@ ^ [X3: b] :
( ( A2 = X3 )
& ( P @ X3 ) ) )
= ( insert_b @ A2 @ bot_bot_set_b ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_b
@ ^ [X3: b] :
( ( A2 = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if2
thf(fact_386_Collect__conv__if,axiom,
! [P: b > $o,A2: b] :
( ( ( P @ A2 )
=> ( ( collect_b
@ ^ [X3: b] :
( ( X3 = A2 )
& ( P @ X3 ) ) )
= ( insert_b @ A2 @ bot_bot_set_b ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_b
@ ^ [X3: b] :
( ( X3 = A2 )
& ( P @ X3 ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if
thf(fact_387_cond__case__prod__eta,axiom,
! [F: dtree_a_b > b > a > a,G2: produc4558475209616630778_a_b_b > a > a] :
( ! [X: dtree_a_b,Y4: b] :
( ( F @ X @ Y4 )
= ( G2 @ ( produc331601717337510060_a_b_b @ X @ Y4 ) ) )
=> ( ( produc2242037354397874494_b_a_a @ F )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_388_case__prod__eta,axiom,
! [F: produc4558475209616630778_a_b_b > a > a] :
( ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,Y5: b] : ( F @ ( produc331601717337510060_a_b_b @ X3 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_389_case__prodE2,axiom,
! [Q: ( a > a ) > $o,P: dtree_a_b > b > a > a,Z2: produc4558475209616630778_a_b_b] :
( ( Q @ ( produc2242037354397874494_b_a_a @ P @ Z2 ) )
=> ~ ! [X: dtree_a_b,Y4: b] :
( ( Z2
= ( produc331601717337510060_a_b_b @ X @ Y4 ) )
=> ~ ( Q @ ( P @ X @ Y4 ) ) ) ) ).
% case_prodE2
thf(fact_390_insert__def,axiom,
( insert_b
= ( ^ [A: b] :
( sup_sup_set_b
@ ( collect_b
@ ^ [X3: b] : ( X3 = A ) ) ) ) ) ).
% insert_def
thf(fact_391_singleton__Un__iff,axiom,
! [X4: b,A3: set_b,B3: set_b] :
( ( ( insert_b @ X4 @ bot_bot_set_b )
= ( sup_sup_set_b @ A3 @ B3 ) )
= ( ( ( A3 = bot_bot_set_b )
& ( B3
= ( insert_b @ X4 @ bot_bot_set_b ) ) )
| ( ( A3
= ( insert_b @ X4 @ bot_bot_set_b ) )
& ( B3 = bot_bot_set_b ) )
| ( ( A3
= ( insert_b @ X4 @ bot_bot_set_b ) )
& ( B3
= ( insert_b @ X4 @ bot_bot_set_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_392_Un__singleton__iff,axiom,
! [A3: set_b,B3: set_b,X4: b] :
( ( ( sup_sup_set_b @ A3 @ B3 )
= ( insert_b @ X4 @ bot_bot_set_b ) )
= ( ( ( A3 = bot_bot_set_b )
& ( B3
= ( insert_b @ X4 @ bot_bot_set_b ) ) )
| ( ( A3
= ( insert_b @ X4 @ bot_bot_set_b ) )
& ( B3 = bot_bot_set_b ) )
| ( ( A3
= ( insert_b @ X4 @ bot_bot_set_b ) )
& ( B3
= ( insert_b @ X4 @ bot_bot_set_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_393_insert__is__Un,axiom,
( insert_b
= ( ^ [A: b] : ( sup_sup_set_b @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% insert_is_Un
thf(fact_394_finsert_Orep__eq,axiom,
! [X4: b,Xa: fset_b] :
( ( fset_b2 @ ( finsert_b @ X4 @ Xa ) )
= ( insert_b @ X4 @ ( fset_b2 @ Xa ) ) ) ).
% finsert.rep_eq
thf(fact_395_finsert_Orep__eq,axiom,
! [X4: produc4558475209616630778_a_b_b,Xa: fset_P5281107635120001194_a_b_b] :
( ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ X4 @ Xa ) )
= ( insert1613891728210272810_a_b_b @ X4 @ ( fset_P783253628892185035_a_b_b @ Xa ) ) ) ).
% finsert.rep_eq
thf(fact_396_union__fset,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Xa: fset_P5281107635120001194_a_b_b] :
( ( fset_P783253628892185035_a_b_b @ ( sup_su860928060825958358_a_b_b @ X4 @ Xa ) )
= ( sup_su2887895092731772380_a_b_b @ ( fset_P783253628892185035_a_b_b @ X4 ) @ ( fset_P783253628892185035_a_b_b @ Xa ) ) ) ).
% union_fset
thf(fact_397_union__fset,axiom,
! [X4: fset_b,Xa: fset_b] :
( ( fset_b2 @ ( sup_sup_fset_b @ X4 @ Xa ) )
= ( sup_sup_set_b @ ( fset_b2 @ X4 ) @ ( fset_b2 @ Xa ) ) ) ).
% union_fset
thf(fact_398_bot__fset_Orep__eq,axiom,
( ( fset_P783253628892185035_a_b_b @ bot_bo2895716411488905534_a_b_b )
= bot_bo3721250822024684356_a_b_b ) ).
% bot_fset.rep_eq
thf(fact_399_bot__fset_Orep__eq,axiom,
( ( fset_b2 @ bot_bot_fset_b )
= bot_bot_set_b ) ).
% bot_fset.rep_eq
thf(fact_400_dhead__ffold__f__alt__commute,axiom,
! [P: fset_P5281107635120001194_a_b_b > $o,R2: a,Q: produc4558475209616630778_a_b_b > a > $o,E4: b,R3: produc4558475209616630778_a_b_b > a > a,Def: b > a,Xs2: fset_P5281107635120001194_a_b_b] :
( ( P
= ( ^ [Xs3: fset_P5281107635120001194_a_b_b] : ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs3 ) ) ) )
=> ( ( Q
= ( produc6139810021161713496_b_a_o
@ ^ [X3: dtree_a_b,E2: b,Uu: a] : ( member_b @ E4 @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) ) ) )
=> ( ( R3
= ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,Uu: a] : ( if_a @ ( E4 = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ Def @ E4 ) ) ) )
=> ( finite414203908571218417_b_b_a
@ ^ [A: produc4558475209616630778_a_b_b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ A @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( Q @ A @ B )
| ~ ( P @ Xs2 ) )
@ B
@ ( R3 @ A @ B ) ) ) ) ) ) ).
% dhead_ffold_f_alt_commute
thf(fact_401_dhead__commute,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,E4: b,R2: a,Def: b > a] :
( finite414203908571218417_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E4 @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( if_a @ ( E4 = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ Def @ E4 ) ) ) ) ) ).
% dhead_commute
thf(fact_402__C2_C,axiom,
ord_le789900035998834954_a_b_b @ xsa @ ( finser8437519239679886002_a_b_b @ x @ xsa ) ).
% "2"
thf(fact_403_dhead__ffold__supset,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,Ys: fset_P5281107635120001194_a_b_b,R2: a,E4: b,Def: b > a] :
( ( ord_le789900035998834954_a_b_b @ Xs2 @ Ys )
=> ( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Ys ) )
=> ( ( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Ys ) )
| ~ ( member_b @ E4 @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Ys ) ) )
@ B
@ ( if_a @ ( E4 = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ Def @ E4 ) ) ) )
@ ( Def @ E4 )
@ Xs2 )
= ( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E4 @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( if_a @ ( E4 = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ Def @ E4 ) ) ) )
@ ( Def @ E4 )
@ Xs2 ) ) ) ) ).
% dhead_ffold_supset
thf(fact_404_dhead__ffold__notelem__eq__def,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,E4: b,Ys: fset_P5281107635120001194_a_b_b,R2: a,Def: b > a] :
( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [T3: dtree_a_b,E12: b] :
( ~ ( member_b @ E4 @ ( darcs_a_b @ T3 ) )
& ( E4 != E12 ) )
@ X ) )
=> ( ( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Ys ) )
| ~ ( member_b @ E4 @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Ys ) ) )
@ B
@ ( if_a @ ( E4 = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ Def @ E4 ) ) ) )
@ ( Def @ E4 )
@ Xs2 )
= ( Def @ E4 ) ) ) ).
% dhead_ffold_notelem_eq_def
thf(fact_405_dhead__commute__aux,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,E4: b,R2: a,Def: b > a,Y: produc4558475209616630778_a_b_b,X4: produc4558475209616630778_a_b_b,Z2: a] :
( ( comp_a_a_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E4 @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( if_a @ ( E4 = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ Def @ E4 ) ) )
@ Y )
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E4 @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( if_a @ ( E4 = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ Def @ E4 ) ) )
@ X4 )
@ Z2 )
= ( comp_a_a_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E4 @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( if_a @ ( E4 = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ Def @ E4 ) ) )
@ X4 )
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E4 @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( if_a @ ( E4 = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ Def @ E4 ) ) )
@ Y )
@ Z2 ) ) ).
% dhead_commute_aux
thf(fact_406_dhead_Opelims,axiom,
! [X4: dtree_a_b,Xa: b > a,Y: b > a] :
( ( ( dhead_a_b @ X4 @ Xa )
= Y )
=> ( ( accp_P1416650344722773512_b_b_a @ dhead_rel_a_b @ ( produc1993688775741047735_b_b_a @ X4 @ Xa ) )
=> ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( ( X4
= ( node_a_b @ R @ Xs ) )
=> ( ( Y
= ( ^ [E: b] :
( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs ) )
| ~ ( member_b @ E @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R @ Xs ) ) )
@ B
@ ( if_a @ ( E = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ Xa @ E ) ) ) )
@ ( Xa @ E )
@ Xs ) ) )
=> ~ ( accp_P1416650344722773512_b_b_a @ dhead_rel_a_b @ ( produc1993688775741047735_b_b_a @ ( node_a_b @ R @ Xs ) @ Xa ) ) ) ) ) ) ).
% dhead.pelims
thf(fact_407_finsert__absorb2,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( finser8437519239679886002_a_b_b @ X4 @ ( finser8437519239679886002_a_b_b @ X4 @ A3 ) )
= ( finser8437519239679886002_a_b_b @ X4 @ A3 ) ) ).
% finsert_absorb2
thf(fact_408_fsubset__antisym,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ B3 )
=> ( ( ord_le789900035998834954_a_b_b @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% fsubset_antisym
thf(fact_409_le__sup__iff,axiom,
! [X4: set_b,Y: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ X4 @ Y ) @ Z2 )
= ( ( ord_less_eq_set_b @ X4 @ Z2 )
& ( ord_less_eq_set_b @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_410_le__sup__iff,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b,Z2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ ( sup_su860928060825958358_a_b_b @ X4 @ Y ) @ Z2 )
= ( ( ord_le789900035998834954_a_b_b @ X4 @ Z2 )
& ( ord_le789900035998834954_a_b_b @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_411_le__sup__iff,axiom,
! [X4: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X4 @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_412_sup_Obounded__iff,axiom,
! [B2: set_b,C2: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq_set_b @ B2 @ A2 )
& ( ord_less_eq_set_b @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_413_sup_Obounded__iff,axiom,
! [B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ ( sup_su860928060825958358_a_b_b @ B2 @ C2 ) @ A2 )
= ( ( ord_le789900035998834954_a_b_b @ B2 @ A2 )
& ( ord_le789900035998834954_a_b_b @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_414_sup_Obounded__iff,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_415_ball__empty,axiom,
! [P: b > $o,X5: b] :
( ( member_b @ X5 @ bot_bot_set_b )
=> ( P @ X5 ) ) ).
% ball_empty
thf(fact_416_finsert__iff,axiom,
! [A2: produc4558475209616630778_a_b_b,B2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ A2 @ ( finser8437519239679886002_a_b_b @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( fmembe3173364709796808819_a_b_b @ A2 @ A3 ) ) ) ).
% finsert_iff
thf(fact_417_finsertCI,axiom,
! [A2: produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b,B2: produc4558475209616630778_a_b_b] :
( ( ~ ( fmembe3173364709796808819_a_b_b @ A2 @ B3 )
=> ( A2 = B2 ) )
=> ( fmembe3173364709796808819_a_b_b @ A2 @ ( finser8437519239679886002_a_b_b @ B2 @ B3 ) ) ) ).
% finsertCI
thf(fact_418_fempty__iff,axiom,
! [C2: produc4558475209616630778_a_b_b] :
~ ( fmembe3173364709796808819_a_b_b @ C2 @ bot_bo2895716411488905534_a_b_b ) ).
% fempty_iff
thf(fact_419_all__not__fin__conv,axiom,
! [A3: fset_P5281107635120001194_a_b_b] :
( ( ! [X3: produc4558475209616630778_a_b_b] :
~ ( fmembe3173364709796808819_a_b_b @ X3 @ A3 ) )
= ( A3 = bot_bo2895716411488905534_a_b_b ) ) ).
% all_not_fin_conv
thf(fact_420_fsubsetI,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ! [X: produc4558475209616630778_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ X @ A3 )
=> ( fmembe3173364709796808819_a_b_b @ X @ B3 ) )
=> ( ord_le789900035998834954_a_b_b @ A3 @ B3 ) ) ).
% fsubsetI
thf(fact_421_funionCI,axiom,
! [C2: produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( ~ ( fmembe3173364709796808819_a_b_b @ C2 @ B3 )
=> ( fmembe3173364709796808819_a_b_b @ C2 @ A3 ) )
=> ( fmembe3173364709796808819_a_b_b @ C2 @ ( sup_su860928060825958358_a_b_b @ A3 @ B3 ) ) ) ).
% funionCI
thf(fact_422_funion__iff,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ C2 @ ( sup_su860928060825958358_a_b_b @ A3 @ B3 ) )
= ( ( fmembe3173364709796808819_a_b_b @ C2 @ A3 )
| ( fmembe3173364709796808819_a_b_b @ C2 @ B3 ) ) ) ).
% funion_iff
thf(fact_423_fsubset__fempty,axiom,
! [A3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ bot_bo2895716411488905534_a_b_b )
= ( A3 = bot_bo2895716411488905534_a_b_b ) ) ).
% fsubset_fempty
thf(fact_424_fempty__fsubsetI,axiom,
! [X4: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ bot_bo2895716411488905534_a_b_b @ X4 ) ).
% fempty_fsubsetI
thf(fact_425_funion__finsert__right,axiom,
! [A3: fset_P5281107635120001194_a_b_b,A2: produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( sup_su860928060825958358_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ A2 @ B3 ) )
= ( finser8437519239679886002_a_b_b @ A2 @ ( sup_su860928060825958358_a_b_b @ A3 @ B3 ) ) ) ).
% funion_finsert_right
thf(fact_426_funion__finsert__left,axiom,
! [A2: produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b,C: fset_P5281107635120001194_a_b_b] :
( ( sup_su860928060825958358_a_b_b @ ( finser8437519239679886002_a_b_b @ A2 @ B3 ) @ C )
= ( finser8437519239679886002_a_b_b @ A2 @ ( sup_su860928060825958358_a_b_b @ B3 @ C ) ) ) ).
% funion_finsert_left
thf(fact_427_finsert__fsubset,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ ( finser8437519239679886002_a_b_b @ X4 @ A3 ) @ B3 )
= ( ( fmembe3173364709796808819_a_b_b @ X4 @ B3 )
& ( ord_le789900035998834954_a_b_b @ A3 @ B3 ) ) ) ).
% finsert_fsubset
thf(fact_428_femptyE,axiom,
! [A2: produc4558475209616630778_a_b_b] :
~ ( fmembe3173364709796808819_a_b_b @ A2 @ bot_bo2895716411488905534_a_b_b ) ).
% femptyE
thf(fact_429_funionE,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ C2 @ ( sup_su860928060825958358_a_b_b @ A3 @ B3 ) )
=> ( ~ ( fmembe3173364709796808819_a_b_b @ C2 @ A3 )
=> ( fmembe3173364709796808819_a_b_b @ C2 @ B3 ) ) ) ).
% funionE
thf(fact_430_Ball__def,axiom,
( ball_P8580587655522039760_a_b_b
= ( ^ [A6: set_Pr3012420139608375472_a_b_b,P3: produc4558475209616630778_a_b_b > $o] :
! [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ A6 )
=> ( P3 @ X3 ) ) ) ) ).
% Ball_def
thf(fact_431_Ball__def,axiom,
( ball_b
= ( ^ [A6: set_b,P3: b > $o] :
! [X3: b] :
( ( member_b @ X3 @ A6 )
=> ( P3 @ X3 ) ) ) ) ).
% Ball_def
thf(fact_432_funionI1,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ C2 @ A3 )
=> ( fmembe3173364709796808819_a_b_b @ C2 @ ( sup_su860928060825958358_a_b_b @ A3 @ B3 ) ) ) ).
% funionI1
thf(fact_433_funionI2,axiom,
! [C2: produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ C2 @ B3 )
=> ( fmembe3173364709796808819_a_b_b @ C2 @ ( sup_su860928060825958358_a_b_b @ A3 @ B3 ) ) ) ).
% funionI2
thf(fact_434_ffold__cong,axiom,
! [F: produc4558475209616630778_a_b_b > a > a,G2: produc4558475209616630778_a_b_b > a > a,A3: fset_P5281107635120001194_a_b_b,S: a,T: a,B3: fset_P5281107635120001194_a_b_b] :
( ( finite414203908571218417_b_b_a @ F )
=> ( ( finite414203908571218417_b_b_a @ G2 )
=> ( ! [X: produc4558475209616630778_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ X @ A3 )
=> ( ( F @ X )
= ( G2 @ X ) ) )
=> ( ( S = T )
=> ( ( A3 = B3 )
=> ( ( ffold_2783168711033344739_b_b_a @ F @ S @ A3 )
= ( ffold_2783168711033344739_b_b_a @ G2 @ T @ B3 ) ) ) ) ) ) ) ).
% ffold_cong
thf(fact_435_ex__fin__conv,axiom,
! [A3: fset_P5281107635120001194_a_b_b] :
( ( ? [X3: produc4558475209616630778_a_b_b] : ( fmembe3173364709796808819_a_b_b @ X3 @ A3 ) )
= ( A3 != bot_bo2895716411488905534_a_b_b ) ) ).
% ex_fin_conv
thf(fact_436_funion__mono,axiom,
! [A3: fset_P5281107635120001194_a_b_b,C: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b,D2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ C )
=> ( ( ord_le789900035998834954_a_b_b @ B3 @ D2 )
=> ( ord_le789900035998834954_a_b_b @ ( sup_su860928060825958358_a_b_b @ A3 @ B3 ) @ ( sup_su860928060825958358_a_b_b @ C @ D2 ) ) ) ) ).
% funion_mono
thf(fact_437_funion__least,axiom,
! [A3: fset_P5281107635120001194_a_b_b,C: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ C )
=> ( ( ord_le789900035998834954_a_b_b @ B3 @ C )
=> ( ord_le789900035998834954_a_b_b @ ( sup_su860928060825958358_a_b_b @ A3 @ B3 ) @ C ) ) ) ).
% funion_least
thf(fact_438_funion__upper1,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ A3 @ ( sup_su860928060825958358_a_b_b @ A3 @ B3 ) ) ).
% funion_upper1
thf(fact_439_funion__upper2,axiom,
! [B3: fset_P5281107635120001194_a_b_b,A3: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ B3 @ ( sup_su860928060825958358_a_b_b @ A3 @ B3 ) ) ).
% funion_upper2
thf(fact_440_equalsffemptyD,axiom,
! [A3: fset_P5281107635120001194_a_b_b,A2: produc4558475209616630778_a_b_b] :
( ( A3 = bot_bo2895716411488905534_a_b_b )
=> ~ ( fmembe3173364709796808819_a_b_b @ A2 @ A3 ) ) ).
% equalsffemptyD
thf(fact_441_equalsffemptyI,axiom,
! [A3: fset_P5281107635120001194_a_b_b] :
( ! [Y4: produc4558475209616630778_a_b_b] :
~ ( fmembe3173364709796808819_a_b_b @ Y4 @ A3 )
=> ( A3 = bot_bo2895716411488905534_a_b_b ) ) ).
% equalsffemptyI
thf(fact_442_funion__absorb1,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ B3 )
=> ( ( sup_su860928060825958358_a_b_b @ A3 @ B3 )
= B3 ) ) ).
% funion_absorb1
thf(fact_443_funion__absorb2,axiom,
! [B3: fset_P5281107635120001194_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ B3 @ A3 )
=> ( ( sup_su860928060825958358_a_b_b @ A3 @ B3 )
= A3 ) ) ).
% funion_absorb2
thf(fact_444_fsubset__finsert,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ~ ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ( ( ord_le789900035998834954_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ X4 @ B3 ) )
= ( ord_le789900035998834954_a_b_b @ A3 @ B3 ) ) ) ).
% fsubset_finsert
thf(fact_445_fsubset__funion__eq,axiom,
( ord_le789900035998834954_a_b_b
= ( ^ [A6: fset_P5281107635120001194_a_b_b,B6: fset_P5281107635120001194_a_b_b] :
( ( sup_su860928060825958358_a_b_b @ A6 @ B6 )
= B6 ) ) ) ).
% fsubset_funion_eq
thf(fact_446_fset__eq__fsubset,axiom,
( ( ^ [Y3: fset_P5281107635120001194_a_b_b,Z: fset_P5281107635120001194_a_b_b] : ( Y3 = Z ) )
= ( ^ [A6: fset_P5281107635120001194_a_b_b,B6: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A6 @ B6 )
& ( ord_le789900035998834954_a_b_b @ B6 @ A6 ) ) ) ) ).
% fset_eq_fsubset
thf(fact_447_eqfelem__imp__iff,axiom,
! [X4: produc4558475209616630778_a_b_b,Y: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( X4 = Y )
=> ( ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
= ( fmembe3173364709796808819_a_b_b @ Y @ A3 ) ) ) ).
% eqfelem_imp_iff
thf(fact_448_if__split__fmem2,axiom,
! [A2: produc4558475209616630778_a_b_b,Q: $o,X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ A2 @ ( if_fse8812573537926886756_a_b_b @ Q @ X4 @ Y ) )
= ( ( Q
=> ( fmembe3173364709796808819_a_b_b @ A2 @ X4 ) )
& ( ~ Q
=> ( fmembe3173364709796808819_a_b_b @ A2 @ Y ) ) ) ) ).
% if_split_fmem2
thf(fact_449_if__split__fmem1,axiom,
! [Q: $o,X4: produc4558475209616630778_a_b_b,Y: produc4558475209616630778_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ ( if_Pro6329973184163622324_a_b_b @ Q @ X4 @ Y ) @ B2 )
= ( ( Q
=> ( fmembe3173364709796808819_a_b_b @ X4 @ B2 ) )
& ( ~ Q
=> ( fmembe3173364709796808819_a_b_b @ Y @ B2 ) ) ) ) ).
% if_split_fmem1
thf(fact_450_eqfset__imp__iff,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b,X4: produc4558475209616630778_a_b_b] :
( ( A3 = B3 )
=> ( ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
= ( fmembe3173364709796808819_a_b_b @ X4 @ B3 ) ) ) ).
% eqfset_imp_iff
thf(fact_451_fsubset__trans,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b,C: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ B3 )
=> ( ( ord_le789900035998834954_a_b_b @ B3 @ C )
=> ( ord_le789900035998834954_a_b_b @ A3 @ C ) ) ) ).
% fsubset_trans
thf(fact_452_eq__fmem__trans,axiom,
! [A2: produc4558475209616630778_a_b_b,B2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( A2 = B2 )
=> ( ( fmembe3173364709796808819_a_b_b @ B2 @ A3 )
=> ( fmembe3173364709796808819_a_b_b @ A2 @ A3 ) ) ) ).
% eq_fmem_trans
thf(fact_453_fsubset__refl,axiom,
! [A3: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ A3 @ A3 ) ).
% fsubset_refl
thf(fact_454_fequalityD2,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( A3 = B3 )
=> ( ord_le789900035998834954_a_b_b @ B3 @ A3 ) ) ).
% fequalityD2
thf(fact_455_fequalityD1,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( A3 = B3 )
=> ( ord_le789900035998834954_a_b_b @ A3 @ B3 ) ) ).
% fequalityD1
thf(fact_456_fequalityCE,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b,C2: produc4558475209616630778_a_b_b] :
( ( A3 = B3 )
=> ( ( ( fmembe3173364709796808819_a_b_b @ C2 @ A3 )
=> ~ ( fmembe3173364709796808819_a_b_b @ C2 @ B3 ) )
=> ~ ( ~ ( fmembe3173364709796808819_a_b_b @ C2 @ A3 )
=> ( fmembe3173364709796808819_a_b_b @ C2 @ B3 ) ) ) ) ).
% fequalityCE
thf(fact_457_fequalityE,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( A3 = B3 )
=> ~ ( ( ord_le789900035998834954_a_b_b @ A3 @ B3 )
=> ~ ( ord_le789900035998834954_a_b_b @ B3 @ A3 ) ) ) ).
% fequalityE
thf(fact_458_fsubsetD,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b,C2: produc4558475209616630778_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ B3 )
=> ( ( fmembe3173364709796808819_a_b_b @ C2 @ A3 )
=> ( fmembe3173364709796808819_a_b_b @ C2 @ B3 ) ) ) ).
% fsubsetD
thf(fact_459_fset__eqI,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ! [X: produc4558475209616630778_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ X @ A3 )
= ( fmembe3173364709796808819_a_b_b @ X @ B3 ) )
=> ( A3 = B3 ) ) ).
% fset_eqI
thf(fact_460_fin__mono,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b,X4: produc4558475209616630778_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ B3 )
=> ( ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ( fmembe3173364709796808819_a_b_b @ X4 @ B3 ) ) ) ).
% fin_mono
thf(fact_461_fmember_Orep__eq,axiom,
( fmember_b
= ( ^ [X3: b,Xa2: fset_b] : ( member_b @ X3 @ ( fset_b2 @ Xa2 ) ) ) ) ).
% fmember.rep_eq
thf(fact_462_fmember_Orep__eq,axiom,
( fmembe3173364709796808819_a_b_b
= ( ^ [X3: produc4558475209616630778_a_b_b,Xa2: fset_P5281107635120001194_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ ( fset_P783253628892185035_a_b_b @ Xa2 ) ) ) ) ).
% fmember.rep_eq
thf(fact_463_notin__fset,axiom,
! [X4: b,S2: fset_b] :
( ( ~ ( fmember_b @ X4 @ S2 ) )
= ( ~ ( member_b @ X4 @ ( fset_b2 @ S2 ) ) ) ) ).
% notin_fset
thf(fact_464_notin__fset,axiom,
! [X4: produc4558475209616630778_a_b_b,S2: fset_P5281107635120001194_a_b_b] :
( ( ~ ( fmembe3173364709796808819_a_b_b @ X4 @ S2 ) )
= ( ~ ( member4380921116106875537_a_b_b @ X4 @ ( fset_P783253628892185035_a_b_b @ S2 ) ) ) ) ).
% notin_fset
thf(fact_465_fmember__iff__member__fset,axiom,
( fmember_b
= ( ^ [X3: b,A6: fset_b] : ( member_b @ X3 @ ( fset_b2 @ A6 ) ) ) ) ).
% fmember_iff_member_fset
thf(fact_466_fmember__iff__member__fset,axiom,
( fmembe3173364709796808819_a_b_b
= ( ^ [X3: produc4558475209616630778_a_b_b,A6: fset_P5281107635120001194_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ ( fset_P783253628892185035_a_b_b @ A6 ) ) ) ) ).
% fmember_iff_member_fset
thf(fact_467_fset__induct__stronger,axiom,
! [P: fset_P5281107635120001194_a_b_b > $o,S2: fset_P5281107635120001194_a_b_b] :
( ( P @ bot_bo2895716411488905534_a_b_b )
=> ( ! [X: produc4558475209616630778_a_b_b,S3: fset_P5281107635120001194_a_b_b] :
( ~ ( fmembe3173364709796808819_a_b_b @ X @ S3 )
=> ( ( P @ S3 )
=> ( P @ ( finser8437519239679886002_a_b_b @ X @ S3 ) ) ) )
=> ( P @ S2 ) ) ) ).
% fset_induct_stronger
thf(fact_468_mk__disjoint__finsert,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ A2 @ A3 )
=> ? [B7: fset_P5281107635120001194_a_b_b] :
( ( A3
= ( finser8437519239679886002_a_b_b @ A2 @ B7 ) )
& ~ ( fmembe3173364709796808819_a_b_b @ A2 @ B7 ) ) ) ).
% mk_disjoint_finsert
thf(fact_469_fset__strong__cases,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b] :
( ( Xs2 != bot_bo2895716411488905534_a_b_b )
=> ~ ! [Ys2: fset_P5281107635120001194_a_b_b,X: produc4558475209616630778_a_b_b] :
( ~ ( fmembe3173364709796808819_a_b_b @ X @ Ys2 )
=> ( Xs2
!= ( finser8437519239679886002_a_b_b @ X @ Ys2 ) ) ) ) ).
% fset_strong_cases
thf(fact_470_fsingleton__iff,axiom,
! [B2: produc4558475209616630778_a_b_b,A2: produc4558475209616630778_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ B2 @ ( finser8437519239679886002_a_b_b @ A2 @ bot_bo2895716411488905534_a_b_b ) )
= ( B2 = A2 ) ) ).
% fsingleton_iff
thf(fact_471_finsert__eq__iff,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B2: produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ~ ( fmembe3173364709796808819_a_b_b @ A2 @ A3 )
=> ( ~ ( fmembe3173364709796808819_a_b_b @ B2 @ B3 )
=> ( ( ( finser8437519239679886002_a_b_b @ A2 @ A3 )
= ( finser8437519239679886002_a_b_b @ B2 @ B3 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B3 ) )
& ( ( A2 != B2 )
=> ? [C3: fset_P5281107635120001194_a_b_b] :
( ( A3
= ( finser8437519239679886002_a_b_b @ B2 @ C3 ) )
& ~ ( fmembe3173364709796808819_a_b_b @ B2 @ C3 )
& ( B3
= ( finser8437519239679886002_a_b_b @ A2 @ C3 ) )
& ~ ( fmembe3173364709796808819_a_b_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% finsert_eq_iff
thf(fact_472_finsert__absorb,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ A2 @ A3 )
=> ( ( finser8437519239679886002_a_b_b @ A2 @ A3 )
= A3 ) ) ).
% finsert_absorb
thf(fact_473_finsert__ident,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ~ ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ( ~ ( fmembe3173364709796808819_a_b_b @ X4 @ B3 )
=> ( ( ( finser8437519239679886002_a_b_b @ X4 @ A3 )
= ( finser8437519239679886002_a_b_b @ X4 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% finsert_ident
thf(fact_474_fset__induct2,axiom,
! [P: fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b > $o,Xsa: fset_P5281107635120001194_a_b_b,Ysa: fset_P5281107635120001194_a_b_b] :
( ( P @ bot_bo2895716411488905534_a_b_b @ bot_bo2895716411488905534_a_b_b )
=> ( ! [X: produc4558475209616630778_a_b_b,Xs: fset_P5281107635120001194_a_b_b] :
( ~ ( fmembe3173364709796808819_a_b_b @ X @ Xs )
=> ( P @ ( finser8437519239679886002_a_b_b @ X @ Xs ) @ bot_bo2895716411488905534_a_b_b ) )
=> ( ! [Y4: produc4558475209616630778_a_b_b,Ys2: fset_P5281107635120001194_a_b_b] :
( ~ ( fmembe3173364709796808819_a_b_b @ Y4 @ Ys2 )
=> ( P @ bot_bo2895716411488905534_a_b_b @ ( finser8437519239679886002_a_b_b @ Y4 @ Ys2 ) ) )
=> ( ! [X: produc4558475209616630778_a_b_b,Xs: fset_P5281107635120001194_a_b_b,Y4: produc4558475209616630778_a_b_b,Ys2: fset_P5281107635120001194_a_b_b] :
( ( P @ Xs @ Ys2 )
=> ( ~ ( fmembe3173364709796808819_a_b_b @ X @ Xs )
=> ( ~ ( fmembe3173364709796808819_a_b_b @ Y4 @ Ys2 )
=> ( P @ ( finser8437519239679886002_a_b_b @ X @ Xs ) @ ( finser8437519239679886002_a_b_b @ Y4 @ Ys2 ) ) ) ) )
=> ( P @ Xsa @ Ysa ) ) ) ) ) ).
% fset_induct2
thf(fact_475_set__finsert,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ~ ! [B7: fset_P5281107635120001194_a_b_b] :
( ( A3
= ( finser8437519239679886002_a_b_b @ X4 @ B7 ) )
=> ( fmembe3173364709796808819_a_b_b @ X4 @ B7 ) ) ) ).
% set_finsert
thf(fact_476_finsertI2,axiom,
! [A2: produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b,B2: produc4558475209616630778_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ A2 @ B3 )
=> ( fmembe3173364709796808819_a_b_b @ A2 @ ( finser8437519239679886002_a_b_b @ B2 @ B3 ) ) ) ).
% finsertI2
thf(fact_477_finsertI1,axiom,
! [A2: produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b] : ( fmembe3173364709796808819_a_b_b @ A2 @ ( finser8437519239679886002_a_b_b @ A2 @ B3 ) ) ).
% finsertI1
thf(fact_478_finsertE,axiom,
! [A2: produc4558475209616630778_a_b_b,B2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ A2 @ ( finser8437519239679886002_a_b_b @ B2 @ A3 ) )
=> ( ( A2 != B2 )
=> ( fmembe3173364709796808819_a_b_b @ A2 @ A3 ) ) ) ).
% finsertE
thf(fact_479_fsubset__fsingletonD,axiom,
! [A3: fset_P5281107635120001194_a_b_b,X4: produc4558475209616630778_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) )
=> ( ( A3 = bot_bo2895716411488905534_a_b_b )
| ( A3
= ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) ) ) ) ).
% fsubset_fsingletonD
thf(fact_480_fsubset__finsertI2,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b,B2: produc4558475209616630778_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ B3 )
=> ( ord_le789900035998834954_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ B2 @ B3 ) ) ) ).
% fsubset_finsertI2
thf(fact_481_fsubset__finsertI,axiom,
! [B3: fset_P5281107635120001194_a_b_b,A2: produc4558475209616630778_a_b_b] : ( ord_le789900035998834954_a_b_b @ B3 @ ( finser8437519239679886002_a_b_b @ A2 @ B3 ) ) ).
% fsubset_finsertI
thf(fact_482_finsert__mono,axiom,
! [C: fset_P5281107635120001194_a_b_b,D2: fset_P5281107635120001194_a_b_b,A2: produc4558475209616630778_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ C @ D2 )
=> ( ord_le789900035998834954_a_b_b @ ( finser8437519239679886002_a_b_b @ A2 @ C ) @ ( finser8437519239679886002_a_b_b @ A2 @ D2 ) ) ) ).
% finsert_mono
thf(fact_483_comp__fun__commute_Offold__finsert2,axiom,
! [F: produc4558475209616630778_a_b_b > a > a,X4: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,Z2: a] :
( ( finite414203908571218417_b_b_a @ F )
=> ( ~ ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ( ( ffold_2783168711033344739_b_b_a @ F @ Z2 @ ( finser8437519239679886002_a_b_b @ X4 @ A3 ) )
= ( ffold_2783168711033344739_b_b_a @ F @ ( F @ X4 @ Z2 ) @ A3 ) ) ) ) ).
% comp_fun_commute.ffold_finsert2
thf(fact_484_comp__fun__commute_Offold__finsert,axiom,
! [F: produc4558475209616630778_a_b_b > a > a,X4: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,Z2: a] :
( ( finite414203908571218417_b_b_a @ F )
=> ( ~ ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ( ( ffold_2783168711033344739_b_b_a @ F @ Z2 @ ( finser8437519239679886002_a_b_b @ X4 @ A3 ) )
= ( F @ X4 @ ( ffold_2783168711033344739_b_b_a @ F @ Z2 @ A3 ) ) ) ) ) ).
% comp_fun_commute.ffold_finsert
thf(fact_485_comp__fun__commute_Offold__fun__left__comm,axiom,
! [F: produc4558475209616630778_a_b_b > a > a,X4: produc4558475209616630778_a_b_b,Z2: a,A3: fset_P5281107635120001194_a_b_b] :
( ( finite414203908571218417_b_b_a @ F )
=> ( ( F @ X4 @ ( ffold_2783168711033344739_b_b_a @ F @ Z2 @ A3 ) )
= ( ffold_2783168711033344739_b_b_a @ F @ ( F @ X4 @ Z2 ) @ A3 ) ) ) ).
% comp_fun_commute.ffold_fun_left_comm
thf(fact_486_inf__sup__ord_I4_J,axiom,
! [Y: set_b,X4: set_b] : ( ord_less_eq_set_b @ Y @ ( sup_sup_set_b @ X4 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_487_inf__sup__ord_I4_J,axiom,
! [Y: fset_P5281107635120001194_a_b_b,X4: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ Y @ ( sup_su860928060825958358_a_b_b @ X4 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_488_inf__sup__ord_I4_J,axiom,
! [Y: nat,X4: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X4 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_489_inf__sup__ord_I3_J,axiom,
! [X4: set_b,Y: set_b] : ( ord_less_eq_set_b @ X4 @ ( sup_sup_set_b @ X4 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_490_inf__sup__ord_I3_J,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ X4 @ ( sup_su860928060825958358_a_b_b @ X4 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_491_inf__sup__ord_I3_J,axiom,
! [X4: nat,Y: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_492_le__supE,axiom,
! [A2: set_b,B2: set_b,X4: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ A2 @ B2 ) @ X4 )
=> ~ ( ( ord_less_eq_set_b @ A2 @ X4 )
=> ~ ( ord_less_eq_set_b @ B2 @ X4 ) ) ) ).
% le_supE
thf(fact_493_le__supE,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,X4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ ( sup_su860928060825958358_a_b_b @ A2 @ B2 ) @ X4 )
=> ~ ( ( ord_le789900035998834954_a_b_b @ A2 @ X4 )
=> ~ ( ord_le789900035998834954_a_b_b @ B2 @ X4 ) ) ) ).
% le_supE
thf(fact_494_le__supE,axiom,
! [A2: nat,B2: nat,X4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X4 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X4 )
=> ~ ( ord_less_eq_nat @ B2 @ X4 ) ) ) ).
% le_supE
thf(fact_495_le__supI,axiom,
! [A2: set_b,X4: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ X4 )
=> ( ( ord_less_eq_set_b @ B2 @ X4 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A2 @ B2 ) @ X4 ) ) ) ).
% le_supI
thf(fact_496_le__supI,axiom,
! [A2: fset_P5281107635120001194_a_b_b,X4: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ X4 )
=> ( ( ord_le789900035998834954_a_b_b @ B2 @ X4 )
=> ( ord_le789900035998834954_a_b_b @ ( sup_su860928060825958358_a_b_b @ A2 @ B2 ) @ X4 ) ) ) ).
% le_supI
thf(fact_497_le__supI,axiom,
! [A2: nat,X4: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X4 )
=> ( ( ord_less_eq_nat @ B2 @ X4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X4 ) ) ) ).
% le_supI
thf(fact_498_sup__ge1,axiom,
! [X4: set_b,Y: set_b] : ( ord_less_eq_set_b @ X4 @ ( sup_sup_set_b @ X4 @ Y ) ) ).
% sup_ge1
thf(fact_499_sup__ge1,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ X4 @ ( sup_su860928060825958358_a_b_b @ X4 @ Y ) ) ).
% sup_ge1
thf(fact_500_sup__ge1,axiom,
! [X4: nat,Y: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y ) ) ).
% sup_ge1
thf(fact_501_sup__ge2,axiom,
! [Y: set_b,X4: set_b] : ( ord_less_eq_set_b @ Y @ ( sup_sup_set_b @ X4 @ Y ) ) ).
% sup_ge2
thf(fact_502_sup__ge2,axiom,
! [Y: fset_P5281107635120001194_a_b_b,X4: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ Y @ ( sup_su860928060825958358_a_b_b @ X4 @ Y ) ) ).
% sup_ge2
thf(fact_503_sup__ge2,axiom,
! [Y: nat,X4: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X4 @ Y ) ) ).
% sup_ge2
thf(fact_504_le__supI1,axiom,
! [X4: set_b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ X4 @ A2 )
=> ( ord_less_eq_set_b @ X4 @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_505_le__supI1,axiom,
! [X4: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X4 @ A2 )
=> ( ord_le789900035998834954_a_b_b @ X4 @ ( sup_su860928060825958358_a_b_b @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_506_le__supI1,axiom,
! [X4: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_507_le__supI2,axiom,
! [X4: set_b,B2: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ X4 @ B2 )
=> ( ord_less_eq_set_b @ X4 @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_508_le__supI2,axiom,
! [X4: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X4 @ B2 )
=> ( ord_le789900035998834954_a_b_b @ X4 @ ( sup_su860928060825958358_a_b_b @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_509_le__supI2,axiom,
! [X4: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X4 @ B2 )
=> ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_510_sup_Omono,axiom,
! [C2: set_b,A2: set_b,D: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ C2 @ A2 )
=> ( ( ord_less_eq_set_b @ D @ B2 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ C2 @ D ) @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_511_sup_Omono,axiom,
! [C2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b,D: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ C2 @ A2 )
=> ( ( ord_le789900035998834954_a_b_b @ D @ B2 )
=> ( ord_le789900035998834954_a_b_b @ ( sup_su860928060825958358_a_b_b @ C2 @ D ) @ ( sup_su860928060825958358_a_b_b @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_512_sup_Omono,axiom,
! [C2: nat,A2: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_513_sup__mono,axiom,
! [A2: set_b,C2: set_b,B2: set_b,D: set_b] :
( ( ord_less_eq_set_b @ A2 @ C2 )
=> ( ( ord_less_eq_set_b @ B2 @ D )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A2 @ B2 ) @ ( sup_sup_set_b @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_514_sup__mono,axiom,
! [A2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,D: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ C2 )
=> ( ( ord_le789900035998834954_a_b_b @ B2 @ D )
=> ( ord_le789900035998834954_a_b_b @ ( sup_su860928060825958358_a_b_b @ A2 @ B2 ) @ ( sup_su860928060825958358_a_b_b @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_515_sup__mono,axiom,
! [A2: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_516_sup__least,axiom,
! [Y: set_b,X4: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ Y @ X4 )
=> ( ( ord_less_eq_set_b @ Z2 @ X4 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ Y @ Z2 ) @ X4 ) ) ) ).
% sup_least
thf(fact_517_sup__least,axiom,
! [Y: fset_P5281107635120001194_a_b_b,X4: fset_P5281107635120001194_a_b_b,Z2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ Y @ X4 )
=> ( ( ord_le789900035998834954_a_b_b @ Z2 @ X4 )
=> ( ord_le789900035998834954_a_b_b @ ( sup_su860928060825958358_a_b_b @ Y @ Z2 ) @ X4 ) ) ) ).
% sup_least
thf(fact_518_sup__least,axiom,
! [Y: nat,X4: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ( ( ord_less_eq_nat @ Z2 @ X4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X4 ) ) ) ).
% sup_least
thf(fact_519_le__iff__sup,axiom,
( ord_less_eq_set_b
= ( ^ [X3: set_b,Y5: set_b] :
( ( sup_sup_set_b @ X3 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_520_le__iff__sup,axiom,
( ord_le789900035998834954_a_b_b
= ( ^ [X3: fset_P5281107635120001194_a_b_b,Y5: fset_P5281107635120001194_a_b_b] :
( ( sup_su860928060825958358_a_b_b @ X3 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_521_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y5: nat] :
( ( sup_sup_nat @ X3 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_522_sup_OorderE,axiom,
! [B2: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_523_sup_OorderE,axiom,
! [B2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ B2 @ A2 )
=> ( A2
= ( sup_su860928060825958358_a_b_b @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_524_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_525_sup_OorderI,axiom,
! [A2: set_b,B2: set_b] :
( ( A2
= ( sup_sup_set_b @ A2 @ B2 ) )
=> ( ord_less_eq_set_b @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_526_sup_OorderI,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( A2
= ( sup_su860928060825958358_a_b_b @ A2 @ B2 ) )
=> ( ord_le789900035998834954_a_b_b @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_527_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_528_sup__unique,axiom,
! [F: set_b > set_b > set_b,X4: set_b,Y: set_b] :
( ! [X: set_b,Y4: set_b] : ( ord_less_eq_set_b @ X @ ( F @ X @ Y4 ) )
=> ( ! [X: set_b,Y4: set_b] : ( ord_less_eq_set_b @ Y4 @ ( F @ X @ Y4 ) )
=> ( ! [X: set_b,Y4: set_b,Z3: set_b] :
( ( ord_less_eq_set_b @ Y4 @ X )
=> ( ( ord_less_eq_set_b @ Z3 @ X )
=> ( ord_less_eq_set_b @ ( F @ Y4 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_set_b @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_529_sup__unique,axiom,
! [F: fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b,X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] :
( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ X @ ( F @ X @ Y4 ) )
=> ( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ Y4 @ ( F @ X @ Y4 ) )
=> ( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b,Z3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ Y4 @ X )
=> ( ( ord_le789900035998834954_a_b_b @ Z3 @ X )
=> ( ord_le789900035998834954_a_b_b @ ( F @ Y4 @ Z3 ) @ X ) ) )
=> ( ( sup_su860928060825958358_a_b_b @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_530_sup__unique,axiom,
! [F: nat > nat > nat,X4: nat,Y: nat] :
( ! [X: nat,Y4: nat] : ( ord_less_eq_nat @ X @ ( F @ X @ Y4 ) )
=> ( ! [X: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X @ Y4 ) )
=> ( ! [X: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y4 @ X )
=> ( ( ord_less_eq_nat @ Z3 @ X )
=> ( ord_less_eq_nat @ ( F @ Y4 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_nat @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_531_sup_Oabsorb1,axiom,
! [B2: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( sup_sup_set_b @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_532_sup_Oabsorb1,axiom,
! [B2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ B2 @ A2 )
=> ( ( sup_su860928060825958358_a_b_b @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_533_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_534_sup_Oabsorb2,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( sup_sup_set_b @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_535_sup_Oabsorb2,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( ( sup_su860928060825958358_a_b_b @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_536_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_537_sup__absorb1,axiom,
! [Y: set_b,X4: set_b] :
( ( ord_less_eq_set_b @ Y @ X4 )
=> ( ( sup_sup_set_b @ X4 @ Y )
= X4 ) ) ).
% sup_absorb1
thf(fact_538_sup__absorb1,axiom,
! [Y: fset_P5281107635120001194_a_b_b,X4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ Y @ X4 )
=> ( ( sup_su860928060825958358_a_b_b @ X4 @ Y )
= X4 ) ) ).
% sup_absorb1
thf(fact_539_sup__absorb1,axiom,
! [Y: nat,X4: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ( ( sup_sup_nat @ X4 @ Y )
= X4 ) ) ).
% sup_absorb1
thf(fact_540_sup__absorb2,axiom,
! [X4: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y )
=> ( ( sup_sup_set_b @ X4 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_541_sup__absorb2,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X4 @ Y )
=> ( ( sup_su860928060825958358_a_b_b @ X4 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_542_sup__absorb2,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( sup_sup_nat @ X4 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_543_sup_OboundedE,axiom,
! [B2: set_b,C2: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ~ ( ord_less_eq_set_b @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_544_sup_OboundedE,axiom,
! [B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ ( sup_su860928060825958358_a_b_b @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_le789900035998834954_a_b_b @ B2 @ A2 )
=> ~ ( ord_le789900035998834954_a_b_b @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_545_sup_OboundedE,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_546_sup_OboundedI,axiom,
! [B2: set_b,A2: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( ord_less_eq_set_b @ C2 @ A2 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_547_sup_OboundedI,axiom,
! [B2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ B2 @ A2 )
=> ( ( ord_le789900035998834954_a_b_b @ C2 @ A2 )
=> ( ord_le789900035998834954_a_b_b @ ( sup_su860928060825958358_a_b_b @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_548_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_549_sup_Oorder__iff,axiom,
( ord_less_eq_set_b
= ( ^ [B: set_b,A: set_b] :
( A
= ( sup_sup_set_b @ A @ B ) ) ) ) ).
% sup.order_iff
thf(fact_550_sup_Oorder__iff,axiom,
( ord_le789900035998834954_a_b_b
= ( ^ [B: fset_P5281107635120001194_a_b_b,A: fset_P5281107635120001194_a_b_b] :
( A
= ( sup_su860928060825958358_a_b_b @ A @ B ) ) ) ) ).
% sup.order_iff
thf(fact_551_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B: nat,A: nat] :
( A
= ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.order_iff
thf(fact_552_sup_Ocobounded1,axiom,
! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ A2 @ ( sup_sup_set_b @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_553_sup_Ocobounded1,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ A2 @ ( sup_su860928060825958358_a_b_b @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_554_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_555_sup_Ocobounded2,axiom,
! [B2: set_b,A2: set_b] : ( ord_less_eq_set_b @ B2 @ ( sup_sup_set_b @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_556_sup_Ocobounded2,axiom,
! [B2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ B2 @ ( sup_su860928060825958358_a_b_b @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_557_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_558_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_b
= ( ^ [B: set_b,A: set_b] :
( ( sup_sup_set_b @ A @ B )
= A ) ) ) ).
% sup.absorb_iff1
thf(fact_559_sup_Oabsorb__iff1,axiom,
( ord_le789900035998834954_a_b_b
= ( ^ [B: fset_P5281107635120001194_a_b_b,A: fset_P5281107635120001194_a_b_b] :
( ( sup_su860928060825958358_a_b_b @ A @ B )
= A ) ) ) ).
% sup.absorb_iff1
thf(fact_560_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B: nat,A: nat] :
( ( sup_sup_nat @ A @ B )
= A ) ) ) ).
% sup.absorb_iff1
thf(fact_561_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_b
= ( ^ [A: set_b,B: set_b] :
( ( sup_sup_set_b @ A @ B )
= B ) ) ) ).
% sup.absorb_iff2
thf(fact_562_sup_Oabsorb__iff2,axiom,
( ord_le789900035998834954_a_b_b
= ( ^ [A: fset_P5281107635120001194_a_b_b,B: fset_P5281107635120001194_a_b_b] :
( ( sup_su860928060825958358_a_b_b @ A @ B )
= B ) ) ) ).
% sup.absorb_iff2
thf(fact_563_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
( ( sup_sup_nat @ A @ B )
= B ) ) ) ).
% sup.absorb_iff2
thf(fact_564_sup_OcoboundedI1,axiom,
! [C2: set_b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ C2 @ A2 )
=> ( ord_less_eq_set_b @ C2 @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_565_sup_OcoboundedI1,axiom,
! [C2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ C2 @ A2 )
=> ( ord_le789900035998834954_a_b_b @ C2 @ ( sup_su860928060825958358_a_b_b @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_566_sup_OcoboundedI1,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_567_sup_OcoboundedI2,axiom,
! [C2: set_b,B2: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ C2 @ B2 )
=> ( ord_less_eq_set_b @ C2 @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_568_sup_OcoboundedI2,axiom,
! [C2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ C2 @ B2 )
=> ( ord_le789900035998834954_a_b_b @ C2 @ ( sup_su860928060825958358_a_b_b @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_569_sup_OcoboundedI2,axiom,
! [C2: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_570_ffold__commute__supset,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,Ys: fset_P5281107635120001194_a_b_b,P: fset_P5281107635120001194_a_b_b > $o,Q: produc4558475209616630778_a_b_b > a > $o,R3: produc4558475209616630778_a_b_b > a > a,Acc: a] :
( ( ord_le789900035998834954_a_b_b @ Xs2 @ Ys )
=> ( ( P @ Ys )
=> ( ! [Ys2: fset_P5281107635120001194_a_b_b,Xs: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ Xs @ Ys2 )
=> ( ( P @ Ys2 )
=> ( P @ Xs ) ) )
=> ( ! [Xs: fset_P5281107635120001194_a_b_b] :
( finite414203908571218417_b_b_a
@ ^ [A: produc4558475209616630778_a_b_b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ A @ ( fset_P783253628892185035_a_b_b @ Xs ) )
| ~ ( Q @ A @ B )
| ~ ( P @ Xs ) )
@ B
@ ( R3 @ A @ B ) ) )
=> ( ( ffold_2783168711033344739_b_b_a
@ ^ [A: produc4558475209616630778_a_b_b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ A @ ( fset_P783253628892185035_a_b_b @ Ys ) )
| ~ ( Q @ A @ B )
| ~ ( P @ Ys ) )
@ B
@ ( R3 @ A @ B ) )
@ Acc
@ Xs2 )
= ( ffold_2783168711033344739_b_b_a
@ ^ [A: produc4558475209616630778_a_b_b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ A @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( Q @ A @ B )
| ~ ( P @ Xs2 ) )
@ B
@ ( R3 @ A @ B ) )
@ Acc
@ Xs2 ) ) ) ) ) ) ).
% ffold_commute_supset
thf(fact_571_wf__darcs__sub,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,Ys: fset_P5281107635120001194_a_b_b,R4: a,R2: a] :
( ( ord_le789900035998834954_a_b_b @ Xs2 @ Ys )
=> ( ( wf_darcs_a_b @ ( node_a_b @ R4 @ Ys ) )
=> ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) ) ) ).
% wf_darcs_sub
thf(fact_572_fset__cong,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] :
( ( ( fset_P783253628892185035_a_b_b @ X4 )
= ( fset_P783253628892185035_a_b_b @ Y ) )
= ( X4 = Y ) ) ).
% fset_cong
thf(fact_573_funion__fsingleton__iff,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b,X4: produc4558475209616630778_a_b_b] :
( ( ( sup_su860928060825958358_a_b_b @ A3 @ B3 )
= ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) )
= ( ( ( A3 = bot_bo2895716411488905534_a_b_b )
& ( B3
= ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) ) )
| ( ( A3
= ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) )
& ( B3 = bot_bo2895716411488905534_a_b_b ) )
| ( ( A3
= ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) )
& ( B3
= ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) ) ) ) ) ).
% funion_fsingleton_iff
thf(fact_574_fsingleton__funion__iff,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b )
= ( sup_su860928060825958358_a_b_b @ A3 @ B3 ) )
= ( ( ( A3 = bot_bo2895716411488905534_a_b_b )
& ( B3
= ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) ) )
| ( ( A3
= ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) )
& ( B3 = bot_bo2895716411488905534_a_b_b ) )
| ( ( A3
= ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) )
& ( B3
= ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) ) ) ) ) ).
% fsingleton_funion_iff
thf(fact_575_finsert__not__fempty,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( finser8437519239679886002_a_b_b @ A2 @ A3 )
!= bot_bo2895716411488905534_a_b_b ) ).
% finsert_not_fempty
thf(fact_576_fsingleton__inject,axiom,
! [A2: produc4558475209616630778_a_b_b,B2: produc4558475209616630778_a_b_b] :
( ( ( finser8437519239679886002_a_b_b @ A2 @ bot_bo2895716411488905534_a_b_b )
= ( finser8437519239679886002_a_b_b @ B2 @ bot_bo2895716411488905534_a_b_b ) )
=> ( A2 = B2 ) ) ).
% fsingleton_inject
thf(fact_577_finsert__is__funion,axiom,
( finser8437519239679886002_a_b_b
= ( ^ [A: produc4558475209616630778_a_b_b] : ( sup_su860928060825958358_a_b_b @ ( finser8437519239679886002_a_b_b @ A @ bot_bo2895716411488905534_a_b_b ) ) ) ) ).
% finsert_is_funion
thf(fact_578_fdoubleton__eq__iff,axiom,
! [A2: produc4558475209616630778_a_b_b,B2: produc4558475209616630778_a_b_b,C2: produc4558475209616630778_a_b_b,D: produc4558475209616630778_a_b_b] :
( ( ( finser8437519239679886002_a_b_b @ A2 @ ( finser8437519239679886002_a_b_b @ B2 @ bot_bo2895716411488905534_a_b_b ) )
= ( finser8437519239679886002_a_b_b @ C2 @ ( finser8437519239679886002_a_b_b @ D @ bot_bo2895716411488905534_a_b_b ) ) )
= ( ( ( A2 = C2 )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C2 ) ) ) ) ).
% fdoubleton_eq_iff
thf(fact_579_finsert__commute,axiom,
! [X4: produc4558475209616630778_a_b_b,Y: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( finser8437519239679886002_a_b_b @ X4 @ ( finser8437519239679886002_a_b_b @ Y @ A3 ) )
= ( finser8437519239679886002_a_b_b @ Y @ ( finser8437519239679886002_a_b_b @ X4 @ A3 ) ) ) ).
% finsert_commute
thf(fact_580_fset__exhaust,axiom,
! [S2: fset_P5281107635120001194_a_b_b] :
( ( S2 != bot_bo2895716411488905534_a_b_b )
=> ~ ! [X: produc4558475209616630778_a_b_b,S4: fset_P5281107635120001194_a_b_b] :
( S2
!= ( finser8437519239679886002_a_b_b @ X @ S4 ) ) ) ).
% fset_exhaust
thf(fact_581_FSet_Ofset__induct,axiom,
! [P: fset_P5281107635120001194_a_b_b > $o,S2: fset_P5281107635120001194_a_b_b] :
( ( P @ bot_bo2895716411488905534_a_b_b )
=> ( ! [X: produc4558475209616630778_a_b_b,S3: fset_P5281107635120001194_a_b_b] :
( ( P @ S3 )
=> ( P @ ( finser8437519239679886002_a_b_b @ X @ S3 ) ) )
=> ( P @ S2 ) ) ) ).
% FSet.fset_induct
thf(fact_582_disjoint__darcs__single,axiom,
! [E4: b,T: dtree_a_b] :
( ( ~ ( member_b @ E4 @ ( darcs_a_b @ T ) ) )
= ( ! [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ ( produc331601717337510060_a_b_b @ T @ E4 ) @ bot_bo2895716411488905534_a_b_b ) ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ ( produc331601717337510060_a_b_b @ T @ E4 ) @ bot_bo2895716411488905534_a_b_b ) ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X3 ) ) ) ) ).
% disjoint_darcs_single
thf(fact_583_wf__darcs__if__darcs_H__aux,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,R2: a] :
( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E: b] : ( wf_darcs_a_b @ Y5 )
@ X ) )
=> ( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X ) )
=> ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) ) ) ).
% wf_darcs_if_darcs'_aux
thf(fact_584_disjoint__darcs__if__wf__xs,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b] :
( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ! [X5: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X5 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X5 ) ) ) ).
% disjoint_darcs_if_wf_xs
thf(fact_585_dtail__ffold__notelem__eq__def,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,E4: b,Ys: fset_P5281107635120001194_a_b_b,R2: a,Def: b > a] :
( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [T3: dtree_a_b,E12: b] :
~ ( member_b @ E4 @ ( darcs_a_b @ T3 ) )
@ X ) )
=> ( ( ffold_8367945289176929151_b_b_a
@ ( produc4313903556115589696_a_b_a
@ ^ [X3: dtree_a_b,E2: b,B: b > a] :
( if_b_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Ys ) )
| ~ ( member_b @ E4 @ ( darcs_a_b @ X3 ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Ys ) ) )
@ B
@ ( dtail_a_b @ X3 @ Def ) ) )
@ Def
@ Xs2 )
= Def ) ) ).
% dtail_ffold_notelem_eq_def
thf(fact_586_disjoint__darcs__subset,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,Ys: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ Xs2 @ Ys )
=> ( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Ys ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Ys ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X ) )
=> ! [X5: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X5 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X5 ) ) ) ) ).
% disjoint_darcs_subset
thf(fact_587_sup__Un__eq,axiom,
! [R3: set_Pr3012420139608375472_a_b_b,S2: set_Pr3012420139608375472_a_b_b] :
( ( sup_su4209747780764569001_b_b_o
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ R3 )
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ S2 ) )
= ( ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ ( sup_su2887895092731772380_a_b_b @ R3 @ S2 ) ) ) ) ).
% sup_Un_eq
thf(fact_588_sup__Un__eq,axiom,
! [R3: set_b,S2: set_b] :
( ( sup_sup_b_o
@ ^ [X3: b] : ( member_b @ X3 @ R3 )
@ ^ [X3: b] : ( member_b @ X3 @ S2 ) )
= ( ^ [X3: b] : ( member_b @ X3 @ ( sup_sup_set_b @ R3 @ S2 ) ) ) ) ).
% sup_Un_eq
thf(fact_589_order__refl,axiom,
! [X4: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ X4 @ X4 ) ).
% order_refl
thf(fact_590_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_591_dual__order_Orefl,axiom,
! [A2: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_592_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_593_empty__subsetI,axiom,
! [A3: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A3 ) ).
% empty_subsetI
thf(fact_594_subset__empty,axiom,
! [A3: set_b] :
( ( ord_less_eq_set_b @ A3 @ bot_bot_set_b )
= ( A3 = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_595_insert__subset,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( ord_le146215904626753808_a_b_b @ ( insert1613891728210272810_a_b_b @ X4 @ A3 ) @ B3 )
= ( ( member4380921116106875537_a_b_b @ X4 @ B3 )
& ( ord_le146215904626753808_a_b_b @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_596_insert__subset,axiom,
! [X4: b,A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X4 @ A3 ) @ B3 )
= ( ( member_b @ X4 @ B3 )
& ( ord_less_eq_set_b @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_597_Int__iff,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ C2 @ ( inf_in6138156342456174402_a_b_b @ A3 @ B3 ) )
= ( ( member4380921116106875537_a_b_b @ C2 @ A3 )
& ( member4380921116106875537_a_b_b @ C2 @ B3 ) ) ) ).
% Int_iff
thf(fact_598_Int__iff,axiom,
! [C2: b,A3: set_b,B3: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A3 @ B3 ) )
= ( ( member_b @ C2 @ A3 )
& ( member_b @ C2 @ B3 ) ) ) ).
% Int_iff
thf(fact_599_IntI,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ C2 @ A3 )
=> ( ( member4380921116106875537_a_b_b @ C2 @ B3 )
=> ( member4380921116106875537_a_b_b @ C2 @ ( inf_in6138156342456174402_a_b_b @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_600_IntI,axiom,
! [C2: b,A3: set_b,B3: set_b] :
( ( member_b @ C2 @ A3 )
=> ( ( member_b @ C2 @ B3 )
=> ( member_b @ C2 @ ( inf_inf_set_b @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_601_Un__subset__iff,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ A3 @ B3 ) @ C )
= ( ( ord_less_eq_set_b @ A3 @ C )
& ( ord_less_eq_set_b @ B3 @ C ) ) ) ).
% Un_subset_iff
thf(fact_602_inf_Obounded__iff,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ ( inf_in7138637532943773244_a_b_b @ B2 @ C2 ) )
= ( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
& ( ord_le789900035998834954_a_b_b @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_603_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_604_le__inf__iff,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b,Z2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X4 @ ( inf_in7138637532943773244_a_b_b @ Y @ Z2 ) )
= ( ( ord_le789900035998834954_a_b_b @ X4 @ Y )
& ( ord_le789900035998834954_a_b_b @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_605_le__inf__iff,axiom,
! [X4: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X4 @ Y )
& ( ord_less_eq_nat @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_606_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X4 )
= bot_bot_set_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_607_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_b] :
( ( inf_inf_set_b @ X4 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_608_inf__bot__right,axiom,
! [X4: set_b] :
( ( inf_inf_set_b @ X4 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% inf_bot_right
thf(fact_609_inf__bot__left,axiom,
! [X4: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X4 )
= bot_bot_set_b ) ).
% inf_bot_left
thf(fact_610_singleton__insert__inj__eq_H,axiom,
! [A2: b,A3: set_b,B2: b] :
( ( ( insert_b @ A2 @ A3 )
= ( insert_b @ B2 @ bot_bot_set_b ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_b @ A3 @ ( insert_b @ B2 @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_611_singleton__insert__inj__eq,axiom,
! [B2: b,A2: b,A3: set_b] :
( ( ( insert_b @ B2 @ bot_bot_set_b )
= ( insert_b @ A2 @ A3 ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_b @ A3 @ ( insert_b @ B2 @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_612_sup__inf__absorb,axiom,
! [X4: set_b,Y: set_b] :
( ( sup_sup_set_b @ X4 @ ( inf_inf_set_b @ X4 @ Y ) )
= X4 ) ).
% sup_inf_absorb
thf(fact_613_inf__sup__absorb,axiom,
! [X4: set_b,Y: set_b] :
( ( inf_inf_set_b @ X4 @ ( sup_sup_set_b @ X4 @ Y ) )
= X4 ) ).
% inf_sup_absorb
thf(fact_614_Int__insert__right__if1,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ A2 @ A3 )
=> ( ( inf_in6138156342456174402_a_b_b @ A3 @ ( insert1613891728210272810_a_b_b @ A2 @ B3 ) )
= ( insert1613891728210272810_a_b_b @ A2 @ ( inf_in6138156342456174402_a_b_b @ A3 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_615_Int__insert__right__if1,axiom,
! [A2: b,A3: set_b,B3: set_b] :
( ( member_b @ A2 @ A3 )
=> ( ( inf_inf_set_b @ A3 @ ( insert_b @ A2 @ B3 ) )
= ( insert_b @ A2 @ ( inf_inf_set_b @ A3 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_616_Int__insert__right__if0,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ~ ( member4380921116106875537_a_b_b @ A2 @ A3 )
=> ( ( inf_in6138156342456174402_a_b_b @ A3 @ ( insert1613891728210272810_a_b_b @ A2 @ B3 ) )
= ( inf_in6138156342456174402_a_b_b @ A3 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_617_Int__insert__right__if0,axiom,
! [A2: b,A3: set_b,B3: set_b] :
( ~ ( member_b @ A2 @ A3 )
=> ( ( inf_inf_set_b @ A3 @ ( insert_b @ A2 @ B3 ) )
= ( inf_inf_set_b @ A3 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_618_insert__inter__insert,axiom,
! [A2: b,A3: set_b,B3: set_b] :
( ( inf_inf_set_b @ ( insert_b @ A2 @ A3 ) @ ( insert_b @ A2 @ B3 ) )
= ( insert_b @ A2 @ ( inf_inf_set_b @ A3 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_619_Int__insert__left__if1,axiom,
! [A2: produc4558475209616630778_a_b_b,C: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ A2 @ C )
=> ( ( inf_in6138156342456174402_a_b_b @ ( insert1613891728210272810_a_b_b @ A2 @ B3 ) @ C )
= ( insert1613891728210272810_a_b_b @ A2 @ ( inf_in6138156342456174402_a_b_b @ B3 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_620_Int__insert__left__if1,axiom,
! [A2: b,C: set_b,B3: set_b] :
( ( member_b @ A2 @ C )
=> ( ( inf_inf_set_b @ ( insert_b @ A2 @ B3 ) @ C )
= ( insert_b @ A2 @ ( inf_inf_set_b @ B3 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_621_Int__insert__left__if0,axiom,
! [A2: produc4558475209616630778_a_b_b,C: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ~ ( member4380921116106875537_a_b_b @ A2 @ C )
=> ( ( inf_in6138156342456174402_a_b_b @ ( insert1613891728210272810_a_b_b @ A2 @ B3 ) @ C )
= ( inf_in6138156342456174402_a_b_b @ B3 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_622_Int__insert__left__if0,axiom,
! [A2: b,C: set_b,B3: set_b] :
( ~ ( member_b @ A2 @ C )
=> ( ( inf_inf_set_b @ ( insert_b @ A2 @ B3 ) @ C )
= ( inf_inf_set_b @ B3 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_623_Un__Int__eq_I1_J,axiom,
! [S2: set_b,T4: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ S2 @ T4 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_624_Un__Int__eq_I2_J,axiom,
! [S2: set_b,T4: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ S2 @ T4 ) @ T4 )
= T4 ) ).
% Un_Int_eq(2)
thf(fact_625_Un__Int__eq_I3_J,axiom,
! [S2: set_b,T4: set_b] :
( ( inf_inf_set_b @ S2 @ ( sup_sup_set_b @ S2 @ T4 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_626_Un__Int__eq_I4_J,axiom,
! [T4: set_b,S2: set_b] :
( ( inf_inf_set_b @ T4 @ ( sup_sup_set_b @ S2 @ T4 ) )
= T4 ) ).
% Un_Int_eq(4)
thf(fact_627_Int__Un__eq_I1_J,axiom,
! [S2: set_b,T4: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ S2 @ T4 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_628_Int__Un__eq_I2_J,axiom,
! [S2: set_b,T4: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ S2 @ T4 ) @ T4 )
= T4 ) ).
% Int_Un_eq(2)
thf(fact_629_Int__Un__eq_I3_J,axiom,
! [S2: set_b,T4: set_b] :
( ( sup_sup_set_b @ S2 @ ( inf_inf_set_b @ S2 @ T4 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_630_Int__Un__eq_I4_J,axiom,
! [T4: set_b,S2: set_b] :
( ( sup_sup_set_b @ T4 @ ( inf_inf_set_b @ S2 @ T4 ) )
= T4 ) ).
% Int_Un_eq(4)
thf(fact_631_insert__disjoint_I1_J,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( ( inf_in6138156342456174402_a_b_b @ ( insert1613891728210272810_a_b_b @ A2 @ A3 ) @ B3 )
= bot_bo3721250822024684356_a_b_b )
= ( ~ ( member4380921116106875537_a_b_b @ A2 @ B3 )
& ( ( inf_in6138156342456174402_a_b_b @ A3 @ B3 )
= bot_bo3721250822024684356_a_b_b ) ) ) ).
% insert_disjoint(1)
thf(fact_632_insert__disjoint_I1_J,axiom,
! [A2: b,A3: set_b,B3: set_b] :
( ( ( inf_inf_set_b @ ( insert_b @ A2 @ A3 ) @ B3 )
= bot_bot_set_b )
= ( ~ ( member_b @ A2 @ B3 )
& ( ( inf_inf_set_b @ A3 @ B3 )
= bot_bot_set_b ) ) ) ).
% insert_disjoint(1)
thf(fact_633_insert__disjoint_I2_J,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( bot_bo3721250822024684356_a_b_b
= ( inf_in6138156342456174402_a_b_b @ ( insert1613891728210272810_a_b_b @ A2 @ A3 ) @ B3 ) )
= ( ~ ( member4380921116106875537_a_b_b @ A2 @ B3 )
& ( bot_bo3721250822024684356_a_b_b
= ( inf_in6138156342456174402_a_b_b @ A3 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_634_insert__disjoint_I2_J,axiom,
! [A2: b,A3: set_b,B3: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ ( insert_b @ A2 @ A3 ) @ B3 ) )
= ( ~ ( member_b @ A2 @ B3 )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A3 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_635_disjoint__insert_I1_J,axiom,
! [B3: set_Pr3012420139608375472_a_b_b,A2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ( ( inf_in6138156342456174402_a_b_b @ B3 @ ( insert1613891728210272810_a_b_b @ A2 @ A3 ) )
= bot_bo3721250822024684356_a_b_b )
= ( ~ ( member4380921116106875537_a_b_b @ A2 @ B3 )
& ( ( inf_in6138156342456174402_a_b_b @ B3 @ A3 )
= bot_bo3721250822024684356_a_b_b ) ) ) ).
% disjoint_insert(1)
thf(fact_636_disjoint__insert_I1_J,axiom,
! [B3: set_b,A2: b,A3: set_b] :
( ( ( inf_inf_set_b @ B3 @ ( insert_b @ A2 @ A3 ) )
= bot_bot_set_b )
= ( ~ ( member_b @ A2 @ B3 )
& ( ( inf_inf_set_b @ B3 @ A3 )
= bot_bot_set_b ) ) ) ).
% disjoint_insert(1)
thf(fact_637_disjoint__insert_I2_J,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,B2: produc4558475209616630778_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( bot_bo3721250822024684356_a_b_b
= ( inf_in6138156342456174402_a_b_b @ A3 @ ( insert1613891728210272810_a_b_b @ B2 @ B3 ) ) )
= ( ~ ( member4380921116106875537_a_b_b @ B2 @ A3 )
& ( bot_bo3721250822024684356_a_b_b
= ( inf_in6138156342456174402_a_b_b @ A3 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_638_disjoint__insert_I2_J,axiom,
! [A3: set_b,B2: b,B3: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ A3 @ ( insert_b @ B2 @ B3 ) ) )
= ( ~ ( member_b @ B2 @ A3 )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A3 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_639_Int__Collect__mono,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b,P: produc4558475209616630778_a_b_b > $o,Q: produc4558475209616630778_a_b_b > $o] :
( ( ord_le146215904626753808_a_b_b @ A3 @ B3 )
=> ( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ A3 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le146215904626753808_a_b_b @ ( inf_in6138156342456174402_a_b_b @ A3 @ ( collec1368399972772960719_a_b_b @ P ) ) @ ( inf_in6138156342456174402_a_b_b @ B3 @ ( collec1368399972772960719_a_b_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_640_Int__Collect__mono,axiom,
! [A3: set_b,B3: set_b,P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ! [X: b] :
( ( member_b @ X @ A3 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A3 @ ( collect_b @ P ) ) @ ( inf_inf_set_b @ B3 @ ( collect_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_641_pred__subset__eq2,axiom,
! [R3: set_Pr3012420139608375472_a_b_b,S2: set_Pr3012420139608375472_a_b_b] :
( ( ord_le2403992017558287159_b_b_o
@ ^ [X3: dtree_a_b,Y5: b] : ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ Y5 ) @ R3 )
@ ^ [X3: dtree_a_b,Y5: b] : ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ Y5 ) @ S2 ) )
= ( ord_le146215904626753808_a_b_b @ R3 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_642_Un__Int__assoc__eq,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( ( sup_sup_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ C )
= ( inf_inf_set_b @ A3 @ ( sup_sup_set_b @ B3 @ C ) ) )
= ( ord_less_eq_set_b @ C @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_643_boolean__algebra_Oconj__disj__distrib,axiom,
! [X4: set_b,Y: set_b,Z2: set_b] :
( ( inf_inf_set_b @ X4 @ ( sup_sup_set_b @ Y @ Z2 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ X4 @ Y ) @ ( inf_inf_set_b @ X4 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_644_boolean__algebra_Odisj__conj__distrib,axiom,
! [X4: set_b,Y: set_b,Z2: set_b] :
( ( sup_sup_set_b @ X4 @ ( inf_inf_set_b @ Y @ Z2 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ X4 @ Y ) @ ( sup_sup_set_b @ X4 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_645_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_b,Z2: set_b,X4: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ Y @ Z2 ) @ X4 )
= ( sup_sup_set_b @ ( inf_inf_set_b @ Y @ X4 ) @ ( inf_inf_set_b @ Z2 @ X4 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_646_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_b,Z2: set_b,X4: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ Y @ Z2 ) @ X4 )
= ( inf_inf_set_b @ ( sup_sup_set_b @ Y @ X4 ) @ ( sup_sup_set_b @ Z2 @ X4 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_647_Int__Collect,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,P: produc4558475209616630778_a_b_b > $o] :
( ( member4380921116106875537_a_b_b @ X4 @ ( inf_in6138156342456174402_a_b_b @ A3 @ ( collec1368399972772960719_a_b_b @ P ) ) )
= ( ( member4380921116106875537_a_b_b @ X4 @ A3 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_648_Int__Collect,axiom,
! [X4: b,A3: set_b,P: b > $o] :
( ( member_b @ X4 @ ( inf_inf_set_b @ A3 @ ( collect_b @ P ) ) )
= ( ( member_b @ X4 @ A3 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_649_Int__def,axiom,
( inf_in6138156342456174402_a_b_b
= ( ^ [A6: set_Pr3012420139608375472_a_b_b,B6: set_Pr3012420139608375472_a_b_b] :
( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ A6 )
& ( member4380921116106875537_a_b_b @ X3 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_650_Int__def,axiom,
( inf_inf_set_b
= ( ^ [A6: set_b,B6: set_b] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A6 )
& ( member_b @ X3 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_651_subrelI,axiom,
! [R2: set_Pr3012420139608375472_a_b_b,S: set_Pr3012420139608375472_a_b_b] :
( ! [X: dtree_a_b,Y4: b] :
( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X @ Y4 ) @ R2 )
=> ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X @ Y4 ) @ S ) )
=> ( ord_le146215904626753808_a_b_b @ R2 @ S ) ) ).
% subrelI
thf(fact_652_IntD2,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ C2 @ ( inf_in6138156342456174402_a_b_b @ A3 @ B3 ) )
=> ( member4380921116106875537_a_b_b @ C2 @ B3 ) ) ).
% IntD2
thf(fact_653_IntD2,axiom,
! [C2: b,A3: set_b,B3: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A3 @ B3 ) )
=> ( member_b @ C2 @ B3 ) ) ).
% IntD2
thf(fact_654_IntD1,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ C2 @ ( inf_in6138156342456174402_a_b_b @ A3 @ B3 ) )
=> ( member4380921116106875537_a_b_b @ C2 @ A3 ) ) ).
% IntD1
thf(fact_655_IntD1,axiom,
! [C2: b,A3: set_b,B3: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A3 @ B3 ) )
=> ( member_b @ C2 @ A3 ) ) ).
% IntD1
thf(fact_656_IntE,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ C2 @ ( inf_in6138156342456174402_a_b_b @ A3 @ B3 ) )
=> ~ ( ( member4380921116106875537_a_b_b @ C2 @ A3 )
=> ~ ( member4380921116106875537_a_b_b @ C2 @ B3 ) ) ) ).
% IntE
thf(fact_657_IntE,axiom,
! [C2: b,A3: set_b,B3: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A3 @ B3 ) )
=> ~ ( ( member_b @ C2 @ A3 )
=> ~ ( member_b @ C2 @ B3 ) ) ) ).
% IntE
thf(fact_658_comp__fun__commute__Image__fold,axiom,
! [S2: set_Pr3012420139608375472_a_b_b] :
( finite4381541246406268242_set_b
@ ( produc9053033572752107902_set_b
@ ^ [X3: produc4558475209616630778_a_b_b,Y5: b,A6: set_b] : ( if_set_b @ ( member4380921116106875537_a_b_b @ X3 @ S2 ) @ ( insert_b @ Y5 @ A6 ) @ A6 ) ) ) ).
% comp_fun_commute_Image_fold
thf(fact_659_comp__fun__commute__Image__fold,axiom,
! [S2: set_b] :
( finite7340995349656252681_set_b
@ ( produc831963642587629969_set_b
@ ^ [X3: b,Y5: b,A6: set_b] : ( if_set_b @ ( member_b @ X3 @ S2 ) @ ( insert_b @ Y5 @ A6 ) @ A6 ) ) ) ).
% comp_fun_commute_Image_fold
thf(fact_660_inf_OcoboundedI2,axiom,
! [B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ B2 @ C2 )
=> ( ord_le789900035998834954_a_b_b @ ( inf_in7138637532943773244_a_b_b @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_661_inf_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_662_inf_OcoboundedI1,axiom,
! [A2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ C2 )
=> ( ord_le789900035998834954_a_b_b @ ( inf_in7138637532943773244_a_b_b @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_663_inf_OcoboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_664_inf_Oabsorb__iff2,axiom,
( ord_le789900035998834954_a_b_b
= ( ^ [B: fset_P5281107635120001194_a_b_b,A: fset_P5281107635120001194_a_b_b] :
( ( inf_in7138637532943773244_a_b_b @ A @ B )
= B ) ) ) ).
% inf.absorb_iff2
thf(fact_665_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B: nat,A: nat] :
( ( inf_inf_nat @ A @ B )
= B ) ) ) ).
% inf.absorb_iff2
thf(fact_666_inf_Oabsorb__iff1,axiom,
( ord_le789900035998834954_a_b_b
= ( ^ [A: fset_P5281107635120001194_a_b_b,B: fset_P5281107635120001194_a_b_b] :
( ( inf_in7138637532943773244_a_b_b @ A @ B )
= A ) ) ) ).
% inf.absorb_iff1
thf(fact_667_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
( ( inf_inf_nat @ A @ B )
= A ) ) ) ).
% inf.absorb_iff1
thf(fact_668_inf_Ocobounded2,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ ( inf_in7138637532943773244_a_b_b @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_669_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_670_inf_Ocobounded1,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ ( inf_in7138637532943773244_a_b_b @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_671_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_672_inf_Oorder__iff,axiom,
( ord_le789900035998834954_a_b_b
= ( ^ [A: fset_P5281107635120001194_a_b_b,B: fset_P5281107635120001194_a_b_b] :
( A
= ( inf_in7138637532943773244_a_b_b @ A @ B ) ) ) ) ).
% inf.order_iff
thf(fact_673_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
( A
= ( inf_inf_nat @ A @ B ) ) ) ) ).
% inf.order_iff
thf(fact_674_inf__greatest,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b,Z2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X4 @ Y )
=> ( ( ord_le789900035998834954_a_b_b @ X4 @ Z2 )
=> ( ord_le789900035998834954_a_b_b @ X4 @ ( inf_in7138637532943773244_a_b_b @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_675_inf__greatest,axiom,
! [X4: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ X4 @ Z2 )
=> ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_676_inf_OboundedI,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( ( ord_le789900035998834954_a_b_b @ A2 @ C2 )
=> ( ord_le789900035998834954_a_b_b @ A2 @ ( inf_in7138637532943773244_a_b_b @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_677_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_678_inf_OboundedE,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ ( inf_in7138637532943773244_a_b_b @ B2 @ C2 ) )
=> ~ ( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ~ ( ord_le789900035998834954_a_b_b @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_679_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_680_inf__absorb2,axiom,
! [Y: fset_P5281107635120001194_a_b_b,X4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ Y @ X4 )
=> ( ( inf_in7138637532943773244_a_b_b @ X4 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_681_inf__absorb2,axiom,
! [Y: nat,X4: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ( ( inf_inf_nat @ X4 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_682_inf__absorb1,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X4 @ Y )
=> ( ( inf_in7138637532943773244_a_b_b @ X4 @ Y )
= X4 ) ) ).
% inf_absorb1
thf(fact_683_inf__absorb1,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( inf_inf_nat @ X4 @ Y )
= X4 ) ) ).
% inf_absorb1
thf(fact_684_inf_Oabsorb2,axiom,
! [B2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ B2 @ A2 )
=> ( ( inf_in7138637532943773244_a_b_b @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_685_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_686_inf_Oabsorb1,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( ( inf_in7138637532943773244_a_b_b @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_687_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_688_le__iff__inf,axiom,
( ord_le789900035998834954_a_b_b
= ( ^ [X3: fset_P5281107635120001194_a_b_b,Y5: fset_P5281107635120001194_a_b_b] :
( ( inf_in7138637532943773244_a_b_b @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_689_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y5: nat] :
( ( inf_inf_nat @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_690_inf__unique,axiom,
! [F: fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b,X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] :
( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ ( F @ X @ Y4 ) @ X )
=> ( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ ( F @ X @ Y4 ) @ Y4 )
=> ( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b,Z3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X @ Y4 )
=> ( ( ord_le789900035998834954_a_b_b @ X @ Z3 )
=> ( ord_le789900035998834954_a_b_b @ X @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_in7138637532943773244_a_b_b @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_691_inf__unique,axiom,
! [F: nat > nat > nat,X4: nat,Y: nat] :
( ! [X: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X @ Y4 ) @ X )
=> ( ! [X: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X @ Y4 ) @ Y4 )
=> ( ! [X: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ord_less_eq_nat @ X @ Z3 )
=> ( ord_less_eq_nat @ X @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_692_inf_OorderI,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( A2
= ( inf_in7138637532943773244_a_b_b @ A2 @ B2 ) )
=> ( ord_le789900035998834954_a_b_b @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_693_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_694_inf_OorderE,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( A2
= ( inf_in7138637532943773244_a_b_b @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_695_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_696_le__infI2,axiom,
! [B2: fset_P5281107635120001194_a_b_b,X4: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ B2 @ X4 )
=> ( ord_le789900035998834954_a_b_b @ ( inf_in7138637532943773244_a_b_b @ A2 @ B2 ) @ X4 ) ) ).
% le_infI2
thf(fact_697_le__infI2,axiom,
! [B2: nat,X4: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X4 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X4 ) ) ).
% le_infI2
thf(fact_698_le__infI1,axiom,
! [A2: fset_P5281107635120001194_a_b_b,X4: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ X4 )
=> ( ord_le789900035998834954_a_b_b @ ( inf_in7138637532943773244_a_b_b @ A2 @ B2 ) @ X4 ) ) ).
% le_infI1
thf(fact_699_le__infI1,axiom,
! [A2: nat,X4: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X4 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X4 ) ) ).
% le_infI1
thf(fact_700_inf__mono,axiom,
! [A2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,D: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ C2 )
=> ( ( ord_le789900035998834954_a_b_b @ B2 @ D )
=> ( ord_le789900035998834954_a_b_b @ ( inf_in7138637532943773244_a_b_b @ A2 @ B2 ) @ ( inf_in7138637532943773244_a_b_b @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_701_inf__mono,axiom,
! [A2: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_702_le__infI,axiom,
! [X4: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X4 @ A2 )
=> ( ( ord_le789900035998834954_a_b_b @ X4 @ B2 )
=> ( ord_le789900035998834954_a_b_b @ X4 @ ( inf_in7138637532943773244_a_b_b @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_703_le__infI,axiom,
! [X4: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X4 @ A2 )
=> ( ( ord_less_eq_nat @ X4 @ B2 )
=> ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_704_le__infE,axiom,
! [X4: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X4 @ ( inf_in7138637532943773244_a_b_b @ A2 @ B2 ) )
=> ~ ( ( ord_le789900035998834954_a_b_b @ X4 @ A2 )
=> ~ ( ord_le789900035998834954_a_b_b @ X4 @ B2 ) ) ) ).
% le_infE
thf(fact_705_le__infE,axiom,
! [X4: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X4 @ A2 )
=> ~ ( ord_less_eq_nat @ X4 @ B2 ) ) ) ).
% le_infE
thf(fact_706_inf__le2,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ ( inf_in7138637532943773244_a_b_b @ X4 @ Y ) @ Y ) ).
% inf_le2
thf(fact_707_inf__le2,axiom,
! [X4: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y ) @ Y ) ).
% inf_le2
thf(fact_708_inf__le1,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ ( inf_in7138637532943773244_a_b_b @ X4 @ Y ) @ X4 ) ).
% inf_le1
thf(fact_709_inf__le1,axiom,
! [X4: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y ) @ X4 ) ).
% inf_le1
thf(fact_710_inf__sup__ord_I1_J,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ ( inf_in7138637532943773244_a_b_b @ X4 @ Y ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_711_inf__sup__ord_I1_J,axiom,
! [X4: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_712_inf__sup__ord_I2_J,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ ( inf_in7138637532943773244_a_b_b @ X4 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_713_inf__sup__ord_I2_J,axiom,
! [X4: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_714_sup__inf__distrib2,axiom,
! [Y: set_b,Z2: set_b,X4: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ Y @ Z2 ) @ X4 )
= ( inf_inf_set_b @ ( sup_sup_set_b @ Y @ X4 ) @ ( sup_sup_set_b @ Z2 @ X4 ) ) ) ).
% sup_inf_distrib2
thf(fact_715_sup__inf__distrib1,axiom,
! [X4: set_b,Y: set_b,Z2: set_b] :
( ( sup_sup_set_b @ X4 @ ( inf_inf_set_b @ Y @ Z2 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ X4 @ Y ) @ ( sup_sup_set_b @ X4 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_716_inf__sup__distrib2,axiom,
! [Y: set_b,Z2: set_b,X4: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ Y @ Z2 ) @ X4 )
= ( sup_sup_set_b @ ( inf_inf_set_b @ Y @ X4 ) @ ( inf_inf_set_b @ Z2 @ X4 ) ) ) ).
% inf_sup_distrib2
thf(fact_717_inf__sup__distrib1,axiom,
! [X4: set_b,Y: set_b,Z2: set_b] :
( ( inf_inf_set_b @ X4 @ ( sup_sup_set_b @ Y @ Z2 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ X4 @ Y ) @ ( inf_inf_set_b @ X4 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_718_distrib__imp2,axiom,
! [X4: set_b,Y: set_b,Z2: set_b] :
( ! [X: set_b,Y4: set_b,Z3: set_b] :
( ( sup_sup_set_b @ X @ ( inf_inf_set_b @ Y4 @ Z3 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ X @ Y4 ) @ ( sup_sup_set_b @ X @ Z3 ) ) )
=> ( ( inf_inf_set_b @ X4 @ ( sup_sup_set_b @ Y @ Z2 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ X4 @ Y ) @ ( inf_inf_set_b @ X4 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_719_distrib__imp1,axiom,
! [X4: set_b,Y: set_b,Z2: set_b] :
( ! [X: set_b,Y4: set_b,Z3: set_b] :
( ( inf_inf_set_b @ X @ ( sup_sup_set_b @ Y4 @ Z3 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ X @ Y4 ) @ ( inf_inf_set_b @ X @ Z3 ) ) )
=> ( ( sup_sup_set_b @ X4 @ ( inf_inf_set_b @ Y @ Z2 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ X4 @ Y ) @ ( sup_sup_set_b @ X4 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_720_disjoint__iff__not__equal,axiom,
! [A3: set_b,B3: set_b] :
( ( ( inf_inf_set_b @ A3 @ B3 )
= bot_bot_set_b )
= ( ! [X3: b] :
( ( member_b @ X3 @ A3 )
=> ! [Y5: b] :
( ( member_b @ Y5 @ B3 )
=> ( X3 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_721_Int__empty__right,axiom,
! [A3: set_b] :
( ( inf_inf_set_b @ A3 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% Int_empty_right
thf(fact_722_Int__empty__left,axiom,
! [B3: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ B3 )
= bot_bot_set_b ) ).
% Int_empty_left
thf(fact_723_disjoint__iff,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( ( inf_in6138156342456174402_a_b_b @ A3 @ B3 )
= bot_bo3721250822024684356_a_b_b )
= ( ! [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ A3 )
=> ~ ( member4380921116106875537_a_b_b @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_724_disjoint__iff,axiom,
! [A3: set_b,B3: set_b] :
( ( ( inf_inf_set_b @ A3 @ B3 )
= bot_bot_set_b )
= ( ! [X3: b] :
( ( member_b @ X3 @ A3 )
=> ~ ( member_b @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_725_Int__emptyI,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ A3 )
=> ~ ( member4380921116106875537_a_b_b @ X @ B3 ) )
=> ( ( inf_in6138156342456174402_a_b_b @ A3 @ B3 )
= bot_bo3721250822024684356_a_b_b ) ) ).
% Int_emptyI
thf(fact_726_Int__emptyI,axiom,
! [A3: set_b,B3: set_b] :
( ! [X: b] :
( ( member_b @ X @ A3 )
=> ~ ( member_b @ X @ B3 ) )
=> ( ( inf_inf_set_b @ A3 @ B3 )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_727_Int__insert__right,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( ( member4380921116106875537_a_b_b @ A2 @ A3 )
=> ( ( inf_in6138156342456174402_a_b_b @ A3 @ ( insert1613891728210272810_a_b_b @ A2 @ B3 ) )
= ( insert1613891728210272810_a_b_b @ A2 @ ( inf_in6138156342456174402_a_b_b @ A3 @ B3 ) ) ) )
& ( ~ ( member4380921116106875537_a_b_b @ A2 @ A3 )
=> ( ( inf_in6138156342456174402_a_b_b @ A3 @ ( insert1613891728210272810_a_b_b @ A2 @ B3 ) )
= ( inf_in6138156342456174402_a_b_b @ A3 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_728_Int__insert__right,axiom,
! [A2: b,A3: set_b,B3: set_b] :
( ( ( member_b @ A2 @ A3 )
=> ( ( inf_inf_set_b @ A3 @ ( insert_b @ A2 @ B3 ) )
= ( insert_b @ A2 @ ( inf_inf_set_b @ A3 @ B3 ) ) ) )
& ( ~ ( member_b @ A2 @ A3 )
=> ( ( inf_inf_set_b @ A3 @ ( insert_b @ A2 @ B3 ) )
= ( inf_inf_set_b @ A3 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_729_Int__insert__left,axiom,
! [A2: produc4558475209616630778_a_b_b,C: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( ( member4380921116106875537_a_b_b @ A2 @ C )
=> ( ( inf_in6138156342456174402_a_b_b @ ( insert1613891728210272810_a_b_b @ A2 @ B3 ) @ C )
= ( insert1613891728210272810_a_b_b @ A2 @ ( inf_in6138156342456174402_a_b_b @ B3 @ C ) ) ) )
& ( ~ ( member4380921116106875537_a_b_b @ A2 @ C )
=> ( ( inf_in6138156342456174402_a_b_b @ ( insert1613891728210272810_a_b_b @ A2 @ B3 ) @ C )
= ( inf_in6138156342456174402_a_b_b @ B3 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_730_Int__insert__left,axiom,
! [A2: b,C: set_b,B3: set_b] :
( ( ( member_b @ A2 @ C )
=> ( ( inf_inf_set_b @ ( insert_b @ A2 @ B3 ) @ C )
= ( insert_b @ A2 @ ( inf_inf_set_b @ B3 @ C ) ) ) )
& ( ~ ( member_b @ A2 @ C )
=> ( ( inf_inf_set_b @ ( insert_b @ A2 @ B3 ) @ C )
= ( inf_inf_set_b @ B3 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_731_Un__Int__distrib2,axiom,
! [B3: set_b,C: set_b,A3: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ B3 @ C ) @ A3 )
= ( inf_inf_set_b @ ( sup_sup_set_b @ B3 @ A3 ) @ ( sup_sup_set_b @ C @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_732_Int__Un__distrib2,axiom,
! [B3: set_b,C: set_b,A3: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ B3 @ C ) @ A3 )
= ( sup_sup_set_b @ ( inf_inf_set_b @ B3 @ A3 ) @ ( inf_inf_set_b @ C @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_733_Un__Int__distrib,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( sup_sup_set_b @ A3 @ ( inf_inf_set_b @ B3 @ C ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ A3 @ B3 ) @ ( sup_sup_set_b @ A3 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_734_Int__Un__distrib,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( inf_inf_set_b @ A3 @ ( sup_sup_set_b @ B3 @ C ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ ( inf_inf_set_b @ A3 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_735_Un__Int__crazy,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ ( inf_inf_set_b @ B3 @ C ) ) @ ( inf_inf_set_b @ C @ A3 ) )
= ( inf_inf_set_b @ ( inf_inf_set_b @ ( sup_sup_set_b @ A3 @ B3 ) @ ( sup_sup_set_b @ B3 @ C ) ) @ ( sup_sup_set_b @ C @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_736_subset__insertI2,axiom,
! [A3: set_b,B3: set_b,B2: b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ord_less_eq_set_b @ A3 @ ( insert_b @ B2 @ B3 ) ) ) ).
% subset_insertI2
thf(fact_737_subset__insertI,axiom,
! [B3: set_b,A2: b] : ( ord_less_eq_set_b @ B3 @ ( insert_b @ A2 @ B3 ) ) ).
% subset_insertI
thf(fact_738_subset__insert,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ~ ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( ( ord_le146215904626753808_a_b_b @ A3 @ ( insert1613891728210272810_a_b_b @ X4 @ B3 ) )
= ( ord_le146215904626753808_a_b_b @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_739_subset__insert,axiom,
! [X4: b,A3: set_b,B3: set_b] :
( ~ ( member_b @ X4 @ A3 )
=> ( ( ord_less_eq_set_b @ A3 @ ( insert_b @ X4 @ B3 ) )
= ( ord_less_eq_set_b @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_740_insert__mono,axiom,
! [C: set_b,D2: set_b,A2: b] :
( ( ord_less_eq_set_b @ C @ D2 )
=> ( ord_less_eq_set_b @ ( insert_b @ A2 @ C ) @ ( insert_b @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_741_subset__Un__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B6: set_b] :
( ( sup_sup_set_b @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_742_subset__UnE,axiom,
! [C: set_b,A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ C @ ( sup_sup_set_b @ A3 @ B3 ) )
=> ~ ! [A7: set_b] :
( ( ord_less_eq_set_b @ A7 @ A3 )
=> ! [B8: set_b] :
( ( ord_less_eq_set_b @ B8 @ B3 )
=> ( C
!= ( sup_sup_set_b @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_743_Un__absorb2,axiom,
! [B3: set_b,A3: set_b] :
( ( ord_less_eq_set_b @ B3 @ A3 )
=> ( ( sup_sup_set_b @ A3 @ B3 )
= A3 ) ) ).
% Un_absorb2
thf(fact_744_Un__absorb1,axiom,
! [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( sup_sup_set_b @ A3 @ B3 )
= B3 ) ) ).
% Un_absorb1
thf(fact_745_Un__upper2,axiom,
! [B3: set_b,A3: set_b] : ( ord_less_eq_set_b @ B3 @ ( sup_sup_set_b @ A3 @ B3 ) ) ).
% Un_upper2
thf(fact_746_Un__upper1,axiom,
! [A3: set_b,B3: set_b] : ( ord_less_eq_set_b @ A3 @ ( sup_sup_set_b @ A3 @ B3 ) ) ).
% Un_upper1
thf(fact_747_Un__least,axiom,
! [A3: set_b,C: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ C )
=> ( ( ord_less_eq_set_b @ B3 @ C )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A3 @ B3 ) @ C ) ) ) ).
% Un_least
thf(fact_748_Un__mono,axiom,
! [A3: set_b,C: set_b,B3: set_b,D2: set_b] :
( ( ord_less_eq_set_b @ A3 @ C )
=> ( ( ord_less_eq_set_b @ B3 @ D2 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A3 @ B3 ) @ ( sup_sup_set_b @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_749_dtail__notelem__eq__def,axiom,
! [E4: b,T: dtree_a_b,Def: b > a] :
( ~ ( member_b @ E4 @ ( darcs_a_b @ T ) )
=> ( ( dtail_a_b @ T @ Def @ E4 )
= ( Def @ E4 ) ) ) ).
% dtail_notelem_eq_def
thf(fact_750_distrib__inf__le,axiom,
! [X4: set_b,Y: set_b,Z2: set_b] : ( ord_less_eq_set_b @ ( sup_sup_set_b @ ( inf_inf_set_b @ X4 @ Y ) @ ( inf_inf_set_b @ X4 @ Z2 ) ) @ ( inf_inf_set_b @ X4 @ ( sup_sup_set_b @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_751_distrib__inf__le,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b,Z2: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ ( sup_su860928060825958358_a_b_b @ ( inf_in7138637532943773244_a_b_b @ X4 @ Y ) @ ( inf_in7138637532943773244_a_b_b @ X4 @ Z2 ) ) @ ( inf_in7138637532943773244_a_b_b @ X4 @ ( sup_su860928060825958358_a_b_b @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_752_distrib__inf__le,axiom,
! [X4: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X4 @ Y ) @ ( inf_inf_nat @ X4 @ Z2 ) ) @ ( inf_inf_nat @ X4 @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_753_distrib__sup__le,axiom,
! [X4: set_b,Y: set_b,Z2: set_b] : ( ord_less_eq_set_b @ ( sup_sup_set_b @ X4 @ ( inf_inf_set_b @ Y @ Z2 ) ) @ ( inf_inf_set_b @ ( sup_sup_set_b @ X4 @ Y ) @ ( sup_sup_set_b @ X4 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_754_distrib__sup__le,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b,Z2: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ ( sup_su860928060825958358_a_b_b @ X4 @ ( inf_in7138637532943773244_a_b_b @ Y @ Z2 ) ) @ ( inf_in7138637532943773244_a_b_b @ ( sup_su860928060825958358_a_b_b @ X4 @ Y ) @ ( sup_su860928060825958358_a_b_b @ X4 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_755_distrib__sup__le,axiom,
! [X4: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X4 @ Y ) @ ( sup_sup_nat @ X4 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_756_subset__singleton__iff,axiom,
! [X6: set_b,A2: b] :
( ( ord_less_eq_set_b @ X6 @ ( insert_b @ A2 @ bot_bot_set_b ) )
= ( ( X6 = bot_bot_set_b )
| ( X6
= ( insert_b @ A2 @ bot_bot_set_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_757_subset__singletonD,axiom,
! [A3: set_b,X4: b] :
( ( ord_less_eq_set_b @ A3 @ ( insert_b @ X4 @ bot_bot_set_b ) )
=> ( ( A3 = bot_bot_set_b )
| ( A3
= ( insert_b @ X4 @ bot_bot_set_b ) ) ) ) ).
% subset_singletonD
thf(fact_758_less__eq__fset_Orep__eq,axiom,
( ord_le789900035998834954_a_b_b
= ( ^ [X3: fset_P5281107635120001194_a_b_b,Xa2: fset_P5281107635120001194_a_b_b] : ( ord_le146215904626753808_a_b_b @ ( fset_P783253628892185035_a_b_b @ X3 ) @ ( fset_P783253628892185035_a_b_b @ Xa2 ) ) ) ) ).
% less_eq_fset.rep_eq
thf(fact_759_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_760_le__cases3,axiom,
! [X4: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_761_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: fset_P5281107635120001194_a_b_b,Z: fset_P5281107635120001194_a_b_b] : ( Y3 = Z ) )
= ( ^ [X3: fset_P5281107635120001194_a_b_b,Y5: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X3 @ Y5 )
& ( ord_le789900035998834954_a_b_b @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_762_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_763_ord__eq__le__trans,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( A2 = B2 )
=> ( ( ord_le789900035998834954_a_b_b @ B2 @ C2 )
=> ( ord_le789900035998834954_a_b_b @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_764_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_765_ord__le__eq__trans,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le789900035998834954_a_b_b @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_766_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_767_order__antisym,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X4 @ Y )
=> ( ( ord_le789900035998834954_a_b_b @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_768_order__antisym,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_769_order_Otrans,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( ( ord_le789900035998834954_a_b_b @ B2 @ C2 )
=> ( ord_le789900035998834954_a_b_b @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_770_order_Otrans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_771_order__trans,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b,Z2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X4 @ Y )
=> ( ( ord_le789900035998834954_a_b_b @ Y @ Z2 )
=> ( ord_le789900035998834954_a_b_b @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_772_order__trans,axiom,
! [X4: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_773_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_774_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: fset_P5281107635120001194_a_b_b,Z: fset_P5281107635120001194_a_b_b] : ( Y3 = Z ) )
= ( ^ [A: fset_P5281107635120001194_a_b_b,B: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ B @ A )
& ( ord_le789900035998834954_a_b_b @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_775_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_776_dual__order_Oantisym,axiom,
! [B2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ B2 @ A2 )
=> ( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_777_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_778_dual__order_Otrans,axiom,
! [B2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ B2 @ A2 )
=> ( ( ord_le789900035998834954_a_b_b @ C2 @ B2 )
=> ( ord_le789900035998834954_a_b_b @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_779_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_780_antisym,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( ( ord_le789900035998834954_a_b_b @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_781_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_782_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: fset_P5281107635120001194_a_b_b,Z: fset_P5281107635120001194_a_b_b] : ( Y3 = Z ) )
= ( ^ [A: fset_P5281107635120001194_a_b_b,B: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A @ B )
& ( ord_le789900035998834954_a_b_b @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_783_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_784_order__subst1,axiom,
! [A2: fset_P5281107635120001194_a_b_b,F: fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ ( F @ B2 ) )
=> ( ( ord_le789900035998834954_a_b_b @ B2 @ C2 )
=> ( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X @ Y4 )
=> ( ord_le789900035998834954_a_b_b @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le789900035998834954_a_b_b @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_785_order__subst1,axiom,
! [A2: fset_P5281107635120001194_a_b_b,F: nat > fset_P5281107635120001194_a_b_b,B2: nat,C2: nat] :
( ( ord_le789900035998834954_a_b_b @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le789900035998834954_a_b_b @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le789900035998834954_a_b_b @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_786_order__subst1,axiom,
! [A2: nat,F: fset_P5281107635120001194_a_b_b > nat,B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le789900035998834954_a_b_b @ B2 @ C2 )
=> ( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_787_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_788_order__subst2,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,F: fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( ( ord_le789900035998834954_a_b_b @ ( F @ B2 ) @ C2 )
=> ( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X @ Y4 )
=> ( ord_le789900035998834954_a_b_b @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le789900035998834954_a_b_b @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_789_order__subst2,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,F: fset_P5281107635120001194_a_b_b > nat,C2: nat] :
( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_790_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le789900035998834954_a_b_b @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le789900035998834954_a_b_b @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le789900035998834954_a_b_b @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_791_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_792_order__eq__refl,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] :
( ( X4 = Y )
=> ( ord_le789900035998834954_a_b_b @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_793_order__eq__refl,axiom,
! [X4: nat,Y: nat] :
( ( X4 = Y )
=> ( ord_less_eq_nat @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_794_linorder__linear,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
| ( ord_less_eq_nat @ Y @ X4 ) ) ).
% linorder_linear
thf(fact_795_ord__eq__le__subst,axiom,
! [A2: fset_P5281107635120001194_a_b_b,F: fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le789900035998834954_a_b_b @ B2 @ C2 )
=> ( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X @ Y4 )
=> ( ord_le789900035998834954_a_b_b @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le789900035998834954_a_b_b @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_796_ord__eq__le__subst,axiom,
! [A2: nat,F: fset_P5281107635120001194_a_b_b > nat,B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le789900035998834954_a_b_b @ B2 @ C2 )
=> ( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_797_ord__eq__le__subst,axiom,
! [A2: fset_P5281107635120001194_a_b_b,F: nat > fset_P5281107635120001194_a_b_b,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le789900035998834954_a_b_b @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le789900035998834954_a_b_b @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_798_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_799_ord__le__eq__subst,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,F: fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X @ Y4 )
=> ( ord_le789900035998834954_a_b_b @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le789900035998834954_a_b_b @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_800_ord__le__eq__subst,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,F: fset_P5281107635120001194_a_b_b > nat,C2: nat] :
( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_801_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le789900035998834954_a_b_b @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le789900035998834954_a_b_b @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_802_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_803_linorder__le__cases,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ Y @ X4 ) ) ).
% linorder_le_cases
thf(fact_804_order__antisym__conv,axiom,
! [Y: fset_P5281107635120001194_a_b_b,X4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ Y @ X4 )
=> ( ( ord_le789900035998834954_a_b_b @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_805_order__antisym__conv,axiom,
! [Y: nat,X4: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_806_boolean__algebra__cancel_Osup2,axiom,
! [B3: set_b,K: set_b,B2: set_b,A2: set_b] :
( ( B3
= ( sup_sup_set_b @ K @ B2 ) )
=> ( ( sup_sup_set_b @ A2 @ B3 )
= ( sup_sup_set_b @ K @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_807_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_b,K: set_b,A2: set_b,B2: set_b] :
( ( A3
= ( sup_sup_set_b @ K @ A2 ) )
=> ( ( sup_sup_set_b @ A3 @ B2 )
= ( sup_sup_set_b @ K @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_808_sup__Un__eq2,axiom,
! [R3: set_Pr3012420139608375472_a_b_b,S2: set_Pr3012420139608375472_a_b_b] :
( ( sup_su6709851091347060739_b_b_o
@ ^ [X3: dtree_a_b,Y5: b] : ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ Y5 ) @ R3 )
@ ^ [X3: dtree_a_b,Y5: b] : ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ Y5 ) @ S2 ) )
= ( ^ [X3: dtree_a_b,Y5: b] : ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ Y5 ) @ ( sup_su2887895092731772380_a_b_b @ R3 @ S2 ) ) ) ) ).
% sup_Un_eq2
thf(fact_809_bot__empty__eq2,axiom,
( bot_bo471016548657204587_b_b_o
= ( ^ [X3: dtree_a_b,Y5: b] : ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ Y5 ) @ bot_bo3721250822024684356_a_b_b ) ) ) ).
% bot_empty_eq2
thf(fact_810_pred__equals__eq2,axiom,
! [R3: set_Pr3012420139608375472_a_b_b,S2: set_Pr3012420139608375472_a_b_b] :
( ( ( ^ [X3: dtree_a_b,Y5: b] : ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ Y5 ) @ R3 ) )
= ( ^ [X3: dtree_a_b,Y5: b] : ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ Y5 ) @ S2 ) ) )
= ( R3 = S2 ) ) ).
% pred_equals_eq2
thf(fact_811_disjoint__darcs__if__wf__aux2,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T1: dtree_a_b,E1: b,T2: dtree_a_b,E22: b] :
( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T1 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T2 @ E22 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( ( ( produc331601717337510060_a_b_b @ T1 @ E1 )
!= ( produc331601717337510060_a_b_b @ T2 @ E22 ) )
=> ( ( inf_inf_set_b @ ( darcs_a_b @ T1 ) @ ( darcs_a_b @ T2 ) )
= bot_bot_set_b ) ) ) ) ) ).
% disjoint_darcs_if_wf_aux2
thf(fact_812_dtail__f__alt__commute,axiom,
! [P: fset_P5281107635120001194_a_b_b > $o,R2: a,Q: produc4558475209616630778_a_b_b > ( b > a ) > $o,E4: b,R3: produc4558475209616630778_a_b_b > ( b > a ) > b > a,Def: b > a,Xs2: fset_P5281107635120001194_a_b_b] :
( ( P
= ( ^ [Xs3: fset_P5281107635120001194_a_b_b] : ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs3 ) ) ) )
=> ( ( Q
= ( produc9194724151488670482_b_a_o
@ ^ [T12: dtree_a_b,E12: b,B: b > a] : ( member_b @ E4 @ ( darcs_a_b @ T12 ) ) ) )
=> ( ( R3
= ( produc4313903556115589696_a_b_a
@ ^ [T12: dtree_a_b,E12: b,B: b > a] : ( dtail_a_b @ T12 @ Def ) ) )
=> ( finite7715548283558590705_b_b_a
@ ^ [A: produc4558475209616630778_a_b_b,B: b > a] :
( if_b_a
@ ( ~ ( member4380921116106875537_a_b_b @ A @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( Q @ A @ B )
| ~ ( P @ Xs2 ) )
@ B
@ ( R3 @ A @ B ) ) ) ) ) ) ).
% dtail_f_alt_commute
thf(fact_813_bot_Oextremum__uniqueI,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
=> ( A2 = bot_bot_set_b ) ) ).
% bot.extremum_uniqueI
thf(fact_814_bot_Oextremum__uniqueI,axiom,
! [A2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ bot_bo2895716411488905534_a_b_b )
=> ( A2 = bot_bo2895716411488905534_a_b_b ) ) ).
% bot.extremum_uniqueI
thf(fact_815_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_816_bot_Oextremum__unique,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
= ( A2 = bot_bot_set_b ) ) ).
% bot.extremum_unique
thf(fact_817_bot_Oextremum__unique,axiom,
! [A2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ bot_bo2895716411488905534_a_b_b )
= ( A2 = bot_bo2895716411488905534_a_b_b ) ) ).
% bot.extremum_unique
thf(fact_818_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_819_bot_Oextremum,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% bot.extremum
thf(fact_820_bot_Oextremum,axiom,
! [A2: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ bot_bo2895716411488905534_a_b_b @ A2 ) ).
% bot.extremum
thf(fact_821_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_822_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_b] :
( ( sup_sup_set_b @ X4 @ bot_bot_set_b )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_823_dtail__in__child__eq__child,axiom,
! [T: dtree_a_b,E1: b,Xs2: fset_P5281107635120001194_a_b_b,E4: b,R2: a,Def: b > a] :
( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( ( member_b @ E4 @ ( darcs_a_b @ T ) )
=> ( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ( ( dtail_a_b @ ( node_a_b @ R2 @ Xs2 ) @ Def @ E4 )
= ( dtail_a_b @ T @ Def @ E4 ) ) ) ) ) ).
% dtail_in_child_eq_child
thf(fact_824_dtail__f__alt,axiom,
! [P: fset_P5281107635120001194_a_b_b > $o,R2: a,Q: produc4558475209616630778_a_b_b > ( b > a ) > $o,E4: b,R3: produc4558475209616630778_a_b_b > ( b > a ) > b > a,Def: b > a,Xs2: fset_P5281107635120001194_a_b_b] :
( ( P
= ( ^ [Xs3: fset_P5281107635120001194_a_b_b] : ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs3 ) ) ) )
=> ( ( Q
= ( produc9194724151488670482_b_a_o
@ ^ [T12: dtree_a_b,E12: b,B: b > a] : ( member_b @ E4 @ ( darcs_a_b @ T12 ) ) ) )
=> ( ( R3
= ( produc4313903556115589696_a_b_a
@ ^ [T12: dtree_a_b,E12: b,B: b > a] : ( dtail_a_b @ T12 @ Def ) ) )
=> ( ( produc4313903556115589696_a_b_a
@ ^ [T12: dtree_a_b,E12: b,B: b > a] :
( if_b_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T12 @ E12 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E4 @ ( darcs_a_b @ T12 ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( dtail_a_b @ T12 @ Def ) ) )
= ( ^ [A: produc4558475209616630778_a_b_b,B: b > a] :
( if_b_a
@ ( ~ ( member4380921116106875537_a_b_b @ A @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( Q @ A @ B )
| ~ ( P @ Xs2 ) )
@ B
@ ( R3 @ A @ B ) ) ) ) ) ) ) ).
% dtail_f_alt
thf(fact_825_dtail__in__child__eq__child__ffold,axiom,
! [T: dtree_a_b,E1: b,Xs2: fset_P5281107635120001194_a_b_b,E4: b,R2: a,Def: b > a] :
( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( ( member_b @ E4 @ ( darcs_a_b @ T ) )
=> ( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ( ( ffold_8367945289176929151_b_b_a
@ ( produc4313903556115589696_a_b_a
@ ^ [X3: dtree_a_b,E2: b,B: b > a] :
( if_b_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E4 @ ( darcs_a_b @ X3 ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( dtail_a_b @ X3 @ Def ) ) )
@ Def
@ Xs2 )
= ( dtail_a_b @ T @ Def ) ) ) ) ) ).
% dtail_in_child_eq_child_ffold
thf(fact_826_dtail__commute,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,E4: b,R2: a,Def: b > a] :
( finite7715548283558590705_b_b_a
@ ( produc4313903556115589696_a_b_a
@ ^ [X3: dtree_a_b,E2: b,B: b > a] :
( if_b_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E4 @ ( darcs_a_b @ X3 ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( dtail_a_b @ X3 @ Def ) ) ) ) ).
% dtail_commute
thf(fact_827_dtail__commute__aux,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,E4: b,R2: a,Def: b > a,Y: produc4558475209616630778_a_b_b,X4: produc4558475209616630778_a_b_b,Z2: b > a] :
( ( comp_b_a_b_a_b_a
@ ( produc4313903556115589696_a_b_a
@ ^ [X3: dtree_a_b,E2: b,B: b > a] :
( if_b_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E4 @ ( darcs_a_b @ X3 ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( dtail_a_b @ X3 @ Def ) )
@ Y )
@ ( produc4313903556115589696_a_b_a
@ ^ [X3: dtree_a_b,E2: b,B: b > a] :
( if_b_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E4 @ ( darcs_a_b @ X3 ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( dtail_a_b @ X3 @ Def ) )
@ X4 )
@ Z2 )
= ( comp_b_a_b_a_b_a
@ ( produc4313903556115589696_a_b_a
@ ^ [X3: dtree_a_b,E2: b,B: b > a] :
( if_b_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E4 @ ( darcs_a_b @ X3 ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( dtail_a_b @ X3 @ Def ) )
@ X4 )
@ ( produc4313903556115589696_a_b_a
@ ^ [X3: dtree_a_b,E2: b,B: b > a] :
( if_b_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E4 @ ( darcs_a_b @ X3 ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( dtail_a_b @ X3 @ Def ) )
@ Y )
@ Z2 ) ) ).
% dtail_commute_aux
thf(fact_828_bot__empty__eq,axiom,
( bot_bo7321339186913516097_b_b_o
= ( ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ bot_bo3721250822024684356_a_b_b ) ) ) ).
% bot_empty_eq
thf(fact_829_bot__empty__eq,axiom,
( bot_bot_b_o
= ( ^ [X3: b] : ( member_b @ X3 @ bot_bot_set_b ) ) ) ).
% bot_empty_eq
thf(fact_830_disjoint__darcs__if__wf__aux5,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T1: dtree_a_b,E1: b,T2: dtree_a_b,E22: b] :
( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T1 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T2 @ E22 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( ( ( produc331601717337510060_a_b_b @ T1 @ E1 )
!= ( produc331601717337510060_a_b_b @ T2 @ E22 ) )
=> ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ T1 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ T2 ) @ ( insert_b @ E22 @ bot_bot_set_b ) ) )
= bot_bot_set_b ) ) ) ) ) ).
% disjoint_darcs_if_wf_aux5
thf(fact_831_disjoint__darcs__simp,axiom,
! [T1: dtree_a_b,E1: b,Xs2: fset_P5281107635120001194_a_b_b,T2: dtree_a_b,E22: b] :
( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T1 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T2 @ E22 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( ( ( produc331601717337510060_a_b_b @ T1 @ E1 )
!= ( produc331601717337510060_a_b_b @ T2 @ E22 ) )
=> ( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X ) )
=> ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ T1 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ T2 ) @ ( insert_b @ E22 @ bot_bot_set_b ) ) )
= bot_bot_set_b ) ) ) ) ) ).
% disjoint_darcs_simp
thf(fact_832_dtail__ffold__supset,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,Ys: fset_P5281107635120001194_a_b_b,R2: a,E4: b,Def: b > a] :
( ( ord_le789900035998834954_a_b_b @ Xs2 @ Ys )
=> ( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Ys ) )
=> ( ( ffold_8367945289176929151_b_b_a
@ ( produc4313903556115589696_a_b_a
@ ^ [X3: dtree_a_b,E2: b,B: b > a] :
( if_b_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Ys ) )
| ~ ( member_b @ E4 @ ( darcs_a_b @ X3 ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Ys ) ) )
@ B
@ ( dtail_a_b @ X3 @ Def ) ) )
@ Def
@ Xs2 )
= ( ffold_8367945289176929151_b_b_a
@ ( produc4313903556115589696_a_b_a
@ ^ [X3: dtree_a_b,E2: b,B: b > a] :
( if_b_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E4 @ ( darcs_a_b @ X3 ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( dtail_a_b @ X3 @ Def ) ) )
@ Def
@ Xs2 ) ) ) ) ).
% dtail_ffold_supset
thf(fact_833_disjoint__darcs__insert,axiom,
! [X4: produc4558475209616630778_a_b_b,Xs2: fset_P5281107635120001194_a_b_b] :
( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ X4 @ Xs2 ) ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ ( finser8437519239679886002_a_b_b @ X4 @ Xs2 ) ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X ) )
=> ! [X5: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X5 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X5 ) ) ) ).
% disjoint_darcs_insert
thf(fact_834_wf__darcs_H_Osimps,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b] :
( ( wf_darcs_a_b2 @ ( node_a_b @ R2 @ Xs2 ) )
= ( ! [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X3 ) )
& ! [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E: b] : ( wf_darcs_a_b2 @ Y5 )
@ X3 ) ) ) ) ).
% wf_darcs'.simps
thf(fact_835_wf__darcs_H_Oelims_I1_J,axiom,
! [X4: dtree_a_b,Y: $o] :
( ( ( wf_darcs_a_b2 @ X4 )
= Y )
=> ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( ( X4
= ( node_a_b @ R @ Xs ) )
=> ( Y
= ( ~ ( ! [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X3 ) )
& ! [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E: b] : ( wf_darcs_a_b2 @ Y5 )
@ X3 ) ) ) ) ) ) ) ).
% wf_darcs'.elims(1)
thf(fact_836_wf__darcs_H_Oelims_I2_J,axiom,
! [X4: dtree_a_b] :
( ( wf_darcs_a_b2 @ X4 )
=> ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( ( X4
= ( node_a_b @ R @ Xs ) )
=> ~ ( ! [X5: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X5 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X5 ) )
& ! [X5: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X5 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E: b] : ( wf_darcs_a_b2 @ Y5 )
@ X5 ) ) ) ) ) ).
% wf_darcs'.elims(2)
thf(fact_837_wf__darcs_H_Oelims_I3_J,axiom,
! [X4: dtree_a_b] :
( ~ ( wf_darcs_a_b2 @ X4 )
=> ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( ( X4
= ( node_a_b @ R @ Xs ) )
=> ( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X ) )
& ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E: b] : ( wf_darcs_a_b2 @ Y5 )
@ X ) ) ) ) ) ).
% wf_darcs'.elims(3)
thf(fact_838_disjoint__darcs__if__wf,axiom,
! [T: dtree_a_b] :
( ( wf_darcs_a_b @ T )
=> ! [X5: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X5 @ ( fset_P783253628892185035_a_b_b @ ( sucs_a_b @ T ) ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ ( sucs_a_b @ T ) ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X5 ) ) ) ).
% disjoint_darcs_if_wf
thf(fact_839_fthe__felem__eq,axiom,
! [X4: produc4558475209616630778_a_b_b] :
( ( fthe_e7442499522476018237_a_b_b @ ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) )
= X4 ) ).
% fthe_felem_eq
thf(fact_840_subsetI,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ A3 )
=> ( member4380921116106875537_a_b_b @ X @ B3 ) )
=> ( ord_le146215904626753808_a_b_b @ A3 @ B3 ) ) ).
% subsetI
thf(fact_841_subsetI,axiom,
! [A3: set_b,B3: set_b] :
( ! [X: b] :
( ( member_b @ X @ A3 )
=> ( member_b @ X @ B3 ) )
=> ( ord_less_eq_set_b @ A3 @ B3 ) ) ).
% subsetI
thf(fact_842_finter__iff,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ C2 @ ( inf_in7138637532943773244_a_b_b @ A3 @ B3 ) )
= ( ( fmembe3173364709796808819_a_b_b @ C2 @ A3 )
& ( fmembe3173364709796808819_a_b_b @ C2 @ B3 ) ) ) ).
% finter_iff
thf(fact_843_finterI,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ C2 @ A3 )
=> ( ( fmembe3173364709796808819_a_b_b @ C2 @ B3 )
=> ( fmembe3173364709796808819_a_b_b @ C2 @ ( inf_in7138637532943773244_a_b_b @ A3 @ B3 ) ) ) ) ).
% finterI
thf(fact_844_finsert__inter__finsert,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( inf_in7138637532943773244_a_b_b @ ( finser8437519239679886002_a_b_b @ A2 @ A3 ) @ ( finser8437519239679886002_a_b_b @ A2 @ B3 ) )
= ( finser8437519239679886002_a_b_b @ A2 @ ( inf_in7138637532943773244_a_b_b @ A3 @ B3 ) ) ) ).
% finsert_inter_finsert
thf(fact_845_inter__fset,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Xa: fset_P5281107635120001194_a_b_b] :
( ( fset_P783253628892185035_a_b_b @ ( inf_in7138637532943773244_a_b_b @ X4 @ Xa ) )
= ( inf_in6138156342456174402_a_b_b @ ( fset_P783253628892185035_a_b_b @ X4 ) @ ( fset_P783253628892185035_a_b_b @ Xa ) ) ) ).
% inter_fset
thf(fact_846_finter__finsert__left__if1,axiom,
! [A2: produc4558475209616630778_a_b_b,C: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ A2 @ C )
=> ( ( inf_in7138637532943773244_a_b_b @ ( finser8437519239679886002_a_b_b @ A2 @ B3 ) @ C )
= ( finser8437519239679886002_a_b_b @ A2 @ ( inf_in7138637532943773244_a_b_b @ B3 @ C ) ) ) ) ).
% finter_finsert_left_if1
thf(fact_847_finter__finsert__right__if1,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ A2 @ A3 )
=> ( ( inf_in7138637532943773244_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ A2 @ B3 ) )
= ( finser8437519239679886002_a_b_b @ A2 @ ( inf_in7138637532943773244_a_b_b @ A3 @ B3 ) ) ) ) ).
% finter_finsert_right_if1
thf(fact_848_finter__finsert__left__ifffempty,axiom,
! [A2: produc4558475209616630778_a_b_b,C: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ~ ( fmembe3173364709796808819_a_b_b @ A2 @ C )
=> ( ( inf_in7138637532943773244_a_b_b @ ( finser8437519239679886002_a_b_b @ A2 @ B3 ) @ C )
= ( inf_in7138637532943773244_a_b_b @ B3 @ C ) ) ) ).
% finter_finsert_left_ifffempty
thf(fact_849_finter__finsert__right__ifffempty,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ~ ( fmembe3173364709796808819_a_b_b @ A2 @ A3 )
=> ( ( inf_in7138637532943773244_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ A2 @ B3 ) )
= ( inf_in7138637532943773244_a_b_b @ A3 @ B3 ) ) ) ).
% finter_finsert_right_ifffempty
thf(fact_850_dtree_Ocollapse,axiom,
! [Dtree: dtree_a_b] :
( ( node_a_b @ ( root_a_b @ Dtree ) @ ( sucs_a_b @ Dtree ) )
= Dtree ) ).
% dtree.collapse
thf(fact_851_in__mono,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b,X4: produc4558475209616630778_a_b_b] :
( ( ord_le146215904626753808_a_b_b @ A3 @ B3 )
=> ( ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( member4380921116106875537_a_b_b @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_852_in__mono,axiom,
! [A3: set_b,B3: set_b,X4: b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( member_b @ X4 @ A3 )
=> ( member_b @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_853_subsetD,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b,C2: produc4558475209616630778_a_b_b] :
( ( ord_le146215904626753808_a_b_b @ A3 @ B3 )
=> ( ( member4380921116106875537_a_b_b @ C2 @ A3 )
=> ( member4380921116106875537_a_b_b @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_854_subsetD,axiom,
! [A3: set_b,B3: set_b,C2: b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( member_b @ C2 @ A3 )
=> ( member_b @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_855_subset__eq,axiom,
( ord_le146215904626753808_a_b_b
= ( ^ [A6: set_Pr3012420139608375472_a_b_b,B6: set_Pr3012420139608375472_a_b_b] :
! [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ A6 )
=> ( member4380921116106875537_a_b_b @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_856_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B6: set_b] :
! [X3: b] :
( ( member_b @ X3 @ A6 )
=> ( member_b @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_857_subset__iff,axiom,
( ord_le146215904626753808_a_b_b
= ( ^ [A6: set_Pr3012420139608375472_a_b_b,B6: set_Pr3012420139608375472_a_b_b] :
! [T3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ T3 @ A6 )
=> ( member4380921116106875537_a_b_b @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_858_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B6: set_b] :
! [T3: b] :
( ( member_b @ T3 @ A6 )
=> ( member_b @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_859_less__eq__set__def,axiom,
( ord_le146215904626753808_a_b_b
= ( ^ [A6: set_Pr3012420139608375472_a_b_b,B6: set_Pr3012420139608375472_a_b_b] :
( ord_le8988533026730861429_b_b_o
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ A6 )
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_860_less__eq__set__def,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B6: set_b] :
( ord_less_eq_b_o
@ ^ [X3: b] : ( member_b @ X3 @ A6 )
@ ^ [X3: b] : ( member_b @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_861_pred__subset__eq,axiom,
! [R3: set_Pr3012420139608375472_a_b_b,S2: set_Pr3012420139608375472_a_b_b] :
( ( ord_le8988533026730861429_b_b_o
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ R3 )
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ S2 ) )
= ( ord_le146215904626753808_a_b_b @ R3 @ S2 ) ) ).
% pred_subset_eq
thf(fact_862_pred__subset__eq,axiom,
! [R3: set_b,S2: set_b] :
( ( ord_less_eq_b_o
@ ^ [X3: b] : ( member_b @ X3 @ R3 )
@ ^ [X3: b] : ( member_b @ X3 @ S2 ) )
= ( ord_less_eq_set_b @ R3 @ S2 ) ) ).
% pred_subset_eq
thf(fact_863_Collect__subset,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,P: produc4558475209616630778_a_b_b > $o] :
( ord_le146215904626753808_a_b_b
@ ( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ A3 )
& ( P @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_864_Collect__subset,axiom,
! [A3: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A3 )
& ( P @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_865_finterD2,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ C2 @ ( inf_in7138637532943773244_a_b_b @ A3 @ B3 ) )
=> ( fmembe3173364709796808819_a_b_b @ C2 @ B3 ) ) ).
% finterD2
thf(fact_866_finterD1,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ C2 @ ( inf_in7138637532943773244_a_b_b @ A3 @ B3 ) )
=> ( fmembe3173364709796808819_a_b_b @ C2 @ A3 ) ) ).
% finterD1
thf(fact_867_finterE,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ C2 @ ( inf_in7138637532943773244_a_b_b @ A3 @ B3 ) )
=> ~ ( ( fmembe3173364709796808819_a_b_b @ C2 @ A3 )
=> ~ ( fmembe3173364709796808819_a_b_b @ C2 @ B3 ) ) ) ).
% finterE
thf(fact_868_finter__greatest,axiom,
! [C: fset_P5281107635120001194_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ C @ A3 )
=> ( ( ord_le789900035998834954_a_b_b @ C @ B3 )
=> ( ord_le789900035998834954_a_b_b @ C @ ( inf_in7138637532943773244_a_b_b @ A3 @ B3 ) ) ) ) ).
% finter_greatest
thf(fact_869_finter__absorb2,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ B3 )
=> ( ( inf_in7138637532943773244_a_b_b @ A3 @ B3 )
= A3 ) ) ).
% finter_absorb2
thf(fact_870_finter__absorb1,axiom,
! [B3: fset_P5281107635120001194_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ B3 @ A3 )
=> ( ( inf_in7138637532943773244_a_b_b @ A3 @ B3 )
= B3 ) ) ).
% finter_absorb1
thf(fact_871_finter__lower2,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ ( inf_in7138637532943773244_a_b_b @ A3 @ B3 ) @ B3 ) ).
% finter_lower2
thf(fact_872_finter__lower1,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ ( inf_in7138637532943773244_a_b_b @ A3 @ B3 ) @ A3 ) ).
% finter_lower1
thf(fact_873_finter__mono,axiom,
! [A3: fset_P5281107635120001194_a_b_b,C: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b,D2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ C )
=> ( ( ord_le789900035998834954_a_b_b @ B3 @ D2 )
=> ( ord_le789900035998834954_a_b_b @ ( inf_in7138637532943773244_a_b_b @ A3 @ B3 ) @ ( inf_in7138637532943773244_a_b_b @ C @ D2 ) ) ) ) ).
% finter_mono
thf(fact_874_dtree_Osel_I2_J,axiom,
! [X1: a,X2: fset_P5281107635120001194_a_b_b] :
( ( sucs_a_b @ ( node_a_b @ X1 @ X2 ) )
= X2 ) ).
% dtree.sel(2)
thf(fact_875_inf__Int__eq2,axiom,
! [R3: set_Pr3012420139608375472_a_b_b,S2: set_Pr3012420139608375472_a_b_b] :
( ( inf_in8207984165653407081_b_b_o
@ ^ [X3: dtree_a_b,Y5: b] : ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ Y5 ) @ R3 )
@ ^ [X3: dtree_a_b,Y5: b] : ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ Y5 ) @ S2 ) )
= ( ^ [X3: dtree_a_b,Y5: b] : ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ Y5 ) @ ( inf_in6138156342456174402_a_b_b @ R3 @ S2 ) ) ) ) ).
% inf_Int_eq2
thf(fact_876_inf__set__def,axiom,
( inf_in6138156342456174402_a_b_b
= ( ^ [A6: set_Pr3012420139608375472_a_b_b,B6: set_Pr3012420139608375472_a_b_b] :
( collec1368399972772960719_a_b_b
@ ( inf_in55627642082981827_b_b_o
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ A6 )
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_877_inf__set__def,axiom,
( inf_inf_set_b
= ( ^ [A6: set_b,B6: set_b] :
( collect_b
@ ( inf_inf_b_o
@ ^ [X3: b] : ( member_b @ X3 @ A6 )
@ ^ [X3: b] : ( member_b @ X3 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_878_inf__Int__eq,axiom,
! [R3: set_Pr3012420139608375472_a_b_b,S2: set_Pr3012420139608375472_a_b_b] :
( ( inf_in55627642082981827_b_b_o
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ R3 )
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ S2 ) )
= ( ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ ( inf_in6138156342456174402_a_b_b @ R3 @ S2 ) ) ) ) ).
% inf_Int_eq
thf(fact_879_inf__Int__eq,axiom,
! [R3: set_b,S2: set_b] :
( ( inf_inf_b_o
@ ^ [X3: b] : ( member_b @ X3 @ R3 )
@ ^ [X3: b] : ( member_b @ X3 @ S2 ) )
= ( ^ [X3: b] : ( member_b @ X3 @ ( inf_inf_set_b @ R3 @ S2 ) ) ) ) ).
% inf_Int_eq
thf(fact_880_dtree_Oexpand,axiom,
! [Dtree: dtree_a_b,Dtree2: dtree_a_b] :
( ( ( ( root_a_b @ Dtree )
= ( root_a_b @ Dtree2 ) )
& ( ( sucs_a_b @ Dtree )
= ( sucs_a_b @ Dtree2 ) ) )
=> ( Dtree = Dtree2 ) ) ).
% dtree.expand
thf(fact_881_wf__darcs_H__if__darcs,axiom,
! [T: dtree_a_b] :
( ( wf_darcs_a_b @ T )
=> ( wf_darcs_a_b2 @ T ) ) ).
% wf_darcs'_if_darcs
thf(fact_882_wf__darcs__if__darcs_H,axiom,
! [T: dtree_a_b] :
( ( wf_darcs_a_b2 @ T )
=> ( wf_darcs_a_b @ T ) ) ).
% wf_darcs_if_darcs'
thf(fact_883_wf__darcs__iff__darcs_H,axiom,
wf_darcs_a_b = wf_darcs_a_b2 ).
% wf_darcs_iff_darcs'
thf(fact_884_finter__finsert__left,axiom,
! [A2: produc4558475209616630778_a_b_b,C: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( ( fmembe3173364709796808819_a_b_b @ A2 @ C )
=> ( ( inf_in7138637532943773244_a_b_b @ ( finser8437519239679886002_a_b_b @ A2 @ B3 ) @ C )
= ( finser8437519239679886002_a_b_b @ A2 @ ( inf_in7138637532943773244_a_b_b @ B3 @ C ) ) ) )
& ( ~ ( fmembe3173364709796808819_a_b_b @ A2 @ C )
=> ( ( inf_in7138637532943773244_a_b_b @ ( finser8437519239679886002_a_b_b @ A2 @ B3 ) @ C )
= ( inf_in7138637532943773244_a_b_b @ B3 @ C ) ) ) ) ).
% finter_finsert_left
thf(fact_885_finter__finsert__right,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( ( fmembe3173364709796808819_a_b_b @ A2 @ A3 )
=> ( ( inf_in7138637532943773244_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ A2 @ B3 ) )
= ( finser8437519239679886002_a_b_b @ A2 @ ( inf_in7138637532943773244_a_b_b @ A3 @ B3 ) ) ) )
& ( ~ ( fmembe3173364709796808819_a_b_b @ A2 @ A3 )
=> ( ( inf_in7138637532943773244_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ A2 @ B3 ) )
= ( inf_in7138637532943773244_a_b_b @ A3 @ B3 ) ) ) ) ).
% finter_finsert_right
thf(fact_886_funion__finter__assoc__eq,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b,C: fset_P5281107635120001194_a_b_b] :
( ( ( sup_su860928060825958358_a_b_b @ ( inf_in7138637532943773244_a_b_b @ A3 @ B3 ) @ C )
= ( inf_in7138637532943773244_a_b_b @ A3 @ ( sup_su860928060825958358_a_b_b @ B3 @ C ) ) )
= ( ord_le789900035998834954_a_b_b @ C @ A3 ) ) ).
% funion_finter_assoc_eq
thf(fact_887_dtree_Oexhaust__sel,axiom,
! [Dtree: dtree_a_b] :
( Dtree
= ( node_a_b @ ( root_a_b @ Dtree ) @ ( sucs_a_b @ Dtree ) ) ) ).
% dtree.exhaust_sel
thf(fact_888_singleton__uneq_H,axiom,
! [R2: a,T: dtree_a_b,E4: b,V: a] :
( ( node_a_b @ R2 @ ( finser8437519239679886002_a_b_b @ ( produc331601717337510060_a_b_b @ T @ E4 ) @ bot_bo2895716411488905534_a_b_b ) )
!= ( node_a_b @ V @ ( sucs_a_b @ T ) ) ) ).
% singleton_uneq'
thf(fact_889_wf__darcs__sucs,axiom,
! [T: dtree_a_b,X4: produc4558475209616630778_a_b_b,R2: a] :
( ( wf_darcs_a_b @ T )
=> ( ( member4380921116106875537_a_b_b @ X4 @ ( fset_P783253628892185035_a_b_b @ ( sucs_a_b @ T ) ) )
=> ( wf_darcs_a_b @ ( node_a_b @ R2 @ ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) ) ) ) ) ).
% wf_darcs_sucs
thf(fact_890_comp__fun__commute__filter__fold,axiom,
! [P: b > $o] :
( finite4863250414163961073_set_b
@ ^ [X3: b,A8: set_b] : ( if_set_b @ ( P @ X3 ) @ ( insert_b @ X3 @ A8 ) @ A8 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_891_disjoint__darcs__img,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,F: dtree_a_b > dtree_a_b] :
( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X ) )
=> ( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [T3: dtree_a_b,E: b] : ( ord_less_eq_set_b @ ( darcs_a_b @ ( F @ T3 ) ) @ ( darcs_a_b @ T3 ) )
@ X ) )
=> ! [X5: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X5
@ ( fset_P783253628892185035_a_b_b
@ ( fimage7457256623133068659_a_b_b
@ ( produc5460679229782211283_a_b_b
@ ^ [T3: dtree_a_b] : ( produc331601717337510060_a_b_b @ ( F @ T3 ) ) )
@ Xs2 ) ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4
@ ( fset_P783253628892185035_a_b_b
@ ( fimage7457256623133068659_a_b_b
@ ( produc5460679229782211283_a_b_b
@ ^ [T3: dtree_a_b] : ( produc331601717337510060_a_b_b @ ( F @ T3 ) ) )
@ Xs2 ) ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X5 ) ) ) ) ).
% disjoint_darcs_img
thf(fact_892_case__prod__app,axiom,
( produc2242037354397874494_b_a_a
= ( ^ [F2: dtree_a_b > b > a > a,X3: produc4558475209616630778_a_b_b,Y5: a] :
( produc3664522937540588133_b_b_a
@ ^ [L: dtree_a_b,R5: b] : ( F2 @ L @ R5 @ Y5 )
@ X3 ) ) ) ).
% case_prod_app
thf(fact_893_fimage__eqI,axiom,
! [B2: produc4558475209616630778_a_b_b,F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b,X4: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( B2
= ( F @ X4 ) )
=> ( ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ( fmembe3173364709796808819_a_b_b @ B2 @ ( fimage7457256623133068659_a_b_b @ F @ A3 ) ) ) ) ).
% fimage_eqI
thf(fact_894_fimage__finsert,axiom,
! [F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b,A2: produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( fimage7457256623133068659_a_b_b @ F @ ( finser8437519239679886002_a_b_b @ A2 @ B3 ) )
= ( finser8437519239679886002_a_b_b @ ( F @ A2 ) @ ( fimage7457256623133068659_a_b_b @ F @ B3 ) ) ) ).
% fimage_finsert
thf(fact_895_finsert__fimage,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ( ( finser8437519239679886002_a_b_b @ ( F @ X4 ) @ ( fimage7457256623133068659_a_b_b @ F @ A3 ) )
= ( fimage7457256623133068659_a_b_b @ F @ A3 ) ) ) ).
% finsert_fimage
thf(fact_896_fset_Omap__ident__strong,axiom,
! [T: fset_b,F: b > b] :
( ! [Z3: b] :
( ( member_b @ Z3 @ ( fset_b2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( fimage_b_b @ F @ T )
= T ) ) ).
% fset.map_ident_strong
thf(fact_897_fset_Omap__ident__strong,axiom,
! [T: fset_P5281107635120001194_a_b_b,F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b] :
( ! [Z3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z3 @ ( fset_P783253628892185035_a_b_b @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( fimage7457256623133068659_a_b_b @ F @ T )
= T ) ) ).
% fset.map_ident_strong
thf(fact_898_rev__fimage__eqI,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B2: produc4558475209616630778_a_b_b,F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ( ( B2
= ( F @ X4 ) )
=> ( fmembe3173364709796808819_a_b_b @ B2 @ ( fimage7457256623133068659_a_b_b @ F @ A3 ) ) ) ) ).
% rev_fimage_eqI
thf(fact_899_fimageI,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ( fmembe3173364709796808819_a_b_b @ ( F @ X4 ) @ ( fimage7457256623133068659_a_b_b @ F @ A3 ) ) ) ).
% fimageI
thf(fact_900_fimageE,axiom,
! [B2: produc4558475209616630778_a_b_b,F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ B2 @ ( fimage7457256623133068659_a_b_b @ F @ A3 ) )
=> ~ ! [X: produc4558475209616630778_a_b_b] :
( ( B2
= ( F @ X ) )
=> ~ ( fmembe3173364709796808819_a_b_b @ X @ A3 ) ) ) ).
% fimageE
thf(fact_901_subset__fimage__iff,axiom,
! [B3: fset_P5281107635120001194_a_b_b,F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ B3 @ ( fimage7457256623133068659_a_b_b @ F @ A3 ) )
= ( ? [AA: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ AA @ A3 )
& ( B3
= ( fimage7457256623133068659_a_b_b @ F @ AA ) ) ) ) ) ).
% subset_fimage_iff
thf(fact_902_fimage__mono,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b,F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ B3 )
=> ( ord_le789900035998834954_a_b_b @ ( fimage7457256623133068659_a_b_b @ F @ A3 ) @ ( fimage7457256623133068659_a_b_b @ F @ B3 ) ) ) ).
% fimage_mono
thf(fact_903_fimage__fsubsetI,axiom,
! [A3: fset_P5281107635120001194_a_b_b,F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ! [X: produc4558475209616630778_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ X @ A3 )
=> ( fmembe3173364709796808819_a_b_b @ ( F @ X ) @ B3 ) )
=> ( ord_le789900035998834954_a_b_b @ ( fimage7457256623133068659_a_b_b @ F @ A3 ) @ B3 ) ) ).
% fimage_fsubsetI
thf(fact_904_fimage__constant,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,C2: produc4558475209616630778_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ( ( fimage7457256623133068659_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] : C2
@ A3 )
= ( finser8437519239679886002_a_b_b @ C2 @ bot_bo2895716411488905534_a_b_b ) ) ) ).
% fimage_constant
thf(fact_905_case__prod__Pair__iden,axiom,
! [P2: produc4558475209616630778_a_b_b] :
( ( produc5460679229782211283_a_b_b @ produc331601717337510060_a_b_b @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_906_wf__darcs_H_Opelims_I1_J,axiom,
! [X4: dtree_a_b,Y: $o] :
( ( ( wf_darcs_a_b2 @ X4 )
= Y )
=> ( ( accp_dtree_a_b @ wf_darcs_rel_a_b @ X4 )
=> ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( ( X4
= ( node_a_b @ R @ Xs ) )
=> ( ( Y
= ( ! [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X3 ) )
& ! [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E: b] : ( wf_darcs_a_b2 @ Y5 )
@ X3 ) ) ) )
=> ~ ( accp_dtree_a_b @ wf_darcs_rel_a_b @ ( node_a_b @ R @ Xs ) ) ) ) ) ) ).
% wf_darcs'.pelims(1)
thf(fact_907_wf__darcs_H_Opelims_I2_J,axiom,
! [X4: dtree_a_b] :
( ( wf_darcs_a_b2 @ X4 )
=> ( ( accp_dtree_a_b @ wf_darcs_rel_a_b @ X4 )
=> ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( ( X4
= ( node_a_b @ R @ Xs ) )
=> ( ( accp_dtree_a_b @ wf_darcs_rel_a_b @ ( node_a_b @ R @ Xs ) )
=> ~ ( ! [X5: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X5 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X5 ) )
& ! [X5: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X5 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E: b] : ( wf_darcs_a_b2 @ Y5 )
@ X5 ) ) ) ) ) ) ) ).
% wf_darcs'.pelims(2)
thf(fact_908_wf__darcs_H_Opelims_I3_J,axiom,
! [X4: dtree_a_b] :
( ~ ( wf_darcs_a_b2 @ X4 )
=> ( ( accp_dtree_a_b @ wf_darcs_rel_a_b @ X4 )
=> ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( ( X4
= ( node_a_b @ R @ Xs ) )
=> ( ( accp_dtree_a_b @ wf_darcs_rel_a_b @ ( node_a_b @ R @ Xs ) )
=> ( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_b @ E12 @ ( darcs_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y5 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X ) )
& ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E: b] : ( wf_darcs_a_b2 @ Y5 )
@ X ) ) ) ) ) ) ) ).
% wf_darcs'.pelims(3)
thf(fact_909_Collect__empty__eq__bot,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( P = bot_bot_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_910_is__singletonI,axiom,
! [X4: b] : ( is_singleton_b @ ( insert_b @ X4 @ bot_bot_set_b ) ) ).
% is_singletonI
thf(fact_911_is__singletonI_H,axiom,
! [A3: set_Pr3012420139608375472_a_b_b] :
( ( A3 != bot_bo3721250822024684356_a_b_b )
=> ( ! [X: produc4558475209616630778_a_b_b,Y4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ A3 )
=> ( ( member4380921116106875537_a_b_b @ Y4 @ A3 )
=> ( X = Y4 ) ) )
=> ( is_sin1118336051388392454_a_b_b @ A3 ) ) ) ).
% is_singletonI'
thf(fact_912_is__singletonI_H,axiom,
! [A3: set_b] :
( ( A3 != bot_bot_set_b )
=> ( ! [X: b,Y4: b] :
( ( member_b @ X @ A3 )
=> ( ( member_b @ Y4 @ A3 )
=> ( X = Y4 ) ) )
=> ( is_singleton_b @ A3 ) ) ) ).
% is_singletonI'
thf(fact_913_is__singleton__def,axiom,
( is_singleton_b
= ( ^ [A6: set_b] :
? [X3: b] :
( A6
= ( insert_b @ X3 @ bot_bot_set_b ) ) ) ) ).
% is_singleton_def
thf(fact_914_is__singletonE,axiom,
! [A3: set_b] :
( ( is_singleton_b @ A3 )
=> ~ ! [X: b] :
( A3
!= ( insert_b @ X @ bot_bot_set_b ) ) ) ).
% is_singletonE
thf(fact_915_split__cong,axiom,
! [Q2: produc4558475209616630778_a_b_b,F: dtree_a_b > b > a > a,G2: dtree_a_b > b > a > a,P2: produc4558475209616630778_a_b_b] :
( ! [X: dtree_a_b,Y4: b] :
( ( ( produc331601717337510060_a_b_b @ X @ Y4 )
= Q2 )
=> ( ( F @ X @ Y4 )
= ( G2 @ X @ Y4 ) ) )
=> ( ( P2 = Q2 )
=> ( ( produc2242037354397874494_b_a_a @ F @ P2 )
= ( produc2242037354397874494_b_a_a @ G2 @ Q2 ) ) ) ) ).
% split_cong
thf(fact_916_insert__subsetI,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,X6: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( ( ord_le146215904626753808_a_b_b @ X6 @ A3 )
=> ( ord_le146215904626753808_a_b_b @ ( insert1613891728210272810_a_b_b @ X4 @ X6 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_917_insert__subsetI,axiom,
! [X4: b,A3: set_b,X6: set_b] :
( ( member_b @ X4 @ A3 )
=> ( ( ord_less_eq_set_b @ X6 @ A3 )
=> ( ord_less_eq_set_b @ ( insert_b @ X4 @ X6 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_918_subset__emptyI,axiom,
! [A3: set_Pr3012420139608375472_a_b_b] :
( ! [X: produc4558475209616630778_a_b_b] :
~ ( member4380921116106875537_a_b_b @ X @ A3 )
=> ( ord_le146215904626753808_a_b_b @ A3 @ bot_bo3721250822024684356_a_b_b ) ) ).
% subset_emptyI
thf(fact_919_subset__emptyI,axiom,
! [A3: set_b] :
( ! [X: b] :
~ ( member_b @ X @ A3 )
=> ( ord_less_eq_set_b @ A3 @ bot_bot_set_b ) ) ).
% subset_emptyI
thf(fact_920_ssubst__Pair__rhs,axiom,
! [R2: dtree_a_b,S: b,R3: set_Pr3012420139608375472_a_b_b,S5: b] :
( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ R2 @ S ) @ R3 )
=> ( ( S5 = S )
=> ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ R2 @ S5 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_921_Collect__restrict,axiom,
! [X6: set_Pr3012420139608375472_a_b_b,P: produc4558475209616630778_a_b_b > $o] :
( ord_le146215904626753808_a_b_b
@ ( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ X6 )
& ( P @ X3 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_922_Collect__restrict,axiom,
! [X6: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ X6 )
& ( P @ X3 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_923_prop__restrict,axiom,
! [X4: produc4558475209616630778_a_b_b,Z5: set_Pr3012420139608375472_a_b_b,X6: set_Pr3012420139608375472_a_b_b,P: produc4558475209616630778_a_b_b > $o] :
( ( member4380921116106875537_a_b_b @ X4 @ Z5 )
=> ( ( ord_le146215904626753808_a_b_b @ Z5
@ ( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ X6 )
& ( P @ X3 ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_924_prop__restrict,axiom,
! [X4: b,Z5: set_b,X6: set_b,P: b > $o] :
( ( member_b @ X4 @ Z5 )
=> ( ( ord_less_eq_set_b @ Z5
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ X6 )
& ( P @ X3 ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_925_is__singleton__the__elem,axiom,
( is_singleton_b
= ( ^ [A6: set_b] :
( A6
= ( insert_b @ ( the_elem_b @ A6 ) @ bot_bot_set_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_926_the__elem__eq,axiom,
! [X4: b] :
( ( the_elem_b @ ( insert_b @ X4 @ bot_bot_set_b ) )
= X4 ) ).
% the_elem_eq
thf(fact_927_dtree_Oroot__def,axiom,
( root_a_b
= ( case_dtree_a_b_a
@ ^ [X13: a,X23: fset_P5281107635120001194_a_b_b] : X13 ) ) ).
% dtree.root_def
thf(fact_928_fthe__elem_Orep__eq,axiom,
( fthe_e7442499522476018237_a_b_b
= ( ^ [X3: fset_P5281107635120001194_a_b_b] : ( the_el4127461656392778949_a_b_b @ ( fset_P783253628892185035_a_b_b @ X3 ) ) ) ) ).
% fthe_elem.rep_eq
thf(fact_929_comp__fun__commute_Offold__rec,axiom,
! [F: produc4558475209616630778_a_b_b > a > a,X4: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,Z2: a] :
( ( finite414203908571218417_b_b_a @ F )
=> ( ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ( ( ffold_2783168711033344739_b_b_a @ F @ Z2 @ A3 )
= ( F @ X4 @ ( ffold_2783168711033344739_b_b_a @ F @ Z2 @ ( minus_1250967532242559235_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) ) ) ) ) ) ) ).
% comp_fun_commute.ffold_rec
thf(fact_930_dtail_Opelims,axiom,
! [X4: dtree_a_b,Xa: b > a,Y: b > a] :
( ( ( dtail_a_b @ X4 @ Xa )
= Y )
=> ( ( accp_P1416650344722773512_b_b_a @ dtail_rel_a_b @ ( produc1993688775741047735_b_b_a @ X4 @ Xa ) )
=> ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( ( X4
= ( node_a_b @ R @ Xs ) )
=> ( ( Y
= ( ^ [E: b] :
( if_a @ ( member_b @ E @ ( image_3908709015779211070_b_b_b @ produc5748100250121904638_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs ) ) ) @ R
@ ( ffold_8367945289176929151_b_b_a
@ ( produc4313903556115589696_a_b_a
@ ^ [X3: dtree_a_b,E2: b,B: b > a] :
( if_b_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs ) )
| ~ ( member_b @ E @ ( darcs_a_b @ X3 ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R @ Xs ) ) )
@ B
@ ( dtail_a_b @ X3 @ Xa ) ) )
@ Xa
@ Xs
@ E ) ) ) )
=> ~ ( accp_P1416650344722773512_b_b_a @ dtail_rel_a_b @ ( produc1993688775741047735_b_b_a @ ( node_a_b @ R @ Xs ) @ Xa ) ) ) ) ) ) ).
% dtail.pelims
thf(fact_931_prod__set__simps_I1_J,axiom,
! [X4: dtree_a_b,Y: b] :
( ( basic_7578771248400840636_a_b_b @ ( produc331601717337510060_a_b_b @ X4 @ Y ) )
= ( insert_dtree_a_b @ X4 @ bot_bo8730652382759064772ee_a_b ) ) ).
% prod_set_simps(1)
thf(fact_932_image__eqI,axiom,
! [B2: produc4558475209616630778_a_b_b,F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b,X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ( B2
= ( F @ X4 ) )
=> ( ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( member4380921116106875537_a_b_b @ B2 @ ( image_6081965176830705659_a_b_b @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_933_image__eqI,axiom,
! [B2: b,F: produc4558475209616630778_a_b_b > b,X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ( B2
= ( F @ X4 ) )
=> ( ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( member_b @ B2 @ ( image_3908709015779211070_b_b_b @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_934_image__eqI,axiom,
! [B2: produc4558475209616630778_a_b_b,F: b > produc4558475209616630778_a_b_b,X4: b,A3: set_b] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_b @ X4 @ A3 )
=> ( member4380921116106875537_a_b_b @ B2 @ ( image_7642607452437185460_a_b_b @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_935_image__eqI,axiom,
! [B2: b,F: b > b,X4: b,A3: set_b] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_b @ X4 @ A3 )
=> ( member_b @ B2 @ ( image_b_b @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_936_image__is__empty,axiom,
! [F: b > b,A3: set_b] :
( ( ( image_b_b @ F @ A3 )
= bot_bot_set_b )
= ( A3 = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_937_empty__is__image,axiom,
! [F: b > b,A3: set_b] :
( ( bot_bot_set_b
= ( image_b_b @ F @ A3 ) )
= ( A3 = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_938_image__empty,axiom,
! [F: b > b] :
( ( image_b_b @ F @ bot_bot_set_b )
= bot_bot_set_b ) ).
% image_empty
thf(fact_939_insert__image,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,F: produc4558475209616630778_a_b_b > b] :
( ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( ( insert_b @ ( F @ X4 ) @ ( image_3908709015779211070_b_b_b @ F @ A3 ) )
= ( image_3908709015779211070_b_b_b @ F @ A3 ) ) ) ).
% insert_image
thf(fact_940_insert__image,axiom,
! [X4: b,A3: set_b,F: b > b] :
( ( member_b @ X4 @ A3 )
=> ( ( insert_b @ ( F @ X4 ) @ ( image_b_b @ F @ A3 ) )
= ( image_b_b @ F @ A3 ) ) ) ).
% insert_image
thf(fact_941_image__insert,axiom,
! [F: b > b,A2: b,B3: set_b] :
( ( image_b_b @ F @ ( insert_b @ A2 @ B3 ) )
= ( insert_b @ ( F @ A2 ) @ ( image_b_b @ F @ B3 ) ) ) ).
% image_insert
thf(fact_942_fminusI,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ C2 @ A3 )
=> ( ~ ( fmembe3173364709796808819_a_b_b @ C2 @ B3 )
=> ( fmembe3173364709796808819_a_b_b @ C2 @ ( minus_1250967532242559235_a_b_b @ A3 @ B3 ) ) ) ) ).
% fminusI
thf(fact_943_fminus__iff,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ C2 @ ( minus_1250967532242559235_a_b_b @ A3 @ B3 ) )
= ( ( fmembe3173364709796808819_a_b_b @ C2 @ A3 )
& ~ ( fmembe3173364709796808819_a_b_b @ C2 @ B3 ) ) ) ).
% fminus_iff
thf(fact_944_fset_Oset__map,axiom,
! [F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b,V: fset_P5281107635120001194_a_b_b] :
( ( fset_P783253628892185035_a_b_b @ ( fimage7457256623133068659_a_b_b @ F @ V ) )
= ( image_6081965176830705659_a_b_b @ F @ ( fset_P783253628892185035_a_b_b @ V ) ) ) ).
% fset.set_map
thf(fact_945_fimage_Orep__eq,axiom,
! [X4: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b,Xa: fset_P5281107635120001194_a_b_b] :
( ( fset_P783253628892185035_a_b_b @ ( fimage7457256623133068659_a_b_b @ X4 @ Xa ) )
= ( image_6081965176830705659_a_b_b @ X4 @ ( fset_P783253628892185035_a_b_b @ Xa ) ) ) ).
% fimage.rep_eq
thf(fact_946_finsert__fminus1,axiom,
! [X4: produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ X4 @ B3 )
=> ( ( minus_1250967532242559235_a_b_b @ ( finser8437519239679886002_a_b_b @ X4 @ A3 ) @ B3 )
= ( minus_1250967532242559235_a_b_b @ A3 @ B3 ) ) ) ).
% finsert_fminus1
thf(fact_947_finsert__fminus__single,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( finser8437519239679886002_a_b_b @ A2 @ ( minus_1250967532242559235_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ A2 @ bot_bo2895716411488905534_a_b_b ) ) )
= ( finser8437519239679886002_a_b_b @ A2 @ A3 ) ) ).
% finsert_fminus_single
thf(fact_948_pair__imageI,axiom,
! [A2: dtree_a_b,B2: b,A3: set_Pr3012420139608375472_a_b_b,F: dtree_a_b > b > produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ A2 @ B2 ) @ A3 )
=> ( member4380921116106875537_a_b_b @ ( F @ A2 @ B2 ) @ ( image_6081965176830705659_a_b_b @ ( produc5460679229782211283_a_b_b @ F ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_949_pair__imageI,axiom,
! [A2: dtree_a_b,B2: b,A3: set_Pr3012420139608375472_a_b_b,F: dtree_a_b > b > b] :
( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ A2 @ B2 ) @ A3 )
=> ( member_b @ ( F @ A2 @ B2 ) @ ( image_3908709015779211070_b_b_b @ ( produc3664522937540588134_b_b_b @ F ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_950_pair__imageI,axiom,
! [A2: dtree_a_b,B2: b,A3: set_Pr3012420139608375472_a_b_b,F: dtree_a_b > b > a > a] :
( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ A2 @ B2 ) @ A3 )
=> ( member_a_a @ ( F @ A2 @ B2 ) @ ( image_1490412026869653094_b_a_a @ ( produc2242037354397874494_b_a_a @ F ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_951_image__subsetI,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ A3 )
=> ( member4380921116106875537_a_b_b @ ( F @ X ) @ B3 ) )
=> ( ord_le146215904626753808_a_b_b @ ( image_6081965176830705659_a_b_b @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_952_image__subsetI,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,F: produc4558475209616630778_a_b_b > b,B3: set_b] :
( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ A3 )
=> ( member_b @ ( F @ X ) @ B3 ) )
=> ( ord_less_eq_set_b @ ( image_3908709015779211070_b_b_b @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_953_image__subsetI,axiom,
! [A3: set_b,F: b > produc4558475209616630778_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ! [X: b] :
( ( member_b @ X @ A3 )
=> ( member4380921116106875537_a_b_b @ ( F @ X ) @ B3 ) )
=> ( ord_le146215904626753808_a_b_b @ ( image_7642607452437185460_a_b_b @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_954_image__subsetI,axiom,
! [A3: set_b,F: b > b,B3: set_b] :
( ! [X: b] :
( ( member_b @ X @ A3 )
=> ( member_b @ ( F @ X ) @ B3 ) )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_955_fminus__fsubset,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] : ( ord_le789900035998834954_a_b_b @ ( minus_1250967532242559235_a_b_b @ A3 @ B3 ) @ A3 ) ).
% fminus_fsubset
thf(fact_956_double__fminus,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b,C: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ B3 )
=> ( ( ord_le789900035998834954_a_b_b @ B3 @ C )
=> ( ( minus_1250967532242559235_a_b_b @ B3 @ ( minus_1250967532242559235_a_b_b @ C @ A3 ) )
= A3 ) ) ) ).
% double_fminus
thf(fact_957_fminus__mono,axiom,
! [A3: fset_P5281107635120001194_a_b_b,C: fset_P5281107635120001194_a_b_b,D2: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ C )
=> ( ( ord_le789900035998834954_a_b_b @ D2 @ B3 )
=> ( ord_le789900035998834954_a_b_b @ ( minus_1250967532242559235_a_b_b @ A3 @ B3 ) @ ( minus_1250967532242559235_a_b_b @ C @ D2 ) ) ) ) ).
% fminus_mono
thf(fact_958_Compr__image__eq,axiom,
! [F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,P: produc4558475209616630778_a_b_b > $o] :
( ( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ ( image_6081965176830705659_a_b_b @ F @ A3 ) )
& ( P @ X3 ) ) )
= ( image_6081965176830705659_a_b_b @ F
@ ( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ A3 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_959_Compr__image__eq,axiom,
! [F: b > produc4558475209616630778_a_b_b,A3: set_b,P: produc4558475209616630778_a_b_b > $o] :
( ( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ ( image_7642607452437185460_a_b_b @ F @ A3 ) )
& ( P @ X3 ) ) )
= ( image_7642607452437185460_a_b_b @ F
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A3 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_960_Compr__image__eq,axiom,
! [F: produc4558475209616630778_a_b_b > b,A3: set_Pr3012420139608375472_a_b_b,P: b > $o] :
( ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( image_3908709015779211070_b_b_b @ F @ A3 ) )
& ( P @ X3 ) ) )
= ( image_3908709015779211070_b_b_b @ F
@ ( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ A3 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_961_Compr__image__eq,axiom,
! [F: b > b,A3: set_b,P: b > $o] :
( ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( image_b_b @ F @ A3 ) )
& ( P @ X3 ) ) )
= ( image_b_b @ F
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A3 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_962_imageE,axiom,
! [B2: produc4558475209616630778_a_b_b,F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ B2 @ ( image_6081965176830705659_a_b_b @ F @ A3 ) )
=> ~ ! [X: produc4558475209616630778_a_b_b] :
( ( B2
= ( F @ X ) )
=> ~ ( member4380921116106875537_a_b_b @ X @ A3 ) ) ) ).
% imageE
thf(fact_963_imageE,axiom,
! [B2: produc4558475209616630778_a_b_b,F: b > produc4558475209616630778_a_b_b,A3: set_b] :
( ( member4380921116106875537_a_b_b @ B2 @ ( image_7642607452437185460_a_b_b @ F @ A3 ) )
=> ~ ! [X: b] :
( ( B2
= ( F @ X ) )
=> ~ ( member_b @ X @ A3 ) ) ) ).
% imageE
thf(fact_964_imageE,axiom,
! [B2: b,F: produc4558475209616630778_a_b_b > b,A3: set_Pr3012420139608375472_a_b_b] :
( ( member_b @ B2 @ ( image_3908709015779211070_b_b_b @ F @ A3 ) )
=> ~ ! [X: produc4558475209616630778_a_b_b] :
( ( B2
= ( F @ X ) )
=> ~ ( member4380921116106875537_a_b_b @ X @ A3 ) ) ) ).
% imageE
thf(fact_965_imageE,axiom,
! [B2: b,F: b > b,A3: set_b] :
( ( member_b @ B2 @ ( image_b_b @ F @ A3 ) )
=> ~ ! [X: b] :
( ( B2
= ( F @ X ) )
=> ~ ( member_b @ X @ A3 ) ) ) ).
% imageE
thf(fact_966_imageI,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( member4380921116106875537_a_b_b @ ( F @ X4 ) @ ( image_6081965176830705659_a_b_b @ F @ A3 ) ) ) ).
% imageI
thf(fact_967_imageI,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,F: produc4558475209616630778_a_b_b > b] :
( ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( member_b @ ( F @ X4 ) @ ( image_3908709015779211070_b_b_b @ F @ A3 ) ) ) ).
% imageI
thf(fact_968_imageI,axiom,
! [X4: b,A3: set_b,F: b > produc4558475209616630778_a_b_b] :
( ( member_b @ X4 @ A3 )
=> ( member4380921116106875537_a_b_b @ ( F @ X4 ) @ ( image_7642607452437185460_a_b_b @ F @ A3 ) ) ) ).
% imageI
thf(fact_969_imageI,axiom,
! [X4: b,A3: set_b,F: b > b] :
( ( member_b @ X4 @ A3 )
=> ( member_b @ ( F @ X4 ) @ ( image_b_b @ F @ A3 ) ) ) ).
% imageI
thf(fact_970_rev__image__eqI,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B2: produc4558475209616630778_a_b_b,F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( ( B2
= ( F @ X4 ) )
=> ( member4380921116106875537_a_b_b @ B2 @ ( image_6081965176830705659_a_b_b @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_971_rev__image__eqI,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B2: b,F: produc4558475209616630778_a_b_b > b] :
( ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_b @ B2 @ ( image_3908709015779211070_b_b_b @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_972_rev__image__eqI,axiom,
! [X4: b,A3: set_b,B2: produc4558475209616630778_a_b_b,F: b > produc4558475209616630778_a_b_b] :
( ( member_b @ X4 @ A3 )
=> ( ( B2
= ( F @ X4 ) )
=> ( member4380921116106875537_a_b_b @ B2 @ ( image_7642607452437185460_a_b_b @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_973_rev__image__eqI,axiom,
! [X4: b,A3: set_b,B2: b,F: b > b] :
( ( member_b @ X4 @ A3 )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_b @ B2 @ ( image_b_b @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_974_fminusE,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ C2 @ ( minus_1250967532242559235_a_b_b @ A3 @ B3 ) )
=> ~ ( ( fmembe3173364709796808819_a_b_b @ C2 @ A3 )
=> ( fmembe3173364709796808819_a_b_b @ C2 @ B3 ) ) ) ).
% fminusE
thf(fact_975_fminusD1,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ C2 @ ( minus_1250967532242559235_a_b_b @ A3 @ B3 ) )
=> ( fmembe3173364709796808819_a_b_b @ C2 @ A3 ) ) ).
% fminusD1
thf(fact_976_fminusD2,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ C2 @ ( minus_1250967532242559235_a_b_b @ A3 @ B3 ) )
=> ~ ( fmembe3173364709796808819_a_b_b @ C2 @ B3 ) ) ).
% fminusD2
thf(fact_977_image__Un,axiom,
! [F: b > b,A3: set_b,B3: set_b] :
( ( image_b_b @ F @ ( sup_sup_set_b @ A3 @ B3 ) )
= ( sup_sup_set_b @ ( image_b_b @ F @ A3 ) @ ( image_b_b @ F @ B3 ) ) ) ).
% image_Un
thf(fact_978_snd__conv,axiom,
! [X1: dtree_a_b,X2: b] :
( ( produc5748100250121904638_a_b_b @ ( produc331601717337510060_a_b_b @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_979_snd__eqD,axiom,
! [X4: dtree_a_b,Y: b,A2: b] :
( ( ( produc5748100250121904638_a_b_b @ ( produc331601717337510060_a_b_b @ X4 @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_980_sndI,axiom,
! [X4: produc4558475209616630778_a_b_b,Y: dtree_a_b,Z2: b] :
( ( X4
= ( produc331601717337510060_a_b_b @ Y @ Z2 ) )
=> ( ( produc5748100250121904638_a_b_b @ X4 )
= Z2 ) ) ).
% sndI
thf(fact_981_diff__shunt__var,axiom,
! [X4: set_b,Y: set_b] :
( ( ( minus_minus_set_b @ X4 @ Y )
= bot_bot_set_b )
= ( ord_less_eq_set_b @ X4 @ Y ) ) ).
% diff_shunt_var
thf(fact_982_finsert__fminus__if,axiom,
! [X4: produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( ( fmembe3173364709796808819_a_b_b @ X4 @ B3 )
=> ( ( minus_1250967532242559235_a_b_b @ ( finser8437519239679886002_a_b_b @ X4 @ A3 ) @ B3 )
= ( minus_1250967532242559235_a_b_b @ A3 @ B3 ) ) )
& ( ~ ( fmembe3173364709796808819_a_b_b @ X4 @ B3 )
=> ( ( minus_1250967532242559235_a_b_b @ ( finser8437519239679886002_a_b_b @ X4 @ A3 ) @ B3 )
= ( finser8437519239679886002_a_b_b @ X4 @ ( minus_1250967532242559235_a_b_b @ A3 @ B3 ) ) ) ) ) ).
% finsert_fminus_if
thf(fact_983_fminus__finsert2,axiom,
! [A3: fset_P5281107635120001194_a_b_b,A2: produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( minus_1250967532242559235_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ A2 @ B3 ) )
= ( minus_1250967532242559235_a_b_b @ ( minus_1250967532242559235_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ A2 @ bot_bo2895716411488905534_a_b_b ) ) @ B3 ) ) ).
% fminus_finsert2
thf(fact_984_fminus__finsert,axiom,
! [A3: fset_P5281107635120001194_a_b_b,A2: produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( minus_1250967532242559235_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ A2 @ B3 ) )
= ( minus_1250967532242559235_a_b_b @ ( minus_1250967532242559235_a_b_b @ A3 @ B3 ) @ ( finser8437519239679886002_a_b_b @ A2 @ bot_bo2895716411488905534_a_b_b ) ) ) ).
% fminus_finsert
thf(fact_985_fminus__partition,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ B3 )
=> ( ( sup_su860928060825958358_a_b_b @ A3 @ ( minus_1250967532242559235_a_b_b @ B3 @ A3 ) )
= B3 ) ) ).
% fminus_partition
thf(fact_986_fminus__fsubset__conv,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b,C: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ ( minus_1250967532242559235_a_b_b @ A3 @ B3 ) @ C )
= ( ord_le789900035998834954_a_b_b @ A3 @ ( sup_su860928060825958358_a_b_b @ B3 @ C ) ) ) ).
% fminus_fsubset_conv
thf(fact_987_image__constant,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,C2: b] :
( ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( ( image_3908709015779211070_b_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] : C2
@ A3 )
= ( insert_b @ C2 @ bot_bot_set_b ) ) ) ).
% image_constant
thf(fact_988_image__constant,axiom,
! [X4: b,A3: set_b,C2: b] :
( ( member_b @ X4 @ A3 )
=> ( ( image_b_b
@ ^ [X3: b] : C2
@ A3 )
= ( insert_b @ C2 @ bot_bot_set_b ) ) ) ).
% image_constant
thf(fact_989_image__constant__conv,axiom,
! [A3: set_b,C2: b] :
( ( ( A3 = bot_bot_set_b )
=> ( ( image_b_b
@ ^ [X3: b] : C2
@ A3 )
= bot_bot_set_b ) )
& ( ( A3 != bot_bot_set_b )
=> ( ( image_b_b
@ ^ [X3: b] : C2
@ A3 )
= ( insert_b @ C2 @ bot_bot_set_b ) ) ) ) ).
% image_constant_conv
thf(fact_990_Compr__fimage__eq,axiom,
! [F: produc4558475209616630778_a_b_b > produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b,P: produc4558475209616630778_a_b_b > $o] :
( ( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ X3 @ ( fimage7457256623133068659_a_b_b @ F @ A3 ) )
& ( P @ X3 ) ) )
= ( image_6081965176830705659_a_b_b @ F
@ ( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ X3 @ A3 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_fimage_eq
thf(fact_991_finsert__fminus,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( fmembe3173364709796808819_a_b_b @ A2 @ A3 )
=> ( ( finser8437519239679886002_a_b_b @ A2 @ ( minus_1250967532242559235_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ A2 @ bot_bo2895716411488905534_a_b_b ) ) )
= A3 ) ) ).
% finsert_fminus
thf(fact_992_fminus__finsert__absorb,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ~ ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ( ( minus_1250967532242559235_a_b_b @ ( finser8437519239679886002_a_b_b @ X4 @ A3 ) @ ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) )
= A3 ) ) ).
% fminus_finsert_absorb
thf(fact_993_fminus__single__finsert,axiom,
! [A3: fset_P5281107635120001194_a_b_b,X4: produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ ( minus_1250967532242559235_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) ) @ B3 )
=> ( ord_le789900035998834954_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ X4 @ B3 ) ) ) ).
% fminus_single_finsert
thf(fact_994_dtree_Oset__intros_I3_J,axiom,
! [Ye: produc4558475209616630778_a_b_b,X2: fset_P5281107635120001194_a_b_b,Yf: dtree_a_b,Yh: b,X1: a] :
( ( member4380921116106875537_a_b_b @ Ye @ ( fset_P783253628892185035_a_b_b @ X2 ) )
=> ( ( member_dtree_a_b @ Yf @ ( basic_7578771248400840636_a_b_b @ Ye ) )
=> ( ( member_b @ Yh @ ( darcs_a_b @ Yf ) )
=> ( member_b @ Yh @ ( darcs_a_b @ ( node_a_b @ X1 @ X2 ) ) ) ) ) ) ).
% dtree.set_intros(3)
thf(fact_995_dtree_Oset__sel_I3_J,axiom,
! [Xe: produc4558475209616630778_a_b_b,A2: dtree_a_b,Xf: dtree_a_b,Xh: b] :
( ( member4380921116106875537_a_b_b @ Xe @ ( fset_P783253628892185035_a_b_b @ ( sucs_a_b @ A2 ) ) )
=> ( ( member_dtree_a_b @ Xf @ ( basic_7578771248400840636_a_b_b @ Xe ) )
=> ( ( member_b @ Xh @ ( darcs_a_b @ Xf ) )
=> ( member_b @ Xh @ ( darcs_a_b @ A2 ) ) ) ) ) ).
% dtree.set_sel(3)
thf(fact_996_fsubset__finsert__iff,axiom,
! [A3: fset_P5281107635120001194_a_b_b,X4: produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ X4 @ B3 ) )
= ( ( ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ( ord_le789900035998834954_a_b_b @ ( minus_1250967532242559235_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) ) @ B3 ) )
& ( ~ ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ( ord_le789900035998834954_a_b_b @ A3 @ B3 ) ) ) ) ).
% fsubset_finsert_iff
thf(fact_997_comp__fun__commute_Offold__finsert__fremove,axiom,
! [F: produc4558475209616630778_a_b_b > a > a,Z2: a,X4: produc4558475209616630778_a_b_b,A3: fset_P5281107635120001194_a_b_b] :
( ( finite414203908571218417_b_b_a @ F )
=> ( ( ffold_2783168711033344739_b_b_a @ F @ Z2 @ ( finser8437519239679886002_a_b_b @ X4 @ A3 ) )
= ( F @ X4 @ ( ffold_2783168711033344739_b_b_a @ F @ Z2 @ ( minus_1250967532242559235_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) ) ) ) ) ) ).
% comp_fun_commute.ffold_finsert_fremove
thf(fact_998_dtail_Oelims,axiom,
! [X4: dtree_a_b,Xa: b > a,Y: b > a] :
( ( ( dtail_a_b @ X4 @ Xa )
= Y )
=> ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( ( X4
= ( node_a_b @ R @ Xs ) )
=> ( Y
!= ( ^ [E: b] :
( if_a @ ( member_b @ E @ ( image_3908709015779211070_b_b_b @ produc5748100250121904638_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs ) ) ) @ R
@ ( ffold_8367945289176929151_b_b_a
@ ( produc4313903556115589696_a_b_a
@ ^ [X3: dtree_a_b,E2: b,B: b > a] :
( if_b_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs ) )
| ~ ( member_b @ E @ ( darcs_a_b @ X3 ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R @ Xs ) ) )
@ B
@ ( dtail_a_b @ X3 @ Xa ) ) )
@ Xa
@ Xs
@ E ) ) ) ) ) ) ).
% dtail.elims
thf(fact_999_dtail_Osimps,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,Def: b > a] :
( ( dtail_a_b @ ( node_a_b @ R2 @ Xs2 ) @ Def )
= ( ^ [E: b] :
( if_a @ ( member_b @ E @ ( image_3908709015779211070_b_b_b @ produc5748100250121904638_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) ) @ R2
@ ( ffold_8367945289176929151_b_b_a
@ ( produc4313903556115589696_a_b_a
@ ^ [X3: dtree_a_b,E2: b,B: b > a] :
( if_b_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
| ~ ( member_b @ E @ ( darcs_a_b @ X3 ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
@ B
@ ( dtail_a_b @ X3 @ Def ) ) )
@ Def
@ Xs2
@ E ) ) ) ) ).
% dtail.simps
thf(fact_1000_eq__snd__iff,axiom,
! [B2: b,P2: produc4558475209616630778_a_b_b] :
( ( B2
= ( produc5748100250121904638_a_b_b @ P2 ) )
= ( ? [A: dtree_a_b] :
( P2
= ( produc331601717337510060_a_b_b @ A @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_1001_DiffI,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ C2 @ A3 )
=> ( ~ ( member4380921116106875537_a_b_b @ C2 @ B3 )
=> ( member4380921116106875537_a_b_b @ C2 @ ( minus_1392386589478415753_a_b_b @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_1002_DiffI,axiom,
! [C2: b,A3: set_b,B3: set_b] :
( ( member_b @ C2 @ A3 )
=> ( ~ ( member_b @ C2 @ B3 )
=> ( member_b @ C2 @ ( minus_minus_set_b @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_1003_Diff__iff,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ C2 @ ( minus_1392386589478415753_a_b_b @ A3 @ B3 ) )
= ( ( member4380921116106875537_a_b_b @ C2 @ A3 )
& ~ ( member4380921116106875537_a_b_b @ C2 @ B3 ) ) ) ).
% Diff_iff
thf(fact_1004_Diff__iff,axiom,
! [C2: b,A3: set_b,B3: set_b] :
( ( member_b @ C2 @ ( minus_minus_set_b @ A3 @ B3 ) )
= ( ( member_b @ C2 @ A3 )
& ~ ( member_b @ C2 @ B3 ) ) ) ).
% Diff_iff
thf(fact_1005_Diff__cancel,axiom,
! [A3: set_b] :
( ( minus_minus_set_b @ A3 @ A3 )
= bot_bot_set_b ) ).
% Diff_cancel
thf(fact_1006_empty__Diff,axiom,
! [A3: set_b] :
( ( minus_minus_set_b @ bot_bot_set_b @ A3 )
= bot_bot_set_b ) ).
% empty_Diff
thf(fact_1007_Diff__empty,axiom,
! [A3: set_b] :
( ( minus_minus_set_b @ A3 @ bot_bot_set_b )
= A3 ) ).
% Diff_empty
thf(fact_1008_insert__Diff1,axiom,
! [X4: produc4558475209616630778_a_b_b,B3: set_Pr3012420139608375472_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ X4 @ B3 )
=> ( ( minus_1392386589478415753_a_b_b @ ( insert1613891728210272810_a_b_b @ X4 @ A3 ) @ B3 )
= ( minus_1392386589478415753_a_b_b @ A3 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_1009_insert__Diff1,axiom,
! [X4: b,B3: set_b,A3: set_b] :
( ( member_b @ X4 @ B3 )
=> ( ( minus_minus_set_b @ ( insert_b @ X4 @ A3 ) @ B3 )
= ( minus_minus_set_b @ A3 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_1010_Diff__insert0,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ~ ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( ( minus_1392386589478415753_a_b_b @ A3 @ ( insert1613891728210272810_a_b_b @ X4 @ B3 ) )
= ( minus_1392386589478415753_a_b_b @ A3 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_1011_Diff__insert0,axiom,
! [X4: b,A3: set_b,B3: set_b] :
( ~ ( member_b @ X4 @ A3 )
=> ( ( minus_minus_set_b @ A3 @ ( insert_b @ X4 @ B3 ) )
= ( minus_minus_set_b @ A3 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_1012_Un__Diff__cancel2,axiom,
! [B3: set_b,A3: set_b] :
( ( sup_sup_set_b @ ( minus_minus_set_b @ B3 @ A3 ) @ A3 )
= ( sup_sup_set_b @ B3 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_1013_Un__Diff__cancel,axiom,
! [A3: set_b,B3: set_b] :
( ( sup_sup_set_b @ A3 @ ( minus_minus_set_b @ B3 @ A3 ) )
= ( sup_sup_set_b @ A3 @ B3 ) ) ).
% Un_Diff_cancel
thf(fact_1014_Diff__eq__empty__iff,axiom,
! [A3: set_b,B3: set_b] :
( ( ( minus_minus_set_b @ A3 @ B3 )
= bot_bot_set_b )
= ( ord_less_eq_set_b @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_1015_insert__Diff__single,axiom,
! [A2: b,A3: set_b] :
( ( insert_b @ A2 @ ( minus_minus_set_b @ A3 @ ( insert_b @ A2 @ bot_bot_set_b ) ) )
= ( insert_b @ A2 @ A3 ) ) ).
% insert_Diff_single
thf(fact_1016_Diff__disjoint,axiom,
! [A3: set_b,B3: set_b] :
( ( inf_inf_set_b @ A3 @ ( minus_minus_set_b @ B3 @ A3 ) )
= bot_bot_set_b ) ).
% Diff_disjoint
thf(fact_1017_minus__fset,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Xa: fset_P5281107635120001194_a_b_b] :
( ( fset_P783253628892185035_a_b_b @ ( minus_1250967532242559235_a_b_b @ X4 @ Xa ) )
= ( minus_1392386589478415753_a_b_b @ ( fset_P783253628892185035_a_b_b @ X4 ) @ ( fset_P783253628892185035_a_b_b @ Xa ) ) ) ).
% minus_fset
thf(fact_1018_comp__fun__commute__Pow__fold,axiom,
( finite3301421349078847953_set_b
@ ^ [X3: b,A6: set_set_b] : ( sup_sup_set_set_b @ A6 @ ( image_set_b_set_b @ ( insert_b @ X3 ) @ A6 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_1019_set__diff__eq,axiom,
( minus_1392386589478415753_a_b_b
= ( ^ [A6: set_Pr3012420139608375472_a_b_b,B6: set_Pr3012420139608375472_a_b_b] :
( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ A6 )
& ~ ( member4380921116106875537_a_b_b @ X3 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1020_set__diff__eq,axiom,
( minus_minus_set_b
= ( ^ [A6: set_b,B6: set_b] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A6 )
& ~ ( member_b @ X3 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1021_minus__set__def,axiom,
( minus_1392386589478415753_a_b_b
= ( ^ [A6: set_Pr3012420139608375472_a_b_b,B6: set_Pr3012420139608375472_a_b_b] :
( collec1368399972772960719_a_b_b
@ ( minus_6397467918800550972_b_b_o
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ A6 )
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_1022_minus__set__def,axiom,
( minus_minus_set_b
= ( ^ [A6: set_b,B6: set_b] :
( collect_b
@ ( minus_minus_b_o
@ ^ [X3: b] : ( member_b @ X3 @ A6 )
@ ^ [X3: b] : ( member_b @ X3 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_1023_DiffE,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ C2 @ ( minus_1392386589478415753_a_b_b @ A3 @ B3 ) )
=> ~ ( ( member4380921116106875537_a_b_b @ C2 @ A3 )
=> ( member4380921116106875537_a_b_b @ C2 @ B3 ) ) ) ).
% DiffE
thf(fact_1024_DiffE,axiom,
! [C2: b,A3: set_b,B3: set_b] :
( ( member_b @ C2 @ ( minus_minus_set_b @ A3 @ B3 ) )
=> ~ ( ( member_b @ C2 @ A3 )
=> ( member_b @ C2 @ B3 ) ) ) ).
% DiffE
thf(fact_1025_DiffD1,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ C2 @ ( minus_1392386589478415753_a_b_b @ A3 @ B3 ) )
=> ( member4380921116106875537_a_b_b @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_1026_DiffD1,axiom,
! [C2: b,A3: set_b,B3: set_b] :
( ( member_b @ C2 @ ( minus_minus_set_b @ A3 @ B3 ) )
=> ( member_b @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_1027_DiffD2,axiom,
! [C2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ C2 @ ( minus_1392386589478415753_a_b_b @ A3 @ B3 ) )
=> ~ ( member4380921116106875537_a_b_b @ C2 @ B3 ) ) ).
% DiffD2
thf(fact_1028_DiffD2,axiom,
! [C2: b,A3: set_b,B3: set_b] :
( ( member_b @ C2 @ ( minus_minus_set_b @ A3 @ B3 ) )
=> ~ ( member_b @ C2 @ B3 ) ) ).
% DiffD2
thf(fact_1029_Un__Diff,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( minus_minus_set_b @ ( sup_sup_set_b @ A3 @ B3 ) @ C )
= ( sup_sup_set_b @ ( minus_minus_set_b @ A3 @ C ) @ ( minus_minus_set_b @ B3 @ C ) ) ) ).
% Un_Diff
thf(fact_1030_insert__Diff__if,axiom,
! [X4: produc4558475209616630778_a_b_b,B3: set_Pr3012420139608375472_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ( ( member4380921116106875537_a_b_b @ X4 @ B3 )
=> ( ( minus_1392386589478415753_a_b_b @ ( insert1613891728210272810_a_b_b @ X4 @ A3 ) @ B3 )
= ( minus_1392386589478415753_a_b_b @ A3 @ B3 ) ) )
& ( ~ ( member4380921116106875537_a_b_b @ X4 @ B3 )
=> ( ( minus_1392386589478415753_a_b_b @ ( insert1613891728210272810_a_b_b @ X4 @ A3 ) @ B3 )
= ( insert1613891728210272810_a_b_b @ X4 @ ( minus_1392386589478415753_a_b_b @ A3 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1031_insert__Diff__if,axiom,
! [X4: b,B3: set_b,A3: set_b] :
( ( ( member_b @ X4 @ B3 )
=> ( ( minus_minus_set_b @ ( insert_b @ X4 @ A3 ) @ B3 )
= ( minus_minus_set_b @ A3 @ B3 ) ) )
& ( ~ ( member_b @ X4 @ B3 )
=> ( ( minus_minus_set_b @ ( insert_b @ X4 @ A3 ) @ B3 )
= ( insert_b @ X4 @ ( minus_minus_set_b @ A3 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1032_in__image__insert__iff,axiom,
! [B3: set_se3183138701204633190_a_b_b,X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ! [C4: set_Pr3012420139608375472_a_b_b] :
( ( member7431159781899395911_a_b_b @ C4 @ B3 )
=> ~ ( member4380921116106875537_a_b_b @ X4 @ C4 ) )
=> ( ( member7431159781899395911_a_b_b @ A3 @ ( image_4903599603319290215_a_b_b @ ( insert1613891728210272810_a_b_b @ X4 ) @ B3 ) )
= ( ( member4380921116106875537_a_b_b @ X4 @ A3 )
& ( member7431159781899395911_a_b_b @ ( minus_1392386589478415753_a_b_b @ A3 @ ( insert1613891728210272810_a_b_b @ X4 @ bot_bo3721250822024684356_a_b_b ) ) @ B3 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1033_in__image__insert__iff,axiom,
! [B3: set_set_b,X4: b,A3: set_b] :
( ! [C4: set_b] :
( ( member_set_b @ C4 @ B3 )
=> ~ ( member_b @ X4 @ C4 ) )
=> ( ( member_set_b @ A3 @ ( image_set_b_set_b @ ( insert_b @ X4 ) @ B3 ) )
= ( ( member_b @ X4 @ A3 )
& ( member_set_b @ ( minus_minus_set_b @ A3 @ ( insert_b @ X4 @ bot_bot_set_b ) ) @ B3 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1034_Diff__insert__absorb,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ~ ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( ( minus_1392386589478415753_a_b_b @ ( insert1613891728210272810_a_b_b @ X4 @ A3 ) @ ( insert1613891728210272810_a_b_b @ X4 @ bot_bo3721250822024684356_a_b_b ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1035_Diff__insert__absorb,axiom,
! [X4: b,A3: set_b] :
( ~ ( member_b @ X4 @ A3 )
=> ( ( minus_minus_set_b @ ( insert_b @ X4 @ A3 ) @ ( insert_b @ X4 @ bot_bot_set_b ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1036_Diff__insert2,axiom,
! [A3: set_b,A2: b,B3: set_b] :
( ( minus_minus_set_b @ A3 @ ( insert_b @ A2 @ B3 ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A3 @ ( insert_b @ A2 @ bot_bot_set_b ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_1037_insert__Diff,axiom,
! [A2: produc4558475209616630778_a_b_b,A3: set_Pr3012420139608375472_a_b_b] :
( ( member4380921116106875537_a_b_b @ A2 @ A3 )
=> ( ( insert1613891728210272810_a_b_b @ A2 @ ( minus_1392386589478415753_a_b_b @ A3 @ ( insert1613891728210272810_a_b_b @ A2 @ bot_bo3721250822024684356_a_b_b ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1038_insert__Diff,axiom,
! [A2: b,A3: set_b] :
( ( member_b @ A2 @ A3 )
=> ( ( insert_b @ A2 @ ( minus_minus_set_b @ A3 @ ( insert_b @ A2 @ bot_bot_set_b ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1039_Diff__insert,axiom,
! [A3: set_b,A2: b,B3: set_b] :
( ( minus_minus_set_b @ A3 @ ( insert_b @ A2 @ B3 ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A3 @ B3 ) @ ( insert_b @ A2 @ bot_bot_set_b ) ) ) ).
% Diff_insert
thf(fact_1040_subset__Diff__insert,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b,X4: produc4558475209616630778_a_b_b,C: set_Pr3012420139608375472_a_b_b] :
( ( ord_le146215904626753808_a_b_b @ A3 @ ( minus_1392386589478415753_a_b_b @ B3 @ ( insert1613891728210272810_a_b_b @ X4 @ C ) ) )
= ( ( ord_le146215904626753808_a_b_b @ A3 @ ( minus_1392386589478415753_a_b_b @ B3 @ C ) )
& ~ ( member4380921116106875537_a_b_b @ X4 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1041_subset__Diff__insert,axiom,
! [A3: set_b,B3: set_b,X4: b,C: set_b] :
( ( ord_less_eq_set_b @ A3 @ ( minus_minus_set_b @ B3 @ ( insert_b @ X4 @ C ) ) )
= ( ( ord_less_eq_set_b @ A3 @ ( minus_minus_set_b @ B3 @ C ) )
& ~ ( member_b @ X4 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1042_Int__Diff__disjoint,axiom,
! [A3: set_b,B3: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ ( minus_minus_set_b @ A3 @ B3 ) )
= bot_bot_set_b ) ).
% Int_Diff_disjoint
thf(fact_1043_Diff__triv,axiom,
! [A3: set_b,B3: set_b] :
( ( ( inf_inf_set_b @ A3 @ B3 )
= bot_bot_set_b )
=> ( ( minus_minus_set_b @ A3 @ B3 )
= A3 ) ) ).
% Diff_triv
thf(fact_1044_Diff__subset__conv,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( ord_less_eq_set_b @ ( minus_minus_set_b @ A3 @ B3 ) @ C )
= ( ord_less_eq_set_b @ A3 @ ( sup_sup_set_b @ B3 @ C ) ) ) ).
% Diff_subset_conv
thf(fact_1045_Diff__partition,axiom,
! [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( sup_sup_set_b @ A3 @ ( minus_minus_set_b @ B3 @ A3 ) )
= B3 ) ) ).
% Diff_partition
thf(fact_1046_Un__Diff__Int,axiom,
! [A3: set_b,B3: set_b] :
( ( sup_sup_set_b @ ( minus_minus_set_b @ A3 @ B3 ) @ ( inf_inf_set_b @ A3 @ B3 ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_1047_Int__Diff__Un,axiom,
! [A3: set_b,B3: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ ( minus_minus_set_b @ A3 @ B3 ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_1048_Diff__Int,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( minus_minus_set_b @ A3 @ ( inf_inf_set_b @ B3 @ C ) )
= ( sup_sup_set_b @ ( minus_minus_set_b @ A3 @ B3 ) @ ( minus_minus_set_b @ A3 @ C ) ) ) ).
% Diff_Int
thf(fact_1049_Diff__Un,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( minus_minus_set_b @ A3 @ ( sup_sup_set_b @ B3 @ C ) )
= ( inf_inf_set_b @ ( minus_minus_set_b @ A3 @ B3 ) @ ( minus_minus_set_b @ A3 @ C ) ) ) ).
% Diff_Un
thf(fact_1050_Diff__single__insert,axiom,
! [A3: set_b,X4: b,B3: set_b] :
( ( ord_less_eq_set_b @ ( minus_minus_set_b @ A3 @ ( insert_b @ X4 @ bot_bot_set_b ) ) @ B3 )
=> ( ord_less_eq_set_b @ A3 @ ( insert_b @ X4 @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_1051_subset__insert__iff,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,X4: produc4558475209616630778_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( ord_le146215904626753808_a_b_b @ A3 @ ( insert1613891728210272810_a_b_b @ X4 @ B3 ) )
= ( ( ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( ord_le146215904626753808_a_b_b @ ( minus_1392386589478415753_a_b_b @ A3 @ ( insert1613891728210272810_a_b_b @ X4 @ bot_bo3721250822024684356_a_b_b ) ) @ B3 ) )
& ( ~ ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( ord_le146215904626753808_a_b_b @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_1052_subset__insert__iff,axiom,
! [A3: set_b,X4: b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ ( insert_b @ X4 @ B3 ) )
= ( ( ( member_b @ X4 @ A3 )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A3 @ ( insert_b @ X4 @ bot_bot_set_b ) ) @ B3 ) )
& ( ~ ( member_b @ X4 @ A3 )
=> ( ord_less_eq_set_b @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_1053_subset__CollectI,axiom,
! [B3: set_Pr3012420139608375472_a_b_b,A3: set_Pr3012420139608375472_a_b_b,Q: produc4558475209616630778_a_b_b > $o,P: produc4558475209616630778_a_b_b > $o] :
( ( ord_le146215904626753808_a_b_b @ B3 @ A3 )
=> ( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ B3 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le146215904626753808_a_b_b
@ ( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ B3 )
& ( Q @ X3 ) ) )
@ ( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ A3 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1054_subset__CollectI,axiom,
! [B3: set_b,A3: set_b,Q: b > $o,P: b > $o] :
( ( ord_less_eq_set_b @ B3 @ A3 )
=> ( ! [X: b] :
( ( member_b @ X @ B3 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ B3 )
& ( Q @ X3 ) ) )
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A3 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1055_subset__Collect__iff,axiom,
! [B3: set_Pr3012420139608375472_a_b_b,A3: set_Pr3012420139608375472_a_b_b,P: produc4558475209616630778_a_b_b > $o] :
( ( ord_le146215904626753808_a_b_b @ B3 @ A3 )
=> ( ( ord_le146215904626753808_a_b_b @ B3
@ ( collec1368399972772960719_a_b_b
@ ^ [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ A3 )
& ( P @ X3 ) ) ) )
= ( ! [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ B3 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1056_subset__Collect__iff,axiom,
! [B3: set_b,A3: set_b,P: b > $o] :
( ( ord_less_eq_set_b @ B3 @ A3 )
=> ( ( ord_less_eq_set_b @ B3
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A3 )
& ( P @ X3 ) ) ) )
= ( ! [X3: b] :
( ( member_b @ X3 @ B3 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1057_singleton__suc,axiom,
! [T: dtree_a_b,R2: a,E4: b] : ( member_dtree_a_b @ T @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ ( sucs_a_b @ ( node_a_b @ R2 @ ( finser8437519239679886002_a_b_b @ ( produc331601717337510060_a_b_b @ T @ E4 ) @ bot_bo2895716411488905534_a_b_b ) ) ) ) ) ) ).
% singleton_suc
thf(fact_1058_prod_Ocollapse,axiom,
! [Prod: produc4558475209616630778_a_b_b] :
( ( produc331601717337510060_a_b_b @ ( produc697780174058963904_a_b_b @ Prod ) @ ( produc5748100250121904638_a_b_b @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_1059_child__uneq,axiom,
! [T: dtree_a_b,Xs2: fset_P5281107635120001194_a_b_b,R2: a] :
( ( member_dtree_a_b @ T @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
=> ( ( node_a_b @ R2 @ Xs2 )
!= T ) ) ).
% child_uneq
thf(fact_1060_suc__uneq,axiom,
! [T1: dtree_a_b,T: dtree_a_b] :
( ( member_dtree_a_b @ T1 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ ( sucs_a_b @ T ) ) ) )
=> ( T != T1 ) ) ).
% suc_uneq
thf(fact_1061_case__prod__unfold,axiom,
( produc2242037354397874494_b_a_a
= ( ^ [C5: dtree_a_b > b > a > a,P4: produc4558475209616630778_a_b_b] : ( C5 @ ( produc697780174058963904_a_b_b @ P4 ) @ ( produc5748100250121904638_a_b_b @ P4 ) ) ) ) ).
% case_prod_unfold
thf(fact_1062_case__prod__beta_H,axiom,
( produc2242037354397874494_b_a_a
= ( ^ [F2: dtree_a_b > b > a > a,X3: produc4558475209616630778_a_b_b] : ( F2 @ ( produc697780174058963904_a_b_b @ X3 ) @ ( produc5748100250121904638_a_b_b @ X3 ) ) ) ) ).
% case_prod_beta'
thf(fact_1063_fstI,axiom,
! [X4: produc4558475209616630778_a_b_b,Y: dtree_a_b,Z2: b] :
( ( X4
= ( produc331601717337510060_a_b_b @ Y @ Z2 ) )
=> ( ( produc697780174058963904_a_b_b @ X4 )
= Y ) ) ).
% fstI
thf(fact_1064_fst__eqD,axiom,
! [X4: dtree_a_b,Y: b,A2: dtree_a_b] :
( ( ( produc697780174058963904_a_b_b @ ( produc331601717337510060_a_b_b @ X4 @ Y ) )
= A2 )
=> ( X4 = A2 ) ) ).
% fst_eqD
thf(fact_1065_fst__conv,axiom,
! [X1: dtree_a_b,X2: b] :
( ( produc697780174058963904_a_b_b @ ( produc331601717337510060_a_b_b @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_1066_Product__Type_OCollect__case__prodD,axiom,
! [X4: produc4558475209616630778_a_b_b,A3: dtree_a_b > b > $o] :
( ( member4380921116106875537_a_b_b @ X4 @ ( collec1368399972772960719_a_b_b @ ( produc1325217093046185599_b_b_o @ A3 ) ) )
=> ( A3 @ ( produc697780174058963904_a_b_b @ X4 ) @ ( produc5748100250121904638_a_b_b @ X4 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_1067_split__beta,axiom,
( produc2242037354397874494_b_a_a
= ( ^ [F2: dtree_a_b > b > a > a,Prod2: produc4558475209616630778_a_b_b] : ( F2 @ ( produc697780174058963904_a_b_b @ Prod2 ) @ ( produc5748100250121904638_a_b_b @ Prod2 ) ) ) ) ).
% split_beta
thf(fact_1068_case__prod__beta,axiom,
( produc2242037354397874494_b_a_a
= ( ^ [F2: dtree_a_b > b > a > a,P4: produc4558475209616630778_a_b_b] : ( F2 @ ( produc697780174058963904_a_b_b @ P4 ) @ ( produc5748100250121904638_a_b_b @ P4 ) ) ) ) ).
% case_prod_beta
thf(fact_1069_prod_Oexhaust__sel,axiom,
! [Prod: produc4558475209616630778_a_b_b] :
( Prod
= ( produc331601717337510060_a_b_b @ ( produc697780174058963904_a_b_b @ Prod ) @ ( produc5748100250121904638_a_b_b @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_1070_surjective__pairing,axiom,
! [T: produc4558475209616630778_a_b_b] :
( T
= ( produc331601717337510060_a_b_b @ ( produc697780174058963904_a_b_b @ T ) @ ( produc5748100250121904638_a_b_b @ T ) ) ) ).
% surjective_pairing
thf(fact_1071_eq__fst__iff,axiom,
! [A2: dtree_a_b,P2: produc4558475209616630778_a_b_b] :
( ( A2
= ( produc697780174058963904_a_b_b @ P2 ) )
= ( ? [B: b] :
( P2
= ( produc331601717337510060_a_b_b @ A2 @ B ) ) ) ) ).
% eq_fst_iff
thf(fact_1072_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: dtree_a_b > b > $o,X4: dtree_a_b,Y: b,A2: produc4558475209616630778_a_b_b] :
( ( P @ X4 @ Y )
=> ( ( A2
= ( produc331601717337510060_a_b_b @ X4 @ Y ) )
=> ( P @ ( produc697780174058963904_a_b_b @ A2 ) @ ( produc5748100250121904638_a_b_b @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1073_prod_Osplit__sel__asm,axiom,
! [P: ( a > a ) > $o,F: dtree_a_b > b > a > a,Prod: produc4558475209616630778_a_b_b] :
( ( P @ ( produc2242037354397874494_b_a_a @ F @ Prod ) )
= ( ~ ( ( Prod
= ( produc331601717337510060_a_b_b @ ( produc697780174058963904_a_b_b @ Prod ) @ ( produc5748100250121904638_a_b_b @ Prod ) ) )
& ~ ( P @ ( F @ ( produc697780174058963904_a_b_b @ Prod ) @ ( produc5748100250121904638_a_b_b @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_1074_prod_Osplit__sel,axiom,
! [P: ( a > a ) > $o,F: dtree_a_b > b > a > a,Prod: produc4558475209616630778_a_b_b] :
( ( P @ ( produc2242037354397874494_b_a_a @ F @ Prod ) )
= ( ( Prod
= ( produc331601717337510060_a_b_b @ ( produc697780174058963904_a_b_b @ Prod ) @ ( produc5748100250121904638_a_b_b @ Prod ) ) )
=> ( P @ ( F @ ( produc697780174058963904_a_b_b @ Prod ) @ ( produc5748100250121904638_a_b_b @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_1075_wf__darcs__rec,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T1: dtree_a_b] :
( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ( ( member_dtree_a_b @ T1 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
=> ( wf_darcs_a_b @ T1 ) ) ) ).
% wf_darcs_rec
thf(fact_1076_child__uneq_H,axiom,
! [T: dtree_a_b,Xs2: fset_P5281107635120001194_a_b_b,R2: a,V: a] :
( ( member_dtree_a_b @ T @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
=> ( ( node_a_b @ R2 @ Xs2 )
!= ( node_a_b @ V @ ( sucs_a_b @ T ) ) ) ) ).
% child_uneq'
thf(fact_1077_suc__uneq_H,axiom,
! [T1: dtree_a_b,T: dtree_a_b,V: a] :
( ( member_dtree_a_b @ T1 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ ( sucs_a_b @ T ) ) ) )
=> ( T
!= ( node_a_b @ V @ ( sucs_a_b @ T1 ) ) ) ) ).
% suc_uneq'
thf(fact_1078_darcs__child__subseteq,axiom,
! [X4: dtree_a_b,Xs2: fset_P5281107635120001194_a_b_b,R2: a] :
( ( member_dtree_a_b @ X4 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
=> ( ord_less_eq_set_b @ ( darcs_a_b @ X4 ) @ ( darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) ) ) ).
% darcs_child_subseteq
thf(fact_1079_snd__fst__flip,axiom,
( produc8959184660827480948ee_a_b
= ( comp_P1139271906589003649ee_a_b @ produc697780174058963904_a_b_b
@ ( produc1296939142185513033_a_b_b
@ ^ [X3: b,Y5: dtree_a_b] : ( produc331601717337510060_a_b_b @ Y5 @ X3 ) ) ) ) ).
% snd_fst_flip
thf(fact_1080_fst__snd__flip,axiom,
( produc3908864584764540214ee_a_b
= ( comp_P6702227762116406538ee_a_b @ produc5748100250121904638_a_b_b
@ ( produc1296939142185513033_a_b_b
@ ^ [X3: b,Y5: dtree_a_b] : ( produc331601717337510060_a_b_b @ Y5 @ X3 ) ) ) ) ).
% fst_snd_flip
thf(fact_1081_elem__neq__if__fset__neq,axiom,
! [F: dtree_a_b > dtree_a_b,Xs2: fset_P5281107635120001194_a_b_b] :
( ( ( fimage7457256623133068659_a_b_b
@ ( produc5460679229782211283_a_b_b
@ ^ [T3: dtree_a_b] : ( produc331601717337510060_a_b_b @ ( F @ T3 ) ) )
@ Xs2 )
!= Xs2 )
=> ? [X: dtree_a_b] :
( ( member_dtree_a_b @ X @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
& ( ( F @ X )
!= X ) ) ) ).
% elem_neq_if_fset_neq
thf(fact_1082_exE__realizer,axiom,
! [P: b > dtree_a_b > $o,P2: produc4558475209616630778_a_b_b,Q: ( a > a ) > $o,F: dtree_a_b > b > a > a] :
( ( P @ ( produc5748100250121904638_a_b_b @ P2 ) @ ( produc697780174058963904_a_b_b @ P2 ) )
=> ( ! [X: dtree_a_b,Y4: b] :
( ( P @ Y4 @ X )
=> ( Q @ ( F @ X @ Y4 ) ) )
=> ( Q @ ( produc2242037354397874494_b_a_a @ F @ P2 ) ) ) ) ).
% exE_realizer
thf(fact_1083_conjI__realizer,axiom,
! [P: dtree_a_b > $o,P2: dtree_a_b,Q: b > $o,Q2: b] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc697780174058963904_a_b_b @ ( produc331601717337510060_a_b_b @ P2 @ Q2 ) ) )
& ( Q @ ( produc5748100250121904638_a_b_b @ ( produc331601717337510060_a_b_b @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_1084_exI__realizer,axiom,
! [P: b > dtree_a_b > $o,Y: b,X4: dtree_a_b] :
( ( P @ Y @ X4 )
=> ( P @ ( produc5748100250121904638_a_b_b @ ( produc331601717337510060_a_b_b @ X4 @ Y ) ) @ ( produc697780174058963904_a_b_b @ ( produc331601717337510060_a_b_b @ X4 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_1085_dtree__size__img__le,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,F: dtree_a_b > dtree_a_b,R2: a] :
( ! [X: dtree_a_b] :
( ( member_dtree_a_b @ X @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
=> ( ord_less_eq_nat @ ( size_size_dtree_a_b @ ( F @ X ) ) @ ( size_size_dtree_a_b @ X ) ) )
=> ( ord_less_eq_nat
@ ( size_size_dtree_a_b
@ ( node_a_b @ R2
@ ( fimage7457256623133068659_a_b_b
@ ( produc5460679229782211283_a_b_b
@ ^ [T3: dtree_a_b] : ( produc331601717337510060_a_b_b @ ( F @ T3 ) ) )
@ Xs2 ) ) )
@ ( size_size_dtree_a_b @ ( node_a_b @ R2 @ Xs2 ) ) ) ) ).
% dtree_size_img_le
thf(fact_1086_wf__dverts__rec,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T1: dtree_a_b] :
( ( wf_dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ( ( member_dtree_a_b @ T1 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
=> ( wf_dverts_a_b @ T1 ) ) ) ).
% wf_dverts_rec
thf(fact_1087_dtree__size__eq__root,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,R4: a] :
( ( size_size_dtree_a_b @ ( node_a_b @ R2 @ Xs2 ) )
= ( size_size_dtree_a_b @ ( node_a_b @ R4 @ Xs2 ) ) ) ).
% dtree_size_eq_root
thf(fact_1088_size__le__if__child__subset,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,Ys: fset_P5281107635120001194_a_b_b,R2: a,V: a] :
( ( ord_le789900035998834954_a_b_b @ Xs2 @ Ys )
=> ( ord_less_eq_nat @ ( size_size_dtree_a_b @ ( node_a_b @ R2 @ Xs2 ) ) @ ( size_size_dtree_a_b @ ( node_a_b @ V @ Ys ) ) ) ) ).
% size_le_if_child_subset
thf(fact_1089_size__le__if__sucs__subset,axiom,
! [T1: dtree_a_b,T2: dtree_a_b] :
( ( ord_le789900035998834954_a_b_b @ ( sucs_a_b @ T1 ) @ ( sucs_a_b @ T2 ) )
=> ( ord_less_eq_nat @ ( size_size_dtree_a_b @ T1 ) @ ( size_size_dtree_a_b @ T2 ) ) ) ).
% size_le_if_sucs_subset
thf(fact_1090_wf__dverts__sub,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,Ys: fset_P5281107635120001194_a_b_b,R2: a] :
( ( ord_le789900035998834954_a_b_b @ Xs2 @ Ys )
=> ( ( wf_dverts_a_b @ ( node_a_b @ R2 @ Ys ) )
=> ( wf_dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) ) ) ) ).
% wf_dverts_sub
thf(fact_1091_dtree__size__img__lt,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,F: dtree_a_b > dtree_a_b,R2: a] :
( ! [X: dtree_a_b] :
( ( member_dtree_a_b @ X @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
=> ( ord_less_eq_nat @ ( size_size_dtree_a_b @ ( F @ X ) ) @ ( size_size_dtree_a_b @ X ) ) )
=> ( ? [X5: dtree_a_b] :
( ( member_dtree_a_b @ X5 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
& ( ord_less_nat @ ( size_size_dtree_a_b @ ( F @ X5 ) ) @ ( size_size_dtree_a_b @ X5 ) ) )
=> ( ord_less_nat
@ ( size_size_dtree_a_b
@ ( node_a_b @ R2
@ ( fimage7457256623133068659_a_b_b
@ ( produc5460679229782211283_a_b_b
@ ^ [T3: dtree_a_b] : ( produc331601717337510060_a_b_b @ ( F @ T3 ) ) )
@ Xs2 ) ) )
@ ( size_size_dtree_a_b @ ( node_a_b @ R2 @ Xs2 ) ) ) ) ) ).
% dtree_size_img_lt
thf(fact_1092_disjoint__dverts__if__wf__sucs,axiom,
! [T: dtree_a_b] :
( ( wf_dverts_a_b @ T )
=> ! [X5: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X5 @ ( fset_P783253628892185035_a_b_b @ ( sucs_a_b @ T ) ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ ( sucs_a_b @ T ) ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_a @ ( dverts_a_b @ Y5 ) @ ( dverts_a_b @ Aa ) )
= bot_bot_set_a )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) )
@ X5 ) ) ) ).
% disjoint_dverts_if_wf_sucs
thf(fact_1093_order__le__imp__less__or__eq,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X4 @ Y )
=> ( ( ord_le7001451600920047870_a_b_b @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1094_order__le__imp__less__or__eq,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_nat @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1095_linorder__le__less__linear,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
| ( ord_less_nat @ Y @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_1096_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_le789900035998834954_a_b_b @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le7001451600920047870_a_b_b @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le7001451600920047870_a_b_b @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1097_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1098_order__less__le__subst1,axiom,
! [A2: fset_P5281107635120001194_a_b_b,F: fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_le7001451600920047870_a_b_b @ A2 @ ( F @ B2 ) )
=> ( ( ord_le789900035998834954_a_b_b @ B2 @ C2 )
=> ( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X @ Y4 )
=> ( ord_le789900035998834954_a_b_b @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le7001451600920047870_a_b_b @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1099_order__less__le__subst1,axiom,
! [A2: nat,F: fset_P5281107635120001194_a_b_b > nat,B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le789900035998834954_a_b_b @ B2 @ C2 )
=> ( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1100_order__less__le__subst1,axiom,
! [A2: fset_P5281107635120001194_a_b_b,F: nat > fset_P5281107635120001194_a_b_b,B2: nat,C2: nat] :
( ( ord_le7001451600920047870_a_b_b @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le789900035998834954_a_b_b @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le7001451600920047870_a_b_b @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1101_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1102_order__le__less__subst2,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,F: fset_P5281107635120001194_a_b_b > fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( ( ord_le7001451600920047870_a_b_b @ ( F @ B2 ) @ C2 )
=> ( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X @ Y4 )
=> ( ord_le789900035998834954_a_b_b @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le7001451600920047870_a_b_b @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1103_order__le__less__subst2,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,F: fset_P5281107635120001194_a_b_b > nat,C2: nat] :
( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: fset_P5281107635120001194_a_b_b,Y4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1104_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le7001451600920047870_a_b_b @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le789900035998834954_a_b_b @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le7001451600920047870_a_b_b @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1105_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1106_order__le__less__subst1,axiom,
! [A2: fset_P5281107635120001194_a_b_b,F: nat > fset_P5281107635120001194_a_b_b,B2: nat,C2: nat] :
( ( ord_le789900035998834954_a_b_b @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le7001451600920047870_a_b_b @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le7001451600920047870_a_b_b @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1107_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1108_order__less__le__trans,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b,Z2: fset_P5281107635120001194_a_b_b] :
( ( ord_le7001451600920047870_a_b_b @ X4 @ Y )
=> ( ( ord_le789900035998834954_a_b_b @ Y @ Z2 )
=> ( ord_le7001451600920047870_a_b_b @ X4 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1109_order__less__le__trans,axiom,
! [X4: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X4 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1110_order__le__less__trans,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b,Z2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X4 @ Y )
=> ( ( ord_le7001451600920047870_a_b_b @ Y @ Z2 )
=> ( ord_le7001451600920047870_a_b_b @ X4 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1111_order__le__less__trans,axiom,
! [X4: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X4 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1112_order__neq__le__trans,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( A2 != B2 )
=> ( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( ord_le7001451600920047870_a_b_b @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1113_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1114_order__le__neq__trans,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le7001451600920047870_a_b_b @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1115_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1116_order__less__imp__le,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] :
( ( ord_le7001451600920047870_a_b_b @ X4 @ Y )
=> ( ord_le789900035998834954_a_b_b @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_1117_order__less__imp__le,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_1118_linorder__not__less,axiom,
! [X4: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( ord_less_eq_nat @ Y @ X4 ) ) ).
% linorder_not_less
thf(fact_1119_linorder__not__le,axiom,
! [X4: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X4 @ Y ) )
= ( ord_less_nat @ Y @ X4 ) ) ).
% linorder_not_le
thf(fact_1120_order__less__le,axiom,
( ord_le7001451600920047870_a_b_b
= ( ^ [X3: fset_P5281107635120001194_a_b_b,Y5: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1121_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1122_order__le__less,axiom,
( ord_le789900035998834954_a_b_b
= ( ^ [X3: fset_P5281107635120001194_a_b_b,Y5: fset_P5281107635120001194_a_b_b] :
( ( ord_le7001451600920047870_a_b_b @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1123_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1124_dual__order_Ostrict__implies__order,axiom,
! [B2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b] :
( ( ord_le7001451600920047870_a_b_b @ B2 @ A2 )
=> ( ord_le789900035998834954_a_b_b @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1125_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1126_order_Ostrict__implies__order,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( ord_le7001451600920047870_a_b_b @ A2 @ B2 )
=> ( ord_le789900035998834954_a_b_b @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1127_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1128_dual__order_Ostrict__iff__not,axiom,
( ord_le7001451600920047870_a_b_b
= ( ^ [B: fset_P5281107635120001194_a_b_b,A: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ B @ A )
& ~ ( ord_le789900035998834954_a_b_b @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1129_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
& ~ ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1130_dual__order_Ostrict__trans2,axiom,
! [B2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_le7001451600920047870_a_b_b @ B2 @ A2 )
=> ( ( ord_le789900035998834954_a_b_b @ C2 @ B2 )
=> ( ord_le7001451600920047870_a_b_b @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1131_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1132_dual__order_Ostrict__trans1,axiom,
! [B2: fset_P5281107635120001194_a_b_b,A2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ B2 @ A2 )
=> ( ( ord_le7001451600920047870_a_b_b @ C2 @ B2 )
=> ( ord_le7001451600920047870_a_b_b @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1133_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1134_dual__order_Ostrict__iff__order,axiom,
( ord_le7001451600920047870_a_b_b
= ( ^ [B: fset_P5281107635120001194_a_b_b,A: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1135_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1136_dual__order_Oorder__iff__strict,axiom,
( ord_le789900035998834954_a_b_b
= ( ^ [B: fset_P5281107635120001194_a_b_b,A: fset_P5281107635120001194_a_b_b] :
( ( ord_le7001451600920047870_a_b_b @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1137_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1138_order_Ostrict__iff__not,axiom,
( ord_le7001451600920047870_a_b_b
= ( ^ [A: fset_P5281107635120001194_a_b_b,B: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A @ B )
& ~ ( ord_le789900035998834954_a_b_b @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_1139_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ~ ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_1140_order_Ostrict__trans2,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_le7001451600920047870_a_b_b @ A2 @ B2 )
=> ( ( ord_le789900035998834954_a_b_b @ B2 @ C2 )
=> ( ord_le7001451600920047870_a_b_b @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1141_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1142_order_Ostrict__trans1,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b,C2: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
=> ( ( ord_le7001451600920047870_a_b_b @ B2 @ C2 )
=> ( ord_le7001451600920047870_a_b_b @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1143_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1144_order_Ostrict__iff__order,axiom,
( ord_le7001451600920047870_a_b_b
= ( ^ [A: fset_P5281107635120001194_a_b_b,B: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_1145_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_1146_order_Oorder__iff__strict,axiom,
( ord_le789900035998834954_a_b_b
= ( ^ [A: fset_P5281107635120001194_a_b_b,B: fset_P5281107635120001194_a_b_b] :
( ( ord_le7001451600920047870_a_b_b @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_1147_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_1148_not__le__imp__less,axiom,
! [Y: nat,X4: nat] :
( ~ ( ord_less_eq_nat @ Y @ X4 )
=> ( ord_less_nat @ X4 @ Y ) ) ).
% not_le_imp_less
thf(fact_1149_less__le__not__le,axiom,
( ord_le7001451600920047870_a_b_b
= ( ^ [X3: fset_P5281107635120001194_a_b_b,Y5: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X3 @ Y5 )
& ~ ( ord_le789900035998834954_a_b_b @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_1150_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_1151_antisym__conv2,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ X4 @ Y )
=> ( ( ~ ( ord_le7001451600920047870_a_b_b @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv2
thf(fact_1152_antisym__conv2,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv2
thf(fact_1153_antisym__conv1,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] :
( ~ ( ord_le7001451600920047870_a_b_b @ X4 @ Y )
=> ( ( ord_le789900035998834954_a_b_b @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% antisym_conv1
thf(fact_1154_antisym__conv1,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% antisym_conv1
thf(fact_1155_nless__le,axiom,
! [A2: fset_P5281107635120001194_a_b_b,B2: fset_P5281107635120001194_a_b_b] :
( ( ~ ( ord_le7001451600920047870_a_b_b @ A2 @ B2 ) )
= ( ~ ( ord_le789900035998834954_a_b_b @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1156_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1157_leI,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ Y @ X4 ) ) ).
% leI
thf(fact_1158_leD,axiom,
! [Y: fset_P5281107635120001194_a_b_b,X4: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ Y @ X4 )
=> ~ ( ord_le7001451600920047870_a_b_b @ X4 @ Y ) ) ).
% leD
thf(fact_1159_leD,axiom,
! [Y: nat,X4: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ~ ( ord_less_nat @ X4 @ Y ) ) ).
% leD
thf(fact_1160_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I: nat] :
( ( ord_less_nat @ K2 @ I )
=> ( P @ I ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1161_less__infI1,axiom,
! [A2: nat,X4: nat,B2: nat] :
( ( ord_less_nat @ A2 @ X4 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X4 ) ) ).
% less_infI1
thf(fact_1162_less__infI2,axiom,
! [B2: nat,X4: nat,A2: nat] :
( ( ord_less_nat @ B2 @ X4 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X4 ) ) ).
% less_infI2
thf(fact_1163_inf_Oabsorb3,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_1164_inf_Oabsorb4,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_1165_inf_Ostrict__boundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ A2 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_1166_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
& ( A != B ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1167_inf_Ostrict__coboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_1168_inf_Ostrict__coboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_1169_gt__ex,axiom,
! [X4: nat] :
? [X_1: nat] : ( ord_less_nat @ X4 @ X_1 ) ).
% gt_ex
thf(fact_1170_less__imp__neq,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( X4 != Y ) ) ).
% less_imp_neq
thf(fact_1171_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_1172_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_1173_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_1174_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X )
=> ( P @ Y6 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_1175_antisym__conv3,axiom,
! [Y: nat,X4: nat] :
( ~ ( ord_less_nat @ Y @ X4 )
=> ( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv3
thf(fact_1176_linorder__cases,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_nat @ X4 @ Y )
=> ( ( X4 != Y )
=> ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_cases
thf(fact_1177_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_1178_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_1179_exists__least__iff,axiom,
( ( ^ [P5: nat > $o] :
? [X7: nat] : ( P5 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_1180_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_1181_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_1182_not__less__iff__gr__or__eq,axiom,
! [X4: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( ( ord_less_nat @ Y @ X4 )
| ( X4 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1183_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_1184_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_1185_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1186_linorder__neqE,axiom,
! [X4: nat,Y: nat] :
( ( X4 != Y )
=> ( ~ ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_neqE
thf(fact_1187_order__less__asym,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ~ ( ord_less_nat @ Y @ X4 ) ) ).
% order_less_asym
thf(fact_1188_linorder__neq__iff,axiom,
! [X4: nat,Y: nat] :
( ( X4 != Y )
= ( ( ord_less_nat @ X4 @ Y )
| ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_1189_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_1190_order__less__trans,axiom,
! [X4: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X4 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_1191_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1192_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1193_order__less__irrefl,axiom,
! [X4: nat] :
~ ( ord_less_nat @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_1194_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_1195_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_1196_order__less__not__sym,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ~ ( ord_less_nat @ Y @ X4 ) ) ).
% order_less_not_sym
thf(fact_1197_order__less__imp__triv,axiom,
! [X4: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_nat @ Y @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_1198_linorder__less__linear,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
| ( X4 = Y )
| ( ord_less_nat @ Y @ X4 ) ) ).
% linorder_less_linear
thf(fact_1199_order__less__imp__not__eq,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( X4 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_1200_order__less__imp__not__eq2,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( Y != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_1201_order__less__imp__not__less,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ~ ( ord_less_nat @ Y @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_1202_dtree_Oset__sel_I1_J,axiom,
! [A2: dtree_a_b] : ( member_a @ ( root_a_b @ A2 ) @ ( dverts_a_b @ A2 ) ) ).
% dtree.set_sel(1)
thf(fact_1203_bot_Oextremum__strict,axiom,
! [A2: set_b] :
~ ( ord_less_set_b @ A2 @ bot_bot_set_b ) ).
% bot.extremum_strict
thf(fact_1204_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1205_bot_Onot__eq__extremum,axiom,
! [A2: set_b] :
( ( A2 != bot_bot_set_b )
= ( ord_less_set_b @ bot_bot_set_b @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1206_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1207_dtree_Oset__intros_I1_J,axiom,
! [X1: a,X2: fset_P5281107635120001194_a_b_b] : ( member_a @ X1 @ ( dverts_a_b @ ( node_a_b @ X1 @ X2 ) ) ) ).
% dtree.set_intros(1)
thf(fact_1208_less__supI1,axiom,
! [X4: set_b,A2: set_b,B2: set_b] :
( ( ord_less_set_b @ X4 @ A2 )
=> ( ord_less_set_b @ X4 @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_1209_less__supI1,axiom,
! [X4: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ X4 @ A2 )
=> ( ord_less_nat @ X4 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_1210_less__supI2,axiom,
! [X4: set_b,B2: set_b,A2: set_b] :
( ( ord_less_set_b @ X4 @ B2 )
=> ( ord_less_set_b @ X4 @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_1211_less__supI2,axiom,
! [X4: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ X4 @ B2 )
=> ( ord_less_nat @ X4 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_1212_sup_Oabsorb3,axiom,
! [B2: set_b,A2: set_b] :
( ( ord_less_set_b @ B2 @ A2 )
=> ( ( sup_sup_set_b @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_1213_sup_Oabsorb3,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_1214_sup_Oabsorb4,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ( ( sup_sup_set_b @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_1215_sup_Oabsorb4,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_1216_sup_Ostrict__boundedE,axiom,
! [B2: set_b,C2: set_b,A2: set_b] :
( ( ord_less_set_b @ ( sup_sup_set_b @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_set_b @ B2 @ A2 )
=> ~ ( ord_less_set_b @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_1217_sup_Ostrict__boundedE,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_1218_sup_Ostrict__order__iff,axiom,
( ord_less_set_b
= ( ^ [B: set_b,A: set_b] :
( ( A
= ( sup_sup_set_b @ A @ B ) )
& ( A != B ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1219_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B: nat,A: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
& ( A != B ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1220_sup_Ostrict__coboundedI1,axiom,
! [C2: set_b,A2: set_b,B2: set_b] :
( ( ord_less_set_b @ C2 @ A2 )
=> ( ord_less_set_b @ C2 @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1221_sup_Ostrict__coboundedI1,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ C2 @ A2 )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1222_sup_Ostrict__coboundedI2,axiom,
! [C2: set_b,B2: set_b,A2: set_b] :
( ( ord_less_set_b @ C2 @ B2 )
=> ( ord_less_set_b @ C2 @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1223_sup_Ostrict__coboundedI2,axiom,
! [C2: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1224_fset__linorder__max__induct,axiom,
! [P: fset_nat > $o,S2: fset_nat] :
( ( P @ bot_bot_fset_nat )
=> ( ! [X: nat,S3: fset_nat] :
( ! [Y6: nat] :
( ( fmember_nat @ Y6 @ S3 )
=> ( ord_less_nat @ Y6 @ X ) )
=> ( ( P @ S3 )
=> ( P @ ( finsert_nat @ X @ S3 ) ) ) )
=> ( P @ S2 ) ) ) ).
% fset_linorder_max_induct
thf(fact_1225_fset__linorder__min__induct,axiom,
! [P: fset_nat > $o,S2: fset_nat] :
( ( P @ bot_bot_fset_nat )
=> ( ! [X: nat,S3: fset_nat] :
( ! [Y6: nat] :
( ( fmember_nat @ Y6 @ S3 )
=> ( ord_less_nat @ X @ Y6 ) )
=> ( ( P @ S3 )
=> ( P @ ( finsert_nat @ X @ S3 ) ) ) )
=> ( P @ S2 ) ) ) ).
% fset_linorder_min_induct
thf(fact_1226_dtree_Oset__intros_I2_J,axiom,
! [Y: produc4558475209616630778_a_b_b,X2: fset_P5281107635120001194_a_b_b,Ya: dtree_a_b,Yb: a,X1: a] :
( ( member4380921116106875537_a_b_b @ Y @ ( fset_P783253628892185035_a_b_b @ X2 ) )
=> ( ( member_dtree_a_b @ Ya @ ( basic_7578771248400840636_a_b_b @ Y ) )
=> ( ( member_a @ Yb @ ( dverts_a_b @ Ya ) )
=> ( member_a @ Yb @ ( dverts_a_b @ ( node_a_b @ X1 @ X2 ) ) ) ) ) ) ).
% dtree.set_intros(2)
thf(fact_1227_dtree_Oset__cases_I1_J,axiom,
! [E4: a,A2: dtree_a_b] :
( ( member_a @ E4 @ ( dverts_a_b @ A2 ) )
=> ( ! [Z22: fset_P5281107635120001194_a_b_b] :
( A2
!= ( node_a_b @ E4 @ Z22 ) )
=> ~ ! [Z1: a,Z22: fset_P5281107635120001194_a_b_b] :
( ( A2
= ( node_a_b @ Z1 @ Z22 ) )
=> ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Z22 ) )
=> ! [Xa3: dtree_a_b] :
( ( member_dtree_a_b @ Xa3 @ ( basic_7578771248400840636_a_b_b @ X ) )
=> ~ ( member_a @ E4 @ ( dverts_a_b @ Xa3 ) ) ) ) ) ) ) ).
% dtree.set_cases(1)
thf(fact_1228_dtree_Oset__sel_I2_J,axiom,
! [X4: produc4558475209616630778_a_b_b,A2: dtree_a_b,Xa: dtree_a_b,Xb: a] :
( ( member4380921116106875537_a_b_b @ X4 @ ( fset_P783253628892185035_a_b_b @ ( sucs_a_b @ A2 ) ) )
=> ( ( member_dtree_a_b @ Xa @ ( basic_7578771248400840636_a_b_b @ X4 ) )
=> ( ( member_a @ Xb @ ( dverts_a_b @ Xa ) )
=> ( member_a @ Xb @ ( dverts_a_b @ A2 ) ) ) ) ) ).
% dtree.set_sel(2)
thf(fact_1229_root__not__child__if__wf__dverts,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T1: dtree_a_b,E1: b] :
( ( wf_dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T1 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ~ ( member_a @ R2 @ ( dverts_a_b @ T1 ) ) ) ) ).
% root_not_child_if_wf_dverts
thf(fact_1230_dverts__child__if__not__root,axiom,
! [V: a,R2: a,Xs2: fset_P5281107635120001194_a_b_b] :
( ( member_a @ V @ ( dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
=> ( ( V != R2 )
=> ? [X: dtree_a_b] :
( ( member_dtree_a_b @ X @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
& ( member_a @ V @ ( dverts_a_b @ X ) ) ) ) ) ).
% dverts_child_if_not_root
thf(fact_1231_dtree__size__decr__aux,axiom,
! [X4: dtree_a_b,Y: b,Xs2: fset_P5281107635120001194_a_b_b,R2: a] :
( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X4 @ Y ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( ord_less_nat @ ( size_size_dtree_a_b @ X4 ) @ ( size_size_dtree_a_b @ ( node_a_b @ R2 @ Xs2 ) ) ) ) ).
% dtree_size_decr_aux
thf(fact_1232_dverts__child__subseteq,axiom,
! [X4: dtree_a_b,Xs2: fset_P5281107635120001194_a_b_b,R2: a] :
( ( member_dtree_a_b @ X4 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
=> ( ord_less_eq_set_a @ ( dverts_a_b @ X4 ) @ ( dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) ) ) ) ).
% dverts_child_subseteq
thf(fact_1233_dverts__suc__subseteq,axiom,
! [X4: dtree_a_b,T: dtree_a_b] :
( ( member_dtree_a_b @ X4 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ ( sucs_a_b @ T ) ) ) )
=> ( ord_less_eq_set_a @ ( dverts_a_b @ X4 ) @ ( dverts_a_b @ T ) ) ) ).
% dverts_suc_subseteq
thf(fact_1234_dverts__suc__if__not__root,axiom,
! [V: a,T: dtree_a_b] :
( ( member_a @ V @ ( dverts_a_b @ T ) )
=> ( ( V
!= ( root_a_b @ T ) )
=> ? [X: dtree_a_b] :
( ( member_dtree_a_b @ X @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ ( sucs_a_b @ T ) ) ) )
& ( member_a @ V @ ( dverts_a_b @ X ) ) ) ) ) ).
% dverts_suc_if_not_root
thf(fact_1235_root__not__child__if__wf__dverts_H,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b] :
( ( wf_dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ! [X5: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X5 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [T12: dtree_a_b,E12: b] :
~ ( member_a @ R2 @ ( dverts_a_b @ T12 ) )
@ X5 ) ) ) ).
% root_not_child_if_wf_dverts'
thf(fact_1236_disjoint__dverts__if__wf__aux,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T1: dtree_a_b,E1: b,T2: dtree_a_b,E22: b] :
( ( wf_dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T1 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T2 @ E22 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( ( ( produc331601717337510060_a_b_b @ T1 @ E1 )
!= ( produc331601717337510060_a_b_b @ T2 @ E22 ) )
=> ( ( inf_inf_set_a @ ( dverts_a_b @ T1 ) @ ( dverts_a_b @ T2 ) )
= bot_bot_set_a ) ) ) ) ) ).
% disjoint_dverts_if_wf_aux
thf(fact_1237_dtree__size__decr__aux_H,axiom,
! [T1: dtree_a_b,Xs2: fset_P5281107635120001194_a_b_b,R2: a] :
( ( member_dtree_a_b @ T1 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
=> ( ord_less_nat @ ( size_size_dtree_a_b @ T1 ) @ ( size_size_dtree_a_b @ ( node_a_b @ R2 @ Xs2 ) ) ) ) ).
% dtree_size_decr_aux'
thf(fact_1238_disjoint__dverts__if__wf,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b] :
( ( wf_dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ! [X5: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X5 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_a @ ( dverts_a_b @ Y5 ) @ ( dverts_a_b @ Aa ) )
= bot_bot_set_a )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) )
@ X5 ) ) ) ).
% disjoint_dverts_if_wf
thf(fact_1239_wf__dverts__if__dverts_H__aux,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,R2: a] :
( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E: b] : ( wf_dverts_a_b @ Y5 )
@ X ) )
=> ( ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_a @ R2 @ ( dverts_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_a @ ( dverts_a_b @ Y5 ) @ ( dverts_a_b @ Aa ) )
= bot_bot_set_a )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) ) )
@ X ) )
=> ( wf_dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) ) ) ) ).
% wf_dverts_if_dverts'_aux
thf(fact_1240_wf__dverts_H_Oelims_I3_J,axiom,
! [X4: dtree_a_b] :
( ~ ( wf_dverts_a_b2 @ X4 )
=> ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( ( X4
= ( node_a_b @ R @ Xs ) )
=> ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_a @ R @ ( dverts_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_a @ ( dverts_a_b @ Y5 ) @ ( dverts_a_b @ Aa ) )
= bot_bot_set_a )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) )
& ( wf_dverts_a_b2 @ Y5 ) )
@ X ) ) ) ) ).
% wf_dverts'.elims(3)
thf(fact_1241_wf__dverts_H_Oelims_I2_J,axiom,
! [X4: dtree_a_b] :
( ( wf_dverts_a_b2 @ X4 )
=> ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( ( X4
= ( node_a_b @ R @ Xs ) )
=> ~ ! [X5: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X5 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_a @ R @ ( dverts_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_a @ ( dverts_a_b @ Y5 ) @ ( dverts_a_b @ Aa ) )
= bot_bot_set_a )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) )
& ( wf_dverts_a_b2 @ Y5 ) )
@ X5 ) ) ) ) ).
% wf_dverts'.elims(2)
thf(fact_1242_pfsubset__eq,axiom,
( ord_le7001451600920047870_a_b_b
= ( ^ [A6: fset_P5281107635120001194_a_b_b,B6: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% pfsubset_eq
thf(fact_1243_less__fset__def,axiom,
( ord_le7001451600920047870_a_b_b
= ( ^ [Xs3: fset_P5281107635120001194_a_b_b,Ys3: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ Xs3 @ Ys3 )
& ( Xs3 != Ys3 ) ) ) ) ).
% less_fset_def
thf(fact_1244_pfsubset__imp__fsubset,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( ord_le7001451600920047870_a_b_b @ A3 @ B3 )
=> ( ord_le789900035998834954_a_b_b @ A3 @ B3 ) ) ).
% pfsubset_imp_fsubset
thf(fact_1245_fsubset__not__fsubset__eq,axiom,
( ord_le7001451600920047870_a_b_b
= ( ^ [A6: fset_P5281107635120001194_a_b_b,B6: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A6 @ B6 )
& ~ ( ord_le789900035998834954_a_b_b @ B6 @ A6 ) ) ) ) ).
% fsubset_not_fsubset_eq
thf(fact_1246_fsubset__pfsubset__trans,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b,C: fset_P5281107635120001194_a_b_b] :
( ( ord_le789900035998834954_a_b_b @ A3 @ B3 )
=> ( ( ord_le7001451600920047870_a_b_b @ B3 @ C )
=> ( ord_le7001451600920047870_a_b_b @ A3 @ C ) ) ) ).
% fsubset_pfsubset_trans
thf(fact_1247_pfsubset__fsubset__trans,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b,C: fset_P5281107635120001194_a_b_b] :
( ( ord_le7001451600920047870_a_b_b @ A3 @ B3 )
=> ( ( ord_le789900035998834954_a_b_b @ B3 @ C )
=> ( ord_le7001451600920047870_a_b_b @ A3 @ C ) ) ) ).
% pfsubset_fsubset_trans
thf(fact_1248_fsubset__iff__pfsubset__eq,axiom,
( ord_le789900035998834954_a_b_b
= ( ^ [A6: fset_P5281107635120001194_a_b_b,B6: fset_P5281107635120001194_a_b_b] :
( ( ord_le7001451600920047870_a_b_b @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% fsubset_iff_pfsubset_eq
thf(fact_1249_pfsubsetD,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b,C2: produc4558475209616630778_a_b_b] :
( ( ord_le7001451600920047870_a_b_b @ A3 @ B3 )
=> ( ( fmembe3173364709796808819_a_b_b @ C2 @ A3 )
=> ( fmembe3173364709796808819_a_b_b @ C2 @ B3 ) ) ) ).
% pfsubsetD
thf(fact_1250_not__psubset__empty,axiom,
! [A3: set_b] :
~ ( ord_less_set_b @ A3 @ bot_bot_set_b ) ).
% not_psubset_empty
thf(fact_1251_pfsubset__imp__ex__fmem,axiom,
! [A3: fset_P5281107635120001194_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( ord_le7001451600920047870_a_b_b @ A3 @ B3 )
=> ? [B5: produc4558475209616630778_a_b_b] : ( fmembe3173364709796808819_a_b_b @ B5 @ ( minus_1250967532242559235_a_b_b @ B3 @ A3 ) ) ) ).
% pfsubset_imp_ex_fmem
thf(fact_1252_psubset__imp__ex__mem,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( ord_le3723863380492978948_a_b_b @ A3 @ B3 )
=> ? [B5: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ B5 @ ( minus_1392386589478415753_a_b_b @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1253_psubset__imp__ex__mem,axiom,
! [A3: set_b,B3: set_b] :
( ( ord_less_set_b @ A3 @ B3 )
=> ? [B5: b] : ( member_b @ B5 @ ( minus_minus_set_b @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1254_psubset__insert__iff,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,X4: produc4558475209616630778_a_b_b,B3: set_Pr3012420139608375472_a_b_b] :
( ( ord_le3723863380492978948_a_b_b @ A3 @ ( insert1613891728210272810_a_b_b @ X4 @ B3 ) )
= ( ( ( member4380921116106875537_a_b_b @ X4 @ B3 )
=> ( ord_le3723863380492978948_a_b_b @ A3 @ B3 ) )
& ( ~ ( member4380921116106875537_a_b_b @ X4 @ B3 )
=> ( ( ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( ord_le3723863380492978948_a_b_b @ ( minus_1392386589478415753_a_b_b @ A3 @ ( insert1613891728210272810_a_b_b @ X4 @ bot_bo3721250822024684356_a_b_b ) ) @ B3 ) )
& ( ~ ( member4380921116106875537_a_b_b @ X4 @ A3 )
=> ( ord_le146215904626753808_a_b_b @ A3 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1255_psubset__insert__iff,axiom,
! [A3: set_b,X4: b,B3: set_b] :
( ( ord_less_set_b @ A3 @ ( insert_b @ X4 @ B3 ) )
= ( ( ( member_b @ X4 @ B3 )
=> ( ord_less_set_b @ A3 @ B3 ) )
& ( ~ ( member_b @ X4 @ B3 )
=> ( ( ( member_b @ X4 @ A3 )
=> ( ord_less_set_b @ ( minus_minus_set_b @ A3 @ ( insert_b @ X4 @ bot_bot_set_b ) ) @ B3 ) )
& ( ~ ( member_b @ X4 @ A3 )
=> ( ord_less_eq_set_b @ A3 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1256_pfsubset__finsert__iff,axiom,
! [A3: fset_P5281107635120001194_a_b_b,X4: produc4558475209616630778_a_b_b,B3: fset_P5281107635120001194_a_b_b] :
( ( ord_le7001451600920047870_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ X4 @ B3 ) )
= ( ( ( fmembe3173364709796808819_a_b_b @ X4 @ B3 )
=> ( ord_le7001451600920047870_a_b_b @ A3 @ B3 ) )
& ( ~ ( fmembe3173364709796808819_a_b_b @ X4 @ B3 )
=> ( ( ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ( ord_le7001451600920047870_a_b_b @ ( minus_1250967532242559235_a_b_b @ A3 @ ( finser8437519239679886002_a_b_b @ X4 @ bot_bo2895716411488905534_a_b_b ) ) @ B3 ) )
& ( ~ ( fmembe3173364709796808819_a_b_b @ X4 @ A3 )
=> ( ord_le789900035998834954_a_b_b @ A3 @ B3 ) ) ) ) ) ) ).
% pfsubset_finsert_iff
thf(fact_1257_wf__dverts_H_Osimps,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b] :
( ( wf_dverts_a_b2 @ ( node_a_b @ R2 @ Xs2 ) )
= ( ! [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_a @ R2 @ ( dverts_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_a @ ( dverts_a_b @ Y5 ) @ ( dverts_a_b @ Aa ) )
= bot_bot_set_a )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) )
& ( wf_dverts_a_b2 @ Y5 ) )
@ X3 ) ) ) ) ).
% wf_dverts'.simps
thf(fact_1258_wf__dverts_H_Oelims_I1_J,axiom,
! [X4: dtree_a_b,Y: $o] :
( ( ( wf_dverts_a_b2 @ X4 )
= Y )
=> ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( ( X4
= ( node_a_b @ R @ Xs ) )
=> ( Y
= ( ~ ! [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_a @ R @ ( dverts_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_a @ ( dverts_a_b @ Y5 ) @ ( dverts_a_b @ Aa ) )
= bot_bot_set_a )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) )
& ( wf_dverts_a_b2 @ Y5 ) )
@ X3 ) ) ) ) ) ) ).
% wf_dverts'.elims(1)
thf(fact_1259_wf__dverts_H_Opelims_I1_J,axiom,
! [X4: dtree_a_b,Y: $o] :
( ( ( wf_dverts_a_b2 @ X4 )
= Y )
=> ( ( accp_dtree_a_b @ wf_dverts_rel_a_b @ X4 )
=> ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( ( X4
= ( node_a_b @ R @ Xs ) )
=> ( ( Y
= ( ! [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_a @ R @ ( dverts_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_a @ ( dverts_a_b @ Y5 ) @ ( dverts_a_b @ Aa ) )
= bot_bot_set_a )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) )
& ( wf_dverts_a_b2 @ Y5 ) )
@ X3 ) ) ) )
=> ~ ( accp_dtree_a_b @ wf_dverts_rel_a_b @ ( node_a_b @ R @ Xs ) ) ) ) ) ) ).
% wf_dverts'.pelims(1)
thf(fact_1260_wf__dverts_H_Opelims_I2_J,axiom,
! [X4: dtree_a_b] :
( ( wf_dverts_a_b2 @ X4 )
=> ( ( accp_dtree_a_b @ wf_dverts_rel_a_b @ X4 )
=> ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( ( X4
= ( node_a_b @ R @ Xs ) )
=> ( ( accp_dtree_a_b @ wf_dverts_rel_a_b @ ( node_a_b @ R @ Xs ) )
=> ~ ! [X5: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X5 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_a @ R @ ( dverts_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_a @ ( dverts_a_b @ Y5 ) @ ( dverts_a_b @ Aa ) )
= bot_bot_set_a )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) )
& ( wf_dverts_a_b2 @ Y5 ) )
@ X5 ) ) ) ) ) ) ).
% wf_dverts'.pelims(2)
thf(fact_1261_less__set__def,axiom,
( ord_le3723863380492978948_a_b_b
= ( ^ [A6: set_Pr3012420139608375472_a_b_b,B6: set_Pr3012420139608375472_a_b_b] :
( ord_le2302600385889936001_b_b_o
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ A6 )
@ ^ [X3: produc4558475209616630778_a_b_b] : ( member4380921116106875537_a_b_b @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_1262_less__set__def,axiom,
( ord_less_set_b
= ( ^ [A6: set_b,B6: set_b] :
( ord_less_b_o
@ ^ [X3: b] : ( member_b @ X3 @ A6 )
@ ^ [X3: b] : ( member_b @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_1263_psubsetD,axiom,
! [A3: set_Pr3012420139608375472_a_b_b,B3: set_Pr3012420139608375472_a_b_b,C2: produc4558475209616630778_a_b_b] :
( ( ord_le3723863380492978948_a_b_b @ A3 @ B3 )
=> ( ( member4380921116106875537_a_b_b @ C2 @ A3 )
=> ( member4380921116106875537_a_b_b @ C2 @ B3 ) ) ) ).
% psubsetD
thf(fact_1264_psubsetD,axiom,
! [A3: set_b,B3: set_b,C2: b] :
( ( ord_less_set_b @ A3 @ B3 )
=> ( ( member_b @ C2 @ A3 )
=> ( member_b @ C2 @ B3 ) ) ) ).
% psubsetD
thf(fact_1265_wf__dverts_H_Opelims_I3_J,axiom,
! [X4: dtree_a_b] :
( ~ ( wf_dverts_a_b2 @ X4 )
=> ( ( accp_dtree_a_b @ wf_dverts_rel_a_b @ X4 )
=> ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
( ( X4
= ( node_a_b @ R @ Xs ) )
=> ( ( accp_dtree_a_b @ wf_dverts_rel_a_b @ ( node_a_b @ R @ Xs ) )
=> ! [X: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y5: dtree_a_b,E12: b] :
( ~ ( member_a @ R @ ( dverts_a_b @ Y5 ) )
& ! [Z4: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z4 @ ( fset_P783253628892185035_a_b_b @ Xs ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_a @ ( dverts_a_b @ Y5 ) @ ( dverts_a_b @ Aa ) )
= bot_bot_set_a )
| ( ( produc331601717337510060_a_b_b @ Y5 @ E12 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z4 ) )
& ( wf_dverts_a_b2 @ Y5 ) )
@ X ) ) ) ) ) ) ).
% wf_dverts'.pelims(3)
thf(fact_1266_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1267_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B2 ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1268_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1269_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1270_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1271_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_1272_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1273_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1274_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1275_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1276_diff__le__mono,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% diff_le_mono
% Helper facts (11)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X4: a,Y: a] :
( ( if_a @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X4: a,Y: a] :
( ( if_a @ $true @ X4 @ Y )
= X4 ) ).
thf(help_If_2_1_If_001_062_Itf__b_Mtf__a_J_T,axiom,
! [X4: b > a,Y: b > a] :
( ( if_b_a @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001_062_Itf__b_Mtf__a_J_T,axiom,
! [X4: b > a,Y: b > a] :
( ( if_b_a @ $true @ X4 @ Y )
= X4 ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__b_J_T,axiom,
! [X4: set_b,Y: set_b] :
( ( if_set_b @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__b_J_T,axiom,
! [X4: set_b,Y: set_b] :
( ( if_set_b @ $true @ X4 @ Y )
= X4 ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_T,axiom,
! [X4: produc4558475209616630778_a_b_b,Y: produc4558475209616630778_a_b_b] :
( ( if_Pro6329973184163622324_a_b_b @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_T,axiom,
! [X4: produc4558475209616630778_a_b_b,Y: produc4558475209616630778_a_b_b] :
( ( if_Pro6329973184163622324_a_b_b @ $true @ X4 @ Y )
= X4 ) ).
thf(help_If_3_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J_T,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] :
( ( if_fse8812573537926886756_a_b_b @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_J_T,axiom,
! [X4: fset_P5281107635120001194_a_b_b,Y: fset_P5281107635120001194_a_b_b] :
( ( if_fse8812573537926886756_a_b_b @ $true @ X4 @ Y )
= X4 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ffold_2783168711033344739_b_b_a
@ ( produc2242037354397874494_b_a_a
@ ^ [X3: dtree_a_b,E2: b,B: a] :
( if_a
@ ( ~ ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X3 @ E2 ) @ ( fset_P783253628892185035_a_b_b @ xsa ) )
| ~ ( member_b @ e @ ( sup_sup_set_b @ ( darcs_a_b @ X3 ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
| ~ ( wf_darcs_a_b @ ( node_a_b @ r @ xsa ) ) )
@ B
@ ( if_a @ ( e = E2 ) @ ( root_a_b @ X3 ) @ ( dhead_a_b @ X3 @ def @ e ) ) ) )
@ ( def @ e )
@ xsa )
= ( root_a_b @ t ) ) ).
%------------------------------------------------------------------------------