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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 : FO_Theory_Rewriting/0078_Context_RR2/prob_00118_004968__18838938_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1868 ( 615 unt; 590 typ; 0 def)
% Number of atoms : 3621 (1416 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 11488 ( 276 ~; 34 |; 185 &;9630 @)
% ( 0 <=>;1363 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 87 ( 86 usr)
% Number of type conns : 3181 (3181 >; 0 *; 0 +; 0 <<)
% Number of symbols : 507 ( 504 usr; 28 con; 0-5 aty)
% Number of variables : 4468 ( 736 ^;3666 !; 66 ?;4468 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:02:49.305
%------------------------------------------------------------------------------
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thf(ty_n_t__Set__Oset_It__Ground____Terms__Ogterm_Itf__a_J_J,type,
set_Ground_gterm_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
product_prod_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
product_prod_nat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
set_option_a: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_I_Eo_J_J,type,
set_option_o: $tType ).
thf(ty_n_t__Option__Ooption_It__Set__Oset_I_Eo_J_J,type,
option_set_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
product_prod_o_o: $tType ).
thf(ty_n_t__Ground____Terms__Ogterm_Itf__a_J,type,
ground_gterm_a: $tType ).
thf(ty_n_t__Ground____Ctxt__Ogctxt_Itf__a_J,type,
ground_gctxt_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Option__Ooption_Itf__a_J,type,
option_a: $tType ).
thf(ty_n_t__Option__Ooption_I_Eo_J,type,
option_o: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (504)
thf(sy_c_BNF__Def_OGr_001_Eo_001_Eo,type,
bNF_Gr_o_o: set_o > ( $o > $o ) > set_Product_prod_o_o ).
thf(sy_c_BNF__Def_OGr_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
bNF_Gr2342861688195830050tion_a: set_option_a > ( option_a > option_a ) > set_Pr7585778909603769095tion_a ).
thf(sy_c_BNF__Def_OGr_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001_Eo,type,
bNF_Gr8237857062489970771on_a_o: set_Pr7585778909603769095tion_a > ( produc3509355604313844263tion_a > $o ) > set_Pr7343186322352036408on_a_o ).
thf(sy_c_BNF__Def_OGr_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Nat__Onat,type,
bNF_Gr7863039376269261781_a_nat: set_Pr7585778909603769095tion_a > ( produc3509355604313844263tion_a > nat ) > set_Pr8660066742269822812_a_nat ).
thf(sy_c_BNF__Def_OGr_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_001_Eo,type,
bNF_Gr2973908843748354126_nat_o: set_Pr8660066742269822812_a_nat > ( produc4760857364844987814_a_nat > $o ) > set_Pr2938601653316290973_nat_o ).
thf(sy_c_BNF__Def_OGr_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J,type,
bNF_Gr3785214186540711495_a_nat: set_Pr4934435412358123699_a_nat > ( product_prod_a_nat > produc4760857364844987814_a_nat ) > set_Pr5895937092904826980_a_nat ).
thf(sy_c_BNF__Def_OGr_001tf__a_001t__Nat__Onat,type,
bNF_Gr_a_nat: set_a > ( a > nat ) > set_Pr4934435412358123699_a_nat ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
bNF_Gr2637526933174132606at_nat: set_Pr1493194377889511160_a_nat > ( ( product_prod_a_nat > produc4760857364844987814_a_nat ) > product_prod_a_nat > nat ) > ( product_prod_a_nat > produc4760857364844987814_a_nat ) > ( product_prod_a_nat > nat ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_J,type,
bNF_Gr6306292117806463247tion_a: set_Pr1493194377889511160_a_nat > ( ( product_prod_a_nat > produc4760857364844987814_a_nat ) > product_prod_a_nat > produc3509355604313844263tion_a ) > ( product_prod_a_nat > produc4760857364844987814_a_nat ) > ( product_prod_a_nat > produc3509355604313844263tion_a ) > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
bNF_Gr5654676046521400910tion_a: set_option_a > ( option_a > option_a ) > option_a > option_a > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Nat__Onat,type,
bNF_Gr2566800059905468329_a_nat: set_Pr7585778909603769095tion_a > ( produc3509355604313844263tion_a > nat ) > produc3509355604313844263tion_a > nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Option__Ooption_Itf__a_J,type,
bNF_Gr2851282060293067563tion_a: set_Pr7585778909603769095tion_a > ( produc3509355604313844263tion_a > option_a ) > produc3509355604313844263tion_a > option_a > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bNF_Gr8625867392921115590at_nat: set_Pr8660066742269822812_a_nat > ( produc4760857364844987814_a_nat > nat ) > produc4760857364844987814_a_nat > nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
bNF_Gr2569508567399099313tion_a: set_Pr8660066742269822812_a_nat > ( produc4760857364844987814_a_nat > produc3509355604313844263tion_a ) > produc4760857364844987814_a_nat > produc3509355604313844263tion_a > $o ).
thf(sy_c_BNF__Def_OGrp_001tf__a_001t__Nat__Onat,type,
bNF_Grp_a_nat: set_a > ( a > nat ) > a > nat > $o ).
thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
bNF_co1154721918818078718tion_a: ( produc3509355604313844263tion_a > option_a ) > ( produc3509355604313844263tion_a > option_a ) > produc3509355604313844263tion_a > produc3509355604313844263tion_a ).
thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Nat__Onat,type,
bNF_co3083941901121763833_a_nat: ( produc3509355604313844263tion_a > produc3509355604313844263tion_a ) > ( produc3509355604313844263tion_a > nat ) > produc3509355604313844263tion_a > produc4760857364844987814_a_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001tf__a_001t__Nat__Onat,type,
bNF_co1502894461497240272_a_nat: ( produc3509355604313844263tion_a > a ) > ( produc3509355604313844263tion_a > nat ) > produc3509355604313844263tion_a > product_prod_a_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
bNF_co3392937053677906959tion_a: ( produc4760857364844987814_a_nat > option_a ) > ( produc4760857364844987814_a_nat > option_a ) > produc4760857364844987814_a_nat > produc3509355604313844263tion_a ).
thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Nat__Onat,type,
bNF_co7748663798163203240_a_nat: ( produc4760857364844987814_a_nat > produc3509355604313844263tion_a ) > ( produc4760857364844987814_a_nat > nat ) > produc4760857364844987814_a_nat > produc4760857364844987814_a_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_001tf__a_001t__Nat__Onat,type,
bNF_co1810985475411362175_a_nat: ( produc4760857364844987814_a_nat > a ) > ( produc4760857364844987814_a_nat > nat ) > produc4760857364844987814_a_nat > product_prod_a_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Nat__Onat,type,
bNF_co4564054178660176785_a_nat: ( product_prod_a_nat > produc3509355604313844263tion_a ) > ( product_prod_a_nat > nat ) > product_prod_a_nat > produc4760857364844987814_a_nat ).
thf(sy_c_BNF__Def_OfstOp_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
bNF_fs7976035448042954642tion_a: ( option_a > option_a > $o ) > ( option_a > option_a > $o ) > produc3509355604313844263tion_a > produc3509355604313844263tion_a ).
thf(sy_c_BNF__Def_OfstOp_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_fs4236391834175287831at_nat: ( produc3509355604313844263tion_a > nat > $o ) > ( nat > nat > $o ) > produc4760857364844987814_a_nat > produc4760857364844987814_a_nat ).
thf(sy_c_BNF__Def_OfstOp_001tf__a_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_fstOp_a_nat_nat: ( a > nat > $o ) > ( nat > nat > $o ) > product_prod_a_nat > product_prod_a_nat ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
bNF_pi5353392459224037856tion_a: ( option_a > option_a > $o ) > ( option_a > option_a > $o ) > option_a > option_a > option_a ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_Eo_001_Eo,type,
bNF_rel_fun_o_o_o_o: ( $o > $o > $o ) > ( $o > $o > $o ) > ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_Eo_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
bNF_re8060677170909647724tion_a: ( $o > $o > $o ) > ( $o > produc3509355604313844263tion_a > $o ) > ( $o > $o ) > ( $o > produc3509355604313844263tion_a ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_Eo_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J,type,
bNF_re5119610622004122145_a_nat: ( $o > $o > $o ) > ( $o > produc4760857364844987814_a_nat > $o ) > ( $o > $o ) > ( $o > produc4760857364844987814_a_nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_Eo_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
bNF_re3038775964670433016_a_nat: ( $o > $o > $o ) > ( $o > product_prod_a_nat > $o ) > ( $o > $o ) > ( $o > product_prod_a_nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001_Eo,type,
bNF_re7598680882129772558on_a_o: ( $o > $o > $o ) > ( produc3509355604313844263tion_a > $o > $o ) > ( $o > produc3509355604313844263tion_a ) > ( $o > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001_Eo,type,
bNF_re7998865585964069436_nat_o: ( $o > $o > $o ) > ( product_prod_a_nat > $o > $o ) > ( $o > product_prod_a_nat ) > ( $o > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J_001_062_It__Option__Ooption_Itf__a_J_M_Eo_J_001_062_It__Option__Ooption_Itf__a_J_M_Eo_J,type,
bNF_re1541143192665202257on_a_o: ( option_a > option_a > $o ) > ( ( option_a > $o ) > ( option_a > $o ) > $o ) > ( option_a > option_a > $o ) > ( option_a > option_a > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J_001_062_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
bNF_re3760461907148547146_nat_o: ( option_a > option_a > $o ) > ( ( produc3509355604313844263tion_a > $o ) > ( nat > $o ) > $o ) > ( option_a > produc3509355604313844263tion_a > $o ) > ( option_a > nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J_001_062_Itf__a_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
bNF_re668239450913114657_nat_o: ( option_a > option_a > $o ) > ( ( a > $o ) > ( nat > $o ) > $o ) > ( option_a > a > $o ) > ( option_a > nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J_001_Eo_001_Eo,type,
bNF_re905040088260304477_a_o_o: ( option_a > option_a > $o ) > ( $o > $o > $o ) > ( option_a > $o ) > ( option_a > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Nat__Onat_001_062_It__Option__Ooption_Itf__a_J_M_Eo_J_001_062_It__Option__Ooption_Itf__a_J_M_Eo_J,type,
bNF_re7417368233926557774on_a_o: ( produc3509355604313844263tion_a > nat > $o ) > ( ( option_a > $o ) > ( option_a > $o ) > $o ) > ( produc3509355604313844263tion_a > option_a > $o ) > ( nat > option_a > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
bNF_re5291775304233153037_nat_o: ( produc3509355604313844263tion_a > nat > $o ) > ( ( produc3509355604313844263tion_a > $o ) > ( nat > $o ) > $o ) > ( produc3509355604313844263tion_a > produc3509355604313844263tion_a > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Nat__Onat_001_062_Itf__a_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
bNF_re7266954853851403236_nat_o: ( produc3509355604313844263tion_a > nat > $o ) > ( ( a > $o ) > ( nat > $o ) > $o ) > ( produc3509355604313844263tion_a > a > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Nat__Onat_001_Eo_001_Eo,type,
bNF_re3303494005150464218at_o_o: ( produc3509355604313844263tion_a > nat > $o ) > ( $o > $o > $o ) > ( produc3509355604313844263tion_a > $o ) > ( nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001_Eo_001_Eo,type,
bNF_re8491395028433271709_a_o_o: ( produc3509355604313844263tion_a > produc3509355604313844263tion_a > $o ) > ( $o > $o > $o ) > ( produc3509355604313844263tion_a > $o ) > ( produc3509355604313844263tion_a > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001_Eo_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
bNF_re6940856228500498104_a_nat: ( produc3509355604313844263tion_a > produc3509355604313844263tion_a > $o ) > ( $o > product_prod_a_nat > $o ) > ( produc3509355604313844263tion_a > $o ) > ( produc3509355604313844263tion_a > product_prod_a_nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001_Eo,type,
bNF_re2677573812939358716_nat_o: ( produc3509355604313844263tion_a > produc3509355604313844263tion_a > $o ) > ( product_prod_a_nat > $o > $o ) > ( produc3509355604313844263tion_a > product_prod_a_nat ) > ( produc3509355604313844263tion_a > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
bNF_re7959353272085281067at_o_o: ( produc4760857364844987814_a_nat > produc4760857364844987814_a_nat > $o ) > ( $o > $o > $o ) > ( produc4760857364844987814_a_nat > $o ) > ( produc4760857364844987814_a_nat > $o ) > $o ).
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image_3729436797865474090_a_nat: ( product_prod_a_nat > produc4760857364844987814_a_nat ) > set_Pr4934435412358123699_a_nat > set_Pr8660066742269822812_a_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
image_8708701149064048897_a_nat: ( product_prod_a_nat > product_prod_a_nat ) > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001tf__a,type,
image_6659209163692594554_nat_a: ( product_prod_a_nat > a ) > set_Pr4934435412358123699_a_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_J_J,type,
image_4143212015729212815_a_nat: ( set_Pr8660066742269822812_a_nat > option2973356794535198124_a_nat ) > set_se7986904383068512786_a_nat > set_op4623351964121498594_a_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
image_4776182516663252797_a_nat: ( set_Pr4934435412358123699_a_nat > option5987904771724668803_a_nat ) > set_se6121441497158405097_a_nat > set_op4138567020912711225_a_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Option__Ooption_Itf__a_J,type,
image_a_option_a: ( a > option_a ) > set_a > set_option_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
image_7468189554597481134tion_a: ( a > produc3509355604313844263tion_a ) > set_a > set_Pr7585778909603769095tion_a ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_J_J,type,
insert8722057353638028252_a_nat: option2973356794535198124_a_nat > set_op4623351964121498594_a_nat > set_op4623351964121498594_a_nat ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
insert8208376198844224947_a_nat: option5987904771724668803_a_nat > set_op4138567020912711225_a_nat > set_op4138567020912711225_a_nat ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_Itf__a_J,type,
insert_option_a: option_a > set_option_a > set_option_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
insert1246254401036548087tion_a: produc3509355604313844263tion_a > set_Pr7585778909603769095tion_a > set_Pr7585778909603769095tion_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J,type,
insert8345643852111475670_a_nat: produc4760857364844987814_a_nat > set_Pr8660066742269822812_a_nat > set_Pr8660066742269822812_a_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
insert8054603423593749677_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_J,type,
insert7606643247661561228_a_nat: set_Pr8660066742269822812_a_nat > set_se7986904383068512786_a_nat > set_se7986904383068512786_a_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
insert6577045944563236579_a_nat: set_Pr4934435412358123699_a_nat > set_se6121441497158405097_a_nat > set_se6121441497158405097_a_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001_Eo,type,
is_singleton_o: set_o > $o ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
is_sin6176258158573935899tion_a: set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J,type,
is_sin523092863759094194_a_nat: set_Pr8660066742269822812_a_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
is_sin7638514652509285001_a_nat: set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_Set_Othe__elem_001_Eo,type,
the_elem_o: set_o > $o ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J,type,
the_el944771114868072817_a_nat: set_Pr8660066742269822812_a_nat > produc4760857364844987814_a_nat ).
thf(sy_c_Typedef_Otype__definition_001_Eo_001_Eo,type,
type_definition_o_o: ( $o > $o ) > ( $o > $o ) > set_o > $o ).
thf(sy_c_Typedef_Otype__definition_001_Eo_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
type_d432855040626640571tion_a: ( $o > produc3509355604313844263tion_a ) > ( produc3509355604313844263tion_a > $o ) > set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_Typedef_Otype__definition_001_Eo_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J,type,
type_d3416793964091482130_a_nat: ( $o > produc4760857364844987814_a_nat ) > ( produc4760857364844987814_a_nat > $o ) > set_Pr8660066742269822812_a_nat > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_001_Eo,type,
type_d9194230788701541213on_a_o: ( produc3509355604313844263tion_a > $o ) > ( $o > produc3509355604313844263tion_a ) > set_o > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_001_Eo,type,
type_d7953064638307121348_nat_o: ( produc4760857364844987814_a_nat > $o ) > ( $o > produc4760857364844987814_a_nat ) > set_o > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
type_d6372012530977374289_a_nat: ( produc4760857364844987814_a_nat > product_prod_a_nat ) > ( product_prod_a_nat > produc4760857364844987814_a_nat ) > set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J,type,
type_d3363870972157910353_a_nat: ( product_prod_a_nat > produc4760857364844987814_a_nat ) > ( produc4760857364844987814_a_nat > product_prod_a_nat ) > set_Pr8660066742269822812_a_nat > $o ).
thf(sy_c_member_001_062_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_J,type,
member2063753777887369945_a_nat: ( product_prod_a_nat > produc4760857364844987814_a_nat ) > set_Pr1493194377889511160_a_nat > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_I_Eo_J,type,
member_option_o: option_o > set_option_o > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_J,type,
member7330119243174117782tion_a: option2893053940705734637tion_a > set_op6528554486239853133tion_a > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_J,type,
member7104896805454144013_a_nat: option5973373053609626358_a_nat > set_op8842393985335015468_a_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
member1239542624773947236_a_nat: option5551091909395471437_a_nat > set_op2732609354225412611_a_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_J_J,type,
member1062715794267306179_a_nat: option2973356794535198124_a_nat > set_op4623351964121498594_a_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
member4183983893482796698_a_nat: option5987904771724668803_a_nat > set_op4138567020912711225_a_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_Itf__a_J,type,
member_option_a: option_a > set_option_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_J,type,
member2654175073444177119tion_a: produc4838336064122203702tion_a > set_Pr8496096474564289814tion_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_J,type,
member9172549829962833796_a_nat: produc6654178220768718189_a_nat > set_Pr7637049405689187875_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
member26132018921490651_a_nat: produc2710310701403425476_a_nat > set_Pr6347322337906478074_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_J,type,
member1799035910557443043tion_a: produc5129331709516570188tion_a > set_Pr2852166431588183298tion_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
member8962352052110095674_nat_a: product_prod_nat_a > set_Pr4193341848836149977_nat_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
member5498148017924304208tion_a: produc3509355604313844263tion_a > set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_M_Eo_J,type,
member10471143963986689on_a_o: produc2194632134642013272on_a_o > set_Pr7343186322352036408on_a_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J,type,
member1430561565885860669_a_nat: produc4760857364844987814_a_nat > set_Pr8660066742269822812_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_M_Eo_J,type,
member1850667382371020647_nat_o: produc4534846064852955472_nat_o > set_Pr8703467907450596486_nat_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
member9062615507155100804_a_nat: produc4708774622424448987_a_nat > set_Pr1811044260758604347_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_Mt__Nat__Onat_J_J,type,
member904957861751271667_a_nat: set_Pr8660066742269822812_a_nat > set_se7986904383068512786_a_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
member8034581882086557258_a_nat: set_Pr4934435412358123699_a_nat > set_se6121441497158405097_a_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_C,type,
c: ground4599112814892907741tion_a ).
thf(sy_v__092_060F_062,type,
f: set_Pr4934435412358123699_a_nat ).
% Relevant facts (1276)
thf(fact_0_pair__imageI,axiom,
! [A: option_a,B: option_a,A2: set_Pr7585778909603769095tion_a,F: option_a > option_a > $o] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ A2 )
=> ( member_o @ ( F @ A @ B ) @ ( image_166006535297930294on_a_o @ ( produc7248784972643495020on_a_o @ F ) @ A2 ) ) ) ).
% pair_imageI
thf(fact_1_pair__imageI,axiom,
! [A: produc3509355604313844263tion_a,B: nat,A2: set_Pr8660066742269822812_a_nat,F: produc3509355604313844263tion_a > nat > $o] :
( ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ A @ B ) @ A2 )
=> ( member_o @ ( F @ A @ B ) @ ( image_6768402879995552939_nat_o @ ( produc7988141324749014923_nat_o @ F ) @ A2 ) ) ) ).
% pair_imageI
thf(fact_2_pair__imageI,axiom,
! [A: a,B: nat,A2: set_Pr4934435412358123699_a_nat,F: a > nat > produc4760857364844987814_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B ) @ A2 )
=> ( member1430561565885860669_a_nat @ ( F @ A @ B ) @ ( image_3729436797865474090_a_nat @ ( produc4004610663780792650_a_nat @ F ) @ A2 ) ) ) ).
% pair_imageI
thf(fact_3_pair__imageI,axiom,
! [A: a,B: nat,A2: set_Pr4934435412358123699_a_nat,F: a > nat > $o] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B ) @ A2 )
=> ( member_o @ ( F @ A @ B ) @ ( image_3076334604395105300_nat_o @ ( produc3680711911437148916_nat_o @ F ) @ A2 ) ) ) ).
% pair_imageI
thf(fact_4_pair__imageI,axiom,
! [A: a,B: nat,A2: set_Pr4934435412358123699_a_nat,F: a > nat > product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B ) @ A2 )
=> ( member5724188588386418708_a_nat @ ( F @ A @ B ) @ ( image_8708701149064048897_a_nat @ ( produc8738682720600764961_a_nat @ F ) @ A2 ) ) ) ).
% pair_imageI
thf(fact_5_pair__imageI,axiom,
! [A: option_a,B: option_a,A2: set_Pr7585778909603769095tion_a,F: option_a > option_a > option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ A2 )
=> ( member_option_a @ ( F @ A @ B ) @ ( image_7272866627766095330tion_a @ ( produc6953443519211250200tion_a @ F ) @ A2 ) ) ) ).
% pair_imageI
thf(fact_6_pair__imageI,axiom,
! [A: option_a,B: option_a,A2: set_Pr7585778909603769095tion_a,F: option_a > option_a > product_prod_a_nat] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ A2 )
=> ( member5724188588386418708_a_nat @ ( F @ A @ B ) @ ( image_2274939045924579167_a_nat @ ( produc4612668676680314537_a_nat @ F ) @ A2 ) ) ) ).
% pair_imageI
thf(fact_7_pair__imageI,axiom,
! [A: a,B: nat,A2: set_Pr4934435412358123699_a_nat,F: a > nat > produc3509355604313844263tion_a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B ) @ A2 )
=> ( member5498148017924304208tion_a @ ( F @ A @ B ) @ ( image_5430222630607662243tion_a @ ( produc7832866272818598787tion_a @ F ) @ A2 ) ) ) ).
% pair_imageI
thf(fact_8_pair__imageI,axiom,
! [A: produc3509355604313844263tion_a,B: nat,A2: set_Pr8660066742269822812_a_nat,F: produc3509355604313844263tion_a > nat > product_prod_a_nat] :
( ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ A @ B ) @ A2 )
=> ( member5724188588386418708_a_nat @ ( F @ A @ B ) @ ( image_6737578356684938026_a_nat @ ( produc7713474374650187850_a_nat @ F ) @ A2 ) ) ) ).
% pair_imageI
thf(fact_9_pair__imageI,axiom,
! [A: option_a,B: option_a,A2: set_Pr7585778909603769095tion_a,F: option_a > option_a > produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ A2 )
=> ( member5498148017924304208tion_a @ ( F @ A @ B ) @ ( image_6668519322520640453tion_a @ ( produc2781663484249587963tion_a @ F ) @ A2 ) ) ) ).
% pair_imageI
thf(fact_10_case__prod__conv,axiom,
! [F: option_a > option_a > $o,A: option_a,B: option_a] :
( ( produc7248784972643495020on_a_o @ F @ ( produc9011544418120257559tion_a @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_11_case__prod__conv,axiom,
! [F: produc3509355604313844263tion_a > nat > $o,A: produc3509355604313844263tion_a,B: nat] :
( ( produc7988141324749014923_nat_o @ F @ ( produc5457283509376734880_a_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_12_case__prod__conv,axiom,
! [F: option_a > option_a > produc3509355604313844263tion_a,A: option_a,B: option_a] :
( ( produc2781663484249587963tion_a @ F @ ( produc9011544418120257559tion_a @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_13_case__prod__conv,axiom,
! [F: produc3509355604313844263tion_a > nat > produc4760857364844987814_a_nat,A: produc3509355604313844263tion_a,B: nat] :
( ( produc5088488592959153011_a_nat @ F @ ( produc5457283509376734880_a_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_14_case__prod__conv,axiom,
! [F: option_a > option_a > option_a,A: option_a,B: option_a] :
( ( produc6953443519211250200tion_a @ F @ ( produc9011544418120257559tion_a @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_15_case__prod__conv,axiom,
! [F: a > nat > produc4760857364844987814_a_nat,A: a,B: nat] :
( ( produc4004610663780792650_a_nat @ F @ ( product_Pair_a_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_16_less__eq__option__Some,axiom,
! [X: option5987904771724668803_a_nat,Y: option5987904771724668803_a_nat] :
( ( ord_le6477260885034027827_a_nat @ ( some_o4687019253342298078_a_nat @ X ) @ ( some_o4687019253342298078_a_nat @ Y ) )
= ( ord_le1496697172603502563_a_nat @ X @ Y ) ) ).
% less_eq_option_Some
thf(fact_17_less__eq__option__Some,axiom,
! [X: option2973356794535198124_a_nat,Y: option2973356794535198124_a_nat] :
( ( ord_le1484599589293175004_a_nat @ ( some_o4943705774378422791_a_nat @ X ) @ ( some_o4943705774378422791_a_nat @ Y ) )
= ( ord_le8848857416346193420_a_nat @ X @ Y ) ) ).
% less_eq_option_Some
thf(fact_18_less__eq__option__Some,axiom,
! [X: set_o,Y: set_o] :
( ( ord_le3873616110718653662_set_o @ ( some_set_o @ X ) @ ( some_set_o @ Y ) )
= ( ord_less_eq_set_o @ X @ Y ) ) ).
% less_eq_option_Some
thf(fact_19_less__eq__option__Some,axiom,
! [X: set_Pr8660066742269822812_a_nat,Y: set_Pr8660066742269822812_a_nat] :
( ( ord_le8848857416346193420_a_nat @ ( some_s2314368709036572727_a_nat @ X ) @ ( some_s2314368709036572727_a_nat @ Y ) )
= ( ord_le5649184635953113020_a_nat @ X @ Y ) ) ).
% less_eq_option_Some
thf(fact_20_less__eq__option__Some,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le1496697172603502563_a_nat @ ( some_s1880800141159441806_a_nat @ X ) @ ( some_s1880800141159441806_a_nat @ Y ) )
= ( ord_le8666007276011122963_a_nat @ X @ Y ) ) ).
% less_eq_option_Some
thf(fact_21_image__ident,axiom,
! [Y2: set_o] :
( ( image_o_o
@ ^ [X2: $o] : X2
@ Y2 )
= Y2 ) ).
% image_ident
thf(fact_22_option_Oinject,axiom,
! [X22: set_Pr4934435412358123699_a_nat,Y22: set_Pr4934435412358123699_a_nat] :
( ( ( some_s1880800141159441806_a_nat @ X22 )
= ( some_s1880800141159441806_a_nat @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_23_option_Oinject,axiom,
! [X22: set_Pr8660066742269822812_a_nat,Y22: set_Pr8660066742269822812_a_nat] :
( ( ( some_s2314368709036572727_a_nat @ X22 )
= ( some_s2314368709036572727_a_nat @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_24_option_Oinject,axiom,
! [X22: a,Y22: a] :
( ( ( some_a @ X22 )
= ( some_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_25_subsetI,axiom,
! [A2: set_Pr7585778909603769095tion_a,B2: set_Pr7585778909603769095tion_a] :
( ! [X3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X3 @ A2 )
=> ( member5498148017924304208tion_a @ X3 @ B2 ) )
=> ( ord_le4471550158292877991tion_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_26_subsetI,axiom,
! [A2: set_o,B2: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_o @ X3 @ B2 ) )
=> ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% subsetI
thf(fact_27_subsetI,axiom,
! [A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ! [X3: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X3 @ A2 )
=> ( member1430561565885860669_a_nat @ X3 @ B2 ) )
=> ( ord_le5649184635953113020_a_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_28_subsetI,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A2 )
=> ( member5724188588386418708_a_nat @ X3 @ B2 ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_29_subset__antisym,axiom,
! [A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_30_subset__antisym,axiom,
! [A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ A2 @ B2 )
=> ( ( ord_le5649184635953113020_a_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_31_subset__antisym,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_32_case__prod__Pair__iden,axiom,
! [P: product_prod_a_nat] :
( ( produc8738682720600764961_a_nat @ product_Pair_a_nat @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_33_case__prod__Pair__iden,axiom,
! [P: produc4760857364844987814_a_nat] :
( ( produc5088488592959153011_a_nat @ produc5457283509376734880_a_nat @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_34_case__prod__Pair__iden,axiom,
! [P: produc3509355604313844263tion_a] :
( ( produc2781663484249587963tion_a @ produc9011544418120257559tion_a @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_35_image__eqI,axiom,
! [B: $o,F: $o > $o,X: $o,A2: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member_o @ B @ ( image_o_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_36_image__eqI,axiom,
! [B: produc4760857364844987814_a_nat,F: product_prod_a_nat > produc4760857364844987814_a_nat,X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( B
= ( F @ X ) )
=> ( ( member5724188588386418708_a_nat @ X @ A2 )
=> ( member1430561565885860669_a_nat @ B @ ( image_3729436797865474090_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_37_image__eqI,axiom,
! [B: product_prod_a_nat,F: $o > product_prod_a_nat,X: $o,A2: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member5724188588386418708_a_nat @ B @ ( image_7339617019956244688_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_38_image__eqI,axiom,
! [B: $o,F: product_prod_a_nat > $o,X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( B
= ( F @ X ) )
=> ( ( member5724188588386418708_a_nat @ X @ A2 )
=> ( member_o @ B @ ( image_3076334604395105300_nat_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_39_image__eqI,axiom,
! [B: produc3509355604313844263tion_a,F: $o > produc3509355604313844263tion_a,X: $o,A2: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member5498148017924304208tion_a @ B @ ( image_628002824077805460tion_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_40_image__eqI,axiom,
! [B: product_prod_a_nat,F: product_prod_a_nat > product_prod_a_nat,X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( B
= ( F @ X ) )
=> ( ( member5724188588386418708_a_nat @ X @ A2 )
=> ( member5724188588386418708_a_nat @ B @ ( image_8708701149064048897_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_41_image__eqI,axiom,
! [B: $o,F: produc3509355604313844263tion_a > $o,X: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a] :
( ( B
= ( F @ X ) )
=> ( ( member5498148017924304208tion_a @ X @ A2 )
=> ( member_o @ B @ ( image_166006535297930294on_a_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_42_image__eqI,axiom,
! [B: produc4760857364844987814_a_nat,F: $o > produc4760857364844987814_a_nat,X: $o,A2: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member1430561565885860669_a_nat @ B @ ( image_2232132205779913721_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_43_image__eqI,axiom,
! [B: produc3509355604313844263tion_a,F: product_prod_a_nat > produc3509355604313844263tion_a,X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( B
= ( F @ X ) )
=> ( ( member5724188588386418708_a_nat @ X @ A2 )
=> ( member5498148017924304208tion_a @ B @ ( image_5430222630607662243tion_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_44_image__eqI,axiom,
! [B: $o,F: produc4760857364844987814_a_nat > $o,X: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat] :
( ( B
= ( F @ X ) )
=> ( ( member1430561565885860669_a_nat @ X @ A2 )
=> ( member_o @ B @ ( image_6768402879995552939_nat_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_45_fst__def,axiom,
( produc174239612569448963tion_a
= ( produc6953443519211250200tion_a
@ ^ [X1: option_a,X23: option_a] : X1 ) ) ).
% fst_def
thf(fact_46_prod_Oinject,axiom,
! [X12: produc3509355604313844263tion_a,X22: nat,Y1: produc3509355604313844263tion_a,Y22: nat] :
( ( ( produc5457283509376734880_a_nat @ X12 @ X22 )
= ( produc5457283509376734880_a_nat @ Y1 @ Y22 ) )
= ( ( X12 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_47_prod_Oinject,axiom,
! [X12: option_a,X22: option_a,Y1: option_a,Y22: option_a] :
( ( ( produc9011544418120257559tion_a @ X12 @ X22 )
= ( produc9011544418120257559tion_a @ Y1 @ Y22 ) )
= ( ( X12 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_48_prod_Oinject,axiom,
! [X12: a,X22: nat,Y1: a,Y22: nat] :
( ( ( product_Pair_a_nat @ X12 @ X22 )
= ( product_Pair_a_nat @ Y1 @ Y22 ) )
= ( ( X12 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_49_old_Oprod_Oinject,axiom,
! [A: produc3509355604313844263tion_a,B: nat,A3: produc3509355604313844263tion_a,B3: nat] :
( ( ( produc5457283509376734880_a_nat @ A @ B )
= ( produc5457283509376734880_a_nat @ A3 @ B3 ) )
= ( ( A = A3 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_50_old_Oprod_Oinject,axiom,
! [A: option_a,B: option_a,A3: option_a,B3: option_a] :
( ( ( produc9011544418120257559tion_a @ A @ B )
= ( produc9011544418120257559tion_a @ A3 @ B3 ) )
= ( ( A = A3 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_51_old_Oprod_Oinject,axiom,
! [A: a,B: nat,A3: a,B3: nat] :
( ( ( product_Pair_a_nat @ A @ B )
= ( product_Pair_a_nat @ A3 @ B3 ) )
= ( ( A = A3 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_52_case__prodI2,axiom,
! [P: product_prod_a_nat,C: a > nat > $o] :
( ! [A4: a,B4: nat] :
( ( P
= ( product_Pair_a_nat @ A4 @ B4 ) )
=> ( C @ A4 @ B4 ) )
=> ( produc3680711911437148916_nat_o @ C @ P ) ) ).
% case_prodI2
thf(fact_53_case__prodI2,axiom,
! [P: produc3509355604313844263tion_a,C: option_a > option_a > $o] :
( ! [A4: option_a,B4: option_a] :
( ( P
= ( produc9011544418120257559tion_a @ A4 @ B4 ) )
=> ( C @ A4 @ B4 ) )
=> ( produc7248784972643495020on_a_o @ C @ P ) ) ).
% case_prodI2
thf(fact_54_case__prodI2,axiom,
! [P: produc4760857364844987814_a_nat,C: produc3509355604313844263tion_a > nat > $o] :
( ! [A4: produc3509355604313844263tion_a,B4: nat] :
( ( P
= ( produc5457283509376734880_a_nat @ A4 @ B4 ) )
=> ( C @ A4 @ B4 ) )
=> ( produc7988141324749014923_nat_o @ C @ P ) ) ).
% case_prodI2
thf(fact_55_case__prodI,axiom,
! [F: a > nat > $o,A: a,B: nat] :
( ( F @ A @ B )
=> ( produc3680711911437148916_nat_o @ F @ ( product_Pair_a_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_56_case__prodI,axiom,
! [F: option_a > option_a > $o,A: option_a,B: option_a] :
( ( F @ A @ B )
=> ( produc7248784972643495020on_a_o @ F @ ( produc9011544418120257559tion_a @ A @ B ) ) ) ).
% case_prodI
thf(fact_57_case__prodI,axiom,
! [F: produc3509355604313844263tion_a > nat > $o,A: produc3509355604313844263tion_a,B: nat] :
( ( F @ A @ B )
=> ( produc7988141324749014923_nat_o @ F @ ( produc5457283509376734880_a_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_58_mem__case__prodI2,axiom,
! [P: product_prod_a_nat,Z: $o,C: a > nat > set_o] :
( ! [A4: a,B4: nat] :
( ( P
= ( product_Pair_a_nat @ A4 @ B4 ) )
=> ( member_o @ Z @ ( C @ A4 @ B4 ) ) )
=> ( member_o @ Z @ ( produc7785968526437677524_set_o @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_59_mem__case__prodI2,axiom,
! [P: produc3509355604313844263tion_a,Z: $o,C: option_a > option_a > set_o] :
( ! [A4: option_a,B4: option_a] :
( ( P
= ( produc9011544418120257559tion_a @ A4 @ B4 ) )
=> ( member_o @ Z @ ( C @ A4 @ B4 ) ) )
=> ( member_o @ Z @ ( produc275038863006032716_set_o @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_60_mem__case__prodI2,axiom,
! [P: product_prod_a_nat,Z: product_prod_a_nat,C: a > nat > set_Pr4934435412358123699_a_nat] :
( ! [A4: a,B4: nat] :
( ( P
= ( product_Pair_a_nat @ A4 @ B4 ) )
=> ( member5724188588386418708_a_nat @ Z @ ( C @ A4 @ B4 ) ) )
=> ( member5724188588386418708_a_nat @ Z @ ( produc7499543521670136919_a_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_61_mem__case__prodI2,axiom,
! [P: produc4760857364844987814_a_nat,Z: $o,C: produc3509355604313844263tion_a > nat > set_o] :
( ! [A4: produc3509355604313844263tion_a,B4: nat] :
( ( P
= ( produc5457283509376734880_a_nat @ A4 @ B4 ) )
=> ( member_o @ Z @ ( C @ A4 @ B4 ) ) )
=> ( member_o @ Z @ ( produc2625910688742146667_set_o @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_62_mem__case__prodI2,axiom,
! [P: produc3509355604313844263tion_a,Z: product_prod_a_nat,C: option_a > option_a > set_Pr4934435412358123699_a_nat] :
( ! [A4: option_a,B4: option_a] :
( ( P
= ( produc9011544418120257559tion_a @ A4 @ B4 ) )
=> ( member5724188588386418708_a_nat @ Z @ ( C @ A4 @ B4 ) ) )
=> ( member5724188588386418708_a_nat @ Z @ ( produc6454328207421087967_a_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_63_mem__case__prodI2,axiom,
! [P: product_prod_a_nat,Z: produc3509355604313844263tion_a,C: a > nat > set_Pr7585778909603769095tion_a] :
( ! [A4: a,B4: nat] :
( ( P
= ( product_Pair_a_nat @ A4 @ B4 ) )
=> ( member5498148017924304208tion_a @ Z @ ( C @ A4 @ B4 ) ) )
=> ( member5498148017924304208tion_a @ Z @ ( produc4752055246895195235tion_a @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_64_mem__case__prodI2,axiom,
! [P: produc4760857364844987814_a_nat,Z: product_prod_a_nat,C: produc3509355604313844263tion_a > nat > set_Pr4934435412358123699_a_nat] :
( ! [A4: produc3509355604313844263tion_a,B4: nat] :
( ( P
= ( produc5457283509376734880_a_nat @ A4 @ B4 ) )
=> ( member5724188588386418708_a_nat @ Z @ ( C @ A4 @ B4 ) ) )
=> ( member5724188588386418708_a_nat @ Z @ ( produc199880962849341056_a_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_65_mem__case__prodI2,axiom,
! [P: produc3509355604313844263tion_a,Z: produc3509355604313844263tion_a,C: option_a > option_a > set_Pr7585778909603769095tion_a] :
( ! [A4: option_a,B4: option_a] :
( ( P
= ( produc9011544418120257559tion_a @ A4 @ B4 ) )
=> ( member5498148017924304208tion_a @ Z @ ( C @ A4 @ B4 ) ) )
=> ( member5498148017924304208tion_a @ Z @ ( produc8494404344289423835tion_a @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_66_mem__case__prodI2,axiom,
! [P: product_prod_a_nat,Z: produc4760857364844987814_a_nat,C: a > nat > set_Pr8660066742269822812_a_nat] :
( ! [A4: a,B4: nat] :
( ( P
= ( product_Pair_a_nat @ A4 @ B4 ) )
=> ( member1430561565885860669_a_nat @ Z @ ( C @ A4 @ B4 ) ) )
=> ( member1430561565885860669_a_nat @ Z @ ( produc6296525983053523712_a_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_67_mem__case__prodI2,axiom,
! [P: produc4760857364844987814_a_nat,Z: produc3509355604313844263tion_a,C: produc3509355604313844263tion_a > nat > set_Pr7585778909603769095tion_a] :
( ! [A4: produc3509355604313844263tion_a,B4: nat] :
( ( P
= ( produc5457283509376734880_a_nat @ A4 @ B4 ) )
=> ( member5498148017924304208tion_a @ Z @ ( C @ A4 @ B4 ) ) )
=> ( member5498148017924304208tion_a @ Z @ ( produc1118003391013324154tion_a @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_68_mem__case__prodI,axiom,
! [Z: $o,C: a > nat > set_o,A: a,B: nat] :
( ( member_o @ Z @ ( C @ A @ B ) )
=> ( member_o @ Z @ ( produc7785968526437677524_set_o @ C @ ( product_Pair_a_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_69_mem__case__prodI,axiom,
! [Z: $o,C: option_a > option_a > set_o,A: option_a,B: option_a] :
( ( member_o @ Z @ ( C @ A @ B ) )
=> ( member_o @ Z @ ( produc275038863006032716_set_o @ C @ ( produc9011544418120257559tion_a @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_70_mem__case__prodI,axiom,
! [Z: product_prod_a_nat,C: a > nat > set_Pr4934435412358123699_a_nat,A: a,B: nat] :
( ( member5724188588386418708_a_nat @ Z @ ( C @ A @ B ) )
=> ( member5724188588386418708_a_nat @ Z @ ( produc7499543521670136919_a_nat @ C @ ( product_Pair_a_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_71_mem__case__prodI,axiom,
! [Z: $o,C: produc3509355604313844263tion_a > nat > set_o,A: produc3509355604313844263tion_a,B: nat] :
( ( member_o @ Z @ ( C @ A @ B ) )
=> ( member_o @ Z @ ( produc2625910688742146667_set_o @ C @ ( produc5457283509376734880_a_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_72_mem__case__prodI,axiom,
! [Z: product_prod_a_nat,C: option_a > option_a > set_Pr4934435412358123699_a_nat,A: option_a,B: option_a] :
( ( member5724188588386418708_a_nat @ Z @ ( C @ A @ B ) )
=> ( member5724188588386418708_a_nat @ Z @ ( produc6454328207421087967_a_nat @ C @ ( produc9011544418120257559tion_a @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_73_mem__case__prodI,axiom,
! [Z: produc3509355604313844263tion_a,C: a > nat > set_Pr7585778909603769095tion_a,A: a,B: nat] :
( ( member5498148017924304208tion_a @ Z @ ( C @ A @ B ) )
=> ( member5498148017924304208tion_a @ Z @ ( produc4752055246895195235tion_a @ C @ ( product_Pair_a_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_74_mem__case__prodI,axiom,
! [Z: product_prod_a_nat,C: produc3509355604313844263tion_a > nat > set_Pr4934435412358123699_a_nat,A: produc3509355604313844263tion_a,B: nat] :
( ( member5724188588386418708_a_nat @ Z @ ( C @ A @ B ) )
=> ( member5724188588386418708_a_nat @ Z @ ( produc199880962849341056_a_nat @ C @ ( produc5457283509376734880_a_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_75_mem__case__prodI,axiom,
! [Z: produc3509355604313844263tion_a,C: option_a > option_a > set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( member5498148017924304208tion_a @ Z @ ( C @ A @ B ) )
=> ( member5498148017924304208tion_a @ Z @ ( produc8494404344289423835tion_a @ C @ ( produc9011544418120257559tion_a @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_76_mem__case__prodI,axiom,
! [Z: produc4760857364844987814_a_nat,C: a > nat > set_Pr8660066742269822812_a_nat,A: a,B: nat] :
( ( member1430561565885860669_a_nat @ Z @ ( C @ A @ B ) )
=> ( member1430561565885860669_a_nat @ Z @ ( produc6296525983053523712_a_nat @ C @ ( product_Pair_a_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_77_mem__case__prodI,axiom,
! [Z: produc3509355604313844263tion_a,C: produc3509355604313844263tion_a > nat > set_Pr7585778909603769095tion_a,A: produc3509355604313844263tion_a,B: nat] :
( ( member5498148017924304208tion_a @ Z @ ( C @ A @ B ) )
=> ( member5498148017924304208tion_a @ Z @ ( produc1118003391013324154tion_a @ C @ ( produc5457283509376734880_a_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_78_Collect__case__prod__mono,axiom,
! [A2: a > nat > $o,B2: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ A2 @ B2 )
=> ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ A2 ) ) @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ B2 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_79_Collect__case__prod__mono,axiom,
! [A2: option_a > option_a > $o,B2: option_a > option_a > $o] :
( ( ord_le3058241657886854542on_a_o @ A2 @ B2 )
=> ( ord_le4471550158292877991tion_a @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ A2 ) ) @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ B2 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_80_Collect__case__prod__mono,axiom,
! [A2: produc3509355604313844263tion_a > nat > $o,B2: produc3509355604313844263tion_a > nat > $o] :
( ( ord_le489641559516061149_nat_o @ A2 @ B2 )
=> ( ord_le5649184635953113020_a_nat @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ A2 ) ) @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ B2 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_81_mem__case__prodE,axiom,
! [Z: $o,C: a > nat > set_o,P: product_prod_a_nat] :
( ( member_o @ Z @ ( produc7785968526437677524_set_o @ C @ P ) )
=> ~ ! [X3: a,Y3: nat] :
( ( P
= ( product_Pair_a_nat @ X3 @ Y3 ) )
=> ~ ( member_o @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_82_mem__case__prodE,axiom,
! [Z: $o,C: option_a > option_a > set_o,P: produc3509355604313844263tion_a] :
( ( member_o @ Z @ ( produc275038863006032716_set_o @ C @ P ) )
=> ~ ! [X3: option_a,Y3: option_a] :
( ( P
= ( produc9011544418120257559tion_a @ X3 @ Y3 ) )
=> ~ ( member_o @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_83_mem__case__prodE,axiom,
! [Z: product_prod_a_nat,C: a > nat > set_Pr4934435412358123699_a_nat,P: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Z @ ( produc7499543521670136919_a_nat @ C @ P ) )
=> ~ ! [X3: a,Y3: nat] :
( ( P
= ( product_Pair_a_nat @ X3 @ Y3 ) )
=> ~ ( member5724188588386418708_a_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_84_mem__case__prodE,axiom,
! [Z: $o,C: produc3509355604313844263tion_a > nat > set_o,P: produc4760857364844987814_a_nat] :
( ( member_o @ Z @ ( produc2625910688742146667_set_o @ C @ P ) )
=> ~ ! [X3: produc3509355604313844263tion_a,Y3: nat] :
( ( P
= ( produc5457283509376734880_a_nat @ X3 @ Y3 ) )
=> ~ ( member_o @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_85_mem__case__prodE,axiom,
! [Z: product_prod_a_nat,C: option_a > option_a > set_Pr4934435412358123699_a_nat,P: produc3509355604313844263tion_a] :
( ( member5724188588386418708_a_nat @ Z @ ( produc6454328207421087967_a_nat @ C @ P ) )
=> ~ ! [X3: option_a,Y3: option_a] :
( ( P
= ( produc9011544418120257559tion_a @ X3 @ Y3 ) )
=> ~ ( member5724188588386418708_a_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_86_mem__case__prodE,axiom,
! [Z: produc3509355604313844263tion_a,C: a > nat > set_Pr7585778909603769095tion_a,P: product_prod_a_nat] :
( ( member5498148017924304208tion_a @ Z @ ( produc4752055246895195235tion_a @ C @ P ) )
=> ~ ! [X3: a,Y3: nat] :
( ( P
= ( product_Pair_a_nat @ X3 @ Y3 ) )
=> ~ ( member5498148017924304208tion_a @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_87_mem__case__prodE,axiom,
! [Z: product_prod_a_nat,C: produc3509355604313844263tion_a > nat > set_Pr4934435412358123699_a_nat,P: produc4760857364844987814_a_nat] :
( ( member5724188588386418708_a_nat @ Z @ ( produc199880962849341056_a_nat @ C @ P ) )
=> ~ ! [X3: produc3509355604313844263tion_a,Y3: nat] :
( ( P
= ( produc5457283509376734880_a_nat @ X3 @ Y3 ) )
=> ~ ( member5724188588386418708_a_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_88_mem__case__prodE,axiom,
! [Z: produc3509355604313844263tion_a,C: option_a > option_a > set_Pr7585778909603769095tion_a,P: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ Z @ ( produc8494404344289423835tion_a @ C @ P ) )
=> ~ ! [X3: option_a,Y3: option_a] :
( ( P
= ( produc9011544418120257559tion_a @ X3 @ Y3 ) )
=> ~ ( member5498148017924304208tion_a @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_89_mem__case__prodE,axiom,
! [Z: produc4760857364844987814_a_nat,C: a > nat > set_Pr8660066742269822812_a_nat,P: product_prod_a_nat] :
( ( member1430561565885860669_a_nat @ Z @ ( produc6296525983053523712_a_nat @ C @ P ) )
=> ~ ! [X3: a,Y3: nat] :
( ( P
= ( product_Pair_a_nat @ X3 @ Y3 ) )
=> ~ ( member1430561565885860669_a_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_90_mem__case__prodE,axiom,
! [Z: produc3509355604313844263tion_a,C: produc3509355604313844263tion_a > nat > set_Pr7585778909603769095tion_a,P: produc4760857364844987814_a_nat] :
( ( member5498148017924304208tion_a @ Z @ ( produc1118003391013324154tion_a @ C @ P ) )
=> ~ ! [X3: produc3509355604313844263tion_a,Y3: nat] :
( ( P
= ( produc5457283509376734880_a_nat @ X3 @ Y3 ) )
=> ~ ( member5498148017924304208tion_a @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_91_case__prodE,axiom,
! [C: a > nat > $o,P: product_prod_a_nat] :
( ( produc3680711911437148916_nat_o @ C @ P )
=> ~ ! [X3: a,Y3: nat] :
( ( P
= ( product_Pair_a_nat @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_92_case__prodE,axiom,
! [C: option_a > option_a > $o,P: produc3509355604313844263tion_a] :
( ( produc7248784972643495020on_a_o @ C @ P )
=> ~ ! [X3: option_a,Y3: option_a] :
( ( P
= ( produc9011544418120257559tion_a @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_93_case__prodE,axiom,
! [C: produc3509355604313844263tion_a > nat > $o,P: produc4760857364844987814_a_nat] :
( ( produc7988141324749014923_nat_o @ C @ P )
=> ~ ! [X3: produc3509355604313844263tion_a,Y3: nat] :
( ( P
= ( produc5457283509376734880_a_nat @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_94_case__prodD,axiom,
! [F: a > nat > $o,A: a,B: nat] :
( ( produc3680711911437148916_nat_o @ F @ ( product_Pair_a_nat @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_95_case__prodD,axiom,
! [F: option_a > option_a > $o,A: option_a,B: option_a] :
( ( produc7248784972643495020on_a_o @ F @ ( produc9011544418120257559tion_a @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_96_case__prodD,axiom,
! [F: produc3509355604313844263tion_a > nat > $o,A: produc3509355604313844263tion_a,B: nat] :
( ( produc7988141324749014923_nat_o @ F @ ( produc5457283509376734880_a_nat @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_97_less__eq__set__def,axiom,
( ord_le4471550158292877991tion_a
= ( ^ [A5: set_Pr7585778909603769095tion_a,B5: set_Pr7585778909603769095tion_a] :
( ord_le4863394807095538934on_a_o
@ ^ [X2: produc3509355604313844263tion_a] : ( member5498148017924304208tion_a @ X2 @ A5 )
@ ^ [X2: produc3509355604313844263tion_a] : ( member5498148017924304208tion_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_98_less__eq__set__def,axiom,
( ord_le5649184635953113020_a_nat
= ( ^ [A5: set_Pr8660066742269822812_a_nat,B5: set_Pr8660066742269822812_a_nat] :
( ord_le8675248135204644809_nat_o
@ ^ [X2: produc4760857364844987814_a_nat] : ( member1430561565885860669_a_nat @ X2 @ A5 )
@ ^ [X2: produc4760857364844987814_a_nat] : ( member1430561565885860669_a_nat @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_99_less__eq__set__def,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ord_le8950881081082477362_nat_o
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ A5 )
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_100_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A5: set_o,B5: set_o] :
( ord_less_eq_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ A5 )
@ ^ [X2: $o] : ( member_o @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_101_Pair__inject,axiom,
! [A: produc3509355604313844263tion_a,B: nat,A3: produc3509355604313844263tion_a,B3: nat] :
( ( ( produc5457283509376734880_a_nat @ A @ B )
= ( produc5457283509376734880_a_nat @ A3 @ B3 ) )
=> ~ ( ( A = A3 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_102_Pair__inject,axiom,
! [A: option_a,B: option_a,A3: option_a,B3: option_a] :
( ( ( produc9011544418120257559tion_a @ A @ B )
= ( produc9011544418120257559tion_a @ A3 @ B3 ) )
=> ~ ( ( A = A3 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_103_Pair__inject,axiom,
! [A: a,B: nat,A3: a,B3: nat] :
( ( ( product_Pair_a_nat @ A @ B )
= ( product_Pair_a_nat @ A3 @ B3 ) )
=> ~ ( ( A = A3 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_104_prod__cases,axiom,
! [P2: produc4760857364844987814_a_nat > $o,P: produc4760857364844987814_a_nat] :
( ! [A4: produc3509355604313844263tion_a,B4: nat] : ( P2 @ ( produc5457283509376734880_a_nat @ A4 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_105_prod__cases,axiom,
! [P2: produc3509355604313844263tion_a > $o,P: produc3509355604313844263tion_a] :
( ! [A4: option_a,B4: option_a] : ( P2 @ ( produc9011544418120257559tion_a @ A4 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_106_prod__cases,axiom,
! [P2: product_prod_a_nat > $o,P: product_prod_a_nat] :
( ! [A4: a,B4: nat] : ( P2 @ ( product_Pair_a_nat @ A4 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_107_surj__pair,axiom,
! [P: produc4760857364844987814_a_nat] :
? [X3: produc3509355604313844263tion_a,Y3: nat] :
( P
= ( produc5457283509376734880_a_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_108_surj__pair,axiom,
! [P: produc3509355604313844263tion_a] :
? [X3: option_a,Y3: option_a] :
( P
= ( produc9011544418120257559tion_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_109_surj__pair,axiom,
! [P: product_prod_a_nat] :
? [X3: a,Y3: nat] :
( P
= ( product_Pair_a_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_110_old_Oprod_Oexhaust,axiom,
! [Y: produc4760857364844987814_a_nat] :
~ ! [A4: produc3509355604313844263tion_a,B4: nat] :
( Y
!= ( produc5457283509376734880_a_nat @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_111_old_Oprod_Oexhaust,axiom,
! [Y: produc3509355604313844263tion_a] :
~ ! [A4: option_a,B4: option_a] :
( Y
!= ( produc9011544418120257559tion_a @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_112_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_nat] :
~ ! [A4: a,B4: nat] :
( Y
!= ( product_Pair_a_nat @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_113_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B: $o,F: $o > $o] :
( ( member_o @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_o_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_114_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B: product_prod_a_nat,F: $o > product_prod_a_nat] :
( ( member_o @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member5724188588386418708_a_nat @ B @ ( image_7339617019956244688_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_115_rev__image__eqI,axiom,
! [X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,B: $o,F: product_prod_a_nat > $o] :
( ( member5724188588386418708_a_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_3076334604395105300_nat_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_116_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B: produc3509355604313844263tion_a,F: $o > produc3509355604313844263tion_a] :
( ( member_o @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member5498148017924304208tion_a @ B @ ( image_628002824077805460tion_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_117_rev__image__eqI,axiom,
! [X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat,F: product_prod_a_nat > product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member5724188588386418708_a_nat @ B @ ( image_8708701149064048897_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_118_rev__image__eqI,axiom,
! [X: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a,B: $o,F: produc3509355604313844263tion_a > $o] :
( ( member5498148017924304208tion_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_166006535297930294on_a_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_119_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B: produc4760857364844987814_a_nat,F: $o > produc4760857364844987814_a_nat] :
( ( member_o @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member1430561565885860669_a_nat @ B @ ( image_2232132205779913721_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_120_rev__image__eqI,axiom,
! [X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,B: produc3509355604313844263tion_a,F: product_prod_a_nat > produc3509355604313844263tion_a] :
( ( member5724188588386418708_a_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member5498148017924304208tion_a @ B @ ( image_5430222630607662243tion_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_121_rev__image__eqI,axiom,
! [X: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat,B: $o,F: produc4760857364844987814_a_nat > $o] :
( ( member1430561565885860669_a_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_6768402879995552939_nat_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_122_rev__image__eqI,axiom,
! [X: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a,B: product_prod_a_nat,F: produc3509355604313844263tion_a > product_prod_a_nat] :
( ( member5498148017924304208tion_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member5724188588386418708_a_nat @ B @ ( image_2274939045924579167_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_123_ball__imageD,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,A2: set_Pr4934435412358123699_a_nat,P2: produc4760857364844987814_a_nat > $o] :
( ! [X3: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X3 @ ( image_3729436797865474090_a_nat @ F @ A2 ) )
=> ( P2 @ X3 ) )
=> ! [X4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X4 @ A2 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_124_ball__imageD,axiom,
! [F: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a,P2: $o > $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( image_166006535297930294on_a_o @ F @ A2 ) )
=> ( P2 @ X3 ) )
=> ! [X4: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X4 @ A2 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_125_ball__imageD,axiom,
! [F: produc4760857364844987814_a_nat > $o,A2: set_Pr8660066742269822812_a_nat,P2: $o > $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( image_6768402879995552939_nat_o @ F @ A2 ) )
=> ( P2 @ X3 ) )
=> ! [X4: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X4 @ A2 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_126_ball__imageD,axiom,
! [F: $o > $o,A2: set_o,P2: $o > $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( image_o_o @ F @ A2 ) )
=> ( P2 @ X3 ) )
=> ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_127_image__cong,axiom,
! [M: set_o,N: set_o,F: $o > $o,G: $o > $o] :
( ( M = N )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_o_o @ F @ M )
= ( image_o_o @ G @ N ) ) ) ) ).
% image_cong
thf(fact_128_image__cong,axiom,
! [M: set_Pr4934435412358123699_a_nat,N: set_Pr4934435412358123699_a_nat,F: product_prod_a_nat > produc4760857364844987814_a_nat,G: product_prod_a_nat > produc4760857364844987814_a_nat] :
( ( M = N )
=> ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_3729436797865474090_a_nat @ F @ M )
= ( image_3729436797865474090_a_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_129_image__cong,axiom,
! [M: set_Pr8660066742269822812_a_nat,N: set_Pr8660066742269822812_a_nat,F: produc4760857364844987814_a_nat > $o,G: produc4760857364844987814_a_nat > $o] :
( ( M = N )
=> ( ! [X3: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_6768402879995552939_nat_o @ F @ M )
= ( image_6768402879995552939_nat_o @ G @ N ) ) ) ) ).
% image_cong
thf(fact_130_image__cong,axiom,
! [M: set_Pr7585778909603769095tion_a,N: set_Pr7585778909603769095tion_a,F: produc3509355604313844263tion_a > $o,G: produc3509355604313844263tion_a > $o] :
( ( M = N )
=> ( ! [X3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_166006535297930294on_a_o @ F @ M )
= ( image_166006535297930294on_a_o @ G @ N ) ) ) ) ).
% image_cong
thf(fact_131_bex__imageD,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,A2: set_Pr4934435412358123699_a_nat,P2: produc4760857364844987814_a_nat > $o] :
( ? [X4: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X4 @ ( image_3729436797865474090_a_nat @ F @ A2 ) )
& ( P2 @ X4 ) )
=> ? [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A2 )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_132_bex__imageD,axiom,
! [F: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a,P2: $o > $o] :
( ? [X4: $o] :
( ( member_o @ X4 @ ( image_166006535297930294on_a_o @ F @ A2 ) )
& ( P2 @ X4 ) )
=> ? [X3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X3 @ A2 )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_133_bex__imageD,axiom,
! [F: produc4760857364844987814_a_nat > $o,A2: set_Pr8660066742269822812_a_nat,P2: $o > $o] :
( ? [X4: $o] :
( ( member_o @ X4 @ ( image_6768402879995552939_nat_o @ F @ A2 ) )
& ( P2 @ X4 ) )
=> ? [X3: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X3 @ A2 )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_134_bex__imageD,axiom,
! [F: $o > $o,A2: set_o,P2: $o > $o] :
( ? [X4: $o] :
( ( member_o @ X4 @ ( image_o_o @ F @ A2 ) )
& ( P2 @ X4 ) )
=> ? [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_135_image__iff,axiom,
! [Z: $o,F: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a] :
( ( member_o @ Z @ ( image_166006535297930294on_a_o @ F @ A2 ) )
= ( ? [X2: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_136_image__iff,axiom,
! [Z: $o,F: produc4760857364844987814_a_nat > $o,A2: set_Pr8660066742269822812_a_nat] :
( ( member_o @ Z @ ( image_6768402879995552939_nat_o @ F @ A2 ) )
= ( ? [X2: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_137_image__iff,axiom,
! [Z: $o,F: $o > $o,A2: set_o] :
( ( member_o @ Z @ ( image_o_o @ F @ A2 ) )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_138_image__iff,axiom,
! [Z: produc4760857364844987814_a_nat,F: product_prod_a_nat > produc4760857364844987814_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( member1430561565885860669_a_nat @ Z @ ( image_3729436797865474090_a_nat @ F @ A2 ) )
= ( ? [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_139_imageI,axiom,
! [X: $o,A2: set_o,F: $o > $o] :
( ( member_o @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_o_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_140_imageI,axiom,
! [X: $o,A2: set_o,F: $o > product_prod_a_nat] :
( ( member_o @ X @ A2 )
=> ( member5724188588386418708_a_nat @ ( F @ X ) @ ( image_7339617019956244688_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_141_imageI,axiom,
! [X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,F: product_prod_a_nat > $o] :
( ( member5724188588386418708_a_nat @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_3076334604395105300_nat_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_142_imageI,axiom,
! [X: $o,A2: set_o,F: $o > produc3509355604313844263tion_a] :
( ( member_o @ X @ A2 )
=> ( member5498148017924304208tion_a @ ( F @ X ) @ ( image_628002824077805460tion_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_143_imageI,axiom,
! [X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,F: product_prod_a_nat > product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A2 )
=> ( member5724188588386418708_a_nat @ ( F @ X ) @ ( image_8708701149064048897_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_144_imageI,axiom,
! [X: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a,F: produc3509355604313844263tion_a > $o] :
( ( member5498148017924304208tion_a @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_166006535297930294on_a_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_145_imageI,axiom,
! [X: $o,A2: set_o,F: $o > produc4760857364844987814_a_nat] :
( ( member_o @ X @ A2 )
=> ( member1430561565885860669_a_nat @ ( F @ X ) @ ( image_2232132205779913721_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_146_imageI,axiom,
! [X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,F: product_prod_a_nat > produc3509355604313844263tion_a] :
( ( member5724188588386418708_a_nat @ X @ A2 )
=> ( member5498148017924304208tion_a @ ( F @ X ) @ ( image_5430222630607662243tion_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_147_imageI,axiom,
! [X: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat,F: produc4760857364844987814_a_nat > $o] :
( ( member1430561565885860669_a_nat @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_6768402879995552939_nat_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_148_imageI,axiom,
! [X: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a,F: produc3509355604313844263tion_a > product_prod_a_nat] :
( ( member5498148017924304208tion_a @ X @ A2 )
=> ( member5724188588386418708_a_nat @ ( F @ X ) @ ( image_2274939045924579167_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_149_mem__Collect__eq,axiom,
! [A: $o,P2: $o > $o] :
( ( member_o @ A @ ( collect_o @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_150_mem__Collect__eq,axiom,
! [A: product_prod_a_nat,P2: product_prod_a_nat > $o] :
( ( member5724188588386418708_a_nat @ A @ ( collec4464134535221767506_a_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_151_mem__Collect__eq,axiom,
! [A: produc4760857364844987814_a_nat,P2: produc4760857364844987814_a_nat > $o] :
( ( member1430561565885860669_a_nat @ A @ ( collec7438131855006933115_a_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_152_mem__Collect__eq,axiom,
! [A: produc3509355604313844263tion_a,P2: produc3509355604313844263tion_a > $o] :
( ( member5498148017924304208tion_a @ A @ ( collec4135126896892755346tion_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_153_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_154_Collect__mem__eq,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_155_Collect__mem__eq,axiom,
! [A2: set_Pr8660066742269822812_a_nat] :
( ( collec7438131855006933115_a_nat
@ ^ [X2: produc4760857364844987814_a_nat] : ( member1430561565885860669_a_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_156_Collect__mem__eq,axiom,
! [A2: set_Pr7585778909603769095tion_a] :
( ( collec4135126896892755346tion_a
@ ^ [X2: produc3509355604313844263tion_a] : ( member5498148017924304208tion_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_157_Collect__mono__iff,axiom,
! [P2: produc4760857364844987814_a_nat > $o,Q: produc4760857364844987814_a_nat > $o] :
( ( ord_le5649184635953113020_a_nat @ ( collec7438131855006933115_a_nat @ P2 ) @ ( collec7438131855006933115_a_nat @ Q ) )
= ( ! [X2: produc4760857364844987814_a_nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_158_Collect__mono__iff,axiom,
! [P2: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P2 ) @ ( collec4464134535221767506_a_nat @ Q ) )
= ( ! [X2: product_prod_a_nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_159_Collect__mono__iff,axiom,
! [P2: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ ( collect_o @ P2 ) @ ( collect_o @ Q ) )
= ( ! [X2: $o] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_160_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr8660066742269822812_a_nat,Z2: set_Pr8660066742269822812_a_nat] : ( Y4 = Z2 ) )
= ( ^ [A5: set_Pr8660066742269822812_a_nat,B5: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ A5 @ B5 )
& ( ord_le5649184635953113020_a_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_161_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( Y4 = Z2 ) )
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
& ( ord_le8666007276011122963_a_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_162_set__eq__subset,axiom,
( ( ^ [Y4: set_o,Z2: set_o] : ( Y4 = Z2 ) )
= ( ^ [A5: set_o,B5: set_o] :
( ( ord_less_eq_set_o @ A5 @ B5 )
& ( ord_less_eq_set_o @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_163_subset__trans,axiom,
! [A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat,C2: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ A2 @ B2 )
=> ( ( ord_le5649184635953113020_a_nat @ B2 @ C2 )
=> ( ord_le5649184635953113020_a_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_164_subset__trans,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C2 )
=> ( ord_le8666007276011122963_a_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_165_subset__trans,axiom,
! [A2: set_o,B2: set_o,C2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( ord_less_eq_set_o @ B2 @ C2 )
=> ( ord_less_eq_set_o @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_166_Collect__mono,axiom,
! [P2: produc4760857364844987814_a_nat > $o,Q: produc4760857364844987814_a_nat > $o] :
( ! [X3: produc4760857364844987814_a_nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le5649184635953113020_a_nat @ ( collec7438131855006933115_a_nat @ P2 ) @ ( collec7438131855006933115_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_167_Collect__mono,axiom,
! [P2: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ! [X3: product_prod_a_nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P2 ) @ ( collec4464134535221767506_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_168_Collect__mono,axiom,
! [P2: $o > $o,Q: $o > $o] :
( ! [X3: $o] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_o @ ( collect_o @ P2 ) @ ( collect_o @ Q ) ) ) ).
% Collect_mono
thf(fact_169_subset__refl,axiom,
! [A2: set_Pr8660066742269822812_a_nat] : ( ord_le5649184635953113020_a_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_170_subset__refl,axiom,
! [A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_171_subset__refl,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ A2 ) ).
% subset_refl
thf(fact_172_subset__iff,axiom,
( ord_le4471550158292877991tion_a
= ( ^ [A5: set_Pr7585778909603769095tion_a,B5: set_Pr7585778909603769095tion_a] :
! [T: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ T @ A5 )
=> ( member5498148017924304208tion_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_173_subset__iff,axiom,
( ord_le5649184635953113020_a_nat
= ( ^ [A5: set_Pr8660066742269822812_a_nat,B5: set_Pr8660066742269822812_a_nat] :
! [T: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ T @ A5 )
=> ( member1430561565885860669_a_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_174_subset__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
! [T: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ T @ A5 )
=> ( member5724188588386418708_a_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_175_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A5: set_o,B5: set_o] :
! [T: $o] :
( ( member_o @ T @ A5 )
=> ( member_o @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_176_equalityD2,axiom,
! [A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ( A2 = B2 )
=> ( ord_le5649184635953113020_a_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_177_equalityD2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( A2 = B2 )
=> ( ord_le8666007276011122963_a_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_178_equalityD2,axiom,
! [A2: set_o,B2: set_o] :
( ( A2 = B2 )
=> ( ord_less_eq_set_o @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_179_equalityD1,axiom,
! [A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ( A2 = B2 )
=> ( ord_le5649184635953113020_a_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_180_equalityD1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( A2 = B2 )
=> ( ord_le8666007276011122963_a_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_181_equalityD1,axiom,
! [A2: set_o,B2: set_o] :
( ( A2 = B2 )
=> ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_182_subset__eq,axiom,
( ord_le4471550158292877991tion_a
= ( ^ [A5: set_Pr7585778909603769095tion_a,B5: set_Pr7585778909603769095tion_a] :
! [X2: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X2 @ A5 )
=> ( member5498148017924304208tion_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_183_subset__eq,axiom,
( ord_le5649184635953113020_a_nat
= ( ^ [A5: set_Pr8660066742269822812_a_nat,B5: set_Pr8660066742269822812_a_nat] :
! [X2: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X2 @ A5 )
=> ( member1430561565885860669_a_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_184_subset__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
! [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A5 )
=> ( member5724188588386418708_a_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_185_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A5: set_o,B5: set_o] :
! [X2: $o] :
( ( member_o @ X2 @ A5 )
=> ( member_o @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_186_equalityE,axiom,
! [A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le5649184635953113020_a_nat @ A2 @ B2 )
=> ~ ( ord_le5649184635953113020_a_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_187_equalityE,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ~ ( ord_le8666007276011122963_a_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_188_equalityE,axiom,
! [A2: set_o,B2: set_o] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_o @ A2 @ B2 )
=> ~ ( ord_less_eq_set_o @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_189_subsetD,axiom,
! [A2: set_Pr7585778909603769095tion_a,B2: set_Pr7585778909603769095tion_a,C: produc3509355604313844263tion_a] :
( ( ord_le4471550158292877991tion_a @ A2 @ B2 )
=> ( ( member5498148017924304208tion_a @ C @ A2 )
=> ( member5498148017924304208tion_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_190_subsetD,axiom,
! [A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat,C: produc4760857364844987814_a_nat] :
( ( ord_le5649184635953113020_a_nat @ A2 @ B2 )
=> ( ( member1430561565885860669_a_nat @ C @ A2 )
=> ( member1430561565885860669_a_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_191_subsetD,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( member5724188588386418708_a_nat @ C @ A2 )
=> ( member5724188588386418708_a_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_192_subsetD,axiom,
! [A2: set_o,B2: set_o,C: $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( member_o @ C @ A2 )
=> ( member_o @ C @ B2 ) ) ) ).
% subsetD
thf(fact_193_in__mono,axiom,
! [A2: set_Pr7585778909603769095tion_a,B2: set_Pr7585778909603769095tion_a,X: produc3509355604313844263tion_a] :
( ( ord_le4471550158292877991tion_a @ A2 @ B2 )
=> ( ( member5498148017924304208tion_a @ X @ A2 )
=> ( member5498148017924304208tion_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_194_in__mono,axiom,
! [A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat,X: produc4760857364844987814_a_nat] :
( ( ord_le5649184635953113020_a_nat @ A2 @ B2 )
=> ( ( member1430561565885860669_a_nat @ X @ A2 )
=> ( member1430561565885860669_a_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_195_in__mono,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( member5724188588386418708_a_nat @ X @ A2 )
=> ( member5724188588386418708_a_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_196_in__mono,axiom,
! [A2: set_o,B2: set_o,X: $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( member_o @ X @ A2 )
=> ( member_o @ X @ B2 ) ) ) ).
% in_mono
thf(fact_197_Compr__image__eq,axiom,
! [F: $o > $o,A2: set_o,P2: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_o_o @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_198_Compr__image__eq,axiom,
! [F: product_prod_a_nat > $o,A2: set_Pr4934435412358123699_a_nat,P2: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_3076334604395105300_nat_o @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_3076334604395105300_nat_o @ F
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_199_Compr__image__eq,axiom,
! [F: $o > product_prod_a_nat,A2: set_o,P2: product_prod_a_nat > $o] :
( ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ ( image_7339617019956244688_a_nat @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_7339617019956244688_a_nat @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_200_Compr__image__eq,axiom,
! [F: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a,P2: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_166006535297930294on_a_o @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_166006535297930294on_a_o @ F
@ ( collec4135126896892755346tion_a
@ ^ [X2: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_201_Compr__image__eq,axiom,
! [F: product_prod_a_nat > product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,P2: product_prod_a_nat > $o] :
( ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ ( image_8708701149064048897_a_nat @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_8708701149064048897_a_nat @ F
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_202_Compr__image__eq,axiom,
! [F: $o > produc3509355604313844263tion_a,A2: set_o,P2: produc3509355604313844263tion_a > $o] :
( ( collec4135126896892755346tion_a
@ ^ [X2: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X2 @ ( image_628002824077805460tion_a @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_628002824077805460tion_a @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_203_Compr__image__eq,axiom,
! [F: produc4760857364844987814_a_nat > $o,A2: set_Pr8660066742269822812_a_nat,P2: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_6768402879995552939_nat_o @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_6768402879995552939_nat_o @ F
@ ( collec7438131855006933115_a_nat
@ ^ [X2: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_204_Compr__image__eq,axiom,
! [F: produc3509355604313844263tion_a > product_prod_a_nat,A2: set_Pr7585778909603769095tion_a,P2: product_prod_a_nat > $o] :
( ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ ( image_2274939045924579167_a_nat @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_2274939045924579167_a_nat @ F
@ ( collec4135126896892755346tion_a
@ ^ [X2: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_205_Compr__image__eq,axiom,
! [F: $o > produc4760857364844987814_a_nat,A2: set_o,P2: produc4760857364844987814_a_nat > $o] :
( ( collec7438131855006933115_a_nat
@ ^ [X2: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X2 @ ( image_2232132205779913721_a_nat @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_2232132205779913721_a_nat @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_206_Compr__image__eq,axiom,
! [F: product_prod_a_nat > produc3509355604313844263tion_a,A2: set_Pr4934435412358123699_a_nat,P2: produc3509355604313844263tion_a > $o] :
( ( collec4135126896892755346tion_a
@ ^ [X2: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X2 @ ( image_5430222630607662243tion_a @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_5430222630607662243tion_a @ F
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_207_image__image,axiom,
! [F: $o > $o,G: $o > $o,A2: set_o] :
( ( image_o_o @ F @ ( image_o_o @ G @ A2 ) )
= ( image_o_o
@ ^ [X2: $o] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_208_image__image,axiom,
! [F: produc3509355604313844263tion_a > $o,G: $o > produc3509355604313844263tion_a,A2: set_o] :
( ( image_166006535297930294on_a_o @ F @ ( image_628002824077805460tion_a @ G @ A2 ) )
= ( image_o_o
@ ^ [X2: $o] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_209_image__image,axiom,
! [F: $o > $o,G: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a] :
( ( image_o_o @ F @ ( image_166006535297930294on_a_o @ G @ A2 ) )
= ( image_166006535297930294on_a_o
@ ^ [X2: produc3509355604313844263tion_a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_210_image__image,axiom,
! [F: produc4760857364844987814_a_nat > $o,G: $o > produc4760857364844987814_a_nat,A2: set_o] :
( ( image_6768402879995552939_nat_o @ F @ ( image_2232132205779913721_a_nat @ G @ A2 ) )
= ( image_o_o
@ ^ [X2: $o] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_211_image__image,axiom,
! [F: $o > $o,G: produc4760857364844987814_a_nat > $o,A2: set_Pr8660066742269822812_a_nat] :
( ( image_o_o @ F @ ( image_6768402879995552939_nat_o @ G @ A2 ) )
= ( image_6768402879995552939_nat_o
@ ^ [X2: produc4760857364844987814_a_nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_212_image__image,axiom,
! [F: produc3509355604313844263tion_a > $o,G: produc3509355604313844263tion_a > produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a] :
( ( image_166006535297930294on_a_o @ F @ ( image_6668519322520640453tion_a @ G @ A2 ) )
= ( image_166006535297930294on_a_o
@ ^ [X2: produc3509355604313844263tion_a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_213_image__image,axiom,
! [F: produc4760857364844987814_a_nat > $o,G: product_prod_a_nat > produc4760857364844987814_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( image_6768402879995552939_nat_o @ F @ ( image_3729436797865474090_a_nat @ G @ A2 ) )
= ( image_3076334604395105300_nat_o
@ ^ [X2: product_prod_a_nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_214_image__image,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,G: product_prod_a_nat > product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( image_3729436797865474090_a_nat @ F @ ( image_8708701149064048897_a_nat @ G @ A2 ) )
= ( image_3729436797865474090_a_nat
@ ^ [X2: product_prod_a_nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_215_image__image,axiom,
! [F: produc3509355604313844263tion_a > $o,G: produc4760857364844987814_a_nat > produc3509355604313844263tion_a,A2: set_Pr8660066742269822812_a_nat] :
( ( image_166006535297930294on_a_o @ F @ ( image_1391120077856136634tion_a @ G @ A2 ) )
= ( image_6768402879995552939_nat_o
@ ^ [X2: produc4760857364844987814_a_nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_216_image__image,axiom,
! [F: produc4760857364844987814_a_nat > $o,G: produc3509355604313844263tion_a > produc4760857364844987814_a_nat,A2: set_Pr7585778909603769095tion_a] :
( ( image_6768402879995552939_nat_o @ F @ ( image_8907095238652253832_a_nat @ G @ A2 ) )
= ( image_166006535297930294on_a_o
@ ^ [X2: produc3509355604313844263tion_a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_217_imageE,axiom,
! [B: $o,F: $o > $o,A2: set_o] :
( ( member_o @ B @ ( image_o_o @ F @ A2 ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_218_imageE,axiom,
! [B: $o,F: product_prod_a_nat > $o,A2: set_Pr4934435412358123699_a_nat] :
( ( member_o @ B @ ( image_3076334604395105300_nat_o @ F @ A2 ) )
=> ~ ! [X3: product_prod_a_nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member5724188588386418708_a_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_219_imageE,axiom,
! [B: product_prod_a_nat,F: $o > product_prod_a_nat,A2: set_o] :
( ( member5724188588386418708_a_nat @ B @ ( image_7339617019956244688_a_nat @ F @ A2 ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_220_imageE,axiom,
! [B: $o,F: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a] :
( ( member_o @ B @ ( image_166006535297930294on_a_o @ F @ A2 ) )
=> ~ ! [X3: produc3509355604313844263tion_a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member5498148017924304208tion_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_221_imageE,axiom,
! [B: product_prod_a_nat,F: product_prod_a_nat > product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ B @ ( image_8708701149064048897_a_nat @ F @ A2 ) )
=> ~ ! [X3: product_prod_a_nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member5724188588386418708_a_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_222_imageE,axiom,
! [B: produc3509355604313844263tion_a,F: $o > produc3509355604313844263tion_a,A2: set_o] :
( ( member5498148017924304208tion_a @ B @ ( image_628002824077805460tion_a @ F @ A2 ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_223_imageE,axiom,
! [B: $o,F: produc4760857364844987814_a_nat > $o,A2: set_Pr8660066742269822812_a_nat] :
( ( member_o @ B @ ( image_6768402879995552939_nat_o @ F @ A2 ) )
=> ~ ! [X3: produc4760857364844987814_a_nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member1430561565885860669_a_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_224_imageE,axiom,
! [B: product_prod_a_nat,F: produc3509355604313844263tion_a > product_prod_a_nat,A2: set_Pr7585778909603769095tion_a] :
( ( member5724188588386418708_a_nat @ B @ ( image_2274939045924579167_a_nat @ F @ A2 ) )
=> ~ ! [X3: produc3509355604313844263tion_a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member5498148017924304208tion_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_225_imageE,axiom,
! [B: produc4760857364844987814_a_nat,F: $o > produc4760857364844987814_a_nat,A2: set_o] :
( ( member1430561565885860669_a_nat @ B @ ( image_2232132205779913721_a_nat @ F @ A2 ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_226_imageE,axiom,
! [B: produc3509355604313844263tion_a,F: product_prod_a_nat > produc3509355604313844263tion_a,A2: set_Pr4934435412358123699_a_nat] :
( ( member5498148017924304208tion_a @ B @ ( image_5430222630607662243tion_a @ F @ A2 ) )
=> ~ ! [X3: product_prod_a_nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member5724188588386418708_a_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_227_Collect__subset,axiom,
! [A2: set_Pr7585778909603769095tion_a,P2: produc3509355604313844263tion_a > $o] :
( ord_le4471550158292877991tion_a
@ ( collec4135126896892755346tion_a
@ ^ [X2: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X2 @ A2 )
& ( P2 @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_228_Collect__subset,axiom,
! [A2: set_Pr8660066742269822812_a_nat,P2: produc4760857364844987814_a_nat > $o] :
( ord_le5649184635953113020_a_nat
@ ( collec7438131855006933115_a_nat
@ ^ [X2: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X2 @ A2 )
& ( P2 @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_229_Collect__subset,axiom,
! [A2: set_Pr4934435412358123699_a_nat,P2: product_prod_a_nat > $o] :
( ord_le8666007276011122963_a_nat
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A2 )
& ( P2 @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_230_Collect__subset,axiom,
! [A2: set_o,P2: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P2 @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_231_prod_Ocase__distrib,axiom,
! [H: $o > $o,F: option_a > option_a > $o,Prod: produc3509355604313844263tion_a] :
( ( H @ ( produc7248784972643495020on_a_o @ F @ Prod ) )
= ( produc7248784972643495020on_a_o
@ ^ [X1: option_a,X23: option_a] : ( H @ ( F @ X1 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_232_prod_Ocase__distrib,axiom,
! [H: $o > option_a,F: option_a > option_a > $o,Prod: produc3509355604313844263tion_a] :
( ( H @ ( produc7248784972643495020on_a_o @ F @ Prod ) )
= ( produc6953443519211250200tion_a
@ ^ [X1: option_a,X23: option_a] : ( H @ ( F @ X1 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_233_prod_Ocase__distrib,axiom,
! [H: option_a > $o,F: option_a > option_a > option_a,Prod: produc3509355604313844263tion_a] :
( ( H @ ( produc6953443519211250200tion_a @ F @ Prod ) )
= ( produc7248784972643495020on_a_o
@ ^ [X1: option_a,X23: option_a] : ( H @ ( F @ X1 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_234_prod_Ocase__distrib,axiom,
! [H: $o > $o,F: produc3509355604313844263tion_a > nat > $o,Prod: produc4760857364844987814_a_nat] :
( ( H @ ( produc7988141324749014923_nat_o @ F @ Prod ) )
= ( produc7988141324749014923_nat_o
@ ^ [X1: produc3509355604313844263tion_a,X23: nat] : ( H @ ( F @ X1 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_235_prod_Ocase__distrib,axiom,
! [H: option_a > option_a,F: option_a > option_a > option_a,Prod: produc3509355604313844263tion_a] :
( ( H @ ( produc6953443519211250200tion_a @ F @ Prod ) )
= ( produc6953443519211250200tion_a
@ ^ [X1: option_a,X23: option_a] : ( H @ ( F @ X1 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_236_prod_Ocase__distrib,axiom,
! [H: $o > produc3509355604313844263tion_a,F: option_a > option_a > $o,Prod: produc3509355604313844263tion_a] :
( ( H @ ( produc7248784972643495020on_a_o @ F @ Prod ) )
= ( produc2781663484249587963tion_a
@ ^ [X1: option_a,X23: option_a] : ( H @ ( F @ X1 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_237_prod_Ocase__distrib,axiom,
! [H: produc3509355604313844263tion_a > $o,F: option_a > option_a > produc3509355604313844263tion_a,Prod: produc3509355604313844263tion_a] :
( ( H @ ( produc2781663484249587963tion_a @ F @ Prod ) )
= ( produc7248784972643495020on_a_o
@ ^ [X1: option_a,X23: option_a] : ( H @ ( F @ X1 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_238_prod_Ocase__distrib,axiom,
! [H: produc3509355604313844263tion_a > option_a,F: option_a > option_a > produc3509355604313844263tion_a,Prod: produc3509355604313844263tion_a] :
( ( H @ ( produc2781663484249587963tion_a @ F @ Prod ) )
= ( produc6953443519211250200tion_a
@ ^ [X1: option_a,X23: option_a] : ( H @ ( F @ X1 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_239_prod_Ocase__distrib,axiom,
! [H: option_a > produc3509355604313844263tion_a,F: option_a > option_a > option_a,Prod: produc3509355604313844263tion_a] :
( ( H @ ( produc6953443519211250200tion_a @ F @ Prod ) )
= ( produc2781663484249587963tion_a
@ ^ [X1: option_a,X23: option_a] : ( H @ ( F @ X1 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_240_prod_Ocase__distrib,axiom,
! [H: $o > produc4760857364844987814_a_nat,F: produc3509355604313844263tion_a > nat > $o,Prod: produc4760857364844987814_a_nat] :
( ( H @ ( produc7988141324749014923_nat_o @ F @ Prod ) )
= ( produc5088488592959153011_a_nat
@ ^ [X1: produc3509355604313844263tion_a,X23: nat] : ( H @ ( F @ X1 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_241_subset__image__iff,axiom,
! [B2: set_Pr8660066742269822812_a_nat,F: produc4760857364844987814_a_nat > produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ B2 @ ( image_4137284907621822035_a_nat @ F @ A2 ) )
= ( ? [AA: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ AA @ A2 )
& ( B2
= ( image_4137284907621822035_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_242_subset__image__iff,axiom,
! [B2: set_Pr8660066742269822812_a_nat,F: product_prod_a_nat > produc4760857364844987814_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le5649184635953113020_a_nat @ B2 @ ( image_3729436797865474090_a_nat @ F @ A2 ) )
= ( ? [AA: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ AA @ A2 )
& ( B2
= ( image_3729436797865474090_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_243_subset__image__iff,axiom,
! [B2: set_Pr8660066742269822812_a_nat,F: $o > produc4760857364844987814_a_nat,A2: set_o] :
( ( ord_le5649184635953113020_a_nat @ B2 @ ( image_2232132205779913721_a_nat @ F @ A2 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A2 )
& ( B2
= ( image_2232132205779913721_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_244_subset__image__iff,axiom,
! [B2: set_Pr4934435412358123699_a_nat,F: produc4760857364844987814_a_nat > product_prod_a_nat,A2: set_Pr8660066742269822812_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ ( image_6737578356684938026_a_nat @ F @ A2 ) )
= ( ? [AA: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ AA @ A2 )
& ( B2
= ( image_6737578356684938026_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_245_subset__image__iff,axiom,
! [B2: set_Pr4934435412358123699_a_nat,F: product_prod_a_nat > product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ ( image_8708701149064048897_a_nat @ F @ A2 ) )
= ( ? [AA: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ AA @ A2 )
& ( B2
= ( image_8708701149064048897_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_246_subset__image__iff,axiom,
! [B2: set_Pr4934435412358123699_a_nat,F: $o > product_prod_a_nat,A2: set_o] :
( ( ord_le8666007276011122963_a_nat @ B2 @ ( image_7339617019956244688_a_nat @ F @ A2 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A2 )
& ( B2
= ( image_7339617019956244688_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_247_subset__image__iff,axiom,
! [B2: set_o,F: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a] :
( ( ord_less_eq_set_o @ B2 @ ( image_166006535297930294on_a_o @ F @ A2 ) )
= ( ? [AA: set_Pr7585778909603769095tion_a] :
( ( ord_le4471550158292877991tion_a @ AA @ A2 )
& ( B2
= ( image_166006535297930294on_a_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_248_subset__image__iff,axiom,
! [B2: set_o,F: produc4760857364844987814_a_nat > $o,A2: set_Pr8660066742269822812_a_nat] :
( ( ord_less_eq_set_o @ B2 @ ( image_6768402879995552939_nat_o @ F @ A2 ) )
= ( ? [AA: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ AA @ A2 )
& ( B2
= ( image_6768402879995552939_nat_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_249_subset__image__iff,axiom,
! [B2: set_o,F: product_prod_a_nat > $o,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_set_o @ B2 @ ( image_3076334604395105300_nat_o @ F @ A2 ) )
= ( ? [AA: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ AA @ A2 )
& ( B2
= ( image_3076334604395105300_nat_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_250_subset__image__iff,axiom,
! [B2: set_o,F: $o > $o,A2: set_o] :
( ( ord_less_eq_set_o @ B2 @ ( image_o_o @ F @ A2 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A2 )
& ( B2
= ( image_o_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_251_image__subset__iff,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ ( image_3729436797865474090_a_nat @ F @ A2 ) @ B2 )
= ( ! [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A2 )
=> ( member1430561565885860669_a_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_252_image__subset__iff,axiom,
! [F: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a,B2: set_o] :
( ( ord_less_eq_set_o @ ( image_166006535297930294on_a_o @ F @ A2 ) @ B2 )
= ( ! [X2: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X2 @ A2 )
=> ( member_o @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_253_image__subset__iff,axiom,
! [F: produc4760857364844987814_a_nat > $o,A2: set_Pr8660066742269822812_a_nat,B2: set_o] :
( ( ord_less_eq_set_o @ ( image_6768402879995552939_nat_o @ F @ A2 ) @ B2 )
= ( ! [X2: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X2 @ A2 )
=> ( member_o @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_254_image__subset__iff,axiom,
! [F: $o > $o,A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B2 )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( member_o @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_255_subset__imageE,axiom,
! [B2: set_Pr8660066742269822812_a_nat,F: produc4760857364844987814_a_nat > produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ B2 @ ( image_4137284907621822035_a_nat @ F @ A2 ) )
=> ~ ! [C3: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ C3 @ A2 )
=> ( B2
!= ( image_4137284907621822035_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_256_subset__imageE,axiom,
! [B2: set_Pr8660066742269822812_a_nat,F: product_prod_a_nat > produc4760857364844987814_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le5649184635953113020_a_nat @ B2 @ ( image_3729436797865474090_a_nat @ F @ A2 ) )
=> ~ ! [C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C3 @ A2 )
=> ( B2
!= ( image_3729436797865474090_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_257_subset__imageE,axiom,
! [B2: set_Pr8660066742269822812_a_nat,F: $o > produc4760857364844987814_a_nat,A2: set_o] :
( ( ord_le5649184635953113020_a_nat @ B2 @ ( image_2232132205779913721_a_nat @ F @ A2 ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A2 )
=> ( B2
!= ( image_2232132205779913721_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_258_subset__imageE,axiom,
! [B2: set_Pr4934435412358123699_a_nat,F: produc4760857364844987814_a_nat > product_prod_a_nat,A2: set_Pr8660066742269822812_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ ( image_6737578356684938026_a_nat @ F @ A2 ) )
=> ~ ! [C3: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ C3 @ A2 )
=> ( B2
!= ( image_6737578356684938026_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_259_subset__imageE,axiom,
! [B2: set_Pr4934435412358123699_a_nat,F: product_prod_a_nat > product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ ( image_8708701149064048897_a_nat @ F @ A2 ) )
=> ~ ! [C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C3 @ A2 )
=> ( B2
!= ( image_8708701149064048897_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_260_subset__imageE,axiom,
! [B2: set_Pr4934435412358123699_a_nat,F: $o > product_prod_a_nat,A2: set_o] :
( ( ord_le8666007276011122963_a_nat @ B2 @ ( image_7339617019956244688_a_nat @ F @ A2 ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A2 )
=> ( B2
!= ( image_7339617019956244688_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_261_subset__imageE,axiom,
! [B2: set_o,F: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a] :
( ( ord_less_eq_set_o @ B2 @ ( image_166006535297930294on_a_o @ F @ A2 ) )
=> ~ ! [C3: set_Pr7585778909603769095tion_a] :
( ( ord_le4471550158292877991tion_a @ C3 @ A2 )
=> ( B2
!= ( image_166006535297930294on_a_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_262_subset__imageE,axiom,
! [B2: set_o,F: produc4760857364844987814_a_nat > $o,A2: set_Pr8660066742269822812_a_nat] :
( ( ord_less_eq_set_o @ B2 @ ( image_6768402879995552939_nat_o @ F @ A2 ) )
=> ~ ! [C3: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ C3 @ A2 )
=> ( B2
!= ( image_6768402879995552939_nat_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_263_subset__imageE,axiom,
! [B2: set_o,F: product_prod_a_nat > $o,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_set_o @ B2 @ ( image_3076334604395105300_nat_o @ F @ A2 ) )
=> ~ ! [C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C3 @ A2 )
=> ( B2
!= ( image_3076334604395105300_nat_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_264_subset__imageE,axiom,
! [B2: set_o,F: $o > $o,A2: set_o] :
( ( ord_less_eq_set_o @ B2 @ ( image_o_o @ F @ A2 ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A2 )
=> ( B2
!= ( image_o_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_265_image__subsetI,axiom,
! [A2: set_o,F: $o > $o,B2: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_266_image__subsetI,axiom,
! [A2: set_o,F: $o > product_prod_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member5724188588386418708_a_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le8666007276011122963_a_nat @ ( image_7339617019956244688_a_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_267_image__subsetI,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F: product_prod_a_nat > $o,B2: set_o] :
( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A2 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_3076334604395105300_nat_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_268_image__subsetI,axiom,
! [A2: set_o,F: $o > produc3509355604313844263tion_a,B2: set_Pr7585778909603769095tion_a] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member5498148017924304208tion_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_le4471550158292877991tion_a @ ( image_628002824077805460tion_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_269_image__subsetI,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F: product_prod_a_nat > product_prod_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A2 )
=> ( member5724188588386418708_a_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le8666007276011122963_a_nat @ ( image_8708701149064048897_a_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_270_image__subsetI,axiom,
! [A2: set_Pr7585778909603769095tion_a,F: produc3509355604313844263tion_a > $o,B2: set_o] :
( ! [X3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X3 @ A2 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_166006535297930294on_a_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_271_image__subsetI,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F: product_prod_a_nat > produc3509355604313844263tion_a,B2: set_Pr7585778909603769095tion_a] :
( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A2 )
=> ( member5498148017924304208tion_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_le4471550158292877991tion_a @ ( image_5430222630607662243tion_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_272_image__subsetI,axiom,
! [A2: set_o,F: $o > produc4760857364844987814_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member1430561565885860669_a_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le5649184635953113020_a_nat @ ( image_2232132205779913721_a_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_273_image__subsetI,axiom,
! [A2: set_Pr7585778909603769095tion_a,F: produc3509355604313844263tion_a > product_prod_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ! [X3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X3 @ A2 )
=> ( member5724188588386418708_a_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le8666007276011122963_a_nat @ ( image_2274939045924579167_a_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_274_image__subsetI,axiom,
! [A2: set_Pr8660066742269822812_a_nat,F: produc4760857364844987814_a_nat > $o,B2: set_o] :
( ! [X3: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X3 @ A2 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_6768402879995552939_nat_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_275_image__mono,axiom,
! [A2: set_Pr7585778909603769095tion_a,B2: set_Pr7585778909603769095tion_a,F: produc3509355604313844263tion_a > $o] :
( ( ord_le4471550158292877991tion_a @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( image_166006535297930294on_a_o @ F @ A2 ) @ ( image_166006535297930294on_a_o @ F @ B2 ) ) ) ).
% image_mono
thf(fact_276_image__mono,axiom,
! [A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat,F: produc4760857364844987814_a_nat > produc4760857364844987814_a_nat] :
( ( ord_le5649184635953113020_a_nat @ A2 @ B2 )
=> ( ord_le5649184635953113020_a_nat @ ( image_4137284907621822035_a_nat @ F @ A2 ) @ ( image_4137284907621822035_a_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_277_image__mono,axiom,
! [A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat,F: produc4760857364844987814_a_nat > product_prod_a_nat] :
( ( ord_le5649184635953113020_a_nat @ A2 @ B2 )
=> ( ord_le8666007276011122963_a_nat @ ( image_6737578356684938026_a_nat @ F @ A2 ) @ ( image_6737578356684938026_a_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_278_image__mono,axiom,
! [A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat,F: produc4760857364844987814_a_nat > $o] :
( ( ord_le5649184635953113020_a_nat @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( image_6768402879995552939_nat_o @ F @ A2 ) @ ( image_6768402879995552939_nat_o @ F @ B2 ) ) ) ).
% image_mono
thf(fact_279_image__mono,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F: product_prod_a_nat > produc4760857364844987814_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ord_le5649184635953113020_a_nat @ ( image_3729436797865474090_a_nat @ F @ A2 ) @ ( image_3729436797865474090_a_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_280_image__mono,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F: product_prod_a_nat > product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ord_le8666007276011122963_a_nat @ ( image_8708701149064048897_a_nat @ F @ A2 ) @ ( image_8708701149064048897_a_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_281_image__mono,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( image_3076334604395105300_nat_o @ F @ A2 ) @ ( image_3076334604395105300_nat_o @ F @ B2 ) ) ) ).
% image_mono
thf(fact_282_image__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > produc4760857364844987814_a_nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_le5649184635953113020_a_nat @ ( image_2232132205779913721_a_nat @ F @ A2 ) @ ( image_2232132205779913721_a_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_283_image__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > product_prod_a_nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_le8666007276011122963_a_nat @ ( image_7339617019956244688_a_nat @ F @ A2 ) @ ( image_7339617019956244688_a_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_284_image__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ ( image_o_o @ F @ B2 ) ) ) ).
% image_mono
thf(fact_285_old_Oprod_Ocase,axiom,
! [F: a > nat > produc4760857364844987814_a_nat,X12: a,X22: nat] :
( ( produc4004610663780792650_a_nat @ F @ ( product_Pair_a_nat @ X12 @ X22 ) )
= ( F @ X12 @ X22 ) ) ).
% old.prod.case
thf(fact_286_old_Oprod_Ocase,axiom,
! [F: option_a > option_a > $o,X12: option_a,X22: option_a] :
( ( produc7248784972643495020on_a_o @ F @ ( produc9011544418120257559tion_a @ X12 @ X22 ) )
= ( F @ X12 @ X22 ) ) ).
% old.prod.case
thf(fact_287_old_Oprod_Ocase,axiom,
! [F: produc3509355604313844263tion_a > nat > $o,X12: produc3509355604313844263tion_a,X22: nat] :
( ( produc7988141324749014923_nat_o @ F @ ( produc5457283509376734880_a_nat @ X12 @ X22 ) )
= ( F @ X12 @ X22 ) ) ).
% old.prod.case
thf(fact_288_old_Oprod_Ocase,axiom,
! [F: option_a > option_a > produc3509355604313844263tion_a,X12: option_a,X22: option_a] :
( ( produc2781663484249587963tion_a @ F @ ( produc9011544418120257559tion_a @ X12 @ X22 ) )
= ( F @ X12 @ X22 ) ) ).
% old.prod.case
thf(fact_289_old_Oprod_Ocase,axiom,
! [F: produc3509355604313844263tion_a > nat > produc4760857364844987814_a_nat,X12: produc3509355604313844263tion_a,X22: nat] :
( ( produc5088488592959153011_a_nat @ F @ ( produc5457283509376734880_a_nat @ X12 @ X22 ) )
= ( F @ X12 @ X22 ) ) ).
% old.prod.case
thf(fact_290_old_Oprod_Ocase,axiom,
! [F: option_a > option_a > option_a,X12: option_a,X22: option_a] :
( ( produc6953443519211250200tion_a @ F @ ( produc9011544418120257559tion_a @ X12 @ X22 ) )
= ( F @ X12 @ X22 ) ) ).
% old.prod.case
thf(fact_291_fst__conv,axiom,
! [X12: produc3509355604313844263tion_a,X22: nat] :
( ( produc1094642028913761716_a_nat @ ( produc5457283509376734880_a_nat @ X12 @ X22 ) )
= X12 ) ).
% fst_conv
thf(fact_292_fst__conv,axiom,
! [X12: a,X22: nat] :
( ( product_fst_a_nat @ ( product_Pair_a_nat @ X12 @ X22 ) )
= X12 ) ).
% fst_conv
thf(fact_293_fst__conv,axiom,
! [X12: option_a,X22: option_a] :
( ( produc174239612569448963tion_a @ ( produc9011544418120257559tion_a @ X12 @ X22 ) )
= X12 ) ).
% fst_conv
thf(fact_294_fst__eqD,axiom,
! [X: produc3509355604313844263tion_a,Y: nat,A: produc3509355604313844263tion_a] :
( ( ( produc1094642028913761716_a_nat @ ( produc5457283509376734880_a_nat @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_295_fst__eqD,axiom,
! [X: a,Y: nat,A: a] :
( ( ( product_fst_a_nat @ ( product_Pair_a_nat @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_296_fst__eqD,axiom,
! [X: option_a,Y: option_a,A: option_a] :
( ( ( produc174239612569448963tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_297_option_Osel,axiom,
! [X22: a] :
( ( the_a @ ( some_a @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_298_option_Osel,axiom,
! [X22: set_Pr4934435412358123699_a_nat] :
( ( the_se1682649557849046815_a_nat @ ( some_s1880800141159441806_a_nat @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_299_option_Osel,axiom,
! [X22: set_Pr8660066742269822812_a_nat] :
( ( the_se5449783460274439624_a_nat @ ( some_s2314368709036572727_a_nat @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_300_cond__case__prod__eta,axiom,
! [F: a > nat > produc4760857364844987814_a_nat,G: product_prod_a_nat > produc4760857364844987814_a_nat] :
( ! [X3: a,Y3: nat] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair_a_nat @ X3 @ Y3 ) ) )
=> ( ( produc4004610663780792650_a_nat @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_301_cond__case__prod__eta,axiom,
! [F: option_a > option_a > $o,G: produc3509355604313844263tion_a > $o] :
( ! [X3: option_a,Y3: option_a] :
( ( F @ X3 @ Y3 )
= ( G @ ( produc9011544418120257559tion_a @ X3 @ Y3 ) ) )
=> ( ( produc7248784972643495020on_a_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_302_cond__case__prod__eta,axiom,
! [F: produc3509355604313844263tion_a > nat > $o,G: produc4760857364844987814_a_nat > $o] :
( ! [X3: produc3509355604313844263tion_a,Y3: nat] :
( ( F @ X3 @ Y3 )
= ( G @ ( produc5457283509376734880_a_nat @ X3 @ Y3 ) ) )
=> ( ( produc7988141324749014923_nat_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_303_cond__case__prod__eta,axiom,
! [F: option_a > option_a > produc3509355604313844263tion_a,G: produc3509355604313844263tion_a > produc3509355604313844263tion_a] :
( ! [X3: option_a,Y3: option_a] :
( ( F @ X3 @ Y3 )
= ( G @ ( produc9011544418120257559tion_a @ X3 @ Y3 ) ) )
=> ( ( produc2781663484249587963tion_a @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_304_cond__case__prod__eta,axiom,
! [F: produc3509355604313844263tion_a > nat > produc4760857364844987814_a_nat,G: produc4760857364844987814_a_nat > produc4760857364844987814_a_nat] :
( ! [X3: produc3509355604313844263tion_a,Y3: nat] :
( ( F @ X3 @ Y3 )
= ( G @ ( produc5457283509376734880_a_nat @ X3 @ Y3 ) ) )
=> ( ( produc5088488592959153011_a_nat @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_305_cond__case__prod__eta,axiom,
! [F: option_a > option_a > option_a,G: produc3509355604313844263tion_a > option_a] :
( ! [X3: option_a,Y3: option_a] :
( ( F @ X3 @ Y3 )
= ( G @ ( produc9011544418120257559tion_a @ X3 @ Y3 ) ) )
=> ( ( produc6953443519211250200tion_a @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_306_case__prod__eta,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat] :
( ( produc4004610663780792650_a_nat
@ ^ [X2: a,Y5: nat] : ( F @ ( product_Pair_a_nat @ X2 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_307_case__prod__eta,axiom,
! [F: produc3509355604313844263tion_a > $o] :
( ( produc7248784972643495020on_a_o
@ ^ [X2: option_a,Y5: option_a] : ( F @ ( produc9011544418120257559tion_a @ X2 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_308_case__prod__eta,axiom,
! [F: produc4760857364844987814_a_nat > $o] :
( ( produc7988141324749014923_nat_o
@ ^ [X2: produc3509355604313844263tion_a,Y5: nat] : ( F @ ( produc5457283509376734880_a_nat @ X2 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_309_case__prod__eta,axiom,
! [F: produc3509355604313844263tion_a > produc3509355604313844263tion_a] :
( ( produc2781663484249587963tion_a
@ ^ [X2: option_a,Y5: option_a] : ( F @ ( produc9011544418120257559tion_a @ X2 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_310_case__prod__eta,axiom,
! [F: produc4760857364844987814_a_nat > produc4760857364844987814_a_nat] :
( ( produc5088488592959153011_a_nat
@ ^ [X2: produc3509355604313844263tion_a,Y5: nat] : ( F @ ( produc5457283509376734880_a_nat @ X2 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_311_case__prod__eta,axiom,
! [F: produc3509355604313844263tion_a > option_a] :
( ( produc6953443519211250200tion_a
@ ^ [X2: option_a,Y5: option_a] : ( F @ ( produc9011544418120257559tion_a @ X2 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_312_case__prodE2,axiom,
! [Q: produc4760857364844987814_a_nat > $o,P2: a > nat > produc4760857364844987814_a_nat,Z: product_prod_a_nat] :
( ( Q @ ( produc4004610663780792650_a_nat @ P2 @ Z ) )
=> ~ ! [X3: a,Y3: nat] :
( ( Z
= ( product_Pair_a_nat @ X3 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_313_case__prodE2,axiom,
! [Q: $o > $o,P2: option_a > option_a > $o,Z: produc3509355604313844263tion_a] :
( ( Q @ ( produc7248784972643495020on_a_o @ P2 @ Z ) )
=> ~ ! [X3: option_a,Y3: option_a] :
( ( Z
= ( produc9011544418120257559tion_a @ X3 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_314_case__prodE2,axiom,
! [Q: $o > $o,P2: produc3509355604313844263tion_a > nat > $o,Z: produc4760857364844987814_a_nat] :
( ( Q @ ( produc7988141324749014923_nat_o @ P2 @ Z ) )
=> ~ ! [X3: produc3509355604313844263tion_a,Y3: nat] :
( ( Z
= ( produc5457283509376734880_a_nat @ X3 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_315_case__prodE2,axiom,
! [Q: produc3509355604313844263tion_a > $o,P2: option_a > option_a > produc3509355604313844263tion_a,Z: produc3509355604313844263tion_a] :
( ( Q @ ( produc2781663484249587963tion_a @ P2 @ Z ) )
=> ~ ! [X3: option_a,Y3: option_a] :
( ( Z
= ( produc9011544418120257559tion_a @ X3 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_316_case__prodE2,axiom,
! [Q: produc4760857364844987814_a_nat > $o,P2: produc3509355604313844263tion_a > nat > produc4760857364844987814_a_nat,Z: produc4760857364844987814_a_nat] :
( ( Q @ ( produc5088488592959153011_a_nat @ P2 @ Z ) )
=> ~ ! [X3: produc3509355604313844263tion_a,Y3: nat] :
( ( Z
= ( produc5457283509376734880_a_nat @ X3 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_317_case__prodE2,axiom,
! [Q: option_a > $o,P2: option_a > option_a > option_a,Z: produc3509355604313844263tion_a] :
( ( Q @ ( produc6953443519211250200tion_a @ P2 @ Z ) )
=> ~ ! [X3: option_a,Y3: option_a] :
( ( Z
= ( produc9011544418120257559tion_a @ X3 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_318_dual__order_Orefl,axiom,
! [A: set_Pr8660066742269822812_a_nat] : ( ord_le5649184635953113020_a_nat @ A @ A ) ).
% dual_order.refl
thf(fact_319_dual__order_Orefl,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ A ) ).
% dual_order.refl
thf(fact_320_dual__order_Orefl,axiom,
! [A: option5987904771724668803_a_nat] : ( ord_le1496697172603502563_a_nat @ A @ A ) ).
% dual_order.refl
thf(fact_321_dual__order_Orefl,axiom,
! [A: option2973356794535198124_a_nat] : ( ord_le8848857416346193420_a_nat @ A @ A ) ).
% dual_order.refl
thf(fact_322_dual__order_Orefl,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ A ) ).
% dual_order.refl
thf(fact_323_order__refl,axiom,
! [X: set_Pr8660066742269822812_a_nat] : ( ord_le5649184635953113020_a_nat @ X @ X ) ).
% order_refl
thf(fact_324_order__refl,axiom,
! [X: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ X ) ).
% order_refl
thf(fact_325_order__refl,axiom,
! [X: option5987904771724668803_a_nat] : ( ord_le1496697172603502563_a_nat @ X @ X ) ).
% order_refl
thf(fact_326_order__refl,axiom,
! [X: option2973356794535198124_a_nat] : ( ord_le8848857416346193420_a_nat @ X @ X ) ).
% order_refl
thf(fact_327_order__refl,axiom,
! [X: set_o] : ( ord_less_eq_set_o @ X @ X ) ).
% order_refl
thf(fact_328_pred__subset__eq2,axiom,
! [R: set_Pr7585778909603769095tion_a,S: set_Pr7585778909603769095tion_a] :
( ( ord_le3058241657886854542on_a_o
@ ^ [X2: option_a,Y5: option_a] : ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y5 ) @ R )
@ ^ [X2: option_a,Y5: option_a] : ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y5 ) @ S ) )
= ( ord_le4471550158292877991tion_a @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_329_pred__subset__eq2,axiom,
! [R: set_Pr8660066742269822812_a_nat,S: set_Pr8660066742269822812_a_nat] :
( ( ord_le489641559516061149_nat_o
@ ^ [X2: produc3509355604313844263tion_a,Y5: nat] : ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ X2 @ Y5 ) @ R )
@ ^ [X2: produc3509355604313844263tion_a,Y5: nat] : ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ X2 @ Y5 ) @ S ) )
= ( ord_le5649184635953113020_a_nat @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_330_pred__subset__eq2,axiom,
! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_a_nat_o
@ ^ [X2: a,Y5: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y5 ) @ R )
@ ^ [X2: a,Y5: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y5 ) @ S ) )
= ( ord_le8666007276011122963_a_nat @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_331_image__Collect__subsetI,axiom,
! [P2: product_prod_a_nat > $o,F: product_prod_a_nat > produc4760857364844987814_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ! [X3: product_prod_a_nat] :
( ( P2 @ X3 )
=> ( member1430561565885860669_a_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le5649184635953113020_a_nat @ ( image_3729436797865474090_a_nat @ F @ ( collec4464134535221767506_a_nat @ P2 ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_332_image__Collect__subsetI,axiom,
! [P2: produc3509355604313844263tion_a > $o,F: produc3509355604313844263tion_a > $o,B2: set_o] :
( ! [X3: produc3509355604313844263tion_a] :
( ( P2 @ X3 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_166006535297930294on_a_o @ F @ ( collec4135126896892755346tion_a @ P2 ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_333_image__Collect__subsetI,axiom,
! [P2: produc4760857364844987814_a_nat > $o,F: produc4760857364844987814_a_nat > $o,B2: set_o] :
( ! [X3: produc4760857364844987814_a_nat] :
( ( P2 @ X3 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_6768402879995552939_nat_o @ F @ ( collec7438131855006933115_a_nat @ P2 ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_334_image__Collect__subsetI,axiom,
! [P2: $o > $o,F: $o > $o,B2: set_o] :
( ! [X3: $o] :
( ( P2 @ X3 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ ( collect_o @ P2 ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_335_RR2__gctxt__unlift__lift__id,axiom,
! [C2: ground563114222783697949_a_nat,F2: set_Pr5612011003857075140at_nat] :
( ( ord_le8609766837066453116at_nat @ ( ground1088615816849950403_a_nat @ C2 )
@ ( image_6008886954644801541at_nat
@ ( produc8166206427995057559at_nat
@ ^ [F3: set_Pr4934435412358123699_a_nat] : ( produc6380145652274861664at_nat @ ( produc1207969925626950559_a_nat @ ( some_s1880800141159441806_a_nat @ F3 ) @ ( some_s1880800141159441806_a_nat @ F3 ) ) ) )
@ F2 ) )
=> ( ( ground8611692079647855452_a_nat
@ ( comp_s467579784663650262_a_nat
@ ^ [F3: set_Pr4934435412358123699_a_nat] : ( produc1207969925626950559_a_nat @ ( some_s1880800141159441806_a_nat @ F3 ) @ ( some_s1880800141159441806_a_nat @ F3 ) )
@ ^ [F3: produc7618478223271187943_a_nat] : ( the_se1682649557849046815_a_nat @ ( produc735037054355678091_a_nat @ F3 ) ) )
@ C2 )
= C2 ) ) ).
% RR2_gctxt_unlift_lift_id
thf(fact_336_RR2__gctxt__unlift__lift__id,axiom,
! [C2: ground6218218001256449135_a_nat,F2: set_Pr2737808803024040429at_nat] :
( ( ord_le8394821662120075470at_nat @ ( ground2086754257399109653_a_nat @ C2 )
@ ( image_1902767331843210624at_nat
@ ( produc1847350529710776082at_nat
@ ^ [F3: set_Pr8660066742269822812_a_nat] : ( produc2385804702627905458at_nat @ ( produc8861491750707134833_a_nat @ ( some_s2314368709036572727_a_nat @ F3 ) @ ( some_s2314368709036572727_a_nat @ F3 ) ) ) )
@ F2 ) )
=> ( ( ground4384087302803801820_a_nat
@ ( comp_s8729298985798543999_a_nat
@ ^ [F3: set_Pr8660066742269822812_a_nat] : ( produc8861491750707134833_a_nat @ ( some_s2314368709036572727_a_nat @ F3 ) @ ( some_s2314368709036572727_a_nat @ F3 ) )
@ ^ [F3: produc4107345600652478265_a_nat] : ( the_se5449783460274439624_a_nat @ ( produc4143784723386224989_a_nat @ F3 ) ) )
@ C2 )
= C2 ) ) ).
% RR2_gctxt_unlift_lift_id
thf(fact_337_RR2__gctxt__unlift__lift__id,axiom,
! [C2: ground4599112814892907741tion_a,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le5649184635953113020_a_nat @ ( ground3966248510942147843tion_a @ C2 )
@ ( image_3729436797865474090_a_nat
@ ( produc4004610663780792650_a_nat
@ ^ [F3: a] : ( produc5457283509376734880_a_nat @ ( produc9011544418120257559tion_a @ ( some_a @ F3 ) @ ( some_a @ F3 ) ) ) )
@ F2 ) )
=> ( ( ground7092706702783325788tion_a
@ ( comp_a6894380724335478043tion_a
@ ^ [F3: a] : ( produc9011544418120257559tion_a @ ( some_a @ F3 ) @ ( some_a @ F3 ) )
@ ^ [F3: produc3509355604313844263tion_a] : ( the_a @ ( produc174239612569448963tion_a @ F3 ) ) )
@ C2 )
= C2 ) ) ).
% RR2_gctxt_unlift_lift_id
thf(fact_338_subrelI,axiom,
! [R2: set_Pr7585778909603769095tion_a,S2: set_Pr7585778909603769095tion_a] :
( ! [X3: option_a,Y3: option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y3 ) @ R2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le4471550158292877991tion_a @ R2 @ S2 ) ) ).
% subrelI
thf(fact_339_subrelI,axiom,
! [R2: set_Pr8660066742269822812_a_nat,S2: set_Pr8660066742269822812_a_nat] :
( ! [X3: produc3509355604313844263tion_a,Y3: nat] :
( ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ X3 @ Y3 ) @ R2 )
=> ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le5649184635953113020_a_nat @ R2 @ S2 ) ) ).
% subrelI
thf(fact_340_subrelI,axiom,
! [R2: set_Pr4934435412358123699_a_nat,S2: set_Pr4934435412358123699_a_nat] :
( ! [X3: a,Y3: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ R2 )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le8666007276011122963_a_nat @ R2 @ S2 ) ) ).
% subrelI
thf(fact_341_eq__fst__iff,axiom,
! [A: produc3509355604313844263tion_a,P: produc4760857364844987814_a_nat] :
( ( A
= ( produc1094642028913761716_a_nat @ P ) )
= ( ? [B6: nat] :
( P
= ( produc5457283509376734880_a_nat @ A @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_342_eq__fst__iff,axiom,
! [A: a,P: product_prod_a_nat] :
( ( A
= ( product_fst_a_nat @ P ) )
= ( ? [B6: nat] :
( P
= ( product_Pair_a_nat @ A @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_343_eq__fst__iff,axiom,
! [A: option_a,P: produc3509355604313844263tion_a] :
( ( A
= ( produc174239612569448963tion_a @ P ) )
= ( ? [B6: option_a] :
( P
= ( produc9011544418120257559tion_a @ A @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_344_fstI,axiom,
! [X: produc4760857364844987814_a_nat,Y: produc3509355604313844263tion_a,Z: nat] :
( ( X
= ( produc5457283509376734880_a_nat @ Y @ Z ) )
=> ( ( produc1094642028913761716_a_nat @ X )
= Y ) ) ).
% fstI
thf(fact_345_fstI,axiom,
! [X: product_prod_a_nat,Y: a,Z: nat] :
( ( X
= ( product_Pair_a_nat @ Y @ Z ) )
=> ( ( product_fst_a_nat @ X )
= Y ) ) ).
% fstI
thf(fact_346_fstI,axiom,
! [X: produc3509355604313844263tion_a,Y: option_a,Z: option_a] :
( ( X
= ( produc9011544418120257559tion_a @ Y @ Z ) )
=> ( ( produc174239612569448963tion_a @ X )
= Y ) ) ).
% fstI
thf(fact_347_split__cong,axiom,
! [Q2: product_prod_a_nat,F: a > nat > produc4760857364844987814_a_nat,G: a > nat > produc4760857364844987814_a_nat,P: product_prod_a_nat] :
( ! [X3: a,Y3: nat] :
( ( ( product_Pair_a_nat @ X3 @ Y3 )
= Q2 )
=> ( ( F @ X3 @ Y3 )
= ( G @ X3 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( produc4004610663780792650_a_nat @ F @ P )
= ( produc4004610663780792650_a_nat @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_348_split__cong,axiom,
! [Q2: produc3509355604313844263tion_a,F: option_a > option_a > $o,G: option_a > option_a > $o,P: produc3509355604313844263tion_a] :
( ! [X3: option_a,Y3: option_a] :
( ( ( produc9011544418120257559tion_a @ X3 @ Y3 )
= Q2 )
=> ( ( F @ X3 @ Y3 )
= ( G @ X3 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( produc7248784972643495020on_a_o @ F @ P )
= ( produc7248784972643495020on_a_o @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_349_split__cong,axiom,
! [Q2: produc4760857364844987814_a_nat,F: produc3509355604313844263tion_a > nat > $o,G: produc3509355604313844263tion_a > nat > $o,P: produc4760857364844987814_a_nat] :
( ! [X3: produc3509355604313844263tion_a,Y3: nat] :
( ( ( produc5457283509376734880_a_nat @ X3 @ Y3 )
= Q2 )
=> ( ( F @ X3 @ Y3 )
= ( G @ X3 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( produc7988141324749014923_nat_o @ F @ P )
= ( produc7988141324749014923_nat_o @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_350_split__cong,axiom,
! [Q2: produc3509355604313844263tion_a,F: option_a > option_a > produc3509355604313844263tion_a,G: option_a > option_a > produc3509355604313844263tion_a,P: produc3509355604313844263tion_a] :
( ! [X3: option_a,Y3: option_a] :
( ( ( produc9011544418120257559tion_a @ X3 @ Y3 )
= Q2 )
=> ( ( F @ X3 @ Y3 )
= ( G @ X3 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( produc2781663484249587963tion_a @ F @ P )
= ( produc2781663484249587963tion_a @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_351_split__cong,axiom,
! [Q2: produc4760857364844987814_a_nat,F: produc3509355604313844263tion_a > nat > produc4760857364844987814_a_nat,G: produc3509355604313844263tion_a > nat > produc4760857364844987814_a_nat,P: produc4760857364844987814_a_nat] :
( ! [X3: produc3509355604313844263tion_a,Y3: nat] :
( ( ( produc5457283509376734880_a_nat @ X3 @ Y3 )
= Q2 )
=> ( ( F @ X3 @ Y3 )
= ( G @ X3 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( produc5088488592959153011_a_nat @ F @ P )
= ( produc5088488592959153011_a_nat @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_352_split__cong,axiom,
! [Q2: produc3509355604313844263tion_a,F: option_a > option_a > option_a,G: option_a > option_a > option_a,P: produc3509355604313844263tion_a] :
( ! [X3: option_a,Y3: option_a] :
( ( ( produc9011544418120257559tion_a @ X3 @ Y3 )
= Q2 )
=> ( ( F @ X3 @ Y3 )
= ( G @ X3 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( produc6953443519211250200tion_a @ F @ P )
= ( produc6953443519211250200tion_a @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_353_all__subset__image,axiom,
! [F: produc4760857364844987814_a_nat > produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat,P2: set_Pr8660066742269822812_a_nat > $o] :
( ( ! [B5: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ B5 @ ( image_4137284907621822035_a_nat @ F @ A2 ) )
=> ( P2 @ B5 ) ) )
= ( ! [B5: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ B5 @ A2 )
=> ( P2 @ ( image_4137284907621822035_a_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_354_all__subset__image,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,A2: set_Pr4934435412358123699_a_nat,P2: set_Pr8660066742269822812_a_nat > $o] :
( ( ! [B5: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ B5 @ ( image_3729436797865474090_a_nat @ F @ A2 ) )
=> ( P2 @ B5 ) ) )
= ( ! [B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B5 @ A2 )
=> ( P2 @ ( image_3729436797865474090_a_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_355_all__subset__image,axiom,
! [F: $o > produc4760857364844987814_a_nat,A2: set_o,P2: set_Pr8660066742269822812_a_nat > $o] :
( ( ! [B5: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ B5 @ ( image_2232132205779913721_a_nat @ F @ A2 ) )
=> ( P2 @ B5 ) ) )
= ( ! [B5: set_o] :
( ( ord_less_eq_set_o @ B5 @ A2 )
=> ( P2 @ ( image_2232132205779913721_a_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_356_all__subset__image,axiom,
! [F: produc4760857364844987814_a_nat > product_prod_a_nat,A2: set_Pr8660066742269822812_a_nat,P2: set_Pr4934435412358123699_a_nat > $o] :
( ( ! [B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B5 @ ( image_6737578356684938026_a_nat @ F @ A2 ) )
=> ( P2 @ B5 ) ) )
= ( ! [B5: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ B5 @ A2 )
=> ( P2 @ ( image_6737578356684938026_a_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_357_all__subset__image,axiom,
! [F: product_prod_a_nat > product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,P2: set_Pr4934435412358123699_a_nat > $o] :
( ( ! [B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B5 @ ( image_8708701149064048897_a_nat @ F @ A2 ) )
=> ( P2 @ B5 ) ) )
= ( ! [B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B5 @ A2 )
=> ( P2 @ ( image_8708701149064048897_a_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_358_all__subset__image,axiom,
! [F: $o > product_prod_a_nat,A2: set_o,P2: set_Pr4934435412358123699_a_nat > $o] :
( ( ! [B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B5 @ ( image_7339617019956244688_a_nat @ F @ A2 ) )
=> ( P2 @ B5 ) ) )
= ( ! [B5: set_o] :
( ( ord_less_eq_set_o @ B5 @ A2 )
=> ( P2 @ ( image_7339617019956244688_a_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_359_all__subset__image,axiom,
! [F: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a,P2: set_o > $o] :
( ( ! [B5: set_o] :
( ( ord_less_eq_set_o @ B5 @ ( image_166006535297930294on_a_o @ F @ A2 ) )
=> ( P2 @ B5 ) ) )
= ( ! [B5: set_Pr7585778909603769095tion_a] :
( ( ord_le4471550158292877991tion_a @ B5 @ A2 )
=> ( P2 @ ( image_166006535297930294on_a_o @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_360_all__subset__image,axiom,
! [F: produc4760857364844987814_a_nat > $o,A2: set_Pr8660066742269822812_a_nat,P2: set_o > $o] :
( ( ! [B5: set_o] :
( ( ord_less_eq_set_o @ B5 @ ( image_6768402879995552939_nat_o @ F @ A2 ) )
=> ( P2 @ B5 ) ) )
= ( ! [B5: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ B5 @ A2 )
=> ( P2 @ ( image_6768402879995552939_nat_o @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_361_all__subset__image,axiom,
! [F: product_prod_a_nat > $o,A2: set_Pr4934435412358123699_a_nat,P2: set_o > $o] :
( ( ! [B5: set_o] :
( ( ord_less_eq_set_o @ B5 @ ( image_3076334604395105300_nat_o @ F @ A2 ) )
=> ( P2 @ B5 ) ) )
= ( ! [B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B5 @ A2 )
=> ( P2 @ ( image_3076334604395105300_nat_o @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_362_all__subset__image,axiom,
! [F: $o > $o,A2: set_o,P2: set_o > $o] :
( ( ! [B5: set_o] :
( ( ord_less_eq_set_o @ B5 @ ( image_o_o @ F @ A2 ) )
=> ( P2 @ B5 ) ) )
= ( ! [B5: set_o] :
( ( ord_less_eq_set_o @ B5 @ A2 )
=> ( P2 @ ( image_o_o @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_363_split__part,axiom,
! [P2: $o,Q: option_a > option_a > $o] :
( ( produc7248784972643495020on_a_o
@ ^ [A6: option_a,B6: option_a] :
( P2
& ( Q @ A6 @ B6 ) ) )
= ( ^ [Ab: produc3509355604313844263tion_a] :
( P2
& ( produc7248784972643495020on_a_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_364_split__part,axiom,
! [P2: $o,Q: produc3509355604313844263tion_a > nat > $o] :
( ( produc7988141324749014923_nat_o
@ ^ [A6: produc3509355604313844263tion_a,B6: nat] :
( P2
& ( Q @ A6 @ B6 ) ) )
= ( ^ [Ab: produc4760857364844987814_a_nat] :
( P2
& ( produc7988141324749014923_nat_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_365_gctxt_Omap__comp,axiom,
! [G: a > a,F: produc3509355604313844263tion_a > a,V: ground4599112814892907741tion_a] :
( ( ground_map_gctxt_a_a @ G @ ( ground8556684339173901107on_a_a @ F @ V ) )
= ( ground8556684339173901107on_a_a @ ( comp_a8076126459503553540tion_a @ G @ F ) @ V ) ) ).
% gctxt.map_comp
thf(fact_366_gctxt_Omap__comp,axiom,
! [G: produc3509355604313844263tion_a > a,F: produc3509355604313844263tion_a > produc3509355604313844263tion_a,V: ground4599112814892907741tion_a] :
( ( ground8556684339173901107on_a_a @ G @ ( ground7092706702783325788tion_a @ F @ V ) )
= ( ground8556684339173901107on_a_a @ ( comp_P1840331437378933421tion_a @ G @ F ) @ V ) ) ).
% gctxt.map_comp
thf(fact_367_prod_Odisc__eq__case,axiom,
! [Prod: produc3509355604313844263tion_a] :
( produc7248784972643495020on_a_o
@ ^ [Uu: option_a,Uv: option_a] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_368_prod_Odisc__eq__case,axiom,
! [Prod: produc4760857364844987814_a_nat] :
( produc7988141324749014923_nat_o
@ ^ [Uu: produc3509355604313844263tion_a,Uv: nat] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_369_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr8660066742269822812_a_nat,Z2: set_Pr8660066742269822812_a_nat] : ( Y4 = Z2 ) )
= ( ^ [X2: set_Pr8660066742269822812_a_nat,Y5: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ X2 @ Y5 )
& ( ord_le5649184635953113020_a_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_370_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( Y4 = Z2 ) )
= ( ^ [X2: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X2 @ Y5 )
& ( ord_le8666007276011122963_a_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_371_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: option5987904771724668803_a_nat,Z2: option5987904771724668803_a_nat] : ( Y4 = Z2 ) )
= ( ^ [X2: option5987904771724668803_a_nat,Y5: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ X2 @ Y5 )
& ( ord_le1496697172603502563_a_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_372_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: option2973356794535198124_a_nat,Z2: option2973356794535198124_a_nat] : ( Y4 = Z2 ) )
= ( ^ [X2: option2973356794535198124_a_nat,Y5: option2973356794535198124_a_nat] :
( ( ord_le8848857416346193420_a_nat @ X2 @ Y5 )
& ( ord_le8848857416346193420_a_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_373_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_o,Z2: set_o] : ( Y4 = Z2 ) )
= ( ^ [X2: set_o,Y5: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y5 )
& ( ord_less_eq_set_o @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_374_ord__eq__le__trans,axiom,
! [A: set_Pr8660066742269822812_a_nat,B: set_Pr8660066742269822812_a_nat,C: set_Pr8660066742269822812_a_nat] :
( ( A = B )
=> ( ( ord_le5649184635953113020_a_nat @ B @ C )
=> ( ord_le5649184635953113020_a_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_375_ord__eq__le__trans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A = B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C )
=> ( ord_le8666007276011122963_a_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_376_ord__eq__le__trans,axiom,
! [A: option5987904771724668803_a_nat,B: option5987904771724668803_a_nat,C: option5987904771724668803_a_nat] :
( ( A = B )
=> ( ( ord_le1496697172603502563_a_nat @ B @ C )
=> ( ord_le1496697172603502563_a_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_377_ord__eq__le__trans,axiom,
! [A: option2973356794535198124_a_nat,B: option2973356794535198124_a_nat,C: option2973356794535198124_a_nat] :
( ( A = B )
=> ( ( ord_le8848857416346193420_a_nat @ B @ C )
=> ( ord_le8848857416346193420_a_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_378_ord__eq__le__trans,axiom,
! [A: set_o,B: set_o,C: set_o] :
( ( A = B )
=> ( ( ord_less_eq_set_o @ B @ C )
=> ( ord_less_eq_set_o @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_379_ord__le__eq__trans,axiom,
! [A: set_Pr8660066742269822812_a_nat,B: set_Pr8660066742269822812_a_nat,C: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le5649184635953113020_a_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_380_ord__le__eq__trans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le8666007276011122963_a_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_381_ord__le__eq__trans,axiom,
! [A: option5987904771724668803_a_nat,B: option5987904771724668803_a_nat,C: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le1496697172603502563_a_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_382_ord__le__eq__trans,axiom,
! [A: option2973356794535198124_a_nat,B: option2973356794535198124_a_nat,C: option2973356794535198124_a_nat] :
( ( ord_le8848857416346193420_a_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le8848857416346193420_a_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_383_ord__le__eq__trans,axiom,
! [A: set_o,B: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_o @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_384_order__antisym,axiom,
! [X: set_Pr8660066742269822812_a_nat,Y: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ X @ Y )
=> ( ( ord_le5649184635953113020_a_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_385_order__antisym,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_386_order__antisym,axiom,
! [X: option5987904771724668803_a_nat,Y: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ X @ Y )
=> ( ( ord_le1496697172603502563_a_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_387_order__antisym,axiom,
! [X: option2973356794535198124_a_nat,Y: option2973356794535198124_a_nat] :
( ( ord_le8848857416346193420_a_nat @ X @ Y )
=> ( ( ord_le8848857416346193420_a_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_388_order__antisym,axiom,
! [X: set_o,Y: set_o] :
( ( ord_less_eq_set_o @ X @ Y )
=> ( ( ord_less_eq_set_o @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_389_order_Otrans,axiom,
! [A: set_Pr8660066742269822812_a_nat,B: set_Pr8660066742269822812_a_nat,C: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ A @ B )
=> ( ( ord_le5649184635953113020_a_nat @ B @ C )
=> ( ord_le5649184635953113020_a_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_390_order_Otrans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C )
=> ( ord_le8666007276011122963_a_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_391_order_Otrans,axiom,
! [A: option5987904771724668803_a_nat,B: option5987904771724668803_a_nat,C: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ A @ B )
=> ( ( ord_le1496697172603502563_a_nat @ B @ C )
=> ( ord_le1496697172603502563_a_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_392_order_Otrans,axiom,
! [A: option2973356794535198124_a_nat,B: option2973356794535198124_a_nat,C: option2973356794535198124_a_nat] :
( ( ord_le8848857416346193420_a_nat @ A @ B )
=> ( ( ord_le8848857416346193420_a_nat @ B @ C )
=> ( ord_le8848857416346193420_a_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_393_order_Otrans,axiom,
! [A: set_o,B: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_set_o @ B @ C )
=> ( ord_less_eq_set_o @ A @ C ) ) ) ).
% order.trans
thf(fact_394_order__trans,axiom,
! [X: set_Pr8660066742269822812_a_nat,Y: set_Pr8660066742269822812_a_nat,Z: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ X @ Y )
=> ( ( ord_le5649184635953113020_a_nat @ Y @ Z )
=> ( ord_le5649184635953113020_a_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_395_order__trans,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ Z )
=> ( ord_le8666007276011122963_a_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_396_order__trans,axiom,
! [X: option5987904771724668803_a_nat,Y: option5987904771724668803_a_nat,Z: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ X @ Y )
=> ( ( ord_le1496697172603502563_a_nat @ Y @ Z )
=> ( ord_le1496697172603502563_a_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_397_order__trans,axiom,
! [X: option2973356794535198124_a_nat,Y: option2973356794535198124_a_nat,Z: option2973356794535198124_a_nat] :
( ( ord_le8848857416346193420_a_nat @ X @ Y )
=> ( ( ord_le8848857416346193420_a_nat @ Y @ Z )
=> ( ord_le8848857416346193420_a_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_398_order__trans,axiom,
! [X: set_o,Y: set_o,Z: set_o] :
( ( ord_less_eq_set_o @ X @ Y )
=> ( ( ord_less_eq_set_o @ Y @ Z )
=> ( ord_less_eq_set_o @ X @ Z ) ) ) ).
% order_trans
thf(fact_399_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr8660066742269822812_a_nat,Z2: set_Pr8660066742269822812_a_nat] : ( Y4 = Z2 ) )
= ( ^ [A6: set_Pr8660066742269822812_a_nat,B6: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ B6 @ A6 )
& ( ord_le5649184635953113020_a_nat @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_400_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( Y4 = Z2 ) )
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B6 @ A6 )
& ( ord_le8666007276011122963_a_nat @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_401_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: option5987904771724668803_a_nat,Z2: option5987904771724668803_a_nat] : ( Y4 = Z2 ) )
= ( ^ [A6: option5987904771724668803_a_nat,B6: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ B6 @ A6 )
& ( ord_le1496697172603502563_a_nat @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_402_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: option2973356794535198124_a_nat,Z2: option2973356794535198124_a_nat] : ( Y4 = Z2 ) )
= ( ^ [A6: option2973356794535198124_a_nat,B6: option2973356794535198124_a_nat] :
( ( ord_le8848857416346193420_a_nat @ B6 @ A6 )
& ( ord_le8848857416346193420_a_nat @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_403_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_o,Z2: set_o] : ( Y4 = Z2 ) )
= ( ^ [A6: set_o,B6: set_o] :
( ( ord_less_eq_set_o @ B6 @ A6 )
& ( ord_less_eq_set_o @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_404_dual__order_Oantisym,axiom,
! [B: set_Pr8660066742269822812_a_nat,A: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ B @ A )
=> ( ( ord_le5649184635953113020_a_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_405_dual__order_Oantisym,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_406_dual__order_Oantisym,axiom,
! [B: option5987904771724668803_a_nat,A: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ B @ A )
=> ( ( ord_le1496697172603502563_a_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_407_dual__order_Oantisym,axiom,
! [B: option2973356794535198124_a_nat,A: option2973356794535198124_a_nat] :
( ( ord_le8848857416346193420_a_nat @ B @ A )
=> ( ( ord_le8848857416346193420_a_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_408_dual__order_Oantisym,axiom,
! [B: set_o,A: set_o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_409_dual__order_Otrans,axiom,
! [B: set_Pr8660066742269822812_a_nat,A: set_Pr8660066742269822812_a_nat,C: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ B @ A )
=> ( ( ord_le5649184635953113020_a_nat @ C @ B )
=> ( ord_le5649184635953113020_a_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_410_dual__order_Otrans,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( ord_le8666007276011122963_a_nat @ C @ B )
=> ( ord_le8666007276011122963_a_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_411_dual__order_Otrans,axiom,
! [B: option5987904771724668803_a_nat,A: option5987904771724668803_a_nat,C: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ B @ A )
=> ( ( ord_le1496697172603502563_a_nat @ C @ B )
=> ( ord_le1496697172603502563_a_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_412_dual__order_Otrans,axiom,
! [B: option2973356794535198124_a_nat,A: option2973356794535198124_a_nat,C: option2973356794535198124_a_nat] :
( ( ord_le8848857416346193420_a_nat @ B @ A )
=> ( ( ord_le8848857416346193420_a_nat @ C @ B )
=> ( ord_le8848857416346193420_a_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_413_dual__order_Otrans,axiom,
! [B: set_o,A: set_o,C: set_o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ( ord_less_eq_set_o @ C @ B )
=> ( ord_less_eq_set_o @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_414_antisym,axiom,
! [A: set_Pr8660066742269822812_a_nat,B: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ A @ B )
=> ( ( ord_le5649184635953113020_a_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_415_antisym,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_416_antisym,axiom,
! [A: option5987904771724668803_a_nat,B: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ A @ B )
=> ( ( ord_le1496697172603502563_a_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_417_antisym,axiom,
! [A: option2973356794535198124_a_nat,B: option2973356794535198124_a_nat] :
( ( ord_le8848857416346193420_a_nat @ A @ B )
=> ( ( ord_le8848857416346193420_a_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_418_antisym,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_set_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_419_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr8660066742269822812_a_nat,Z2: set_Pr8660066742269822812_a_nat] : ( Y4 = Z2 ) )
= ( ^ [A6: set_Pr8660066742269822812_a_nat,B6: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ A6 @ B6 )
& ( ord_le5649184635953113020_a_nat @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_420_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( Y4 = Z2 ) )
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A6 @ B6 )
& ( ord_le8666007276011122963_a_nat @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_421_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: option5987904771724668803_a_nat,Z2: option5987904771724668803_a_nat] : ( Y4 = Z2 ) )
= ( ^ [A6: option5987904771724668803_a_nat,B6: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ A6 @ B6 )
& ( ord_le1496697172603502563_a_nat @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_422_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: option2973356794535198124_a_nat,Z2: option2973356794535198124_a_nat] : ( Y4 = Z2 ) )
= ( ^ [A6: option2973356794535198124_a_nat,B6: option2973356794535198124_a_nat] :
( ( ord_le8848857416346193420_a_nat @ A6 @ B6 )
& ( ord_le8848857416346193420_a_nat @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_423_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_o,Z2: set_o] : ( Y4 = Z2 ) )
= ( ^ [A6: set_o,B6: set_o] :
( ( ord_less_eq_set_o @ A6 @ B6 )
& ( ord_less_eq_set_o @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_424_order__subst1,axiom,
! [A: set_o,F: set_o > set_o,B: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_o @ B @ C )
=> ( ! [X3: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_425_order__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F: set_o > set_Pr4934435412358123699_a_nat,B: set_o,C: set_o] :
( ( ord_le8666007276011122963_a_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_o @ B @ C )
=> ( ! [X3: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_426_order__subst1,axiom,
! [A: set_o,F: set_Pr4934435412358123699_a_nat > set_o,B: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_set_o @ A @ ( F @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_427_order__subst1,axiom,
! [A: option5987904771724668803_a_nat,F: set_o > option5987904771724668803_a_nat,B: set_o,C: set_o] :
( ( ord_le1496697172603502563_a_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_o @ B @ C )
=> ( ! [X3: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y3 )
=> ( ord_le1496697172603502563_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1496697172603502563_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_428_order__subst1,axiom,
! [A: set_o,F: option5987904771724668803_a_nat > set_o,B: option5987904771724668803_a_nat,C: option5987904771724668803_a_nat] :
( ( ord_less_eq_set_o @ A @ ( F @ B ) )
=> ( ( ord_le1496697172603502563_a_nat @ B @ C )
=> ( ! [X3: option5987904771724668803_a_nat,Y3: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_429_order__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( F @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_430_order__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F: option5987904771724668803_a_nat > set_Pr4934435412358123699_a_nat,B: option5987904771724668803_a_nat,C: option5987904771724668803_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( F @ B ) )
=> ( ( ord_le1496697172603502563_a_nat @ B @ C )
=> ( ! [X3: option5987904771724668803_a_nat,Y3: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_431_order__subst1,axiom,
! [A: option5987904771724668803_a_nat,F: set_Pr4934435412358123699_a_nat > option5987904771724668803_a_nat,B: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le1496697172603502563_a_nat @ A @ ( F @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le1496697172603502563_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1496697172603502563_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_432_order__subst1,axiom,
! [A: set_Pr8660066742269822812_a_nat,F: set_o > set_Pr8660066742269822812_a_nat,B: set_o,C: set_o] :
( ( ord_le5649184635953113020_a_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_o @ B @ C )
=> ( ! [X3: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y3 )
=> ( ord_le5649184635953113020_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le5649184635953113020_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_433_order__subst1,axiom,
! [A: option5987904771724668803_a_nat,F: option5987904771724668803_a_nat > option5987904771724668803_a_nat,B: option5987904771724668803_a_nat,C: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ A @ ( F @ B ) )
=> ( ( ord_le1496697172603502563_a_nat @ B @ C )
=> ( ! [X3: option5987904771724668803_a_nat,Y3: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ X3 @ Y3 )
=> ( ord_le1496697172603502563_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1496697172603502563_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_434_order__subst2,axiom,
! [A: set_o,B: set_o,F: set_o > set_o,C: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_set_o @ ( F @ B ) @ C )
=> ( ! [X3: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_435_order__subst2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > set_o,C: set_o] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_less_eq_set_o @ ( F @ B ) @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_436_order__subst2,axiom,
! [A: set_o,B: set_o,F: set_o > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_437_order__subst2,axiom,
! [A: option5987904771724668803_a_nat,B: option5987904771724668803_a_nat,F: option5987904771724668803_a_nat > set_o,C: set_o] :
( ( ord_le1496697172603502563_a_nat @ A @ B )
=> ( ( ord_less_eq_set_o @ ( F @ B ) @ C )
=> ( ! [X3: option5987904771724668803_a_nat,Y3: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_438_order__subst2,axiom,
! [A: set_o,B: set_o,F: set_o > option5987904771724668803_a_nat,C: option5987904771724668803_a_nat] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_le1496697172603502563_a_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y3 )
=> ( ord_le1496697172603502563_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1496697172603502563_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_439_order__subst2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_440_order__subst2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > option5987904771724668803_a_nat,C: option5987904771724668803_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le1496697172603502563_a_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le1496697172603502563_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1496697172603502563_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_441_order__subst2,axiom,
! [A: option5987904771724668803_a_nat,B: option5987904771724668803_a_nat,F: option5987904771724668803_a_nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le1496697172603502563_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ ( F @ B ) @ C )
=> ( ! [X3: option5987904771724668803_a_nat,Y3: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_442_order__subst2,axiom,
! [A: set_Pr8660066742269822812_a_nat,B: set_Pr8660066742269822812_a_nat,F: set_Pr8660066742269822812_a_nat > set_o,C: set_o] :
( ( ord_le5649184635953113020_a_nat @ A @ B )
=> ( ( ord_less_eq_set_o @ ( F @ B ) @ C )
=> ( ! [X3: set_Pr8660066742269822812_a_nat,Y3: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_443_order__subst2,axiom,
! [A: option5987904771724668803_a_nat,B: option5987904771724668803_a_nat,F: option5987904771724668803_a_nat > option5987904771724668803_a_nat,C: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ A @ B )
=> ( ( ord_le1496697172603502563_a_nat @ ( F @ B ) @ C )
=> ( ! [X3: option5987904771724668803_a_nat,Y3: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ X3 @ Y3 )
=> ( ord_le1496697172603502563_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1496697172603502563_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_444_order__eq__refl,axiom,
! [X: set_Pr8660066742269822812_a_nat,Y: set_Pr8660066742269822812_a_nat] :
( ( X = Y )
=> ( ord_le5649184635953113020_a_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_445_order__eq__refl,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( X = Y )
=> ( ord_le8666007276011122963_a_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_446_order__eq__refl,axiom,
! [X: option5987904771724668803_a_nat,Y: option5987904771724668803_a_nat] :
( ( X = Y )
=> ( ord_le1496697172603502563_a_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_447_order__eq__refl,axiom,
! [X: option2973356794535198124_a_nat,Y: option2973356794535198124_a_nat] :
( ( X = Y )
=> ( ord_le8848857416346193420_a_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_448_order__eq__refl,axiom,
! [X: set_o,Y: set_o] :
( ( X = Y )
=> ( ord_less_eq_set_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_449_ord__eq__le__subst,axiom,
! [A: set_o,F: set_o > set_o,B: set_o,C: set_o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_o @ B @ C )
=> ( ! [X3: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_450_ord__eq__le__subst,axiom,
! [A: set_o,F: set_Pr4934435412358123699_a_nat > set_o,B: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_451_ord__eq__le__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,F: set_o > set_Pr4934435412358123699_a_nat,B: set_o,C: set_o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_o @ B @ C )
=> ( ! [X3: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_452_ord__eq__le__subst,axiom,
! [A: set_o,F: option5987904771724668803_a_nat > set_o,B: option5987904771724668803_a_nat,C: option5987904771724668803_a_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1496697172603502563_a_nat @ B @ C )
=> ( ! [X3: option5987904771724668803_a_nat,Y3: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_453_ord__eq__le__subst,axiom,
! [A: option5987904771724668803_a_nat,F: set_o > option5987904771724668803_a_nat,B: set_o,C: set_o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_o @ B @ C )
=> ( ! [X3: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y3 )
=> ( ord_le1496697172603502563_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1496697172603502563_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_454_ord__eq__le__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_455_ord__eq__le__subst,axiom,
! [A: option5987904771724668803_a_nat,F: set_Pr4934435412358123699_a_nat > option5987904771724668803_a_nat,B: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le1496697172603502563_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1496697172603502563_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_456_ord__eq__le__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,F: option5987904771724668803_a_nat > set_Pr4934435412358123699_a_nat,B: option5987904771724668803_a_nat,C: option5987904771724668803_a_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1496697172603502563_a_nat @ B @ C )
=> ( ! [X3: option5987904771724668803_a_nat,Y3: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_457_ord__eq__le__subst,axiom,
! [A: set_o,F: set_Pr8660066742269822812_a_nat > set_o,B: set_Pr8660066742269822812_a_nat,C: set_Pr8660066742269822812_a_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5649184635953113020_a_nat @ B @ C )
=> ( ! [X3: set_Pr8660066742269822812_a_nat,Y3: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_458_ord__eq__le__subst,axiom,
! [A: option5987904771724668803_a_nat,F: option5987904771724668803_a_nat > option5987904771724668803_a_nat,B: option5987904771724668803_a_nat,C: option5987904771724668803_a_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1496697172603502563_a_nat @ B @ C )
=> ( ! [X3: option5987904771724668803_a_nat,Y3: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ X3 @ Y3 )
=> ( ord_le1496697172603502563_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1496697172603502563_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_459_ord__le__eq__subst,axiom,
! [A: set_o,B: set_o,F: set_o > set_o,C: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_460_ord__le__eq__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > set_o,C: set_o] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_461_ord__le__eq__subst,axiom,
! [A: set_o,B: set_o,F: set_o > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_462_ord__le__eq__subst,axiom,
! [A: option5987904771724668803_a_nat,B: option5987904771724668803_a_nat,F: option5987904771724668803_a_nat > set_o,C: set_o] :
( ( ord_le1496697172603502563_a_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: option5987904771724668803_a_nat,Y3: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_463_ord__le__eq__subst,axiom,
! [A: set_o,B: set_o,F: set_o > option5987904771724668803_a_nat,C: option5987904771724668803_a_nat] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y3 )
=> ( ord_le1496697172603502563_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1496697172603502563_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_464_ord__le__eq__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_465_ord__le__eq__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > option5987904771724668803_a_nat,C: option5987904771724668803_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le1496697172603502563_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1496697172603502563_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_466_ord__le__eq__subst,axiom,
! [A: option5987904771724668803_a_nat,B: option5987904771724668803_a_nat,F: option5987904771724668803_a_nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le1496697172603502563_a_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: option5987904771724668803_a_nat,Y3: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_467_ord__le__eq__subst,axiom,
! [A: set_Pr8660066742269822812_a_nat,B: set_Pr8660066742269822812_a_nat,F: set_Pr8660066742269822812_a_nat > set_o,C: set_o] :
( ( ord_le5649184635953113020_a_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_Pr8660066742269822812_a_nat,Y3: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_468_ord__le__eq__subst,axiom,
! [A: option5987904771724668803_a_nat,B: option5987904771724668803_a_nat,F: option5987904771724668803_a_nat > option5987904771724668803_a_nat,C: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: option5987904771724668803_a_nat,Y3: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ X3 @ Y3 )
=> ( ord_le1496697172603502563_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1496697172603502563_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_469_order__antisym__conv,axiom,
! [Y: set_Pr8660066742269822812_a_nat,X: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ Y @ X )
=> ( ( ord_le5649184635953113020_a_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_470_order__antisym__conv,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( ( ord_le8666007276011122963_a_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_471_order__antisym__conv,axiom,
! [Y: option5987904771724668803_a_nat,X: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ Y @ X )
=> ( ( ord_le1496697172603502563_a_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_472_order__antisym__conv,axiom,
! [Y: option2973356794535198124_a_nat,X: option2973356794535198124_a_nat] :
( ( ord_le8848857416346193420_a_nat @ Y @ X )
=> ( ( ord_le8848857416346193420_a_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_473_order__antisym__conv,axiom,
! [Y: set_o,X: set_o] :
( ( ord_less_eq_set_o @ Y @ X )
=> ( ( ord_less_eq_set_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_474_ssubst__Pair__rhs,axiom,
! [R2: produc3509355604313844263tion_a,S2: nat,R: set_Pr8660066742269822812_a_nat,S3: nat] :
( ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ R2 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ R2 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_475_ssubst__Pair__rhs,axiom,
! [R2: option_a,S2: option_a,R: set_Pr7585778909603769095tion_a,S3: option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ R2 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ R2 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_476_ssubst__Pair__rhs,axiom,
! [R2: a,S2: nat,R: set_Pr4934435412358123699_a_nat,S3: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ R2 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ R2 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_477_pred__equals__eq2,axiom,
! [R: set_Pr8660066742269822812_a_nat,S: set_Pr8660066742269822812_a_nat] :
( ( ( ^ [X2: produc3509355604313844263tion_a,Y5: nat] : ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ X2 @ Y5 ) @ R ) )
= ( ^ [X2: produc3509355604313844263tion_a,Y5: nat] : ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ X2 @ Y5 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_478_pred__equals__eq2,axiom,
! [R: set_Pr7585778909603769095tion_a,S: set_Pr7585778909603769095tion_a] :
( ( ( ^ [X2: option_a,Y5: option_a] : ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y5 ) @ R ) )
= ( ^ [X2: option_a,Y5: option_a] : ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y5 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_479_pred__equals__eq2,axiom,
! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ( ^ [X2: a,Y5: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y5 ) @ R ) )
= ( ^ [X2: a,Y5: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y5 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_480_prop__restrict,axiom,
! [X: produc3509355604313844263tion_a,Z3: set_Pr7585778909603769095tion_a,X5: set_Pr7585778909603769095tion_a,P2: produc3509355604313844263tion_a > $o] :
( ( member5498148017924304208tion_a @ X @ Z3 )
=> ( ( ord_le4471550158292877991tion_a @ Z3
@ ( collec4135126896892755346tion_a
@ ^ [X2: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X2 @ X5 )
& ( P2 @ X2 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_481_prop__restrict,axiom,
! [X: produc4760857364844987814_a_nat,Z3: set_Pr8660066742269822812_a_nat,X5: set_Pr8660066742269822812_a_nat,P2: produc4760857364844987814_a_nat > $o] :
( ( member1430561565885860669_a_nat @ X @ Z3 )
=> ( ( ord_le5649184635953113020_a_nat @ Z3
@ ( collec7438131855006933115_a_nat
@ ^ [X2: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X2 @ X5 )
& ( P2 @ X2 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_482_prop__restrict,axiom,
! [X: product_prod_a_nat,Z3: set_Pr4934435412358123699_a_nat,X5: set_Pr4934435412358123699_a_nat,P2: product_prod_a_nat > $o] :
( ( member5724188588386418708_a_nat @ X @ Z3 )
=> ( ( ord_le8666007276011122963_a_nat @ Z3
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ X5 )
& ( P2 @ X2 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_483_prop__restrict,axiom,
! [X: $o,Z3: set_o,X5: set_o,P2: $o > $o] :
( ( member_o @ X @ Z3 )
=> ( ( ord_less_eq_set_o @ Z3
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ X5 )
& ( P2 @ X2 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_484_Collect__restrict,axiom,
! [X5: set_Pr7585778909603769095tion_a,P2: produc3509355604313844263tion_a > $o] :
( ord_le4471550158292877991tion_a
@ ( collec4135126896892755346tion_a
@ ^ [X2: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X2 @ X5 )
& ( P2 @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_485_Collect__restrict,axiom,
! [X5: set_Pr8660066742269822812_a_nat,P2: produc4760857364844987814_a_nat > $o] :
( ord_le5649184635953113020_a_nat
@ ( collec7438131855006933115_a_nat
@ ^ [X2: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X2 @ X5 )
& ( P2 @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_486_Collect__restrict,axiom,
! [X5: set_Pr4934435412358123699_a_nat,P2: product_prod_a_nat > $o] :
( ord_le8666007276011122963_a_nat
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ X5 )
& ( P2 @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_487_Collect__restrict,axiom,
! [X5: set_o,P2: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ X5 )
& ( P2 @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_488_pred__subset__eq,axiom,
! [R: set_Pr7585778909603769095tion_a,S: set_Pr7585778909603769095tion_a] :
( ( ord_le4863394807095538934on_a_o
@ ^ [X2: produc3509355604313844263tion_a] : ( member5498148017924304208tion_a @ X2 @ R )
@ ^ [X2: produc3509355604313844263tion_a] : ( member5498148017924304208tion_a @ X2 @ S ) )
= ( ord_le4471550158292877991tion_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_489_pred__subset__eq,axiom,
! [R: set_Pr8660066742269822812_a_nat,S: set_Pr8660066742269822812_a_nat] :
( ( ord_le8675248135204644809_nat_o
@ ^ [X2: produc4760857364844987814_a_nat] : ( member1430561565885860669_a_nat @ X2 @ R )
@ ^ [X2: produc4760857364844987814_a_nat] : ( member1430561565885860669_a_nat @ X2 @ S ) )
= ( ord_le5649184635953113020_a_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_490_pred__subset__eq,axiom,
! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le8950881081082477362_nat_o
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ R )
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ S ) )
= ( ord_le8666007276011122963_a_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_491_pred__subset__eq,axiom,
! [R: set_o,S: set_o] :
( ( ord_less_eq_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ R )
@ ^ [X2: $o] : ( member_o @ X2 @ S ) )
= ( ord_less_eq_set_o @ R @ S ) ) ).
% pred_subset_eq
thf(fact_492_RR2__gterm__unlift__lift__id,axiom,
! [T2: ground4005964956264846616_a_nat,F2: set_Pr5612011003857075140at_nat] :
( ( ord_le8609766837066453116at_nat @ ( ground7958862358551846640_a_nat @ T2 )
@ ( image_6008886954644801541at_nat
@ ( produc8166206427995057559at_nat
@ ^ [F3: set_Pr4934435412358123699_a_nat] : ( produc6380145652274861664at_nat @ ( produc1207969925626950559_a_nat @ ( some_s1880800141159441806_a_nat @ F3 ) @ ( some_s1880800141159441806_a_nat @ F3 ) ) ) )
@ F2 ) )
=> ( ( ground3351558865046840614_a_nat
@ ( comp_s467579784663650262_a_nat
@ ^ [F3: set_Pr4934435412358123699_a_nat] : ( produc1207969925626950559_a_nat @ ( some_s1880800141159441806_a_nat @ F3 ) @ ( some_s1880800141159441806_a_nat @ F3 ) )
@ ^ [F3: produc7618478223271187943_a_nat] : ( the_se1682649557849046815_a_nat @ ( produc735037054355678091_a_nat @ F3 ) ) )
@ T2 )
= T2 ) ) ).
% RR2_gterm_unlift_lift_id
thf(fact_493_RR2__gterm__unlift__lift__id,axiom,
! [T2: ground3180571006387129706_a_nat,F2: set_Pr2737808803024040429at_nat] :
( ( ord_le8394821662120075470at_nat @ ( ground7469869589483366466_a_nat @ T2 )
@ ( image_1902767331843210624at_nat
@ ( produc1847350529710776082at_nat
@ ^ [F3: set_Pr8660066742269822812_a_nat] : ( produc2385804702627905458at_nat @ ( produc8861491750707134833_a_nat @ ( some_s2314368709036572727_a_nat @ F3 ) @ ( some_s2314368709036572727_a_nat @ F3 ) ) ) )
@ F2 ) )
=> ( ( ground7863440295930738854_a_nat
@ ( comp_s8729298985798543999_a_nat
@ ^ [F3: set_Pr8660066742269822812_a_nat] : ( produc8861491750707134833_a_nat @ ( some_s2314368709036572727_a_nat @ F3 ) @ ( some_s2314368709036572727_a_nat @ F3 ) )
@ ^ [F3: produc4107345600652478265_a_nat] : ( the_se5449783460274439624_a_nat @ ( produc4143784723386224989_a_nat @ F3 ) ) )
@ T2 )
= T2 ) ) ).
% RR2_gterm_unlift_lift_id
thf(fact_494_RR2__gterm__unlift__lift__id,axiom,
! [T2: ground5314827683623246424tion_a,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le5649184635953113020_a_nat @ ( ground5706566889460337840tion_a @ T2 )
@ ( image_3729436797865474090_a_nat
@ ( produc4004610663780792650_a_nat
@ ^ [F3: a] : ( produc5457283509376734880_a_nat @ ( produc9011544418120257559tion_a @ ( some_a @ F3 ) @ ( some_a @ F3 ) ) ) )
@ F2 ) )
=> ( ( ground6258472780936250918tion_a
@ ( comp_a6894380724335478043tion_a
@ ^ [F3: a] : ( produc9011544418120257559tion_a @ ( some_a @ F3 ) @ ( some_a @ F3 ) )
@ ^ [F3: produc3509355604313844263tion_a] : ( the_a @ ( produc174239612569448963tion_a @ F3 ) ) )
@ T2 )
= T2 ) ) ).
% RR2_gterm_unlift_lift_id
thf(fact_495_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: $o > $o,F: $o > $o,A2: set_o] :
( ( Inf @ ( image_o_o @ G @ ( image_o_o @ F @ A2 ) ) )
= ( Inf @ ( image_o_o @ ( comp_o_o_o @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_496_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: produc3509355604313844263tion_a > $o,F: $o > produc3509355604313844263tion_a,A2: set_o] :
( ( Inf @ ( image_166006535297930294on_a_o @ G @ ( image_628002824077805460tion_a @ F @ A2 ) ) )
= ( Inf @ ( image_o_o @ ( comp_P4515272243702675076_a_o_o @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_497_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: $o > $o,F: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a] :
( ( Inf @ ( image_o_o @ G @ ( image_166006535297930294on_a_o @ F @ A2 ) ) )
= ( Inf @ ( image_166006535297930294on_a_o @ ( comp_o8812929617658428164tion_a @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_498_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: produc4760857364844987814_a_nat > $o,F: $o > produc4760857364844987814_a_nat,A2: set_o] :
( ( Inf @ ( image_6768402879995552939_nat_o @ G @ ( image_2232132205779913721_a_nat @ F @ A2 ) ) )
= ( Inf @ ( image_o_o @ ( comp_P885853961682478089at_o_o @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_499_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: $o > $o,F: produc4760857364844987814_a_nat > $o,A2: set_Pr8660066742269822812_a_nat] :
( ( Inf @ ( image_o_o @ G @ ( image_6768402879995552939_nat_o @ F @ A2 ) ) )
= ( Inf @ ( image_6768402879995552939_nat_o @ ( comp_o3516818292788251785_a_nat @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_500_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: produc3509355604313844263tion_a > $o,F: produc3509355604313844263tion_a > produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a] :
( ( Inf @ ( image_166006535297930294on_a_o @ G @ ( image_6668519322520640453tion_a @ F @ A2 ) ) )
= ( Inf @ ( image_166006535297930294on_a_o @ ( comp_P3337116118691028755tion_a @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_501_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: produc4760857364844987814_a_nat > $o,F: product_prod_a_nat > produc4760857364844987814_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( Inf @ ( image_6768402879995552939_nat_o @ G @ ( image_3729436797865474090_a_nat @ F @ A2 ) ) )
= ( Inf @ ( image_3076334604395105300_nat_o @ ( comp_P7346075404395887948_a_nat @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_502_Inf_OINF__image,axiom,
! [Inf: set_Pr8660066742269822812_a_nat > produc4760857364844987814_a_nat,G: product_prod_a_nat > produc4760857364844987814_a_nat,F: product_prod_a_nat > product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( Inf @ ( image_3729436797865474090_a_nat @ G @ ( image_8708701149064048897_a_nat @ F @ A2 ) ) )
= ( Inf @ ( image_3729436797865474090_a_nat @ ( comp_P6608656588004095109_a_nat @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_503_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: produc3509355604313844263tion_a > $o,F: produc4760857364844987814_a_nat > produc3509355604313844263tion_a,A2: set_Pr8660066742269822812_a_nat] :
( ( Inf @ ( image_166006535297930294on_a_o @ G @ ( image_1391120077856136634tion_a @ F @ A2 ) ) )
= ( Inf @ ( image_6768402879995552939_nat_o @ ( comp_P4709773760693074362_a_nat @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_504_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: produc4760857364844987814_a_nat > $o,F: produc3509355604313844263tion_a > produc4760857364844987814_a_nat,A2: set_Pr7585778909603769095tion_a] :
( ( Inf @ ( image_6768402879995552939_nat_o @ G @ ( image_8907095238652253832_a_nat @ F @ A2 ) ) )
= ( Inf @ ( image_166006535297930294on_a_o @ ( comp_P8503938455319246360tion_a @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_505_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: $o > $o,F: $o > $o,A2: set_o] :
( ( Sup @ ( image_o_o @ G @ ( image_o_o @ F @ A2 ) ) )
= ( Sup @ ( image_o_o @ ( comp_o_o_o @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_506_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: produc3509355604313844263tion_a > $o,F: $o > produc3509355604313844263tion_a,A2: set_o] :
( ( Sup @ ( image_166006535297930294on_a_o @ G @ ( image_628002824077805460tion_a @ F @ A2 ) ) )
= ( Sup @ ( image_o_o @ ( comp_P4515272243702675076_a_o_o @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_507_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: $o > $o,F: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a] :
( ( Sup @ ( image_o_o @ G @ ( image_166006535297930294on_a_o @ F @ A2 ) ) )
= ( Sup @ ( image_166006535297930294on_a_o @ ( comp_o8812929617658428164tion_a @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_508_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: produc4760857364844987814_a_nat > $o,F: $o > produc4760857364844987814_a_nat,A2: set_o] :
( ( Sup @ ( image_6768402879995552939_nat_o @ G @ ( image_2232132205779913721_a_nat @ F @ A2 ) ) )
= ( Sup @ ( image_o_o @ ( comp_P885853961682478089at_o_o @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_509_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: $o > $o,F: produc4760857364844987814_a_nat > $o,A2: set_Pr8660066742269822812_a_nat] :
( ( Sup @ ( image_o_o @ G @ ( image_6768402879995552939_nat_o @ F @ A2 ) ) )
= ( Sup @ ( image_6768402879995552939_nat_o @ ( comp_o3516818292788251785_a_nat @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_510_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: produc3509355604313844263tion_a > $o,F: produc3509355604313844263tion_a > produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a] :
( ( Sup @ ( image_166006535297930294on_a_o @ G @ ( image_6668519322520640453tion_a @ F @ A2 ) ) )
= ( Sup @ ( image_166006535297930294on_a_o @ ( comp_P3337116118691028755tion_a @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_511_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: produc4760857364844987814_a_nat > $o,F: product_prod_a_nat > produc4760857364844987814_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( Sup @ ( image_6768402879995552939_nat_o @ G @ ( image_3729436797865474090_a_nat @ F @ A2 ) ) )
= ( Sup @ ( image_3076334604395105300_nat_o @ ( comp_P7346075404395887948_a_nat @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_512_Sup_OSUP__image,axiom,
! [Sup: set_Pr8660066742269822812_a_nat > produc4760857364844987814_a_nat,G: product_prod_a_nat > produc4760857364844987814_a_nat,F: product_prod_a_nat > product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( Sup @ ( image_3729436797865474090_a_nat @ G @ ( image_8708701149064048897_a_nat @ F @ A2 ) ) )
= ( Sup @ ( image_3729436797865474090_a_nat @ ( comp_P6608656588004095109_a_nat @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_513_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: produc3509355604313844263tion_a > $o,F: produc4760857364844987814_a_nat > produc3509355604313844263tion_a,A2: set_Pr8660066742269822812_a_nat] :
( ( Sup @ ( image_166006535297930294on_a_o @ G @ ( image_1391120077856136634tion_a @ F @ A2 ) ) )
= ( Sup @ ( image_6768402879995552939_nat_o @ ( comp_P4709773760693074362_a_nat @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_514_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: produc4760857364844987814_a_nat > $o,F: produc3509355604313844263tion_a > produc4760857364844987814_a_nat,A2: set_Pr7585778909603769095tion_a] :
( ( Sup @ ( image_6768402879995552939_nat_o @ G @ ( image_8907095238652253832_a_nat @ F @ A2 ) ) )
= ( Sup @ ( image_166006535297930294on_a_o @ ( comp_P8503938455319246360tion_a @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_515_image__comp,axiom,
! [F: $o > $o,G: $o > $o,R2: set_o] :
( ( image_o_o @ F @ ( image_o_o @ G @ R2 ) )
= ( image_o_o @ ( comp_o_o_o @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_516_image__comp,axiom,
! [F: produc3509355604313844263tion_a > $o,G: $o > produc3509355604313844263tion_a,R2: set_o] :
( ( image_166006535297930294on_a_o @ F @ ( image_628002824077805460tion_a @ G @ R2 ) )
= ( image_o_o @ ( comp_P4515272243702675076_a_o_o @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_517_image__comp,axiom,
! [F: $o > $o,G: produc3509355604313844263tion_a > $o,R2: set_Pr7585778909603769095tion_a] :
( ( image_o_o @ F @ ( image_166006535297930294on_a_o @ G @ R2 ) )
= ( image_166006535297930294on_a_o @ ( comp_o8812929617658428164tion_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_518_image__comp,axiom,
! [F: produc4760857364844987814_a_nat > $o,G: $o > produc4760857364844987814_a_nat,R2: set_o] :
( ( image_6768402879995552939_nat_o @ F @ ( image_2232132205779913721_a_nat @ G @ R2 ) )
= ( image_o_o @ ( comp_P885853961682478089at_o_o @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_519_image__comp,axiom,
! [F: $o > $o,G: produc4760857364844987814_a_nat > $o,R2: set_Pr8660066742269822812_a_nat] :
( ( image_o_o @ F @ ( image_6768402879995552939_nat_o @ G @ R2 ) )
= ( image_6768402879995552939_nat_o @ ( comp_o3516818292788251785_a_nat @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_520_image__comp,axiom,
! [F: produc3509355604313844263tion_a > $o,G: produc3509355604313844263tion_a > produc3509355604313844263tion_a,R2: set_Pr7585778909603769095tion_a] :
( ( image_166006535297930294on_a_o @ F @ ( image_6668519322520640453tion_a @ G @ R2 ) )
= ( image_166006535297930294on_a_o @ ( comp_P3337116118691028755tion_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_521_image__comp,axiom,
! [F: produc4760857364844987814_a_nat > $o,G: product_prod_a_nat > produc4760857364844987814_a_nat,R2: set_Pr4934435412358123699_a_nat] :
( ( image_6768402879995552939_nat_o @ F @ ( image_3729436797865474090_a_nat @ G @ R2 ) )
= ( image_3076334604395105300_nat_o @ ( comp_P7346075404395887948_a_nat @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_522_image__comp,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,G: product_prod_a_nat > product_prod_a_nat,R2: set_Pr4934435412358123699_a_nat] :
( ( image_3729436797865474090_a_nat @ F @ ( image_8708701149064048897_a_nat @ G @ R2 ) )
= ( image_3729436797865474090_a_nat @ ( comp_P6608656588004095109_a_nat @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_523_image__comp,axiom,
! [F: produc3509355604313844263tion_a > $o,G: produc4760857364844987814_a_nat > produc3509355604313844263tion_a,R2: set_Pr8660066742269822812_a_nat] :
( ( image_166006535297930294on_a_o @ F @ ( image_1391120077856136634tion_a @ G @ R2 ) )
= ( image_6768402879995552939_nat_o @ ( comp_P4709773760693074362_a_nat @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_524_image__comp,axiom,
! [F: produc4760857364844987814_a_nat > $o,G: produc3509355604313844263tion_a > produc4760857364844987814_a_nat,R2: set_Pr7585778909603769095tion_a] :
( ( image_6768402879995552939_nat_o @ F @ ( image_8907095238652253832_a_nat @ G @ R2 ) )
= ( image_166006535297930294on_a_o @ ( comp_P8503938455319246360tion_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_525_image__eq__imp__comp,axiom,
! [F: $o > $o,A2: set_o,G: $o > $o,B2: set_o,H: $o > $o] :
( ( ( image_o_o @ F @ A2 )
= ( image_o_o @ G @ B2 ) )
=> ( ( image_o_o @ ( comp_o_o_o @ H @ F ) @ A2 )
= ( image_o_o @ ( comp_o_o_o @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_526_image__eq__imp__comp,axiom,
! [F: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a,G: $o > $o,B2: set_o,H: $o > $o] :
( ( ( image_166006535297930294on_a_o @ F @ A2 )
= ( image_o_o @ G @ B2 ) )
=> ( ( image_166006535297930294on_a_o @ ( comp_o8812929617658428164tion_a @ H @ F ) @ A2 )
= ( image_o_o @ ( comp_o_o_o @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_527_image__eq__imp__comp,axiom,
! [F: $o > $o,A2: set_o,G: produc3509355604313844263tion_a > $o,B2: set_Pr7585778909603769095tion_a,H: $o > $o] :
( ( ( image_o_o @ F @ A2 )
= ( image_166006535297930294on_a_o @ G @ B2 ) )
=> ( ( image_o_o @ ( comp_o_o_o @ H @ F ) @ A2 )
= ( image_166006535297930294on_a_o @ ( comp_o8812929617658428164tion_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_528_image__eq__imp__comp,axiom,
! [F: produc4760857364844987814_a_nat > $o,A2: set_Pr8660066742269822812_a_nat,G: $o > $o,B2: set_o,H: $o > $o] :
( ( ( image_6768402879995552939_nat_o @ F @ A2 )
= ( image_o_o @ G @ B2 ) )
=> ( ( image_6768402879995552939_nat_o @ ( comp_o3516818292788251785_a_nat @ H @ F ) @ A2 )
= ( image_o_o @ ( comp_o_o_o @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_529_image__eq__imp__comp,axiom,
! [F: $o > $o,A2: set_o,G: produc4760857364844987814_a_nat > $o,B2: set_Pr8660066742269822812_a_nat,H: $o > $o] :
( ( ( image_o_o @ F @ A2 )
= ( image_6768402879995552939_nat_o @ G @ B2 ) )
=> ( ( image_o_o @ ( comp_o_o_o @ H @ F ) @ A2 )
= ( image_6768402879995552939_nat_o @ ( comp_o3516818292788251785_a_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_530_image__eq__imp__comp,axiom,
! [F: $o > produc4760857364844987814_a_nat,A2: set_o,G: product_prod_a_nat > produc4760857364844987814_a_nat,B2: set_Pr4934435412358123699_a_nat,H: produc4760857364844987814_a_nat > $o] :
( ( ( image_2232132205779913721_a_nat @ F @ A2 )
= ( image_3729436797865474090_a_nat @ G @ B2 ) )
=> ( ( image_o_o @ ( comp_P885853961682478089at_o_o @ H @ F ) @ A2 )
= ( image_3076334604395105300_nat_o @ ( comp_P7346075404395887948_a_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_531_image__eq__imp__comp,axiom,
! [F: product_prod_a_nat > $o,A2: set_Pr4934435412358123699_a_nat,G: $o > $o,B2: set_o,H: $o > produc4760857364844987814_a_nat] :
( ( ( image_3076334604395105300_nat_o @ F @ A2 )
= ( image_o_o @ G @ B2 ) )
=> ( ( image_3729436797865474090_a_nat @ ( comp_o2208771042594549402_a_nat @ H @ F ) @ A2 )
= ( image_2232132205779913721_a_nat @ ( comp_o8053088967003891003_nat_o @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_532_image__eq__imp__comp,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,A2: set_Pr4934435412358123699_a_nat,G: $o > produc4760857364844987814_a_nat,B2: set_o,H: produc4760857364844987814_a_nat > $o] :
( ( ( image_3729436797865474090_a_nat @ F @ A2 )
= ( image_2232132205779913721_a_nat @ G @ B2 ) )
=> ( ( image_3076334604395105300_nat_o @ ( comp_P7346075404395887948_a_nat @ H @ F ) @ A2 )
= ( image_o_o @ ( comp_P885853961682478089at_o_o @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_533_image__eq__imp__comp,axiom,
! [F: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a,G: produc3509355604313844263tion_a > $o,B2: set_Pr7585778909603769095tion_a,H: $o > $o] :
( ( ( image_166006535297930294on_a_o @ F @ A2 )
= ( image_166006535297930294on_a_o @ G @ B2 ) )
=> ( ( image_166006535297930294on_a_o @ ( comp_o8812929617658428164tion_a @ H @ F ) @ A2 )
= ( image_166006535297930294on_a_o @ ( comp_o8812929617658428164tion_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_534_image__eq__imp__comp,axiom,
! [F: $o > $o,A2: set_o,G: product_prod_a_nat > $o,B2: set_Pr4934435412358123699_a_nat,H: $o > produc4760857364844987814_a_nat] :
( ( ( image_o_o @ F @ A2 )
= ( image_3076334604395105300_nat_o @ G @ B2 ) )
=> ( ( image_2232132205779913721_a_nat @ ( comp_o8053088967003891003_nat_o @ H @ F ) @ A2 )
= ( image_3729436797865474090_a_nat @ ( comp_o2208771042594549402_a_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_535_flip__pred,axiom,
! [A2: set_Pr4193341848836149977_nat_a,R: a > nat > $o] :
( ( ord_le7924913712489149241_nat_a @ A2 @ ( collec7702297998945444472_nat_a @ ( produc2746933349376800278at_a_o @ ( conversep_a_nat @ R ) ) ) )
=> ( ord_le8666007276011122963_a_nat
@ ( image_5148427718660454439_a_nat
@ ( produc732906326552059263_a_nat
@ ^ [X2: nat,Y5: a] : ( product_Pair_a_nat @ Y5 @ X2 ) )
@ A2 )
@ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ R ) ) ) ) ).
% flip_pred
thf(fact_536_flip__pred,axiom,
! [A2: set_Pr4934435412358123699_a_nat,R: nat > a > $o] :
( ( ord_le8666007276011122963_a_nat @ A2 @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ ( conversep_nat_a @ R ) ) ) )
=> ( ord_le7924913712489149241_nat_a
@ ( image_2723492575932950055_nat_a
@ ( produc2753474147469666119_nat_a
@ ^ [X2: a,Y5: nat] : ( product_Pair_nat_a @ Y5 @ X2 ) )
@ A2 )
@ ( collec7702297998945444472_nat_a @ ( produc2746933349376800278at_a_o @ R ) ) ) ) ).
% flip_pred
thf(fact_537_flip__pred,axiom,
! [A2: set_Pr2852166431588183298tion_a,R: produc3509355604313844263tion_a > nat > $o] :
( ( ord_le9064656362126249314tion_a @ A2 @ ( collec7806606199678515489tion_a @ ( produc8689481078867993261on_a_o @ ( conver1870972795803939522_a_nat @ R ) ) ) )
=> ( ord_le5649184635953113020_a_nat
@ ( image_9075947210971128057_a_nat
@ ( produc9185384946573455057_a_nat
@ ^ [X2: nat,Y5: produc3509355604313844263tion_a] : ( produc5457283509376734880_a_nat @ Y5 @ X2 ) )
@ A2 )
@ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ R ) ) ) ) ).
% flip_pred
thf(fact_538_flip__pred,axiom,
! [A2: set_Pr7585778909603769095tion_a,R: option_a > option_a > $o] :
( ( ord_le4471550158292877991tion_a @ A2 @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ ( conver2091984906772465269tion_a @ R ) ) ) )
=> ( ord_le4471550158292877991tion_a
@ ( image_6668519322520640453tion_a
@ ( produc2781663484249587963tion_a
@ ^ [X2: option_a,Y5: option_a] : ( produc9011544418120257559tion_a @ Y5 @ X2 ) )
@ A2 )
@ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ R ) ) ) ) ).
% flip_pred
thf(fact_539_flip__pred,axiom,
! [A2: set_Pr8660066742269822812_a_nat,R: nat > produc3509355604313844263tion_a > $o] :
( ( ord_le5649184635953113020_a_nat @ A2 @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ ( conver8875248373682674720tion_a @ R ) ) ) )
=> ( ord_le9064656362126249314tion_a
@ ( image_4505759252293404409tion_a
@ ( produc5456962937630735385tion_a
@ ^ [X2: produc3509355604313844263tion_a,Y5: nat] : ( produc3238187050400694270tion_a @ Y5 @ X2 ) )
@ A2 )
@ ( collec7806606199678515489tion_a @ ( produc8689481078867993261on_a_o @ R ) ) ) ) ).
% flip_pred
thf(fact_540_internal__case__prod__def,axiom,
produc2952256763413369076_a_nat = produc4004610663780792650_a_nat ).
% internal_case_prod_def
thf(fact_541_internal__case__prod__def,axiom,
produc7617790198452412438on_a_o = produc7248784972643495020on_a_o ).
% internal_case_prod_def
thf(fact_542_internal__case__prod__def,axiom,
produc1776589373858715489_nat_o = produc7988141324749014923_nat_o ).
% internal_case_prod_def
thf(fact_543_internal__case__prod__def,axiom,
produc4798496092881172389tion_a = produc2781663484249587963tion_a ).
% internal_case_prod_def
thf(fact_544_internal__case__prod__def,axiom,
produc1110342753507423005_a_nat = produc5088488592959153011_a_nat ).
% internal_case_prod_def
thf(fact_545_internal__case__prod__def,axiom,
produc3532408494181128642tion_a = produc6953443519211250200tion_a ).
% internal_case_prod_def
thf(fact_546_Sup_OSUP__cong,axiom,
! [A2: set_o,B2: set_o,C2: $o > $o,D: $o > $o,Sup: set_o > $o] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_o_o @ C2 @ A2 ) )
= ( Sup @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_547_Sup_OSUP__cong,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C2: product_prod_a_nat > produc4760857364844987814_a_nat,D: product_prod_a_nat > produc4760857364844987814_a_nat,Sup: set_Pr8660066742269822812_a_nat > produc4760857364844987814_a_nat] :
( ( A2 = B2 )
=> ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_3729436797865474090_a_nat @ C2 @ A2 ) )
= ( Sup @ ( image_3729436797865474090_a_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_548_Sup_OSUP__cong,axiom,
! [A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat,C2: produc4760857364844987814_a_nat > $o,D: produc4760857364844987814_a_nat > $o,Sup: set_o > $o] :
( ( A2 = B2 )
=> ( ! [X3: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_6768402879995552939_nat_o @ C2 @ A2 ) )
= ( Sup @ ( image_6768402879995552939_nat_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_549_Sup_OSUP__cong,axiom,
! [A2: set_Pr7585778909603769095tion_a,B2: set_Pr7585778909603769095tion_a,C2: produc3509355604313844263tion_a > $o,D: produc3509355604313844263tion_a > $o,Sup: set_o > $o] :
( ( A2 = B2 )
=> ( ! [X3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_166006535297930294on_a_o @ C2 @ A2 ) )
= ( Sup @ ( image_166006535297930294on_a_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_550_Inf_OINF__cong,axiom,
! [A2: set_o,B2: set_o,C2: $o > $o,D: $o > $o,Inf: set_o > $o] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_o_o @ C2 @ A2 ) )
= ( Inf @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_551_Inf_OINF__cong,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C2: product_prod_a_nat > produc4760857364844987814_a_nat,D: product_prod_a_nat > produc4760857364844987814_a_nat,Inf: set_Pr8660066742269822812_a_nat > produc4760857364844987814_a_nat] :
( ( A2 = B2 )
=> ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_3729436797865474090_a_nat @ C2 @ A2 ) )
= ( Inf @ ( image_3729436797865474090_a_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_552_Inf_OINF__cong,axiom,
! [A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat,C2: produc4760857364844987814_a_nat > $o,D: produc4760857364844987814_a_nat > $o,Inf: set_o > $o] :
( ( A2 = B2 )
=> ( ! [X3: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_6768402879995552939_nat_o @ C2 @ A2 ) )
= ( Inf @ ( image_6768402879995552939_nat_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_553_Inf_OINF__cong,axiom,
! [A2: set_Pr7585778909603769095tion_a,B2: set_Pr7585778909603769095tion_a,C2: produc3509355604313844263tion_a > $o,D: produc3509355604313844263tion_a > $o,Inf: set_o > $o] :
( ( A2 = B2 )
=> ( ! [X3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_166006535297930294on_a_o @ C2 @ A2 ) )
= ( Inf @ ( image_166006535297930294on_a_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_554_gterm__funas__lift__RR2__funas,axiom,
! [T2: ground7172055985132536664_a_nat,F2: set_Pr5612011003857075140at_nat] :
( ( ord_le3946384098489951076at_nat @ ( ground2178884368406916842_a_nat @ T2 ) @ F2 )
=> ( ord_le8609766837066453116at_nat
@ ( ground7958862358551846640_a_nat
@ ( ground2964810367549707404_a_nat
@ ^ [F3: set_Pr4934435412358123699_a_nat] : ( produc1207969925626950559_a_nat @ ( some_s1880800141159441806_a_nat @ F3 ) @ ( some_s1880800141159441806_a_nat @ F3 ) )
@ T2 ) )
@ ( image_6008886954644801541at_nat
@ ( produc8166206427995057559at_nat
@ ^ [F3: set_Pr4934435412358123699_a_nat] : ( produc6380145652274861664at_nat @ ( produc1207969925626950559_a_nat @ ( some_s1880800141159441806_a_nat @ F3 ) @ ( some_s1880800141159441806_a_nat @ F3 ) ) ) )
@ F2 ) ) ) ).
% gterm_funas_lift_RR2_funas
thf(fact_555_gterm__funas__lift__RR2__funas,axiom,
! [T2: ground8501651587922439553_a_nat,F2: set_Pr2737808803024040429at_nat] :
( ( ord_le221046818917144973at_nat @ ( ground2506338072319351187_a_nat @ T2 ) @ F2 )
=> ( ord_le8394821662120075470at_nat
@ ( ground7469869589483366466_a_nat
@ ( ground1353630888789153013_a_nat
@ ^ [F3: set_Pr8660066742269822812_a_nat] : ( produc8861491750707134833_a_nat @ ( some_s2314368709036572727_a_nat @ F3 ) @ ( some_s2314368709036572727_a_nat @ F3 ) )
@ T2 ) )
@ ( image_1902767331843210624at_nat
@ ( produc1847350529710776082at_nat
@ ^ [F3: set_Pr8660066742269822812_a_nat] : ( produc2385804702627905458at_nat @ ( produc8861491750707134833_a_nat @ ( some_s2314368709036572727_a_nat @ F3 ) @ ( some_s2314368709036572727_a_nat @ F3 ) ) ) )
@ F2 ) ) ) ).
% gterm_funas_lift_RR2_funas
thf(fact_556_gterm__funas__lift__RR2__funas,axiom,
! [T2: ground5314827683623246424tion_a,F2: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ ( ground5706566889460337840tion_a @ T2 ) @ F2 )
=> ( ord_le6272267606287294716_a_nat
@ ( ground2948544073636793840tion_a
@ ( ground6366415769211361126tion_a
@ ^ [F3: produc3509355604313844263tion_a] : ( produc3538477569256109527tion_a @ ( some_P7306582813780734732tion_a @ F3 ) @ ( some_P7306582813780734732tion_a @ F3 ) )
@ T2 ) )
@ ( image_3379191556159813523_a_nat
@ ( produc1049598122668174515_a_nat
@ ^ [F3: produc3509355604313844263tion_a] : ( produc4015641737766485984_a_nat @ ( produc3538477569256109527tion_a @ ( some_P7306582813780734732tion_a @ F3 ) @ ( some_P7306582813780734732tion_a @ F3 ) ) ) )
@ F2 ) ) ) ).
% gterm_funas_lift_RR2_funas
thf(fact_557_gterm__funas__lift__RR2__funas,axiom,
! [T2: ground_gterm_a,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( ground_funas_gterm_a @ T2 ) @ F2 )
=> ( ord_le5649184635953113020_a_nat
@ ( ground5706566889460337840tion_a
@ ( ground3050120939345217807tion_a
@ ^ [F3: a] : ( produc9011544418120257559tion_a @ ( some_a @ F3 ) @ ( some_a @ F3 ) )
@ T2 ) )
@ ( image_3729436797865474090_a_nat
@ ( produc4004610663780792650_a_nat
@ ^ [F3: a] : ( produc5457283509376734880_a_nat @ ( produc9011544418120257559tion_a @ ( some_a @ F3 ) @ ( some_a @ F3 ) ) ) )
@ F2 ) ) ) ).
% gterm_funas_lift_RR2_funas
thf(fact_558_RR2__gterm__unlift__funas,axiom,
! [T2: ground4005964956264846616_a_nat,F2: set_Pr5612011003857075140at_nat] :
( ( ord_le8609766837066453116at_nat @ ( ground7958862358551846640_a_nat @ T2 )
@ ( image_6008886954644801541at_nat
@ ( produc8166206427995057559at_nat
@ ^ [F3: set_Pr4934435412358123699_a_nat] : ( produc6380145652274861664at_nat @ ( produc1207969925626950559_a_nat @ ( some_s1880800141159441806_a_nat @ F3 ) @ ( some_s1880800141159441806_a_nat @ F3 ) ) ) )
@ F2 ) )
=> ( ord_le3946384098489951076at_nat
@ ( ground2178884368406916842_a_nat
@ ( ground9183630286821479476_a_nat
@ ^ [F3: produc7618478223271187943_a_nat] : ( the_se1682649557849046815_a_nat @ ( produc735037054355678091_a_nat @ F3 ) )
@ T2 ) )
@ F2 ) ) ).
% RR2_gterm_unlift_funas
thf(fact_559_RR2__gterm__unlift__funas,axiom,
! [T2: ground3180571006387129706_a_nat,F2: set_Pr2737808803024040429at_nat] :
( ( ord_le8394821662120075470at_nat @ ( ground7469869589483366466_a_nat @ T2 )
@ ( image_1902767331843210624at_nat
@ ( produc1847350529710776082at_nat
@ ^ [F3: set_Pr8660066742269822812_a_nat] : ( produc2385804702627905458at_nat @ ( produc8861491750707134833_a_nat @ ( some_s2314368709036572727_a_nat @ F3 ) @ ( some_s2314368709036572727_a_nat @ F3 ) ) ) )
@ F2 ) )
=> ( ord_le221046818917144973at_nat
@ ( ground2506338072319351187_a_nat
@ ( ground9154320000392521263_a_nat
@ ^ [F3: produc4107345600652478265_a_nat] : ( the_se5449783460274439624_a_nat @ ( produc4143784723386224989_a_nat @ F3 ) )
@ T2 ) )
@ F2 ) ) ).
% RR2_gterm_unlift_funas
thf(fact_560_RR2__gterm__unlift__funas,axiom,
! [T2: ground4831483382674469784tion_a,F2: set_Pr8660066742269822812_a_nat] :
( ( ord_le6272267606287294716_a_nat @ ( ground2948544073636793840tion_a @ T2 )
@ ( image_3379191556159813523_a_nat
@ ( produc1049598122668174515_a_nat
@ ^ [F3: produc3509355604313844263tion_a] : ( produc4015641737766485984_a_nat @ ( produc3538477569256109527tion_a @ ( some_P7306582813780734732tion_a @ F3 ) @ ( some_P7306582813780734732tion_a @ F3 ) ) ) )
@ F2 ) )
=> ( ord_le5649184635953113020_a_nat
@ ( ground5706566889460337840tion_a
@ ( ground2612807628724573414tion_a
@ ^ [F3: produc314695012608889703tion_a] : ( the_Pr8472423629061403323tion_a @ ( produc6169883020650379203tion_a @ F3 ) )
@ T2 ) )
@ F2 ) ) ).
% RR2_gterm_unlift_funas
thf(fact_561_RR2__gterm__unlift__funas,axiom,
! [T2: ground5314827683623246424tion_a,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le5649184635953113020_a_nat @ ( ground5706566889460337840tion_a @ T2 )
@ ( image_3729436797865474090_a_nat
@ ( produc4004610663780792650_a_nat
@ ^ [F3: a] : ( produc5457283509376734880_a_nat @ ( produc9011544418120257559tion_a @ ( some_a @ F3 ) @ ( some_a @ F3 ) ) ) )
@ F2 ) )
=> ( ord_le8666007276011122963_a_nat
@ ( ground_funas_gterm_a
@ ( ground161688327512779517on_a_a
@ ^ [F3: produc3509355604313844263tion_a] : ( the_a @ ( produc174239612569448963tion_a @ F3 ) )
@ T2 ) )
@ F2 ) ) ).
% RR2_gterm_unlift_funas
thf(fact_562_Sup_OSUP__identity__eq,axiom,
! [Sup: set_o > $o,A2: set_o] :
( ( Sup
@ ( image_o_o
@ ^ [X2: $o] : X2
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_563_Inf_OINF__identity__eq,axiom,
! [Inf: set_o > $o,A2: set_o] :
( ( Inf
@ ( image_o_o
@ ^ [X2: $o] : X2
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_564_funas__gterm__map__gterm,axiom,
! [T2: ground5314827683623246424tion_a,F2: set_Pr8660066742269822812_a_nat,F: produc3509355604313844263tion_a > a] :
( ( ord_le5649184635953113020_a_nat @ ( ground5706566889460337840tion_a @ T2 ) @ F2 )
=> ( ord_le8666007276011122963_a_nat @ ( ground_funas_gterm_a @ ( ground161688327512779517on_a_a @ F @ T2 ) )
@ ( image_6737578356684938026_a_nat
@ ( produc7713474374650187850_a_nat
@ ^ [H2: produc3509355604313844263tion_a] : ( product_Pair_a_nat @ ( F @ H2 ) ) )
@ F2 ) ) ) ).
% funas_gterm_map_gterm
thf(fact_565_funas__gterm__map__gterm,axiom,
! [T2: ground_gterm_a,F2: set_Pr4934435412358123699_a_nat,F: a > a] :
( ( ord_le8666007276011122963_a_nat @ ( ground_funas_gterm_a @ T2 ) @ F2 )
=> ( ord_le8666007276011122963_a_nat @ ( ground_funas_gterm_a @ ( ground_map_gterm_a_a @ F @ T2 ) )
@ ( image_8708701149064048897_a_nat
@ ( produc8738682720600764961_a_nat
@ ^ [H2: a] : ( product_Pair_a_nat @ ( F @ H2 ) ) )
@ F2 ) ) ) ).
% funas_gterm_map_gterm
thf(fact_566_funas__gterm__map__gterm,axiom,
! [T2: ground_gterm_a,F2: set_Pr4934435412358123699_a_nat,F: a > produc3509355604313844263tion_a] :
( ( ord_le8666007276011122963_a_nat @ ( ground_funas_gterm_a @ T2 ) @ F2 )
=> ( ord_le5649184635953113020_a_nat @ ( ground5706566889460337840tion_a @ ( ground3050120939345217807tion_a @ F @ T2 ) )
@ ( image_3729436797865474090_a_nat
@ ( produc4004610663780792650_a_nat
@ ^ [H2: a] : ( produc5457283509376734880_a_nat @ ( F @ H2 ) ) )
@ F2 ) ) ) ).
% funas_gterm_map_gterm
thf(fact_567_funas__gterm__map__gterm,axiom,
! [T2: ground5314827683623246424tion_a,F2: set_Pr8660066742269822812_a_nat,F: produc3509355604313844263tion_a > produc3509355604313844263tion_a] :
( ( ord_le5649184635953113020_a_nat @ ( ground5706566889460337840tion_a @ T2 ) @ F2 )
=> ( ord_le5649184635953113020_a_nat @ ( ground5706566889460337840tion_a @ ( ground6258472780936250918tion_a @ F @ T2 ) )
@ ( image_4137284907621822035_a_nat
@ ( produc5088488592959153011_a_nat
@ ^ [H2: produc3509355604313844263tion_a] : ( produc5457283509376734880_a_nat @ ( F @ H2 ) ) )
@ F2 ) ) ) ).
% funas_gterm_map_gterm
thf(fact_568_fst__diag__fst,axiom,
( ( comp_P3618168059068810855tion_a @ produc174239612569448963tion_a
@ ( comp_o4150946686980370657tion_a
@ ^ [X2: option_a] : ( produc9011544418120257559tion_a @ X2 @ X2 )
@ produc174239612569448963tion_a ) )
= produc174239612569448963tion_a ) ).
% fst_diag_fst
thf(fact_569_gterms__mono,axiom,
! [G2: set_Pr8660066742269822812_a_nat,F2: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ G2 @ F2 )
=> ( ord_le1985992432954399960tion_a
@ ( collec8661210613136333635tion_a
@ ^ [S4: ground5314827683623246424tion_a] : ( ord_le5649184635953113020_a_nat @ ( ground5706566889460337840tion_a @ S4 ) @ G2 ) )
@ ( collec8661210613136333635tion_a
@ ^ [S4: ground5314827683623246424tion_a] : ( ord_le5649184635953113020_a_nat @ ( ground5706566889460337840tion_a @ S4 ) @ F2 ) ) ) ) ).
% gterms_mono
thf(fact_570_gterms__mono,axiom,
! [G2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ G2 @ F2 )
=> ( ord_le2380367239515552431term_a
@ ( collec3522850604242857114term_a
@ ^ [S4: ground_gterm_a] : ( ord_le8666007276011122963_a_nat @ ( ground_funas_gterm_a @ S4 ) @ G2 ) )
@ ( collec3522850604242857114term_a
@ ^ [S4: ground_gterm_a] : ( ord_le8666007276011122963_a_nat @ ( ground_funas_gterm_a @ S4 ) @ F2 ) ) ) ) ).
% gterms_mono
thf(fact_571_subset__Collect__iff,axiom,
! [B2: set_Pr7585778909603769095tion_a,A2: set_Pr7585778909603769095tion_a,P2: produc3509355604313844263tion_a > $o] :
( ( ord_le4471550158292877991tion_a @ B2 @ A2 )
=> ( ( ord_le4471550158292877991tion_a @ B2
@ ( collec4135126896892755346tion_a
@ ^ [X2: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X2 @ A2 )
& ( P2 @ X2 ) ) ) )
= ( ! [X2: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X2 @ B2 )
=> ( P2 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_572_subset__Collect__iff,axiom,
! [B2: set_Pr8660066742269822812_a_nat,A2: set_Pr8660066742269822812_a_nat,P2: produc4760857364844987814_a_nat > $o] :
( ( ord_le5649184635953113020_a_nat @ B2 @ A2 )
=> ( ( ord_le5649184635953113020_a_nat @ B2
@ ( collec7438131855006933115_a_nat
@ ^ [X2: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X2 @ A2 )
& ( P2 @ X2 ) ) ) )
= ( ! [X2: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X2 @ B2 )
=> ( P2 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_573_subset__Collect__iff,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,P2: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A2 )
& ( P2 @ X2 ) ) ) )
= ( ! [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ B2 )
=> ( P2 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_574_subset__Collect__iff,axiom,
! [B2: set_o,A2: set_o,P2: $o > $o] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ( ord_less_eq_set_o @ B2
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P2 @ X2 ) ) ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ B2 )
=> ( P2 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_575_subset__CollectI,axiom,
! [B2: set_Pr7585778909603769095tion_a,A2: set_Pr7585778909603769095tion_a,Q: produc3509355604313844263tion_a > $o,P2: produc3509355604313844263tion_a > $o] :
( ( ord_le4471550158292877991tion_a @ B2 @ A2 )
=> ( ! [X3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_le4471550158292877991tion_a
@ ( collec4135126896892755346tion_a
@ ^ [X2: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X2 @ B2 )
& ( Q @ X2 ) ) )
@ ( collec4135126896892755346tion_a
@ ^ [X2: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X2 @ A2 )
& ( P2 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_576_subset__CollectI,axiom,
! [B2: set_Pr8660066742269822812_a_nat,A2: set_Pr8660066742269822812_a_nat,Q: produc4760857364844987814_a_nat > $o,P2: produc4760857364844987814_a_nat > $o] :
( ( ord_le5649184635953113020_a_nat @ B2 @ A2 )
=> ( ! [X3: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_le5649184635953113020_a_nat
@ ( collec7438131855006933115_a_nat
@ ^ [X2: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X2 @ B2 )
& ( Q @ X2 ) ) )
@ ( collec7438131855006933115_a_nat
@ ^ [X2: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X2 @ A2 )
& ( P2 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_577_subset__CollectI,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,Q: product_prod_a_nat > $o,P2: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_le8666007276011122963_a_nat
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ B2 )
& ( Q @ X2 ) ) )
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A2 )
& ( P2 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_578_subset__CollectI,axiom,
! [B2: set_o,A2: set_o,Q: $o > $o,P2: $o > $o] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ B2 )
& ( Q @ X2 ) ) )
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P2 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_579_conj__subset__def,axiom,
! [A2: set_Pr8660066742269822812_a_nat,P2: produc4760857364844987814_a_nat > $o,Q: produc4760857364844987814_a_nat > $o] :
( ( ord_le5649184635953113020_a_nat @ A2
@ ( collec7438131855006933115_a_nat
@ ^ [X2: produc4760857364844987814_a_nat] :
( ( P2 @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le5649184635953113020_a_nat @ A2 @ ( collec7438131855006933115_a_nat @ P2 ) )
& ( ord_le5649184635953113020_a_nat @ A2 @ ( collec7438131855006933115_a_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_580_conj__subset__def,axiom,
! [A2: set_Pr4934435412358123699_a_nat,P2: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ A2
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( P2 @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le8666007276011122963_a_nat @ A2 @ ( collec4464134535221767506_a_nat @ P2 ) )
& ( ord_le8666007276011122963_a_nat @ A2 @ ( collec4464134535221767506_a_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_581_conj__subset__def,axiom,
! [A2: set_o,P2: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A2
@ ( collect_o
@ ^ [X2: $o] :
( ( P2 @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_o @ A2 @ ( collect_o @ P2 ) )
& ( ord_less_eq_set_o @ A2 @ ( collect_o @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_582_image__paired__Times,axiom,
! [F: $o > $o,G: $o > $o,A2: set_o,B2: set_o] :
( ( image_9131363867636255685od_o_o
@ ( produc7436348682273225467od_o_o
@ ^ [X2: $o,Y5: $o] : ( product_Pair_o_o @ ( F @ X2 ) @ ( G @ Y5 ) ) )
@ ( product_Sigma_o_o @ A2
@ ^ [Uu: $o] : B2 ) )
= ( product_Sigma_o_o @ ( image_o_o @ F @ A2 )
@ ^ [Uu: $o] : ( image_o_o @ G @ B2 ) ) ) ).
% image_paired_Times
thf(fact_583_image__paired__Times,axiom,
! [F: produc3509355604313844263tion_a > $o,G: $o > $o,A2: set_Pr7585778909603769095tion_a,B2: set_o] :
( ( image_1602282915056835540od_o_o
@ ( produc2888798419338443658od_o_o
@ ^ [X2: produc3509355604313844263tion_a,Y5: $o] : ( product_Pair_o_o @ ( F @ X2 ) @ ( G @ Y5 ) ) )
@ ( produc8823275029784582905on_a_o @ A2
@ ^ [Uu: produc3509355604313844263tion_a] : B2 ) )
= ( product_Sigma_o_o @ ( image_166006535297930294on_a_o @ F @ A2 )
@ ^ [Uu: $o] : ( image_o_o @ G @ B2 ) ) ) ).
% image_paired_Times
thf(fact_584_image__paired__Times,axiom,
! [F: $o > $o,G: produc3509355604313844263tion_a > $o,A2: set_o,B2: set_Pr7585778909603769095tion_a] :
( ( image_7124636254000440694od_o_o
@ ( produc8450651747251506092od_o_o
@ ^ [X2: $o,Y5: produc3509355604313844263tion_a] : ( product_Pair_o_o @ ( F @ X2 ) @ ( G @ Y5 ) ) )
@ ( produc61899281709682263tion_a @ A2
@ ^ [Uu: $o] : B2 ) )
= ( product_Sigma_o_o @ ( image_o_o @ F @ A2 )
@ ^ [Uu: $o] : ( image_166006535297930294on_a_o @ G @ B2 ) ) ) ).
% image_paired_Times
thf(fact_585_image__paired__Times,axiom,
! [F: a > produc3509355604313844263tion_a,G: nat > nat,A2: set_a,B2: set_nat] :
( ( image_3729436797865474090_a_nat
@ ( produc4004610663780792650_a_nat
@ ^ [X2: a,Y5: nat] : ( produc5457283509376734880_a_nat @ ( F @ X2 ) @ ( G @ Y5 ) ) )
@ ( product_Sigma_a_nat @ A2
@ ^ [Uu: a] : B2 ) )
= ( produc2066629144795197103_a_nat @ ( image_7468189554597481134tion_a @ F @ A2 )
@ ^ [Uu: produc3509355604313844263tion_a] : ( image_nat_nat @ G @ B2 ) ) ) ).
% image_paired_Times
thf(fact_586_image__paired__Times,axiom,
! [F: option_a > option_a,G: option_a > option_a,A2: set_option_a,B2: set_option_a] :
( ( image_6668519322520640453tion_a
@ ( produc2781663484249587963tion_a
@ ^ [X2: option_a,Y5: option_a] : ( produc9011544418120257559tion_a @ ( F @ X2 ) @ ( G @ Y5 ) ) )
@ ( produc269287337874323144tion_a @ A2
@ ^ [Uu: option_a] : B2 ) )
= ( produc269287337874323144tion_a @ ( image_7439109396645324421tion_a @ F @ A2 )
@ ^ [Uu: option_a] : ( image_7439109396645324421tion_a @ G @ B2 ) ) ) ).
% image_paired_Times
thf(fact_587_image__paired__Times,axiom,
! [F: produc4760857364844987814_a_nat > $o,G: $o > $o,A2: set_Pr8660066742269822812_a_nat,B2: set_o] :
( ( image_8314802100926819769od_o_o
@ ( produc2029383703971365601od_o_o
@ ^ [X2: produc4760857364844987814_a_nat,Y5: $o] : ( product_Pair_o_o @ ( F @ X2 ) @ ( G @ Y5 ) ) )
@ ( produc7403436832878706216_nat_o @ A2
@ ^ [Uu: produc4760857364844987814_a_nat] : B2 ) )
= ( product_Sigma_o_o @ ( image_6768402879995552939_nat_o @ F @ A2 )
@ ^ [Uu: $o] : ( image_o_o @ G @ B2 ) ) ) ).
% image_paired_Times
thf(fact_588_image__paired__Times,axiom,
! [F: $o > $o,G: produc4760857364844987814_a_nat > $o,A2: set_o,B2: set_Pr8660066742269822812_a_nat] :
( ( image_4350507979490927539od_o_o
@ ( produc7267942115636050707od_o_o
@ ^ [X2: $o,Y5: produc4760857364844987814_a_nat] : ( product_Pair_o_o @ ( F @ X2 ) @ ( G @ Y5 ) ) )
@ ( produc2867166158663066998_a_nat @ A2
@ ^ [Uu: $o] : B2 ) )
= ( product_Sigma_o_o @ ( image_o_o @ F @ A2 )
@ ^ [Uu: $o] : ( image_6768402879995552939_nat_o @ G @ B2 ) ) ) ).
% image_paired_Times
thf(fact_589_image__paired__Times,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,G: $o > $o,A2: set_Pr4934435412358123699_a_nat,B2: set_o] :
( ( image_2753023396963955134_nat_o
@ ( produc4223001627282770582_nat_o
@ ^ [X2: product_prod_a_nat,Y5: $o] : ( produc9152484117443933593_nat_o @ ( F @ X2 ) @ ( G @ Y5 ) ) )
@ ( produc8052552943834900881_nat_o @ A2
@ ^ [Uu: product_prod_a_nat] : B2 ) )
= ( produc7403436832878706216_nat_o @ ( image_3729436797865474090_a_nat @ F @ A2 )
@ ^ [Uu: produc4760857364844987814_a_nat] : ( image_o_o @ G @ B2 ) ) ) ).
% image_paired_Times
thf(fact_590_image__paired__Times,axiom,
! [F: produc3509355604313844263tion_a > $o,G: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a,B2: set_Pr7585778909603769095tion_a] :
( ( image_646863362436313605od_o_o
@ ( produc4615123733411498171od_o_o
@ ^ [X2: produc3509355604313844263tion_a,Y5: produc3509355604313844263tion_a] : ( product_Pair_o_o @ ( F @ X2 ) @ ( G @ Y5 ) ) )
@ ( produc7093159808355054856tion_a @ A2
@ ^ [Uu: produc3509355604313844263tion_a] : B2 ) )
= ( product_Sigma_o_o @ ( image_166006535297930294on_a_o @ F @ A2 )
@ ^ [Uu: $o] : ( image_166006535297930294on_a_o @ G @ B2 ) ) ) ).
% image_paired_Times
thf(fact_591_image__paired__Times,axiom,
! [F: $o > $o,G: product_prod_a_nat > produc4760857364844987814_a_nat,A2: set_o,B2: set_Pr4934435412358123699_a_nat] :
( ( image_7697852917414831288_a_nat
@ ( produc3055971631918832856_a_nat
@ ^ [X2: $o,Y5: product_prod_a_nat] : ( produc4616213443228294375_a_nat @ ( F @ X2 ) @ ( G @ Y5 ) ) )
@ ( produc3092463322541264461_a_nat @ A2
@ ^ [Uu: $o] : B2 ) )
= ( produc2867166158663066998_a_nat @ ( image_o_o @ F @ A2 )
@ ^ [Uu: $o] : ( image_3729436797865474090_a_nat @ G @ B2 ) ) ) ).
% image_paired_Times
thf(fact_592_SigmaI,axiom,
! [A: $o,A2: set_o,B: $o,B2: $o > set_o] :
( ( member_o @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A2 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_593_SigmaI,axiom,
! [A: a,A2: set_a,B: nat,B2: a > set_nat] :
( ( member_a @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B ) @ ( product_Sigma_a_nat @ A2 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_594_SigmaI,axiom,
! [A: $o,A2: set_o,B: product_prod_a_nat,B2: $o > set_Pr4934435412358123699_a_nat] :
( ( member_o @ A @ A2 )
=> ( ( member5724188588386418708_a_nat @ B @ ( B2 @ A ) )
=> ( member26132018921490651_a_nat @ ( produc7367187609789241790_a_nat @ A @ B ) @ ( produc3092463322541264461_a_nat @ A2 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_595_SigmaI,axiom,
! [A: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,B: $o,B2: product_prod_a_nat > set_o] :
( ( member5724188588386418708_a_nat @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member1850667382371020647_nat_o @ ( produc3103905194228102402_nat_o @ A @ B ) @ ( produc8052552943834900881_nat_o @ A2 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_596_SigmaI,axiom,
! [A: option_a,A2: set_option_a,B: option_a,B2: option_a > set_option_a] :
( ( member_option_a @ A @ A2 )
=> ( ( member_option_a @ B @ ( B2 @ A ) )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ ( produc269287337874323144tion_a @ A2 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_597_SigmaI,axiom,
! [A: $o,A2: set_o,B: produc3509355604313844263tion_a,B2: $o > set_Pr7585778909603769095tion_a] :
( ( member_o @ A @ A2 )
=> ( ( member5498148017924304208tion_a @ B @ ( B2 @ A ) )
=> ( member2654175073444177119tion_a @ ( produc1801025925631506470tion_a @ A @ B ) @ ( produc61899281709682263tion_a @ A2 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_598_SigmaI,axiom,
! [A: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat,B2: product_prod_a_nat > set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A @ A2 )
=> ( ( member5724188588386418708_a_nat @ B @ ( B2 @ A ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( produc3832836940518797764_a_nat @ A2 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_599_SigmaI,axiom,
! [A: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a,B: $o,B2: produc3509355604313844263tion_a > set_o] :
( ( member5498148017924304208tion_a @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member10471143963986689on_a_o @ ( produc1339029636851631304on_a_o @ A @ B ) @ ( produc8823275029784582905on_a_o @ A2 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_600_SigmaI,axiom,
! [A: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a,B: nat,B2: produc3509355604313844263tion_a > set_nat] :
( ( member5498148017924304208tion_a @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ A @ B ) @ ( produc2066629144795197103_a_nat @ A2 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_601_SigmaI,axiom,
! [A: $o,A2: set_o,B: produc4760857364844987814_a_nat,B2: $o > set_Pr8660066742269822812_a_nat] :
( ( member_o @ A @ A2 )
=> ( ( member1430561565885860669_a_nat @ B @ ( B2 @ A ) )
=> ( member9172549829962833796_a_nat @ ( produc4616213443228294375_a_nat @ A @ B ) @ ( produc2867166158663066998_a_nat @ A2 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_602_mem__Sigma__iff,axiom,
! [A: $o,B: $o,A2: set_o,B2: $o > set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A2 @ B2 ) )
= ( ( member_o @ A @ A2 )
& ( member_o @ B @ ( B2 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_603_mem__Sigma__iff,axiom,
! [A: a,B: nat,A2: set_a,B2: a > set_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B ) @ ( product_Sigma_a_nat @ A2 @ B2 ) )
= ( ( member_a @ A @ A2 )
& ( member_nat @ B @ ( B2 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_604_mem__Sigma__iff,axiom,
! [A: $o,B: product_prod_a_nat,A2: set_o,B2: $o > set_Pr4934435412358123699_a_nat] :
( ( member26132018921490651_a_nat @ ( produc7367187609789241790_a_nat @ A @ B ) @ ( produc3092463322541264461_a_nat @ A2 @ B2 ) )
= ( ( member_o @ A @ A2 )
& ( member5724188588386418708_a_nat @ B @ ( B2 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_605_mem__Sigma__iff,axiom,
! [A: product_prod_a_nat,B: $o,A2: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat > set_o] :
( ( member1850667382371020647_nat_o @ ( produc3103905194228102402_nat_o @ A @ B ) @ ( produc8052552943834900881_nat_o @ A2 @ B2 ) )
= ( ( member5724188588386418708_a_nat @ A @ A2 )
& ( member_o @ B @ ( B2 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_606_mem__Sigma__iff,axiom,
! [A: option_a,B: option_a,A2: set_option_a,B2: option_a > set_option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ ( produc269287337874323144tion_a @ A2 @ B2 ) )
= ( ( member_option_a @ A @ A2 )
& ( member_option_a @ B @ ( B2 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_607_mem__Sigma__iff,axiom,
! [A: $o,B: produc3509355604313844263tion_a,A2: set_o,B2: $o > set_Pr7585778909603769095tion_a] :
( ( member2654175073444177119tion_a @ ( produc1801025925631506470tion_a @ A @ B ) @ ( produc61899281709682263tion_a @ A2 @ B2 ) )
= ( ( member_o @ A @ A2 )
& ( member5498148017924304208tion_a @ B @ ( B2 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_608_mem__Sigma__iff,axiom,
! [A: product_prod_a_nat,B: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat > set_Pr4934435412358123699_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( produc3832836940518797764_a_nat @ A2 @ B2 ) )
= ( ( member5724188588386418708_a_nat @ A @ A2 )
& ( member5724188588386418708_a_nat @ B @ ( B2 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_609_mem__Sigma__iff,axiom,
! [A: produc3509355604313844263tion_a,B: $o,A2: set_Pr7585778909603769095tion_a,B2: produc3509355604313844263tion_a > set_o] :
( ( member10471143963986689on_a_o @ ( produc1339029636851631304on_a_o @ A @ B ) @ ( produc8823275029784582905on_a_o @ A2 @ B2 ) )
= ( ( member5498148017924304208tion_a @ A @ A2 )
& ( member_o @ B @ ( B2 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_610_mem__Sigma__iff,axiom,
! [A: produc3509355604313844263tion_a,B: nat,A2: set_Pr7585778909603769095tion_a,B2: produc3509355604313844263tion_a > set_nat] :
( ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ A @ B ) @ ( produc2066629144795197103_a_nat @ A2 @ B2 ) )
= ( ( member5498148017924304208tion_a @ A @ A2 )
& ( member_nat @ B @ ( B2 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_611_mem__Sigma__iff,axiom,
! [A: $o,B: produc4760857364844987814_a_nat,A2: set_o,B2: $o > set_Pr8660066742269822812_a_nat] :
( ( member9172549829962833796_a_nat @ ( produc4616213443228294375_a_nat @ A @ B ) @ ( produc2867166158663066998_a_nat @ A2 @ B2 ) )
= ( ( member_o @ A @ A2 )
& ( member1430561565885860669_a_nat @ B @ ( B2 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_612_Collect__case__prod,axiom,
! [P2: option_a > $o,Q: option_a > $o] :
( ( collec4135126896892755346tion_a
@ ( produc7248784972643495020on_a_o
@ ^ [A6: option_a,B6: option_a] :
( ( P2 @ A6 )
& ( Q @ B6 ) ) ) )
= ( produc269287337874323144tion_a @ ( collect_option_a @ P2 )
@ ^ [Uu: option_a] : ( collect_option_a @ Q ) ) ) ).
% Collect_case_prod
thf(fact_613_Collect__case__prod,axiom,
! [P2: produc3509355604313844263tion_a > $o,Q: nat > $o] :
( ( collec7438131855006933115_a_nat
@ ( produc7988141324749014923_nat_o
@ ^ [A6: produc3509355604313844263tion_a,B6: nat] :
( ( P2 @ A6 )
& ( Q @ B6 ) ) ) )
= ( produc2066629144795197103_a_nat @ ( collec4135126896892755346tion_a @ P2 )
@ ^ [Uu: produc3509355604313844263tion_a] : ( collect_nat @ Q ) ) ) ).
% Collect_case_prod
thf(fact_614_SigmaE,axiom,
! [C: product_prod_o_o,A2: set_o,B2: $o > set_o] :
( ( member7466972457876170832od_o_o @ C @ ( product_Sigma_o_o @ A2 @ B2 ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ! [Y3: $o] :
( ( member_o @ Y3 @ ( B2 @ X3 ) )
=> ( C
!= ( product_Pair_o_o @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_615_SigmaE,axiom,
! [C: product_prod_a_nat,A2: set_a,B2: a > set_nat] :
( ( member5724188588386418708_a_nat @ C @ ( product_Sigma_a_nat @ A2 @ B2 ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( B2 @ X3 ) )
=> ( C
!= ( product_Pair_a_nat @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_616_SigmaE,axiom,
! [C: produc2710310701403425476_a_nat,A2: set_o,B2: $o > set_Pr4934435412358123699_a_nat] :
( ( member26132018921490651_a_nat @ C @ ( produc3092463322541264461_a_nat @ A2 @ B2 ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ! [Y3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Y3 @ ( B2 @ X3 ) )
=> ( C
!= ( produc7367187609789241790_a_nat @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_617_SigmaE,axiom,
! [C: produc4534846064852955472_nat_o,A2: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat > set_o] :
( ( member1850667382371020647_nat_o @ C @ ( produc8052552943834900881_nat_o @ A2 @ B2 ) )
=> ~ ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A2 )
=> ! [Y3: $o] :
( ( member_o @ Y3 @ ( B2 @ X3 ) )
=> ( C
!= ( produc3103905194228102402_nat_o @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_618_SigmaE,axiom,
! [C: produc3509355604313844263tion_a,A2: set_option_a,B2: option_a > set_option_a] :
( ( member5498148017924304208tion_a @ C @ ( produc269287337874323144tion_a @ A2 @ B2 ) )
=> ~ ! [X3: option_a] :
( ( member_option_a @ X3 @ A2 )
=> ! [Y3: option_a] :
( ( member_option_a @ Y3 @ ( B2 @ X3 ) )
=> ( C
!= ( produc9011544418120257559tion_a @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_619_SigmaE,axiom,
! [C: produc4838336064122203702tion_a,A2: set_o,B2: $o > set_Pr7585778909603769095tion_a] :
( ( member2654175073444177119tion_a @ C @ ( produc61899281709682263tion_a @ A2 @ B2 ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ! [Y3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ Y3 @ ( B2 @ X3 ) )
=> ( C
!= ( produc1801025925631506470tion_a @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_620_SigmaE,axiom,
! [C: produc4708774622424448987_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat > set_Pr4934435412358123699_a_nat] :
( ( member9062615507155100804_a_nat @ C @ ( produc3832836940518797764_a_nat @ A2 @ B2 ) )
=> ~ ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A2 )
=> ! [Y3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Y3 @ ( B2 @ X3 ) )
=> ( C
!= ( produc2026711137822539155_a_nat @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_621_SigmaE,axiom,
! [C: produc2194632134642013272on_a_o,A2: set_Pr7585778909603769095tion_a,B2: produc3509355604313844263tion_a > set_o] :
( ( member10471143963986689on_a_o @ C @ ( produc8823275029784582905on_a_o @ A2 @ B2 ) )
=> ~ ! [X3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X3 @ A2 )
=> ! [Y3: $o] :
( ( member_o @ Y3 @ ( B2 @ X3 ) )
=> ( C
!= ( produc1339029636851631304on_a_o @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_622_SigmaE,axiom,
! [C: produc4760857364844987814_a_nat,A2: set_Pr7585778909603769095tion_a,B2: produc3509355604313844263tion_a > set_nat] :
( ( member1430561565885860669_a_nat @ C @ ( produc2066629144795197103_a_nat @ A2 @ B2 ) )
=> ~ ! [X3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X3 @ A2 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( B2 @ X3 ) )
=> ( C
!= ( produc5457283509376734880_a_nat @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_623_SigmaE,axiom,
! [C: produc6654178220768718189_a_nat,A2: set_o,B2: $o > set_Pr8660066742269822812_a_nat] :
( ( member9172549829962833796_a_nat @ C @ ( produc2867166158663066998_a_nat @ A2 @ B2 ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ! [Y3: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ Y3 @ ( B2 @ X3 ) )
=> ( C
!= ( produc4616213443228294375_a_nat @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_624_SigmaD1,axiom,
! [A: produc3509355604313844263tion_a,B: nat,A2: set_Pr7585778909603769095tion_a,B2: produc3509355604313844263tion_a > set_nat] :
( ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ A @ B ) @ ( produc2066629144795197103_a_nat @ A2 @ B2 ) )
=> ( member5498148017924304208tion_a @ A @ A2 ) ) ).
% SigmaD1
thf(fact_625_SigmaD1,axiom,
! [A: option_a,B: option_a,A2: set_option_a,B2: option_a > set_option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ ( produc269287337874323144tion_a @ A2 @ B2 ) )
=> ( member_option_a @ A @ A2 ) ) ).
% SigmaD1
thf(fact_626_SigmaD1,axiom,
! [A: a,B: nat,A2: set_a,B2: a > set_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B ) @ ( product_Sigma_a_nat @ A2 @ B2 ) )
=> ( member_a @ A @ A2 ) ) ).
% SigmaD1
thf(fact_627_SigmaD2,axiom,
! [A: produc3509355604313844263tion_a,B: nat,A2: set_Pr7585778909603769095tion_a,B2: produc3509355604313844263tion_a > set_nat] :
( ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ A @ B ) @ ( produc2066629144795197103_a_nat @ A2 @ B2 ) )
=> ( member_nat @ B @ ( B2 @ A ) ) ) ).
% SigmaD2
thf(fact_628_SigmaD2,axiom,
! [A: option_a,B: option_a,A2: set_option_a,B2: option_a > set_option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ ( produc269287337874323144tion_a @ A2 @ B2 ) )
=> ( member_option_a @ B @ ( B2 @ A ) ) ) ).
% SigmaD2
thf(fact_629_SigmaD2,axiom,
! [A: a,B: nat,A2: set_a,B2: a > set_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B ) @ ( product_Sigma_a_nat @ A2 @ B2 ) )
=> ( member_nat @ B @ ( B2 @ A ) ) ) ).
% SigmaD2
thf(fact_630_SigmaE2,axiom,
! [A: $o,B: $o,A2: set_o,B2: $o > set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A2 @ B2 ) )
=> ~ ( ( member_o @ A @ A2 )
=> ~ ( member_o @ B @ ( B2 @ A ) ) ) ) ).
% SigmaE2
thf(fact_631_SigmaE2,axiom,
! [A: a,B: nat,A2: set_a,B2: a > set_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B ) @ ( product_Sigma_a_nat @ A2 @ B2 ) )
=> ~ ( ( member_a @ A @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ A ) ) ) ) ).
% SigmaE2
thf(fact_632_SigmaE2,axiom,
! [A: $o,B: product_prod_a_nat,A2: set_o,B2: $o > set_Pr4934435412358123699_a_nat] :
( ( member26132018921490651_a_nat @ ( produc7367187609789241790_a_nat @ A @ B ) @ ( produc3092463322541264461_a_nat @ A2 @ B2 ) )
=> ~ ( ( member_o @ A @ A2 )
=> ~ ( member5724188588386418708_a_nat @ B @ ( B2 @ A ) ) ) ) ).
% SigmaE2
thf(fact_633_SigmaE2,axiom,
! [A: product_prod_a_nat,B: $o,A2: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat > set_o] :
( ( member1850667382371020647_nat_o @ ( produc3103905194228102402_nat_o @ A @ B ) @ ( produc8052552943834900881_nat_o @ A2 @ B2 ) )
=> ~ ( ( member5724188588386418708_a_nat @ A @ A2 )
=> ~ ( member_o @ B @ ( B2 @ A ) ) ) ) ).
% SigmaE2
thf(fact_634_SigmaE2,axiom,
! [A: option_a,B: option_a,A2: set_option_a,B2: option_a > set_option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ ( produc269287337874323144tion_a @ A2 @ B2 ) )
=> ~ ( ( member_option_a @ A @ A2 )
=> ~ ( member_option_a @ B @ ( B2 @ A ) ) ) ) ).
% SigmaE2
thf(fact_635_SigmaE2,axiom,
! [A: $o,B: produc3509355604313844263tion_a,A2: set_o,B2: $o > set_Pr7585778909603769095tion_a] :
( ( member2654175073444177119tion_a @ ( produc1801025925631506470tion_a @ A @ B ) @ ( produc61899281709682263tion_a @ A2 @ B2 ) )
=> ~ ( ( member_o @ A @ A2 )
=> ~ ( member5498148017924304208tion_a @ B @ ( B2 @ A ) ) ) ) ).
% SigmaE2
thf(fact_636_SigmaE2,axiom,
! [A: product_prod_a_nat,B: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat > set_Pr4934435412358123699_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( produc3832836940518797764_a_nat @ A2 @ B2 ) )
=> ~ ( ( member5724188588386418708_a_nat @ A @ A2 )
=> ~ ( member5724188588386418708_a_nat @ B @ ( B2 @ A ) ) ) ) ).
% SigmaE2
thf(fact_637_SigmaE2,axiom,
! [A: produc3509355604313844263tion_a,B: $o,A2: set_Pr7585778909603769095tion_a,B2: produc3509355604313844263tion_a > set_o] :
( ( member10471143963986689on_a_o @ ( produc1339029636851631304on_a_o @ A @ B ) @ ( produc8823275029784582905on_a_o @ A2 @ B2 ) )
=> ~ ( ( member5498148017924304208tion_a @ A @ A2 )
=> ~ ( member_o @ B @ ( B2 @ A ) ) ) ) ).
% SigmaE2
thf(fact_638_SigmaE2,axiom,
! [A: produc3509355604313844263tion_a,B: nat,A2: set_Pr7585778909603769095tion_a,B2: produc3509355604313844263tion_a > set_nat] :
( ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ A @ B ) @ ( produc2066629144795197103_a_nat @ A2 @ B2 ) )
=> ~ ( ( member5498148017924304208tion_a @ A @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ A ) ) ) ) ).
% SigmaE2
thf(fact_639_SigmaE2,axiom,
! [A: $o,B: produc4760857364844987814_a_nat,A2: set_o,B2: $o > set_Pr8660066742269822812_a_nat] :
( ( member9172549829962833796_a_nat @ ( produc4616213443228294375_a_nat @ A @ B ) @ ( produc2867166158663066998_a_nat @ A2 @ B2 ) )
=> ~ ( ( member_o @ A @ A2 )
=> ~ ( member1430561565885860669_a_nat @ B @ ( B2 @ A ) ) ) ) ).
% SigmaE2
thf(fact_640_Collect__case__prod__Sigma,axiom,
! [P2: option_a > $o,Q: option_a > option_a > $o] :
( ( collec4135126896892755346tion_a
@ ( produc7248784972643495020on_a_o
@ ^ [X2: option_a,Y5: option_a] :
( ( P2 @ X2 )
& ( Q @ X2 @ Y5 ) ) ) )
= ( produc269287337874323144tion_a @ ( collect_option_a @ P2 )
@ ^ [X2: option_a] : ( collect_option_a @ ( Q @ X2 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_641_Collect__case__prod__Sigma,axiom,
! [P2: produc3509355604313844263tion_a > $o,Q: produc3509355604313844263tion_a > nat > $o] :
( ( collec7438131855006933115_a_nat
@ ( produc7988141324749014923_nat_o
@ ^ [X2: produc3509355604313844263tion_a,Y5: nat] :
( ( P2 @ X2 )
& ( Q @ X2 @ Y5 ) ) ) )
= ( produc2066629144795197103_a_nat @ ( collec4135126896892755346tion_a @ P2 )
@ ^ [X2: produc3509355604313844263tion_a] : ( collect_nat @ ( Q @ X2 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_642_Sigma__mono,axiom,
! [A2: set_a,C2: set_a,B2: a > set_nat,D: a > set_nat] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( product_Sigma_a_nat @ A2 @ B2 ) @ ( product_Sigma_a_nat @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_643_Sigma__mono,axiom,
! [A2: set_o,C2: set_o,B2: $o > set_o,D: $o > set_o] :
( ( ord_less_eq_set_o @ A2 @ C2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le192195778054999463od_o_o @ ( product_Sigma_o_o @ A2 @ B2 ) @ ( product_Sigma_o_o @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_644_Sigma__mono,axiom,
! [A2: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat > set_o,D: product_prod_a_nat > set_o] :
( ( ord_le8666007276011122963_a_nat @ A2 @ C2 )
=> ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le5431503644745719014_nat_o @ ( produc8052552943834900881_nat_o @ A2 @ B2 ) @ ( produc8052552943834900881_nat_o @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_645_Sigma__mono,axiom,
! [A2: set_o,C2: set_o,B2: $o > set_Pr4934435412358123699_a_nat,D: $o > set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_set_o @ A2 @ C2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_le8666007276011122963_a_nat @ ( B2 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le3075358075201600602_a_nat @ ( produc3092463322541264461_a_nat @ A2 @ B2 ) @ ( produc3092463322541264461_a_nat @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_646_Sigma__mono,axiom,
! [A2: set_Pr7585778909603769095tion_a,C2: set_Pr7585778909603769095tion_a,B2: produc3509355604313844263tion_a > set_nat,D: produc3509355604313844263tion_a > set_nat] :
( ( ord_le4471550158292877991tion_a @ A2 @ C2 )
=> ( ! [X3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le5649184635953113020_a_nat @ ( produc2066629144795197103_a_nat @ A2 @ B2 ) @ ( produc2066629144795197103_a_nat @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_647_Sigma__mono,axiom,
! [A2: set_Pr7585778909603769095tion_a,C2: set_Pr7585778909603769095tion_a,B2: produc3509355604313844263tion_a > set_o,D: produc3509355604313844263tion_a > set_o] :
( ( ord_le4471550158292877991tion_a @ A2 @ C2 )
=> ( ! [X3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le6981016051002401240on_a_o @ ( produc8823275029784582905on_a_o @ A2 @ B2 ) @ ( produc8823275029784582905on_a_o @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_648_Sigma__mono,axiom,
! [A2: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat > set_Pr4934435412358123699_a_nat,D: product_prod_a_nat > set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ C2 )
=> ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A2 )
=> ( ord_le8666007276011122963_a_nat @ ( B2 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le1018141841545957851_a_nat @ ( produc3832836940518797764_a_nat @ A2 @ B2 ) @ ( produc3832836940518797764_a_nat @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_649_Sigma__mono,axiom,
! [A2: set_Pr7585778909603769095tion_a,C2: set_Pr7585778909603769095tion_a,B2: produc3509355604313844263tion_a > set_Pr4934435412358123699_a_nat,D: produc3509355604313844263tion_a > set_Pr4934435412358123699_a_nat] :
( ( ord_le4471550158292877991tion_a @ A2 @ C2 )
=> ( ! [X3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X3 @ A2 )
=> ( ord_le8666007276011122963_a_nat @ ( B2 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le2854690519574212457_a_nat @ ( produc1276521302575469148_a_nat @ A2 @ B2 ) @ ( produc1276521302575469148_a_nat @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_650_Sigma__mono,axiom,
! [A2: set_Pr8660066742269822812_a_nat,C2: set_Pr8660066742269822812_a_nat,B2: produc4760857364844987814_a_nat > set_o,D: produc4760857364844987814_a_nat > set_o] :
( ( ord_le5649184635953113020_a_nat @ A2 @ C2 )
=> ( ! [X3: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le8405921517615413757_nat_o @ ( produc7403436832878706216_nat_o @ A2 @ B2 ) @ ( produc7403436832878706216_nat_o @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_651_Sigma__mono,axiom,
! [A2: set_o,C2: set_o,B2: $o > set_Pr8660066742269822812_a_nat,D: $o > set_Pr8660066742269822812_a_nat] :
( ( ord_less_eq_set_o @ A2 @ C2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_le5649184635953113020_a_nat @ ( B2 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le3880997233133534851_a_nat @ ( produc2867166158663066998_a_nat @ A2 @ B2 ) @ ( produc2867166158663066998_a_nat @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_652_Times__subset__cancel2,axiom,
! [X: $o,C2: set_o,A2: set_o,B2: set_o] :
( ( member_o @ X @ C2 )
=> ( ( ord_le192195778054999463od_o_o
@ ( product_Sigma_o_o @ A2
@ ^ [Uu: $o] : C2 )
@ ( product_Sigma_o_o @ B2
@ ^ [Uu: $o] : C2 ) )
= ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_653_Times__subset__cancel2,axiom,
! [X: nat,C2: set_nat,A2: set_a,B2: set_a] :
( ( member_nat @ X @ C2 )
=> ( ( ord_le8666007276011122963_a_nat
@ ( product_Sigma_a_nat @ A2
@ ^ [Uu: a] : C2 )
@ ( product_Sigma_a_nat @ B2
@ ^ [Uu: a] : C2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_654_Times__subset__cancel2,axiom,
! [X: $o,C2: set_o,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( member_o @ X @ C2 )
=> ( ( ord_le5431503644745719014_nat_o
@ ( produc8052552943834900881_nat_o @ A2
@ ^ [Uu: product_prod_a_nat] : C2 )
@ ( produc8052552943834900881_nat_o @ B2
@ ^ [Uu: product_prod_a_nat] : C2 ) )
= ( ord_le8666007276011122963_a_nat @ A2 @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_655_Times__subset__cancel2,axiom,
! [X: product_prod_a_nat,C2: set_Pr4934435412358123699_a_nat,A2: set_o,B2: set_o] :
( ( member5724188588386418708_a_nat @ X @ C2 )
=> ( ( ord_le3075358075201600602_a_nat
@ ( produc3092463322541264461_a_nat @ A2
@ ^ [Uu: $o] : C2 )
@ ( produc3092463322541264461_a_nat @ B2
@ ^ [Uu: $o] : C2 ) )
= ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_656_Times__subset__cancel2,axiom,
! [X: product_prod_a_nat,C2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X @ C2 )
=> ( ( ord_le1018141841545957851_a_nat
@ ( produc3832836940518797764_a_nat @ A2
@ ^ [Uu: product_prod_a_nat] : C2 )
@ ( produc3832836940518797764_a_nat @ B2
@ ^ [Uu: product_prod_a_nat] : C2 ) )
= ( ord_le8666007276011122963_a_nat @ A2 @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_657_Times__subset__cancel2,axiom,
! [X: produc3509355604313844263tion_a,C2: set_Pr7585778909603769095tion_a,A2: set_o,B2: set_o] :
( ( member5498148017924304208tion_a @ X @ C2 )
=> ( ( ord_le8133926203214654646tion_a
@ ( produc61899281709682263tion_a @ A2
@ ^ [Uu: $o] : C2 )
@ ( produc61899281709682263tion_a @ B2
@ ^ [Uu: $o] : C2 ) )
= ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_658_Times__subset__cancel2,axiom,
! [X: nat,C2: set_nat,A2: set_Pr7585778909603769095tion_a,B2: set_Pr7585778909603769095tion_a] :
( ( member_nat @ X @ C2 )
=> ( ( ord_le5649184635953113020_a_nat
@ ( produc2066629144795197103_a_nat @ A2
@ ^ [Uu: produc3509355604313844263tion_a] : C2 )
@ ( produc2066629144795197103_a_nat @ B2
@ ^ [Uu: produc3509355604313844263tion_a] : C2 ) )
= ( ord_le4471550158292877991tion_a @ A2 @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_659_Times__subset__cancel2,axiom,
! [X: $o,C2: set_o,A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ( member_o @ X @ C2 )
=> ( ( ord_le8405921517615413757_nat_o
@ ( produc7403436832878706216_nat_o @ A2
@ ^ [Uu: produc4760857364844987814_a_nat] : C2 )
@ ( produc7403436832878706216_nat_o @ B2
@ ^ [Uu: produc4760857364844987814_a_nat] : C2 ) )
= ( ord_le5649184635953113020_a_nat @ A2 @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_660_Times__subset__cancel2,axiom,
! [X: produc3509355604313844263tion_a,C2: set_Pr7585778909603769095tion_a,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( member5498148017924304208tion_a @ X @ C2 )
=> ( ( ord_le7563727043670204533tion_a
@ ( produc4431804887258552224tion_a @ A2
@ ^ [Uu: product_prod_a_nat] : C2 )
@ ( produc4431804887258552224tion_a @ B2
@ ^ [Uu: product_prod_a_nat] : C2 ) )
= ( ord_le8666007276011122963_a_nat @ A2 @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_661_Times__subset__cancel2,axiom,
! [X: produc4760857364844987814_a_nat,C2: set_Pr8660066742269822812_a_nat,A2: set_o,B2: set_o] :
( ( member1430561565885860669_a_nat @ X @ C2 )
=> ( ( ord_le3880997233133534851_a_nat
@ ( produc2867166158663066998_a_nat @ A2
@ ^ [Uu: $o] : C2 )
@ ( produc2867166158663066998_a_nat @ B2
@ ^ [Uu: $o] : C2 ) )
= ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_662_swap__product,axiom,
! [A2: set_nat,B2: set_Pr7585778909603769095tion_a] :
( ( image_9075947210971128057_a_nat
@ ( produc9185384946573455057_a_nat
@ ^ [I: nat,J: produc3509355604313844263tion_a] : ( produc5457283509376734880_a_nat @ J @ I ) )
@ ( produc9070904722673932301tion_a @ A2
@ ^ [Uu: nat] : B2 ) )
= ( produc2066629144795197103_a_nat @ B2
@ ^ [Uu: produc3509355604313844263tion_a] : A2 ) ) ).
% swap_product
thf(fact_663_swap__product,axiom,
! [A2: set_nat,B2: set_a] :
( ( image_5148427718660454439_a_nat
@ ( produc732906326552059263_a_nat
@ ^ [I: nat,J: a] : ( product_Pair_a_nat @ J @ I ) )
@ ( product_Sigma_nat_a @ A2
@ ^ [Uu: nat] : B2 ) )
= ( product_Sigma_a_nat @ B2
@ ^ [Uu: a] : A2 ) ) ).
% swap_product
thf(fact_664_swap__product,axiom,
! [A2: set_option_a,B2: set_option_a] :
( ( image_6668519322520640453tion_a
@ ( produc2781663484249587963tion_a
@ ^ [I: option_a,J: option_a] : ( produc9011544418120257559tion_a @ J @ I ) )
@ ( produc269287337874323144tion_a @ A2
@ ^ [Uu: option_a] : B2 ) )
= ( produc269287337874323144tion_a @ B2
@ ^ [Uu: option_a] : A2 ) ) ).
% swap_product
thf(fact_665_Gr__incl,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F: product_prod_a_nat > produc4760857364844987814_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ( ord_le4239986153908076548_a_nat @ ( bNF_Gr3785214186540711495_a_nat @ A2 @ F )
@ ( produc8234572646870318317_a_nat @ A2
@ ^ [Uu: product_prod_a_nat] : B2 ) )
= ( ord_le5649184635953113020_a_nat @ ( image_3729436797865474090_a_nat @ F @ A2 ) @ B2 ) ) ).
% Gr_incl
thf(fact_666_Gr__incl,axiom,
! [A2: set_Pr7585778909603769095tion_a,F: produc3509355604313844263tion_a > $o,B2: set_o] :
( ( ord_le6981016051002401240on_a_o @ ( bNF_Gr8237857062489970771on_a_o @ A2 @ F )
@ ( produc8823275029784582905on_a_o @ A2
@ ^ [Uu: produc3509355604313844263tion_a] : B2 ) )
= ( ord_less_eq_set_o @ ( image_166006535297930294on_a_o @ F @ A2 ) @ B2 ) ) ).
% Gr_incl
thf(fact_667_Gr__incl,axiom,
! [A2: set_Pr8660066742269822812_a_nat,F: produc4760857364844987814_a_nat > $o,B2: set_o] :
( ( ord_le8405921517615413757_nat_o @ ( bNF_Gr2973908843748354126_nat_o @ A2 @ F )
@ ( produc7403436832878706216_nat_o @ A2
@ ^ [Uu: produc4760857364844987814_a_nat] : B2 ) )
= ( ord_less_eq_set_o @ ( image_6768402879995552939_nat_o @ F @ A2 ) @ B2 ) ) ).
% Gr_incl
thf(fact_668_Gr__incl,axiom,
! [A2: set_o,F: $o > $o,B2: set_o] :
( ( ord_le192195778054999463od_o_o @ ( bNF_Gr_o_o @ A2 @ F )
@ ( product_Sigma_o_o @ A2
@ ^ [Uu: $o] : B2 ) )
= ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B2 ) ) ).
% Gr_incl
thf(fact_669_Gr__incl,axiom,
! [A2: set_Pr7585778909603769095tion_a,F: produc3509355604313844263tion_a > nat,B2: set_nat] :
( ( ord_le5649184635953113020_a_nat @ ( bNF_Gr7863039376269261781_a_nat @ A2 @ F )
@ ( produc2066629144795197103_a_nat @ A2
@ ^ [Uu: produc3509355604313844263tion_a] : B2 ) )
= ( ord_less_eq_set_nat @ ( image_5787351454808800946_a_nat @ F @ A2 ) @ B2 ) ) ).
% Gr_incl
thf(fact_670_Gr__incl,axiom,
! [A2: set_a,F: a > nat,B2: set_nat] :
( ( ord_le8666007276011122963_a_nat @ ( bNF_Gr_a_nat @ A2 @ F )
@ ( product_Sigma_a_nat @ A2
@ ^ [Uu: a] : B2 ) )
= ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A2 ) @ B2 ) ) ).
% Gr_incl
thf(fact_671_fst__fstOp,axiom,
! [P2: option_a > option_a > $o,Q: option_a > option_a > $o] :
( produc174239612569448963tion_a
= ( comp_P3618168059068810855tion_a @ produc174239612569448963tion_a @ ( bNF_fs7976035448042954642tion_a @ P2 @ Q ) ) ) ).
% fst_fstOp
thf(fact_672_fst__comp__apfst,axiom,
! [F: option_a > option_a] :
( ( comp_P3618168059068810855tion_a @ produc174239612569448963tion_a @ ( produc1685019323022372872tion_a @ F ) )
= ( comp_o511183382729970692tion_a @ F @ produc174239612569448963tion_a ) ) ).
% fst_comp_apfst
thf(fact_673_member__product,axiom,
! [X: product_prod_a_nat,A2: set_a,B2: set_nat] :
( ( member5724188588386418708_a_nat @ X @ ( produc6032010372060166652_a_nat @ A2 @ B2 ) )
= ( member5724188588386418708_a_nat @ X
@ ( product_Sigma_a_nat @ A2
@ ^ [Uu: a] : B2 ) ) ) ).
% member_product
thf(fact_674_member__product,axiom,
! [X: produc4760857364844987814_a_nat,A2: set_Pr7585778909603769095tion_a,B2: set_nat] :
( ( member1430561565885860669_a_nat @ X @ ( produc167207870343747877_a_nat @ A2 @ B2 ) )
= ( member1430561565885860669_a_nat @ X
@ ( produc2066629144795197103_a_nat @ A2
@ ^ [Uu: produc3509355604313844263tion_a] : B2 ) ) ) ).
% member_product
thf(fact_675_member__product,axiom,
! [X: produc3509355604313844263tion_a,A2: set_option_a,B2: set_option_a] :
( ( member5498148017924304208tion_a @ X @ ( produc4142764644767291346tion_a @ A2 @ B2 ) )
= ( member5498148017924304208tion_a @ X
@ ( produc269287337874323144tion_a @ A2
@ ^ [Uu: option_a] : B2 ) ) ) ).
% member_product
thf(fact_676_snd__diag__fst,axiom,
( ( comp_P3618168059068810855tion_a @ produc8952039381521079109tion_a
@ ( comp_o4150946686980370657tion_a
@ ^ [X2: option_a] : ( produc9011544418120257559tion_a @ X2 @ X2 )
@ produc174239612569448963tion_a ) )
= produc174239612569448963tion_a ) ).
% snd_diag_fst
thf(fact_677_fst__image__times,axiom,
! [B2: set_option_a,A2: set_option_a] :
( ( ( B2 = bot_bot_set_option_a )
=> ( ( image_7272866627766095330tion_a @ produc174239612569448963tion_a
@ ( produc269287337874323144tion_a @ A2
@ ^ [Uu: option_a] : B2 ) )
= bot_bot_set_option_a ) )
& ( ( B2 != bot_bot_set_option_a )
=> ( ( image_7272866627766095330tion_a @ produc174239612569448963tion_a
@ ( produc269287337874323144tion_a @ A2
@ ^ [Uu: option_a] : B2 ) )
= A2 ) ) ) ).
% fst_image_times
thf(fact_678_all__not__in__conv,axiom,
! [A2: set_o] :
( ( ! [X2: $o] :
~ ( member_o @ X2 @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_679_all__not__in__conv,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( ! [X2: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ X2 @ A2 ) )
= ( A2 = bot_bo9049108969261143879_a_nat ) ) ).
% all_not_in_conv
thf(fact_680_all__not__in__conv,axiom,
! [A2: set_Pr8660066742269822812_a_nat] :
( ( ! [X2: produc4760857364844987814_a_nat] :
~ ( member1430561565885860669_a_nat @ X2 @ A2 ) )
= ( A2 = bot_bo8420108762324073968_a_nat ) ) ).
% all_not_in_conv
thf(fact_681_all__not__in__conv,axiom,
! [A2: set_Pr7585778909603769095tion_a] :
( ( ! [X2: produc3509355604313844263tion_a] :
~ ( member5498148017924304208tion_a @ X2 @ A2 ) )
= ( A2 = bot_bo235252021745139059tion_a ) ) ).
% all_not_in_conv
thf(fact_682_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_683_empty__iff,axiom,
! [C: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ C @ bot_bo9049108969261143879_a_nat ) ).
% empty_iff
thf(fact_684_empty__iff,axiom,
! [C: produc4760857364844987814_a_nat] :
~ ( member1430561565885860669_a_nat @ C @ bot_bo8420108762324073968_a_nat ) ).
% empty_iff
thf(fact_685_empty__iff,axiom,
! [C: produc3509355604313844263tion_a] :
~ ( member5498148017924304208tion_a @ C @ bot_bo235252021745139059tion_a ) ).
% empty_iff
thf(fact_686_image__is__empty,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ( image_3729436797865474090_a_nat @ F @ A2 )
= bot_bo8420108762324073968_a_nat )
= ( A2 = bot_bo9049108969261143879_a_nat ) ) ).
% image_is_empty
thf(fact_687_image__is__empty,axiom,
! [F: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a] :
( ( ( image_166006535297930294on_a_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bo235252021745139059tion_a ) ) ).
% image_is_empty
thf(fact_688_image__is__empty,axiom,
! [F: produc4760857364844987814_a_nat > $o,A2: set_Pr8660066742269822812_a_nat] :
( ( ( image_6768402879995552939_nat_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bo8420108762324073968_a_nat ) ) ).
% image_is_empty
thf(fact_689_image__is__empty,axiom,
! [F: $o > $o,A2: set_o] :
( ( ( image_o_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_690_empty__is__image,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( bot_bo8420108762324073968_a_nat
= ( image_3729436797865474090_a_nat @ F @ A2 ) )
= ( A2 = bot_bo9049108969261143879_a_nat ) ) ).
% empty_is_image
thf(fact_691_empty__is__image,axiom,
! [F: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a] :
( ( bot_bot_set_o
= ( image_166006535297930294on_a_o @ F @ A2 ) )
= ( A2 = bot_bo235252021745139059tion_a ) ) ).
% empty_is_image
thf(fact_692_empty__is__image,axiom,
! [F: produc4760857364844987814_a_nat > $o,A2: set_Pr8660066742269822812_a_nat] :
( ( bot_bot_set_o
= ( image_6768402879995552939_nat_o @ F @ A2 ) )
= ( A2 = bot_bo8420108762324073968_a_nat ) ) ).
% empty_is_image
thf(fact_693_empty__is__image,axiom,
! [F: $o > $o,A2: set_o] :
( ( bot_bot_set_o
= ( image_o_o @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_694_image__empty,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat] :
( ( image_3729436797865474090_a_nat @ F @ bot_bo9049108969261143879_a_nat )
= bot_bo8420108762324073968_a_nat ) ).
% image_empty
thf(fact_695_image__empty,axiom,
! [F: produc3509355604313844263tion_a > $o] :
( ( image_166006535297930294on_a_o @ F @ bot_bo235252021745139059tion_a )
= bot_bot_set_o ) ).
% image_empty
thf(fact_696_image__empty,axiom,
! [F: produc4760857364844987814_a_nat > $o] :
( ( image_6768402879995552939_nat_o @ F @ bot_bo8420108762324073968_a_nat )
= bot_bot_set_o ) ).
% image_empty
thf(fact_697_image__empty,axiom,
! [F: $o > $o] :
( ( image_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_698_empty__subsetI,axiom,
! [A2: set_Pr8660066742269822812_a_nat] : ( ord_le5649184635953113020_a_nat @ bot_bo8420108762324073968_a_nat @ A2 ) ).
% empty_subsetI
thf(fact_699_empty__subsetI,axiom,
! [A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A2 ) ).
% empty_subsetI
thf(fact_700_empty__subsetI,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% empty_subsetI
thf(fact_701_subset__empty,axiom,
! [A2: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ A2 @ bot_bo8420108762324073968_a_nat )
= ( A2 = bot_bo8420108762324073968_a_nat ) ) ).
% subset_empty
thf(fact_702_subset__empty,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ bot_bo9049108969261143879_a_nat )
= ( A2 = bot_bo9049108969261143879_a_nat ) ) ).
% subset_empty
thf(fact_703_subset__empty,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_704_apfst__conv,axiom,
! [F: produc3509355604313844263tion_a > produc3509355604313844263tion_a,X: produc3509355604313844263tion_a,Y: nat] :
( ( produc1522032229839045964_a_nat @ F @ ( produc5457283509376734880_a_nat @ X @ Y ) )
= ( produc5457283509376734880_a_nat @ ( F @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_705_apfst__conv,axiom,
! [F: produc3509355604313844263tion_a > a,X: produc3509355604313844263tion_a,Y: nat] :
( ( produc1282494340376965667_a_nat @ F @ ( produc5457283509376734880_a_nat @ X @ Y ) )
= ( product_Pair_a_nat @ ( F @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_706_apfst__conv,axiom,
! [F: option_a > option_a,X: option_a,Y: option_a] :
( ( produc1685019323022372872tion_a @ F @ ( produc9011544418120257559tion_a @ X @ Y ) )
= ( produc9011544418120257559tion_a @ ( F @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_707_apfst__conv,axiom,
! [F: a > produc3509355604313844263tion_a,X: a,Y: nat] :
( ( produc6204860511972624437_a_nat @ F @ ( product_Pair_a_nat @ X @ Y ) )
= ( produc5457283509376734880_a_nat @ ( F @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_708_apfst__conv,axiom,
! [F: a > a,X: a,Y: nat] :
( ( produc7773453608010410252_a_nat @ F @ ( product_Pair_a_nat @ X @ Y ) )
= ( product_Pair_a_nat @ ( F @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_709_fst__apfst,axiom,
! [F: option_a > option_a,X: produc3509355604313844263tion_a] :
( ( produc174239612569448963tion_a @ ( produc1685019323022372872tion_a @ F @ X ) )
= ( F @ ( produc174239612569448963tion_a @ X ) ) ) ).
% fst_apfst
thf(fact_710_prod_Ocollapse,axiom,
! [Prod: produc4760857364844987814_a_nat] :
( ( produc5457283509376734880_a_nat @ ( produc1094642028913761716_a_nat @ Prod ) @ ( produc8051327305810483698_a_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_711_prod_Ocollapse,axiom,
! [Prod: product_prod_a_nat] :
( ( product_Pair_a_nat @ ( product_fst_a_nat @ Prod ) @ ( product_snd_a_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_712_prod_Ocollapse,axiom,
! [Prod: produc3509355604313844263tion_a] :
( ( produc9011544418120257559tion_a @ ( produc174239612569448963tion_a @ Prod ) @ ( produc8952039381521079109tion_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_713_ex__in__conv,axiom,
! [A2: set_o] :
( ( ? [X2: $o] : ( member_o @ X2 @ A2 ) )
= ( A2 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_714_ex__in__conv,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( ? [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ A2 ) )
= ( A2 != bot_bo9049108969261143879_a_nat ) ) ).
% ex_in_conv
thf(fact_715_ex__in__conv,axiom,
! [A2: set_Pr8660066742269822812_a_nat] :
( ( ? [X2: produc4760857364844987814_a_nat] : ( member1430561565885860669_a_nat @ X2 @ A2 ) )
= ( A2 != bot_bo8420108762324073968_a_nat ) ) ).
% ex_in_conv
thf(fact_716_ex__in__conv,axiom,
! [A2: set_Pr7585778909603769095tion_a] :
( ( ? [X2: produc3509355604313844263tion_a] : ( member5498148017924304208tion_a @ X2 @ A2 ) )
= ( A2 != bot_bo235252021745139059tion_a ) ) ).
% ex_in_conv
thf(fact_717_equals0I,axiom,
! [A2: set_o] :
( ! [Y3: $o] :
~ ( member_o @ Y3 @ A2 )
=> ( A2 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_718_equals0I,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ! [Y3: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ Y3 @ A2 )
=> ( A2 = bot_bo9049108969261143879_a_nat ) ) ).
% equals0I
thf(fact_719_equals0I,axiom,
! [A2: set_Pr8660066742269822812_a_nat] :
( ! [Y3: produc4760857364844987814_a_nat] :
~ ( member1430561565885860669_a_nat @ Y3 @ A2 )
=> ( A2 = bot_bo8420108762324073968_a_nat ) ) ).
% equals0I
thf(fact_720_equals0I,axiom,
! [A2: set_Pr7585778909603769095tion_a] :
( ! [Y3: produc3509355604313844263tion_a] :
~ ( member5498148017924304208tion_a @ Y3 @ A2 )
=> ( A2 = bot_bo235252021745139059tion_a ) ) ).
% equals0I
thf(fact_721_equals0D,axiom,
! [A2: set_o,A: $o] :
( ( A2 = bot_bot_set_o )
=> ~ ( member_o @ A @ A2 ) ) ).
% equals0D
thf(fact_722_equals0D,axiom,
! [A2: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat] :
( ( A2 = bot_bo9049108969261143879_a_nat )
=> ~ ( member5724188588386418708_a_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_723_equals0D,axiom,
! [A2: set_Pr8660066742269822812_a_nat,A: produc4760857364844987814_a_nat] :
( ( A2 = bot_bo8420108762324073968_a_nat )
=> ~ ( member1430561565885860669_a_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_724_equals0D,axiom,
! [A2: set_Pr7585778909603769095tion_a,A: produc3509355604313844263tion_a] :
( ( A2 = bot_bo235252021745139059tion_a )
=> ~ ( member5498148017924304208tion_a @ A @ A2 ) ) ).
% equals0D
thf(fact_725_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_726_emptyE,axiom,
! [A: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ A @ bot_bo9049108969261143879_a_nat ) ).
% emptyE
thf(fact_727_emptyE,axiom,
! [A: produc4760857364844987814_a_nat] :
~ ( member1430561565885860669_a_nat @ A @ bot_bo8420108762324073968_a_nat ) ).
% emptyE
thf(fact_728_emptyE,axiom,
! [A: produc3509355604313844263tion_a] :
~ ( member5498148017924304208tion_a @ A @ bot_bo235252021745139059tion_a ) ).
% emptyE
thf(fact_729_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ A @ bot_bo8420108762324073968_a_nat )
=> ( A = bot_bo8420108762324073968_a_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_730_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ bot_bo9049108969261143879_a_nat )
=> ( A = bot_bo9049108969261143879_a_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_731_bot_Oextremum__uniqueI,axiom,
! [A: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ A @ bot_bo3602513548093573143_a_nat )
=> ( A = bot_bo3602513548093573143_a_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_732_bot_Oextremum__uniqueI,axiom,
! [A: option2973356794535198124_a_nat] :
( ( ord_le8848857416346193420_a_nat @ A @ bot_bo2302117019856176704_a_nat )
=> ( A = bot_bo2302117019856176704_a_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_733_bot_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
=> ( A = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
thf(fact_734_bot_Oextremum__unique,axiom,
! [A: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ A @ bot_bo8420108762324073968_a_nat )
= ( A = bot_bo8420108762324073968_a_nat ) ) ).
% bot.extremum_unique
thf(fact_735_bot_Oextremum__unique,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ bot_bo9049108969261143879_a_nat )
= ( A = bot_bo9049108969261143879_a_nat ) ) ).
% bot.extremum_unique
thf(fact_736_bot_Oextremum__unique,axiom,
! [A: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ A @ bot_bo3602513548093573143_a_nat )
= ( A = bot_bo3602513548093573143_a_nat ) ) ).
% bot.extremum_unique
thf(fact_737_bot_Oextremum__unique,axiom,
! [A: option2973356794535198124_a_nat] :
( ( ord_le8848857416346193420_a_nat @ A @ bot_bo2302117019856176704_a_nat )
= ( A = bot_bo2302117019856176704_a_nat ) ) ).
% bot.extremum_unique
thf(fact_738_bot_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% bot.extremum_unique
thf(fact_739_bot_Oextremum,axiom,
! [A: set_Pr8660066742269822812_a_nat] : ( ord_le5649184635953113020_a_nat @ bot_bo8420108762324073968_a_nat @ A ) ).
% bot.extremum
thf(fact_740_bot_Oextremum,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A ) ).
% bot.extremum
thf(fact_741_bot_Oextremum,axiom,
! [A: option5987904771724668803_a_nat] : ( ord_le1496697172603502563_a_nat @ bot_bo3602513548093573143_a_nat @ A ) ).
% bot.extremum
thf(fact_742_bot_Oextremum,axiom,
! [A: option2973356794535198124_a_nat] : ( ord_le8848857416346193420_a_nat @ bot_bo2302117019856176704_a_nat @ A ) ).
% bot.extremum
thf(fact_743_bot_Oextremum,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% bot.extremum
thf(fact_744_snd__conv,axiom,
! [X12: produc3509355604313844263tion_a,X22: nat] :
( ( produc8051327305810483698_a_nat @ ( produc5457283509376734880_a_nat @ X12 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_745_snd__conv,axiom,
! [X12: option_a,X22: option_a] :
( ( produc8952039381521079109tion_a @ ( produc9011544418120257559tion_a @ X12 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_746_snd__conv,axiom,
! [X12: a,X22: nat] :
( ( product_snd_a_nat @ ( product_Pair_a_nat @ X12 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_747_snd__eqD,axiom,
! [X: produc3509355604313844263tion_a,Y: nat,A: nat] :
( ( ( produc8051327305810483698_a_nat @ ( produc5457283509376734880_a_nat @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_748_snd__eqD,axiom,
! [X: option_a,Y: option_a,A: option_a] :
( ( ( produc8952039381521079109tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_749_snd__eqD,axiom,
! [X: a,Y: nat,A: nat] :
( ( ( product_snd_a_nat @ ( product_Pair_a_nat @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_750_eq__snd__iff,axiom,
! [B: nat,P: produc4760857364844987814_a_nat] :
( ( B
= ( produc8051327305810483698_a_nat @ P ) )
= ( ? [A6: produc3509355604313844263tion_a] :
( P
= ( produc5457283509376734880_a_nat @ A6 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_751_eq__snd__iff,axiom,
! [B: option_a,P: produc3509355604313844263tion_a] :
( ( B
= ( produc8952039381521079109tion_a @ P ) )
= ( ? [A6: option_a] :
( P
= ( produc9011544418120257559tion_a @ A6 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_752_eq__snd__iff,axiom,
! [B: nat,P: product_prod_a_nat] :
( ( B
= ( product_snd_a_nat @ P ) )
= ( ? [A6: a] :
( P
= ( product_Pair_a_nat @ A6 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_753_sndI,axiom,
! [X: produc4760857364844987814_a_nat,Y: produc3509355604313844263tion_a,Z: nat] :
( ( X
= ( produc5457283509376734880_a_nat @ Y @ Z ) )
=> ( ( produc8051327305810483698_a_nat @ X )
= Z ) ) ).
% sndI
thf(fact_754_sndI,axiom,
! [X: produc3509355604313844263tion_a,Y: option_a,Z: option_a] :
( ( X
= ( produc9011544418120257559tion_a @ Y @ Z ) )
=> ( ( produc8952039381521079109tion_a @ X )
= Z ) ) ).
% sndI
thf(fact_755_sndI,axiom,
! [X: product_prod_a_nat,Y: a,Z: nat] :
( ( X
= ( product_Pair_a_nat @ Y @ Z ) )
=> ( ( product_snd_a_nat @ X )
= Z ) ) ).
% sndI
thf(fact_756_prod__eq__iff,axiom,
( ( ^ [Y4: produc3509355604313844263tion_a,Z2: produc3509355604313844263tion_a] : ( Y4 = Z2 ) )
= ( ^ [S4: produc3509355604313844263tion_a,T: produc3509355604313844263tion_a] :
( ( ( produc174239612569448963tion_a @ S4 )
= ( produc174239612569448963tion_a @ T ) )
& ( ( produc8952039381521079109tion_a @ S4 )
= ( produc8952039381521079109tion_a @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_757_prod__eqI,axiom,
! [P: produc3509355604313844263tion_a,Q2: produc3509355604313844263tion_a] :
( ( ( produc174239612569448963tion_a @ P )
= ( produc174239612569448963tion_a @ Q2 ) )
=> ( ( ( produc8952039381521079109tion_a @ P )
= ( produc8952039381521079109tion_a @ Q2 ) )
=> ( P = Q2 ) ) ) ).
% prod_eqI
thf(fact_758_prod_Oexpand,axiom,
! [Prod: produc3509355604313844263tion_a,Prod2: produc3509355604313844263tion_a] :
( ( ( ( produc174239612569448963tion_a @ Prod )
= ( produc174239612569448963tion_a @ Prod2 ) )
& ( ( produc8952039381521079109tion_a @ Prod )
= ( produc8952039381521079109tion_a @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_759_subset__emptyI,axiom,
! [A2: set_Pr7585778909603769095tion_a] :
( ! [X3: produc3509355604313844263tion_a] :
~ ( member5498148017924304208tion_a @ X3 @ A2 )
=> ( ord_le4471550158292877991tion_a @ A2 @ bot_bo235252021745139059tion_a ) ) ).
% subset_emptyI
thf(fact_760_subset__emptyI,axiom,
! [A2: set_Pr8660066742269822812_a_nat] :
( ! [X3: produc4760857364844987814_a_nat] :
~ ( member1430561565885860669_a_nat @ X3 @ A2 )
=> ( ord_le5649184635953113020_a_nat @ A2 @ bot_bo8420108762324073968_a_nat ) ) ).
% subset_emptyI
thf(fact_761_subset__emptyI,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ! [X3: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ X3 @ A2 )
=> ( ord_le8666007276011122963_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) ).
% subset_emptyI
thf(fact_762_subset__emptyI,axiom,
! [A2: set_o] :
( ! [X3: $o] :
~ ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_o @ A2 @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_763_snd__def,axiom,
( produc8952039381521079109tion_a
= ( produc6953443519211250200tion_a
@ ^ [X1: option_a,X23: option_a] : X23 ) ) ).
% snd_def
thf(fact_764_GrD2,axiom,
! [X: produc3509355604313844263tion_a,Fx: nat,A2: set_Pr7585778909603769095tion_a,F: produc3509355604313844263tion_a > nat] :
( ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ X @ Fx ) @ ( bNF_Gr7863039376269261781_a_nat @ A2 @ F ) )
=> ( ( F @ X )
= Fx ) ) ).
% GrD2
thf(fact_765_GrD2,axiom,
! [X: option_a,Fx: option_a,A2: set_option_a,F: option_a > option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Fx ) @ ( bNF_Gr2342861688195830050tion_a @ A2 @ F ) )
=> ( ( F @ X )
= Fx ) ) ).
% GrD2
thf(fact_766_GrD2,axiom,
! [X: a,Fx: nat,A2: set_a,F: a > nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Fx ) @ ( bNF_Gr_a_nat @ A2 @ F ) )
=> ( ( F @ X )
= Fx ) ) ).
% GrD2
thf(fact_767_GrD1,axiom,
! [X: produc3509355604313844263tion_a,Fx: nat,A2: set_Pr7585778909603769095tion_a,F: produc3509355604313844263tion_a > nat] :
( ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ X @ Fx ) @ ( bNF_Gr7863039376269261781_a_nat @ A2 @ F ) )
=> ( member5498148017924304208tion_a @ X @ A2 ) ) ).
% GrD1
thf(fact_768_GrD1,axiom,
! [X: option_a,Fx: option_a,A2: set_option_a,F: option_a > option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Fx ) @ ( bNF_Gr2342861688195830050tion_a @ A2 @ F ) )
=> ( member_option_a @ X @ A2 ) ) ).
% GrD1
thf(fact_769_GrD1,axiom,
! [X: a,Fx: nat,A2: set_a,F: a > nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Fx ) @ ( bNF_Gr_a_nat @ A2 @ F ) )
=> ( member_a @ X @ A2 ) ) ).
% GrD1
thf(fact_770_surjective__pairing,axiom,
! [T2: produc4760857364844987814_a_nat] :
( T2
= ( produc5457283509376734880_a_nat @ ( produc1094642028913761716_a_nat @ T2 ) @ ( produc8051327305810483698_a_nat @ T2 ) ) ) ).
% surjective_pairing
thf(fact_771_surjective__pairing,axiom,
! [T2: product_prod_a_nat] :
( T2
= ( product_Pair_a_nat @ ( product_fst_a_nat @ T2 ) @ ( product_snd_a_nat @ T2 ) ) ) ).
% surjective_pairing
thf(fact_772_surjective__pairing,axiom,
! [T2: produc3509355604313844263tion_a] :
( T2
= ( produc9011544418120257559tion_a @ ( produc174239612569448963tion_a @ T2 ) @ ( produc8952039381521079109tion_a @ T2 ) ) ) ).
% surjective_pairing
thf(fact_773_prod_Oexhaust__sel,axiom,
! [Prod: produc4760857364844987814_a_nat] :
( Prod
= ( produc5457283509376734880_a_nat @ ( produc1094642028913761716_a_nat @ Prod ) @ ( produc8051327305810483698_a_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_774_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_a_nat] :
( Prod
= ( product_Pair_a_nat @ ( product_fst_a_nat @ Prod ) @ ( product_snd_a_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_775_prod_Oexhaust__sel,axiom,
! [Prod: produc3509355604313844263tion_a] :
( Prod
= ( produc9011544418120257559tion_a @ ( produc174239612569448963tion_a @ Prod ) @ ( produc8952039381521079109tion_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_776_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: produc3509355604313844263tion_a > nat > $o,X: produc3509355604313844263tion_a,Y: nat,A: produc4760857364844987814_a_nat] :
( ( P2 @ X @ Y )
=> ( ( A
= ( produc5457283509376734880_a_nat @ X @ Y ) )
=> ( P2 @ ( produc1094642028913761716_a_nat @ A ) @ ( produc8051327305810483698_a_nat @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_777_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: a > nat > $o,X: a,Y: nat,A: product_prod_a_nat] :
( ( P2 @ X @ Y )
=> ( ( A
= ( product_Pair_a_nat @ X @ Y ) )
=> ( P2 @ ( product_fst_a_nat @ A ) @ ( product_snd_a_nat @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_778_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: option_a > option_a > $o,X: option_a,Y: option_a,A: produc3509355604313844263tion_a] :
( ( P2 @ X @ Y )
=> ( ( A
= ( produc9011544418120257559tion_a @ X @ Y ) )
=> ( P2 @ ( produc174239612569448963tion_a @ A ) @ ( produc8952039381521079109tion_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_779_case__prod__beta,axiom,
( produc4004610663780792650_a_nat
= ( ^ [F3: a > nat > produc4760857364844987814_a_nat,P3: product_prod_a_nat] : ( F3 @ ( product_fst_a_nat @ P3 ) @ ( product_snd_a_nat @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_780_case__prod__beta,axiom,
( produc7248784972643495020on_a_o
= ( ^ [F3: option_a > option_a > $o,P3: produc3509355604313844263tion_a] : ( F3 @ ( produc174239612569448963tion_a @ P3 ) @ ( produc8952039381521079109tion_a @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_781_case__prod__beta,axiom,
( produc7988141324749014923_nat_o
= ( ^ [F3: produc3509355604313844263tion_a > nat > $o,P3: produc4760857364844987814_a_nat] : ( F3 @ ( produc1094642028913761716_a_nat @ P3 ) @ ( produc8051327305810483698_a_nat @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_782_case__prod__beta,axiom,
( produc2781663484249587963tion_a
= ( ^ [F3: option_a > option_a > produc3509355604313844263tion_a,P3: produc3509355604313844263tion_a] : ( F3 @ ( produc174239612569448963tion_a @ P3 ) @ ( produc8952039381521079109tion_a @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_783_case__prod__beta,axiom,
( produc5088488592959153011_a_nat
= ( ^ [F3: produc3509355604313844263tion_a > nat > produc4760857364844987814_a_nat,P3: produc4760857364844987814_a_nat] : ( F3 @ ( produc1094642028913761716_a_nat @ P3 ) @ ( produc8051327305810483698_a_nat @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_784_case__prod__beta,axiom,
( produc6953443519211250200tion_a
= ( ^ [F3: option_a > option_a > option_a,P3: produc3509355604313844263tion_a] : ( F3 @ ( produc174239612569448963tion_a @ P3 ) @ ( produc8952039381521079109tion_a @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_785_split__beta,axiom,
( produc4004610663780792650_a_nat
= ( ^ [F3: a > nat > produc4760857364844987814_a_nat,Prod3: product_prod_a_nat] : ( F3 @ ( product_fst_a_nat @ Prod3 ) @ ( product_snd_a_nat @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_786_split__beta,axiom,
( produc7248784972643495020on_a_o
= ( ^ [F3: option_a > option_a > $o,Prod3: produc3509355604313844263tion_a] : ( F3 @ ( produc174239612569448963tion_a @ Prod3 ) @ ( produc8952039381521079109tion_a @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_787_split__beta,axiom,
( produc7988141324749014923_nat_o
= ( ^ [F3: produc3509355604313844263tion_a > nat > $o,Prod3: produc4760857364844987814_a_nat] : ( F3 @ ( produc1094642028913761716_a_nat @ Prod3 ) @ ( produc8051327305810483698_a_nat @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_788_split__beta,axiom,
( produc2781663484249587963tion_a
= ( ^ [F3: option_a > option_a > produc3509355604313844263tion_a,Prod3: produc3509355604313844263tion_a] : ( F3 @ ( produc174239612569448963tion_a @ Prod3 ) @ ( produc8952039381521079109tion_a @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_789_split__beta,axiom,
( produc5088488592959153011_a_nat
= ( ^ [F3: produc3509355604313844263tion_a > nat > produc4760857364844987814_a_nat,Prod3: produc4760857364844987814_a_nat] : ( F3 @ ( produc1094642028913761716_a_nat @ Prod3 ) @ ( produc8051327305810483698_a_nat @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_790_split__beta,axiom,
( produc6953443519211250200tion_a
= ( ^ [F3: option_a > option_a > option_a,Prod3: produc3509355604313844263tion_a] : ( F3 @ ( produc174239612569448963tion_a @ Prod3 ) @ ( produc8952039381521079109tion_a @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_791_Product__Type_OCollect__case__prodD,axiom,
! [X: product_prod_a_nat,A2: a > nat > $o] :
( ( member5724188588386418708_a_nat @ X @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ A2 ) ) )
=> ( A2 @ ( product_fst_a_nat @ X ) @ ( product_snd_a_nat @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_792_Product__Type_OCollect__case__prodD,axiom,
! [X: produc3509355604313844263tion_a,A2: option_a > option_a > $o] :
( ( member5498148017924304208tion_a @ X @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ A2 ) ) )
=> ( A2 @ ( produc174239612569448963tion_a @ X ) @ ( produc8952039381521079109tion_a @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_793_Product__Type_OCollect__case__prodD,axiom,
! [X: produc4760857364844987814_a_nat,A2: produc3509355604313844263tion_a > nat > $o] :
( ( member1430561565885860669_a_nat @ X @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ A2 ) ) )
=> ( A2 @ ( produc1094642028913761716_a_nat @ X ) @ ( produc8051327305810483698_a_nat @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_794_case__prod__unfold,axiom,
( produc4004610663780792650_a_nat
= ( ^ [C4: a > nat > produc4760857364844987814_a_nat,P3: product_prod_a_nat] : ( C4 @ ( product_fst_a_nat @ P3 ) @ ( product_snd_a_nat @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_795_case__prod__unfold,axiom,
( produc7248784972643495020on_a_o
= ( ^ [C4: option_a > option_a > $o,P3: produc3509355604313844263tion_a] : ( C4 @ ( produc174239612569448963tion_a @ P3 ) @ ( produc8952039381521079109tion_a @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_796_case__prod__unfold,axiom,
( produc7988141324749014923_nat_o
= ( ^ [C4: produc3509355604313844263tion_a > nat > $o,P3: produc4760857364844987814_a_nat] : ( C4 @ ( produc1094642028913761716_a_nat @ P3 ) @ ( produc8051327305810483698_a_nat @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_797_case__prod__unfold,axiom,
( produc2781663484249587963tion_a
= ( ^ [C4: option_a > option_a > produc3509355604313844263tion_a,P3: produc3509355604313844263tion_a] : ( C4 @ ( produc174239612569448963tion_a @ P3 ) @ ( produc8952039381521079109tion_a @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_798_case__prod__unfold,axiom,
( produc5088488592959153011_a_nat
= ( ^ [C4: produc3509355604313844263tion_a > nat > produc4760857364844987814_a_nat,P3: produc4760857364844987814_a_nat] : ( C4 @ ( produc1094642028913761716_a_nat @ P3 ) @ ( produc8051327305810483698_a_nat @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_799_case__prod__unfold,axiom,
( produc6953443519211250200tion_a
= ( ^ [C4: option_a > option_a > option_a,P3: produc3509355604313844263tion_a] : ( C4 @ ( produc174239612569448963tion_a @ P3 ) @ ( produc8952039381521079109tion_a @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_800_case__prod__beta_H,axiom,
( produc4004610663780792650_a_nat
= ( ^ [F3: a > nat > produc4760857364844987814_a_nat,X2: product_prod_a_nat] : ( F3 @ ( product_fst_a_nat @ X2 ) @ ( product_snd_a_nat @ X2 ) ) ) ) ).
% case_prod_beta'
thf(fact_801_case__prod__beta_H,axiom,
( produc7248784972643495020on_a_o
= ( ^ [F3: option_a > option_a > $o,X2: produc3509355604313844263tion_a] : ( F3 @ ( produc174239612569448963tion_a @ X2 ) @ ( produc8952039381521079109tion_a @ X2 ) ) ) ) ).
% case_prod_beta'
thf(fact_802_case__prod__beta_H,axiom,
( produc7988141324749014923_nat_o
= ( ^ [F3: produc3509355604313844263tion_a > nat > $o,X2: produc4760857364844987814_a_nat] : ( F3 @ ( produc1094642028913761716_a_nat @ X2 ) @ ( produc8051327305810483698_a_nat @ X2 ) ) ) ) ).
% case_prod_beta'
thf(fact_803_case__prod__beta_H,axiom,
( produc2781663484249587963tion_a
= ( ^ [F3: option_a > option_a > produc3509355604313844263tion_a,X2: produc3509355604313844263tion_a] : ( F3 @ ( produc174239612569448963tion_a @ X2 ) @ ( produc8952039381521079109tion_a @ X2 ) ) ) ) ).
% case_prod_beta'
thf(fact_804_case__prod__beta_H,axiom,
( produc5088488592959153011_a_nat
= ( ^ [F3: produc3509355604313844263tion_a > nat > produc4760857364844987814_a_nat,X2: produc4760857364844987814_a_nat] : ( F3 @ ( produc1094642028913761716_a_nat @ X2 ) @ ( produc8051327305810483698_a_nat @ X2 ) ) ) ) ).
% case_prod_beta'
thf(fact_805_case__prod__beta_H,axiom,
( produc6953443519211250200tion_a
= ( ^ [F3: option_a > option_a > option_a,X2: produc3509355604313844263tion_a] : ( F3 @ ( produc174239612569448963tion_a @ X2 ) @ ( produc8952039381521079109tion_a @ X2 ) ) ) ) ).
% case_prod_beta'
thf(fact_806_mem__Times__iff,axiom,
! [X: product_prod_o_o,A2: set_o,B2: set_o] :
( ( member7466972457876170832od_o_o @ X
@ ( product_Sigma_o_o @ A2
@ ^ [Uu: $o] : B2 ) )
= ( ( member_o @ ( product_fst_o_o @ X ) @ A2 )
& ( member_o @ ( product_snd_o_o @ X ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_807_mem__Times__iff,axiom,
! [X: product_prod_a_nat,A2: set_a,B2: set_nat] :
( ( member5724188588386418708_a_nat @ X
@ ( product_Sigma_a_nat @ A2
@ ^ [Uu: a] : B2 ) )
= ( ( member_a @ ( product_fst_a_nat @ X ) @ A2 )
& ( member_nat @ ( product_snd_a_nat @ X ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_808_mem__Times__iff,axiom,
! [X: produc2710310701403425476_a_nat,A2: set_o,B2: set_Pr4934435412358123699_a_nat] :
( ( member26132018921490651_a_nat @ X
@ ( produc3092463322541264461_a_nat @ A2
@ ^ [Uu: $o] : B2 ) )
= ( ( member_o @ ( produc1832116388144416466_a_nat @ X ) @ A2 )
& ( member5724188588386418708_a_nat @ ( produc2498418525026724880_a_nat @ X ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_809_mem__Times__iff,axiom,
! [X: produc4534846064852955472_nat_o,A2: set_Pr4934435412358123699_a_nat,B2: set_o] :
( ( member1850667382371020647_nat_o @ X
@ ( produc8052552943834900881_nat_o @ A2
@ ^ [Uu: product_prod_a_nat] : B2 ) )
= ( ( member5724188588386418708_a_nat @ ( produc6792206009438052886_nat_o @ X ) @ A2 )
& ( member_o @ ( produc7458508146320361300_nat_o @ X ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_810_mem__Times__iff,axiom,
! [X: produc3509355604313844263tion_a,A2: set_option_a,B2: set_option_a] :
( ( member5498148017924304208tion_a @ X
@ ( produc269287337874323144tion_a @ A2
@ ^ [Uu: option_a] : B2 ) )
= ( ( member_option_a @ ( produc174239612569448963tion_a @ X ) @ A2 )
& ( member_option_a @ ( produc8952039381521079109tion_a @ X ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_811_mem__Times__iff,axiom,
! [X: produc4838336064122203702tion_a,A2: set_o,B2: set_Pr7585778909603769095tion_a] :
( ( member2654175073444177119tion_a @ X
@ ( produc61899281709682263tion_a @ A2
@ ^ [Uu: $o] : B2 ) )
= ( ( member_o @ ( produc2912122044608807954tion_a @ X ) @ A2 )
& ( member5498148017924304208tion_a @ ( produc6215565060926053460tion_a @ X ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_812_mem__Times__iff,axiom,
! [X: produc4708774622424448987_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( member9062615507155100804_a_nat @ X
@ ( produc3832836940518797764_a_nat @ A2
@ ^ [Uu: product_prod_a_nat] : B2 ) )
= ( ( member5724188588386418708_a_nat @ ( produc315936122747728255_a_nat @ X ) @ A2 )
& ( member5724188588386418708_a_nat @ ( produc897854599763871169_a_nat @ X ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_813_mem__Times__iff,axiom,
! [X: produc2194632134642013272on_a_o,A2: set_Pr7585778909603769095tion_a,B2: set_o] :
( ( member10471143963986689on_a_o @ X
@ ( produc8823275029784582905on_a_o @ A2
@ ^ [Uu: produc3509355604313844263tion_a] : B2 ) )
= ( ( member5498148017924304208tion_a @ ( produc2450125755828932788on_a_o @ X ) @ A2 )
& ( member_o @ ( produc5753568772146178294on_a_o @ X ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_814_mem__Times__iff,axiom,
! [X: produc4760857364844987814_a_nat,A2: set_Pr7585778909603769095tion_a,B2: set_nat] :
( ( member1430561565885860669_a_nat @ X
@ ( produc2066629144795197103_a_nat @ A2
@ ^ [Uu: produc3509355604313844263tion_a] : B2 ) )
= ( ( member5498148017924304208tion_a @ ( produc1094642028913761716_a_nat @ X ) @ A2 )
& ( member_nat @ ( produc8051327305810483698_a_nat @ X ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_815_mem__Times__iff,axiom,
! [X: produc6654178220768718189_a_nat,A2: set_o,B2: set_Pr8660066742269822812_a_nat] :
( ( member9172549829962833796_a_nat @ X
@ ( produc2867166158663066998_a_nat @ A2
@ ^ [Uu: $o] : B2 ) )
= ( ( member_o @ ( produc4588323236588632571_a_nat @ X ) @ A2 )
& ( member1430561565885860669_a_nat @ ( produc2116952056027252537_a_nat @ X ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_816_prod_Osplit__sel__asm,axiom,
! [P2: produc4760857364844987814_a_nat > $o,F: a > nat > produc4760857364844987814_a_nat,Prod: product_prod_a_nat] :
( ( P2 @ ( produc4004610663780792650_a_nat @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair_a_nat @ ( product_fst_a_nat @ Prod ) @ ( product_snd_a_nat @ Prod ) ) )
& ~ ( P2 @ ( F @ ( product_fst_a_nat @ Prod ) @ ( product_snd_a_nat @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_817_prod_Osplit__sel__asm,axiom,
! [P2: $o > $o,F: option_a > option_a > $o,Prod: produc3509355604313844263tion_a] :
( ( P2 @ ( produc7248784972643495020on_a_o @ F @ Prod ) )
= ( ~ ( ( Prod
= ( produc9011544418120257559tion_a @ ( produc174239612569448963tion_a @ Prod ) @ ( produc8952039381521079109tion_a @ Prod ) ) )
& ~ ( P2 @ ( F @ ( produc174239612569448963tion_a @ Prod ) @ ( produc8952039381521079109tion_a @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_818_prod_Osplit__sel__asm,axiom,
! [P2: $o > $o,F: produc3509355604313844263tion_a > nat > $o,Prod: produc4760857364844987814_a_nat] :
( ( P2 @ ( produc7988141324749014923_nat_o @ F @ Prod ) )
= ( ~ ( ( Prod
= ( produc5457283509376734880_a_nat @ ( produc1094642028913761716_a_nat @ Prod ) @ ( produc8051327305810483698_a_nat @ Prod ) ) )
& ~ ( P2 @ ( F @ ( produc1094642028913761716_a_nat @ Prod ) @ ( produc8051327305810483698_a_nat @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_819_prod_Osplit__sel__asm,axiom,
! [P2: produc3509355604313844263tion_a > $o,F: option_a > option_a > produc3509355604313844263tion_a,Prod: produc3509355604313844263tion_a] :
( ( P2 @ ( produc2781663484249587963tion_a @ F @ Prod ) )
= ( ~ ( ( Prod
= ( produc9011544418120257559tion_a @ ( produc174239612569448963tion_a @ Prod ) @ ( produc8952039381521079109tion_a @ Prod ) ) )
& ~ ( P2 @ ( F @ ( produc174239612569448963tion_a @ Prod ) @ ( produc8952039381521079109tion_a @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_820_prod_Osplit__sel__asm,axiom,
! [P2: produc4760857364844987814_a_nat > $o,F: produc3509355604313844263tion_a > nat > produc4760857364844987814_a_nat,Prod: produc4760857364844987814_a_nat] :
( ( P2 @ ( produc5088488592959153011_a_nat @ F @ Prod ) )
= ( ~ ( ( Prod
= ( produc5457283509376734880_a_nat @ ( produc1094642028913761716_a_nat @ Prod ) @ ( produc8051327305810483698_a_nat @ Prod ) ) )
& ~ ( P2 @ ( F @ ( produc1094642028913761716_a_nat @ Prod ) @ ( produc8051327305810483698_a_nat @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_821_prod_Osplit__sel__asm,axiom,
! [P2: option_a > $o,F: option_a > option_a > option_a,Prod: produc3509355604313844263tion_a] :
( ( P2 @ ( produc6953443519211250200tion_a @ F @ Prod ) )
= ( ~ ( ( Prod
= ( produc9011544418120257559tion_a @ ( produc174239612569448963tion_a @ Prod ) @ ( produc8952039381521079109tion_a @ Prod ) ) )
& ~ ( P2 @ ( F @ ( produc174239612569448963tion_a @ Prod ) @ ( produc8952039381521079109tion_a @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_822_prod_Osplit__sel,axiom,
! [P2: produc4760857364844987814_a_nat > $o,F: a > nat > produc4760857364844987814_a_nat,Prod: product_prod_a_nat] :
( ( P2 @ ( produc4004610663780792650_a_nat @ F @ Prod ) )
= ( ( Prod
= ( product_Pair_a_nat @ ( product_fst_a_nat @ Prod ) @ ( product_snd_a_nat @ Prod ) ) )
=> ( P2 @ ( F @ ( product_fst_a_nat @ Prod ) @ ( product_snd_a_nat @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_823_prod_Osplit__sel,axiom,
! [P2: $o > $o,F: option_a > option_a > $o,Prod: produc3509355604313844263tion_a] :
( ( P2 @ ( produc7248784972643495020on_a_o @ F @ Prod ) )
= ( ( Prod
= ( produc9011544418120257559tion_a @ ( produc174239612569448963tion_a @ Prod ) @ ( produc8952039381521079109tion_a @ Prod ) ) )
=> ( P2 @ ( F @ ( produc174239612569448963tion_a @ Prod ) @ ( produc8952039381521079109tion_a @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_824_prod_Osplit__sel,axiom,
! [P2: $o > $o,F: produc3509355604313844263tion_a > nat > $o,Prod: produc4760857364844987814_a_nat] :
( ( P2 @ ( produc7988141324749014923_nat_o @ F @ Prod ) )
= ( ( Prod
= ( produc5457283509376734880_a_nat @ ( produc1094642028913761716_a_nat @ Prod ) @ ( produc8051327305810483698_a_nat @ Prod ) ) )
=> ( P2 @ ( F @ ( produc1094642028913761716_a_nat @ Prod ) @ ( produc8051327305810483698_a_nat @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_825_prod_Osplit__sel,axiom,
! [P2: produc3509355604313844263tion_a > $o,F: option_a > option_a > produc3509355604313844263tion_a,Prod: produc3509355604313844263tion_a] :
( ( P2 @ ( produc2781663484249587963tion_a @ F @ Prod ) )
= ( ( Prod
= ( produc9011544418120257559tion_a @ ( produc174239612569448963tion_a @ Prod ) @ ( produc8952039381521079109tion_a @ Prod ) ) )
=> ( P2 @ ( F @ ( produc174239612569448963tion_a @ Prod ) @ ( produc8952039381521079109tion_a @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_826_prod_Osplit__sel,axiom,
! [P2: produc4760857364844987814_a_nat > $o,F: produc3509355604313844263tion_a > nat > produc4760857364844987814_a_nat,Prod: produc4760857364844987814_a_nat] :
( ( P2 @ ( produc5088488592959153011_a_nat @ F @ Prod ) )
= ( ( Prod
= ( produc5457283509376734880_a_nat @ ( produc1094642028913761716_a_nat @ Prod ) @ ( produc8051327305810483698_a_nat @ Prod ) ) )
=> ( P2 @ ( F @ ( produc1094642028913761716_a_nat @ Prod ) @ ( produc8051327305810483698_a_nat @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_827_prod_Osplit__sel,axiom,
! [P2: option_a > $o,F: option_a > option_a > option_a,Prod: produc3509355604313844263tion_a] :
( ( P2 @ ( produc6953443519211250200tion_a @ F @ Prod ) )
= ( ( Prod
= ( produc9011544418120257559tion_a @ ( produc174239612569448963tion_a @ Prod ) @ ( produc8952039381521079109tion_a @ Prod ) ) )
=> ( P2 @ ( F @ ( produc174239612569448963tion_a @ Prod ) @ ( produc8952039381521079109tion_a @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_828_times__subset__iff,axiom,
! [A2: set_o,C2: set_o,B2: set_o,D: set_o] :
( ( ord_le192195778054999463od_o_o
@ ( product_Sigma_o_o @ A2
@ ^ [Uu: $o] : C2 )
@ ( product_Sigma_o_o @ B2
@ ^ [Uu: $o] : D ) )
= ( ( A2 = bot_bot_set_o )
| ( C2 = bot_bot_set_o )
| ( ( ord_less_eq_set_o @ A2 @ B2 )
& ( ord_less_eq_set_o @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_829_times__subset__iff,axiom,
! [A2: set_a,C2: set_nat,B2: set_a,D: set_nat] :
( ( ord_le8666007276011122963_a_nat
@ ( product_Sigma_a_nat @ A2
@ ^ [Uu: a] : C2 )
@ ( product_Sigma_a_nat @ B2
@ ^ [Uu: a] : D ) )
= ( ( A2 = bot_bot_set_a )
| ( C2 = bot_bot_set_nat )
| ( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_830_times__subset__iff,axiom,
! [A2: set_Pr4934435412358123699_a_nat,C2: set_o,B2: set_Pr4934435412358123699_a_nat,D: set_o] :
( ( ord_le5431503644745719014_nat_o
@ ( produc8052552943834900881_nat_o @ A2
@ ^ [Uu: product_prod_a_nat] : C2 )
@ ( produc8052552943834900881_nat_o @ B2
@ ^ [Uu: product_prod_a_nat] : D ) )
= ( ( A2 = bot_bo9049108969261143879_a_nat )
| ( C2 = bot_bot_set_o )
| ( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
& ( ord_less_eq_set_o @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_831_times__subset__iff,axiom,
! [A2: set_o,C2: set_Pr4934435412358123699_a_nat,B2: set_o,D: set_Pr4934435412358123699_a_nat] :
( ( ord_le3075358075201600602_a_nat
@ ( produc3092463322541264461_a_nat @ A2
@ ^ [Uu: $o] : C2 )
@ ( produc3092463322541264461_a_nat @ B2
@ ^ [Uu: $o] : D ) )
= ( ( A2 = bot_bot_set_o )
| ( C2 = bot_bo9049108969261143879_a_nat )
| ( ( ord_less_eq_set_o @ A2 @ B2 )
& ( ord_le8666007276011122963_a_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_832_times__subset__iff,axiom,
! [A2: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat] :
( ( ord_le1018141841545957851_a_nat
@ ( produc3832836940518797764_a_nat @ A2
@ ^ [Uu: product_prod_a_nat] : C2 )
@ ( produc3832836940518797764_a_nat @ B2
@ ^ [Uu: product_prod_a_nat] : D ) )
= ( ( A2 = bot_bo9049108969261143879_a_nat )
| ( C2 = bot_bo9049108969261143879_a_nat )
| ( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
& ( ord_le8666007276011122963_a_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_833_times__subset__iff,axiom,
! [A2: set_Pr7585778909603769095tion_a,C2: set_nat,B2: set_Pr7585778909603769095tion_a,D: set_nat] :
( ( ord_le5649184635953113020_a_nat
@ ( produc2066629144795197103_a_nat @ A2
@ ^ [Uu: produc3509355604313844263tion_a] : C2 )
@ ( produc2066629144795197103_a_nat @ B2
@ ^ [Uu: produc3509355604313844263tion_a] : D ) )
= ( ( A2 = bot_bo235252021745139059tion_a )
| ( C2 = bot_bot_set_nat )
| ( ( ord_le4471550158292877991tion_a @ A2 @ B2 )
& ( ord_less_eq_set_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_834_times__subset__iff,axiom,
! [A2: set_Pr8660066742269822812_a_nat,C2: set_o,B2: set_Pr8660066742269822812_a_nat,D: set_o] :
( ( ord_le8405921517615413757_nat_o
@ ( produc7403436832878706216_nat_o @ A2
@ ^ [Uu: produc4760857364844987814_a_nat] : C2 )
@ ( produc7403436832878706216_nat_o @ B2
@ ^ [Uu: produc4760857364844987814_a_nat] : D ) )
= ( ( A2 = bot_bo8420108762324073968_a_nat )
| ( C2 = bot_bot_set_o )
| ( ( ord_le5649184635953113020_a_nat @ A2 @ B2 )
& ( ord_less_eq_set_o @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_835_times__subset__iff,axiom,
! [A2: set_o,C2: set_Pr8660066742269822812_a_nat,B2: set_o,D: set_Pr8660066742269822812_a_nat] :
( ( ord_le3880997233133534851_a_nat
@ ( produc2867166158663066998_a_nat @ A2
@ ^ [Uu: $o] : C2 )
@ ( produc2867166158663066998_a_nat @ B2
@ ^ [Uu: $o] : D ) )
= ( ( A2 = bot_bot_set_o )
| ( C2 = bot_bo8420108762324073968_a_nat )
| ( ( ord_less_eq_set_o @ A2 @ B2 )
& ( ord_le5649184635953113020_a_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_836_times__subset__iff,axiom,
! [A2: set_Pr8660066742269822812_a_nat,C2: set_Pr4934435412358123699_a_nat,B2: set_Pr8660066742269822812_a_nat,D: set_Pr4934435412358123699_a_nat] :
( ( ord_le4664675211497543556_a_nat
@ ( produc2019342168835006445_a_nat @ A2
@ ^ [Uu: produc4760857364844987814_a_nat] : C2 )
@ ( produc2019342168835006445_a_nat @ B2
@ ^ [Uu: produc4760857364844987814_a_nat] : D ) )
= ( ( A2 = bot_bo8420108762324073968_a_nat )
| ( C2 = bot_bo9049108969261143879_a_nat )
| ( ( ord_le5649184635953113020_a_nat @ A2 @ B2 )
& ( ord_le8666007276011122963_a_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_837_times__subset__iff,axiom,
! [A2: set_Pr4934435412358123699_a_nat,C2: set_Pr8660066742269822812_a_nat,B2: set_Pr4934435412358123699_a_nat,D: set_Pr8660066742269822812_a_nat] :
( ( ord_le4239986153908076548_a_nat
@ ( produc8234572646870318317_a_nat @ A2
@ ^ [Uu: product_prod_a_nat] : C2 )
@ ( produc8234572646870318317_a_nat @ B2
@ ^ [Uu: product_prod_a_nat] : D ) )
= ( ( A2 = bot_bo9049108969261143879_a_nat )
| ( C2 = bot_bo8420108762324073968_a_nat )
| ( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
& ( ord_le5649184635953113020_a_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_838_fst__image__Sigma,axiom,
! [A2: set_option_a,B2: option_a > set_option_a] :
( ( image_7272866627766095330tion_a @ produc174239612569448963tion_a @ ( produc269287337874323144tion_a @ A2 @ B2 ) )
= ( collect_option_a
@ ^ [X2: option_a] :
( ( member_option_a @ X2 @ A2 )
& ( ( B2 @ X2 )
!= bot_bot_set_option_a ) ) ) ) ).
% fst_image_Sigma
thf(fact_839_Collect__split__mono__strong,axiom,
! [X5: set_a,A2: set_Pr4934435412358123699_a_nat,Y2: set_nat,P2: a > nat > $o,Q: a > nat > $o] :
( ( X5
= ( image_6659209163692594554_nat_a @ product_fst_a_nat @ A2 ) )
=> ( ( Y2
= ( image_838238048506182228at_nat @ product_snd_a_nat @ A2 ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ X5 )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ Y2 )
=> ( ( P2 @ X3 @ Xa )
=> ( Q @ X3 @ Xa ) ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ A2 @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ P2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_840_Collect__split__mono__strong,axiom,
! [X5: set_option_a,A2: set_Pr7585778909603769095tion_a,Y2: set_option_a,P2: option_a > option_a > $o,Q: option_a > option_a > $o] :
( ( X5
= ( image_7272866627766095330tion_a @ produc174239612569448963tion_a @ A2 ) )
=> ( ( Y2
= ( image_7272866627766095330tion_a @ produc8952039381521079109tion_a @ A2 ) )
=> ( ! [X3: option_a] :
( ( member_option_a @ X3 @ X5 )
=> ! [Xa: option_a] :
( ( member_option_a @ Xa @ Y2 )
=> ( ( P2 @ X3 @ Xa )
=> ( Q @ X3 @ Xa ) ) ) )
=> ( ( ord_le4471550158292877991tion_a @ A2 @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ P2 ) ) )
=> ( ord_le4471550158292877991tion_a @ A2 @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_841_Collect__split__mono__strong,axiom,
! [X5: set_Pr7585778909603769095tion_a,A2: set_Pr8660066742269822812_a_nat,Y2: set_nat,P2: produc3509355604313844263tion_a > nat > $o,Q: produc3509355604313844263tion_a > nat > $o] :
( ( X5
= ( image_1391120077856136634tion_a @ produc1094642028913761716_a_nat @ A2 ) )
=> ( ( Y2
= ( image_7587833714734314877at_nat @ produc8051327305810483698_a_nat @ A2 ) )
=> ( ! [X3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X3 @ X5 )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ Y2 )
=> ( ( P2 @ X3 @ Xa )
=> ( Q @ X3 @ Xa ) ) ) )
=> ( ( ord_le5649184635953113020_a_nat @ A2 @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ P2 ) ) )
=> ( ord_le5649184635953113020_a_nat @ A2 @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_842_subset__fst__snd,axiom,
! [A2: set_Pr7585778909603769095tion_a] :
( ord_le4471550158292877991tion_a @ A2
@ ( produc269287337874323144tion_a @ ( image_7272866627766095330tion_a @ produc174239612569448963tion_a @ A2 )
@ ^ [Uu: option_a] : ( image_7272866627766095330tion_a @ produc8952039381521079109tion_a @ A2 ) ) ) ).
% subset_fst_snd
thf(fact_843_subset__fst__snd,axiom,
! [A2: set_Pr8660066742269822812_a_nat] :
( ord_le5649184635953113020_a_nat @ A2
@ ( produc2066629144795197103_a_nat @ ( image_1391120077856136634tion_a @ produc1094642028913761716_a_nat @ A2 )
@ ^ [Uu: produc3509355604313844263tion_a] : ( image_7587833714734314877at_nat @ produc8051327305810483698_a_nat @ A2 ) ) ) ).
% subset_fst_snd
thf(fact_844_subset__fst__snd,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ord_le8666007276011122963_a_nat @ A2
@ ( product_Sigma_a_nat @ ( image_6659209163692594554_nat_a @ product_fst_a_nat @ A2 )
@ ^ [Uu: a] : ( image_838238048506182228at_nat @ product_snd_a_nat @ A2 ) ) ) ).
% subset_fst_snd
thf(fact_845_snd__fst__flip,axiom,
( produc5832230846834443088tion_a
= ( comp_P8481598692407811498tion_a @ produc1094642028913761716_a_nat
@ ( produc9185384946573455057_a_nat
@ ^ [X2: nat,Y5: produc3509355604313844263tion_a] : ( produc5457283509376734880_a_nat @ Y5 @ X2 ) ) ) ) ).
% snd_fst_flip
thf(fact_846_snd__fst__flip,axiom,
( product_snd_nat_a
= ( comp_P2661830749605898945_nat_a @ product_fst_a_nat
@ ( produc732906326552059263_a_nat
@ ^ [X2: nat,Y5: a] : ( product_Pair_a_nat @ Y5 @ X2 ) ) ) ) ).
% snd_fst_flip
thf(fact_847_snd__fst__flip,axiom,
( produc8952039381521079109tion_a
= ( comp_P3618168059068810855tion_a @ produc174239612569448963tion_a
@ ( produc2781663484249587963tion_a
@ ^ [X2: option_a,Y5: option_a] : ( produc9011544418120257559tion_a @ Y5 @ X2 ) ) ) ) ).
% snd_fst_flip
thf(fact_848_fst__snd__flip,axiom,
( produc8098917606792496914tion_a
= ( comp_P1040842001491653543tion_a @ produc8051327305810483698_a_nat
@ ( produc9185384946573455057_a_nat
@ ^ [X2: nat,Y5: produc3509355604313844263tion_a] : ( produc5457283509376734880_a_nat @ Y5 @ X2 ) ) ) ) ).
% fst_snd_flip
thf(fact_849_fst__snd__flip,axiom,
( product_fst_nat_a
= ( comp_P5720018585459053269_nat_a @ product_snd_a_nat
@ ( produc732906326552059263_a_nat
@ ^ [X2: nat,Y5: a] : ( product_Pair_a_nat @ Y5 @ X2 ) ) ) ) ).
% fst_snd_flip
thf(fact_850_fst__snd__flip,axiom,
( produc174239612569448963tion_a
= ( comp_P3618168059068810855tion_a @ produc8952039381521079109tion_a
@ ( produc2781663484249587963tion_a
@ ^ [X2: option_a,Y5: option_a] : ( produc9011544418120257559tion_a @ Y5 @ X2 ) ) ) ) ).
% fst_snd_flip
thf(fact_851_exE__realizer,axiom,
! [P2: nat > a > $o,P: product_prod_a_nat,Q: produc4760857364844987814_a_nat > $o,F: a > nat > produc4760857364844987814_a_nat] :
( ( P2 @ ( product_snd_a_nat @ P ) @ ( product_fst_a_nat @ P ) )
=> ( ! [X3: a,Y3: nat] :
( ( P2 @ Y3 @ X3 )
=> ( Q @ ( F @ X3 @ Y3 ) ) )
=> ( Q @ ( produc4004610663780792650_a_nat @ F @ P ) ) ) ) ).
% exE_realizer
thf(fact_852_exE__realizer,axiom,
! [P2: option_a > option_a > $o,P: produc3509355604313844263tion_a,Q: $o > $o,F: option_a > option_a > $o] :
( ( P2 @ ( produc8952039381521079109tion_a @ P ) @ ( produc174239612569448963tion_a @ P ) )
=> ( ! [X3: option_a,Y3: option_a] :
( ( P2 @ Y3 @ X3 )
=> ( Q @ ( F @ X3 @ Y3 ) ) )
=> ( Q @ ( produc7248784972643495020on_a_o @ F @ P ) ) ) ) ).
% exE_realizer
thf(fact_853_exE__realizer,axiom,
! [P2: nat > produc3509355604313844263tion_a > $o,P: produc4760857364844987814_a_nat,Q: $o > $o,F: produc3509355604313844263tion_a > nat > $o] :
( ( P2 @ ( produc8051327305810483698_a_nat @ P ) @ ( produc1094642028913761716_a_nat @ P ) )
=> ( ! [X3: produc3509355604313844263tion_a,Y3: nat] :
( ( P2 @ Y3 @ X3 )
=> ( Q @ ( F @ X3 @ Y3 ) ) )
=> ( Q @ ( produc7988141324749014923_nat_o @ F @ P ) ) ) ) ).
% exE_realizer
thf(fact_854_exE__realizer,axiom,
! [P2: option_a > option_a > $o,P: produc3509355604313844263tion_a,Q: produc3509355604313844263tion_a > $o,F: option_a > option_a > produc3509355604313844263tion_a] :
( ( P2 @ ( produc8952039381521079109tion_a @ P ) @ ( produc174239612569448963tion_a @ P ) )
=> ( ! [X3: option_a,Y3: option_a] :
( ( P2 @ Y3 @ X3 )
=> ( Q @ ( F @ X3 @ Y3 ) ) )
=> ( Q @ ( produc2781663484249587963tion_a @ F @ P ) ) ) ) ).
% exE_realizer
thf(fact_855_exE__realizer,axiom,
! [P2: nat > produc3509355604313844263tion_a > $o,P: produc4760857364844987814_a_nat,Q: produc4760857364844987814_a_nat > $o,F: produc3509355604313844263tion_a > nat > produc4760857364844987814_a_nat] :
( ( P2 @ ( produc8051327305810483698_a_nat @ P ) @ ( produc1094642028913761716_a_nat @ P ) )
=> ( ! [X3: produc3509355604313844263tion_a,Y3: nat] :
( ( P2 @ Y3 @ X3 )
=> ( Q @ ( F @ X3 @ Y3 ) ) )
=> ( Q @ ( produc5088488592959153011_a_nat @ F @ P ) ) ) ) ).
% exE_realizer
thf(fact_856_exE__realizer,axiom,
! [P2: option_a > option_a > $o,P: produc3509355604313844263tion_a,Q: option_a > $o,F: option_a > option_a > option_a] :
( ( P2 @ ( produc8952039381521079109tion_a @ P ) @ ( produc174239612569448963tion_a @ P ) )
=> ( ! [X3: option_a,Y3: option_a] :
( ( P2 @ Y3 @ X3 )
=> ( Q @ ( F @ X3 @ Y3 ) ) )
=> ( Q @ ( produc6953443519211250200tion_a @ F @ P ) ) ) ) ).
% exE_realizer
thf(fact_857_fstOp__def,axiom,
( bNF_fs7976035448042954642tion_a
= ( ^ [P4: option_a > option_a > $o,Q3: option_a > option_a > $o,Ac: produc3509355604313844263tion_a] : ( produc9011544418120257559tion_a @ ( produc174239612569448963tion_a @ Ac ) @ ( bNF_pi5353392459224037856tion_a @ P4 @ Q3 @ ( produc174239612569448963tion_a @ Ac ) @ ( produc8952039381521079109tion_a @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_858_conjI__realizer,axiom,
! [P2: produc3509355604313844263tion_a > $o,P: produc3509355604313844263tion_a,Q: nat > $o,Q2: nat] :
( ( P2 @ P )
=> ( ( Q @ Q2 )
=> ( ( P2 @ ( produc1094642028913761716_a_nat @ ( produc5457283509376734880_a_nat @ P @ Q2 ) ) )
& ( Q @ ( produc8051327305810483698_a_nat @ ( produc5457283509376734880_a_nat @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_859_conjI__realizer,axiom,
! [P2: a > $o,P: a,Q: nat > $o,Q2: nat] :
( ( P2 @ P )
=> ( ( Q @ Q2 )
=> ( ( P2 @ ( product_fst_a_nat @ ( product_Pair_a_nat @ P @ Q2 ) ) )
& ( Q @ ( product_snd_a_nat @ ( product_Pair_a_nat @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_860_conjI__realizer,axiom,
! [P2: option_a > $o,P: option_a,Q: option_a > $o,Q2: option_a] :
( ( P2 @ P )
=> ( ( Q @ Q2 )
=> ( ( P2 @ ( produc174239612569448963tion_a @ ( produc9011544418120257559tion_a @ P @ Q2 ) ) )
& ( Q @ ( produc8952039381521079109tion_a @ ( produc9011544418120257559tion_a @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_861_exI__realizer,axiom,
! [P2: nat > produc3509355604313844263tion_a > $o,Y: nat,X: produc3509355604313844263tion_a] :
( ( P2 @ Y @ X )
=> ( P2 @ ( produc8051327305810483698_a_nat @ ( produc5457283509376734880_a_nat @ X @ Y ) ) @ ( produc1094642028913761716_a_nat @ ( produc5457283509376734880_a_nat @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_862_exI__realizer,axiom,
! [P2: nat > a > $o,Y: nat,X: a] :
( ( P2 @ Y @ X )
=> ( P2 @ ( product_snd_a_nat @ ( product_Pair_a_nat @ X @ Y ) ) @ ( product_fst_a_nat @ ( product_Pair_a_nat @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_863_exI__realizer,axiom,
! [P2: option_a > option_a > $o,Y: option_a,X: option_a] :
( ( P2 @ Y @ X )
=> ( P2 @ ( produc8952039381521079109tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) ) @ ( produc174239612569448963tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_864_apfst__convE,axiom,
! [Q2: produc4760857364844987814_a_nat,F: produc3509355604313844263tion_a > produc3509355604313844263tion_a,P: produc4760857364844987814_a_nat] :
( ( Q2
= ( produc1522032229839045964_a_nat @ F @ P ) )
=> ~ ! [X3: produc3509355604313844263tion_a,Y3: nat] :
( ( P
= ( produc5457283509376734880_a_nat @ X3 @ Y3 ) )
=> ( Q2
!= ( produc5457283509376734880_a_nat @ ( F @ X3 ) @ Y3 ) ) ) ) ).
% apfst_convE
thf(fact_865_apfst__convE,axiom,
! [Q2: product_prod_a_nat,F: produc3509355604313844263tion_a > a,P: produc4760857364844987814_a_nat] :
( ( Q2
= ( produc1282494340376965667_a_nat @ F @ P ) )
=> ~ ! [X3: produc3509355604313844263tion_a,Y3: nat] :
( ( P
= ( produc5457283509376734880_a_nat @ X3 @ Y3 ) )
=> ( Q2
!= ( product_Pair_a_nat @ ( F @ X3 ) @ Y3 ) ) ) ) ).
% apfst_convE
thf(fact_866_apfst__convE,axiom,
! [Q2: produc3509355604313844263tion_a,F: option_a > option_a,P: produc3509355604313844263tion_a] :
( ( Q2
= ( produc1685019323022372872tion_a @ F @ P ) )
=> ~ ! [X3: option_a,Y3: option_a] :
( ( P
= ( produc9011544418120257559tion_a @ X3 @ Y3 ) )
=> ( Q2
!= ( produc9011544418120257559tion_a @ ( F @ X3 ) @ Y3 ) ) ) ) ).
% apfst_convE
thf(fact_867_apfst__convE,axiom,
! [Q2: produc4760857364844987814_a_nat,F: a > produc3509355604313844263tion_a,P: product_prod_a_nat] :
( ( Q2
= ( produc6204860511972624437_a_nat @ F @ P ) )
=> ~ ! [X3: a,Y3: nat] :
( ( P
= ( product_Pair_a_nat @ X3 @ Y3 ) )
=> ( Q2
!= ( produc5457283509376734880_a_nat @ ( F @ X3 ) @ Y3 ) ) ) ) ).
% apfst_convE
thf(fact_868_apfst__convE,axiom,
! [Q2: product_prod_a_nat,F: a > a,P: product_prod_a_nat] :
( ( Q2
= ( produc7773453608010410252_a_nat @ F @ P ) )
=> ~ ! [X3: a,Y3: nat] :
( ( P
= ( product_Pair_a_nat @ X3 @ Y3 ) )
=> ( Q2
!= ( product_Pair_a_nat @ ( F @ X3 ) @ Y3 ) ) ) ) ).
% apfst_convE
thf(fact_869_apsnd__apfst,axiom,
! [F: option_a > nat,G: option_a > produc3509355604313844263tion_a,X: produc3509355604313844263tion_a] :
( ( produc1897394789254173455tion_a @ F @ ( produc2088028666236427237tion_a @ G @ X ) )
= ( produc5457283509376734880_a_nat @ ( G @ ( produc174239612569448963tion_a @ X ) ) @ ( F @ ( produc8952039381521079109tion_a @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_870_apsnd__apfst,axiom,
! [F: option_a > option_a,G: option_a > option_a,X: produc3509355604313844263tion_a] :
( ( produc8446506941092750918tion_a @ F @ ( produc1685019323022372872tion_a @ G @ X ) )
= ( produc9011544418120257559tion_a @ ( G @ ( produc174239612569448963tion_a @ X ) ) @ ( F @ ( produc8952039381521079109tion_a @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_871_apsnd__apfst,axiom,
! [F: option_a > nat,G: option_a > a,X: produc3509355604313844263tion_a] :
( ( produc6561150746041953254_nat_a @ F @ ( produc4086220583163799886tion_a @ G @ X ) )
= ( product_Pair_a_nat @ ( G @ ( produc174239612569448963tion_a @ X ) ) @ ( F @ ( produc8952039381521079109tion_a @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_872_apfst__apsnd,axiom,
! [F: option_a > produc3509355604313844263tion_a,G: option_a > nat,X: produc3509355604313844263tion_a] :
( ( produc585380348467062127_a_nat @ F @ ( produc2390184552787727916tion_a @ G @ X ) )
= ( produc5457283509376734880_a_nat @ ( F @ ( produc174239612569448963tion_a @ X ) ) @ ( G @ ( produc8952039381521079109tion_a @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_873_apfst__apsnd,axiom,
! [F: option_a > option_a,G: option_a > option_a,X: produc3509355604313844263tion_a] :
( ( produc1685019323022372872tion_a @ F @ ( produc8446506941092750918tion_a @ G @ X ) )
= ( produc9011544418120257559tion_a @ ( F @ ( produc174239612569448963tion_a @ X ) ) @ ( G @ ( produc8952039381521079109tion_a @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_874_apfst__apsnd,axiom,
! [F: option_a > a,G: option_a > nat,X: produc3509355604313844263tion_a] :
( ( produc7608241796298018374_a_nat @ F @ ( produc2390184552787727916tion_a @ G @ X ) )
= ( product_Pair_a_nat @ ( F @ ( produc174239612569448963tion_a @ X ) ) @ ( G @ ( produc8952039381521079109tion_a @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_875_exE__realizer_H,axiom,
! [P2: option_a > option_a > $o,P: produc3509355604313844263tion_a] :
( ( P2 @ ( produc8952039381521079109tion_a @ P ) @ ( produc174239612569448963tion_a @ P ) )
=> ~ ! [X3: option_a,Y3: option_a] :
~ ( P2 @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_876_apsnd__conv,axiom,
! [F: nat > nat,X: produc3509355604313844263tion_a,Y: nat] :
( ( produc3436106325092296583tion_a @ F @ ( produc5457283509376734880_a_nat @ X @ Y ) )
= ( produc5457283509376734880_a_nat @ X @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_877_apsnd__conv,axiom,
! [F: option_a > option_a,X: option_a,Y: option_a] :
( ( produc8446506941092750918tion_a @ F @ ( produc9011544418120257559tion_a @ X @ Y ) )
= ( produc9011544418120257559tion_a @ X @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_878_apsnd__conv,axiom,
! [F: nat > nat,X: a,Y: nat] :
( ( produc8521403278789009502_nat_a @ F @ ( product_Pair_a_nat @ X @ Y ) )
= ( product_Pair_a_nat @ X @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_879_fst__apsnd,axiom,
! [F: option_a > option_a,X: produc3509355604313844263tion_a] :
( ( produc174239612569448963tion_a @ ( produc8446506941092750918tion_a @ F @ X ) )
= ( produc174239612569448963tion_a @ X ) ) ).
% fst_apsnd
thf(fact_880_fst__comp__apsnd,axiom,
! [F: option_a > option_a] :
( ( comp_P3618168059068810855tion_a @ produc174239612569448963tion_a @ ( produc8446506941092750918tion_a @ F ) )
= produc174239612569448963tion_a ) ).
% fst_comp_apsnd
thf(fact_881_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X2: $o] : ( member_o @ X2 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_882_bot__empty__eq,axiom,
( bot_bo1388487281569869310_nat_o
= ( ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ bot_bo9049108969261143879_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_883_bot__empty__eq,axiom,
( bot_bo210245961477745301_nat_o
= ( ^ [X2: produc4760857364844987814_a_nat] : ( member1430561565885860669_a_nat @ X2 @ bot_bo8420108762324073968_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_884_bot__empty__eq,axiom,
( bot_bo8217098413137715498on_a_o
= ( ^ [X2: produc3509355604313844263tion_a] : ( member5498148017924304208tion_a @ X2 @ bot_bo235252021745139059tion_a ) ) ) ).
% bot_empty_eq
thf(fact_885_bot__empty__eq2,axiom,
( bot_bo2240373790306724881_nat_o
= ( ^ [X2: produc3509355604313844263tion_a,Y5: nat] : ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ X2 @ Y5 ) @ bot_bo8420108762324073968_a_nat ) ) ) ).
% bot_empty_eq2
thf(fact_886_bot__empty__eq2,axiom,
( bot_bo2944298884891136090on_a_o
= ( ^ [X2: option_a,Y5: option_a] : ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y5 ) @ bot_bo235252021745139059tion_a ) ) ) ).
% bot_empty_eq2
thf(fact_887_bot__empty__eq2,axiom,
( bot_bot_a_nat_o
= ( ^ [X2: a,Y5: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y5 ) @ bot_bo9049108969261143879_a_nat ) ) ) ).
% bot_empty_eq2
thf(fact_888_sndOp__def,axiom,
( bNF_sn4238875959383316176tion_a
= ( ^ [P4: option_a > option_a > $o,Q3: option_a > option_a > $o,Ac: produc3509355604313844263tion_a] : ( produc9011544418120257559tion_a @ ( bNF_pi5353392459224037856tion_a @ P4 @ Q3 @ ( produc174239612569448963tion_a @ Ac ) @ ( produc8952039381521079109tion_a @ Ac ) ) @ ( produc8952039381521079109tion_a @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_889_predicate2__transferD,axiom,
! [R1: a > nat > $o,R22: a > nat > $o,P2: a > a > $o,Q: nat > nat > $o,A: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( bNF_re296739048540615757_nat_o @ R1
@ ( bNF_re214721413657058609at_o_o @ R22
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
@ P2
@ Q )
=> ( ( member5724188588386418708_a_nat @ A @ A2 )
=> ( ( member5724188588386418708_a_nat @ B @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ A2 @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ R1 ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ R22 ) ) )
=> ( ( P2 @ ( product_fst_a_nat @ A ) @ ( product_fst_a_nat @ B ) )
= ( Q @ ( product_snd_a_nat @ A ) @ ( product_snd_a_nat @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_890_predicate2__transferD,axiom,
! [R1: a > nat > $o,R22: option_a > option_a > $o,P2: a > option_a > $o,Q: nat > option_a > $o,A: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,B: produc3509355604313844263tion_a,B2: set_Pr7585778909603769095tion_a] :
( ( bNF_re4132041373010397989on_a_o @ R1
@ ( bNF_re905040088260304477_a_o_o @ R22
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
@ P2
@ Q )
=> ( ( member5724188588386418708_a_nat @ A @ A2 )
=> ( ( member5498148017924304208tion_a @ B @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ A2 @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ R1 ) ) )
=> ( ( ord_le4471550158292877991tion_a @ B2 @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ R22 ) ) )
=> ( ( P2 @ ( product_fst_a_nat @ A ) @ ( produc174239612569448963tion_a @ B ) )
= ( Q @ ( product_snd_a_nat @ A ) @ ( produc8952039381521079109tion_a @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_891_predicate2__transferD,axiom,
! [R1: a > nat > $o,R22: produc3509355604313844263tion_a > nat > $o,P2: a > produc3509355604313844263tion_a > $o,Q: nat > nat > $o,A: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,B: produc4760857364844987814_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ( bNF_re4840581437969981046_nat_o @ R1
@ ( bNF_re3303494005150464218at_o_o @ R22
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
@ P2
@ Q )
=> ( ( member5724188588386418708_a_nat @ A @ A2 )
=> ( ( member1430561565885860669_a_nat @ B @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ A2 @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ R1 ) ) )
=> ( ( ord_le5649184635953113020_a_nat @ B2 @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ R22 ) ) )
=> ( ( P2 @ ( product_fst_a_nat @ A ) @ ( produc1094642028913761716_a_nat @ B ) )
= ( Q @ ( product_snd_a_nat @ A ) @ ( produc8051327305810483698_a_nat @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_892_predicate2__transferD,axiom,
! [R1: option_a > option_a > $o,R22: a > nat > $o,P2: option_a > a > $o,Q: option_a > nat > $o,A: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a,B: product_prod_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( bNF_re668239450913114657_nat_o @ R1
@ ( bNF_re214721413657058609at_o_o @ R22
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
@ P2
@ Q )
=> ( ( member5498148017924304208tion_a @ A @ A2 )
=> ( ( member5724188588386418708_a_nat @ B @ B2 )
=> ( ( ord_le4471550158292877991tion_a @ A2 @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ R1 ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ R22 ) ) )
=> ( ( P2 @ ( produc174239612569448963tion_a @ A ) @ ( product_fst_a_nat @ B ) )
= ( Q @ ( produc8952039381521079109tion_a @ A ) @ ( product_snd_a_nat @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_893_predicate2__transferD,axiom,
! [R1: option_a > option_a > $o,R22: option_a > option_a > $o,P2: option_a > option_a > $o,Q: option_a > option_a > $o,A: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a,B: produc3509355604313844263tion_a,B2: set_Pr7585778909603769095tion_a] :
( ( bNF_re1541143192665202257on_a_o @ R1
@ ( bNF_re905040088260304477_a_o_o @ R22
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
@ P2
@ Q )
=> ( ( member5498148017924304208tion_a @ A @ A2 )
=> ( ( member5498148017924304208tion_a @ B @ B2 )
=> ( ( ord_le4471550158292877991tion_a @ A2 @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ R1 ) ) )
=> ( ( ord_le4471550158292877991tion_a @ B2 @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ R22 ) ) )
=> ( ( P2 @ ( produc174239612569448963tion_a @ A ) @ ( produc174239612569448963tion_a @ B ) )
= ( Q @ ( produc8952039381521079109tion_a @ A ) @ ( produc8952039381521079109tion_a @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_894_predicate2__transferD,axiom,
! [R1: option_a > option_a > $o,R22: produc3509355604313844263tion_a > nat > $o,P2: option_a > produc3509355604313844263tion_a > $o,Q: option_a > nat > $o,A: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a,B: produc4760857364844987814_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ( bNF_re3760461907148547146_nat_o @ R1
@ ( bNF_re3303494005150464218at_o_o @ R22
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
@ P2
@ Q )
=> ( ( member5498148017924304208tion_a @ A @ A2 )
=> ( ( member1430561565885860669_a_nat @ B @ B2 )
=> ( ( ord_le4471550158292877991tion_a @ A2 @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ R1 ) ) )
=> ( ( ord_le5649184635953113020_a_nat @ B2 @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ R22 ) ) )
=> ( ( P2 @ ( produc174239612569448963tion_a @ A ) @ ( produc1094642028913761716_a_nat @ B ) )
= ( Q @ ( produc8952039381521079109tion_a @ A ) @ ( produc8051327305810483698_a_nat @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_895_predicate2__transferD,axiom,
! [R1: produc3509355604313844263tion_a > nat > $o,R22: a > nat > $o,P2: produc3509355604313844263tion_a > a > $o,Q: nat > nat > $o,A: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat,B: product_prod_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( bNF_re7266954853851403236_nat_o @ R1
@ ( bNF_re214721413657058609at_o_o @ R22
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
@ P2
@ Q )
=> ( ( member1430561565885860669_a_nat @ A @ A2 )
=> ( ( member5724188588386418708_a_nat @ B @ B2 )
=> ( ( ord_le5649184635953113020_a_nat @ A2 @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ R1 ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ R22 ) ) )
=> ( ( P2 @ ( produc1094642028913761716_a_nat @ A ) @ ( product_fst_a_nat @ B ) )
= ( Q @ ( produc8051327305810483698_a_nat @ A ) @ ( product_snd_a_nat @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_896_predicate2__transferD,axiom,
! [R1: produc3509355604313844263tion_a > nat > $o,R22: option_a > option_a > $o,P2: produc3509355604313844263tion_a > option_a > $o,Q: nat > option_a > $o,A: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat,B: produc3509355604313844263tion_a,B2: set_Pr7585778909603769095tion_a] :
( ( bNF_re7417368233926557774on_a_o @ R1
@ ( bNF_re905040088260304477_a_o_o @ R22
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
@ P2
@ Q )
=> ( ( member1430561565885860669_a_nat @ A @ A2 )
=> ( ( member5498148017924304208tion_a @ B @ B2 )
=> ( ( ord_le5649184635953113020_a_nat @ A2 @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ R1 ) ) )
=> ( ( ord_le4471550158292877991tion_a @ B2 @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ R22 ) ) )
=> ( ( P2 @ ( produc1094642028913761716_a_nat @ A ) @ ( produc174239612569448963tion_a @ B ) )
= ( Q @ ( produc8051327305810483698_a_nat @ A ) @ ( produc8952039381521079109tion_a @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_897_predicate2__transferD,axiom,
! [R1: produc3509355604313844263tion_a > nat > $o,R22: produc3509355604313844263tion_a > nat > $o,P2: produc3509355604313844263tion_a > produc3509355604313844263tion_a > $o,Q: nat > nat > $o,A: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat,B: produc4760857364844987814_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ( bNF_re5291775304233153037_nat_o @ R1
@ ( bNF_re3303494005150464218at_o_o @ R22
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
@ P2
@ Q )
=> ( ( member1430561565885860669_a_nat @ A @ A2 )
=> ( ( member1430561565885860669_a_nat @ B @ B2 )
=> ( ( ord_le5649184635953113020_a_nat @ A2 @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ R1 ) ) )
=> ( ( ord_le5649184635953113020_a_nat @ B2 @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ R22 ) ) )
=> ( ( P2 @ ( produc1094642028913761716_a_nat @ A ) @ ( produc1094642028913761716_a_nat @ B ) )
= ( Q @ ( produc8051327305810483698_a_nat @ A ) @ ( produc8051327305810483698_a_nat @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_898_fun_Oin__rel,axiom,
! [R: produc3509355604313844263tion_a > nat > $o,A: product_prod_a_nat > produc3509355604313844263tion_a,B: product_prod_a_nat > nat] :
( ( bNF_re5878663065082895966_a_nat
@ ^ [Y4: product_prod_a_nat,Z2: product_prod_a_nat] : ( Y4 = Z2 )
@ R
@ A
@ B )
= ( ? [Z4: product_prod_a_nat > produc4760857364844987814_a_nat] :
( ( member2063753777887369945_a_nat @ Z4
@ ( collec4943286744929717527_a_nat
@ ^ [X2: product_prod_a_nat > produc4760857364844987814_a_nat] : ( ord_le5649184635953113020_a_nat @ ( image_3729436797865474090_a_nat @ X2 @ top_to3353692345378799459_a_nat ) @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ R ) ) ) ) )
& ( ( comp_P6873071508423116251_a_nat @ produc1094642028913761716_a_nat @ Z4 )
= A )
& ( ( comp_P6605823348199399384_a_nat @ produc8051327305810483698_a_nat @ Z4 )
= B ) ) ) ) ).
% fun.in_rel
thf(fact_899_convol__image__vimage2p,axiom,
! [F: a > produc3509355604313844263tion_a,G: nat > nat,R: produc3509355604313844263tion_a > nat > $o] : ( ord_le5649184635953113020_a_nat @ ( image_3729436797865474090_a_nat @ ( bNF_co4564054178660176785_a_nat @ ( comp_a5132706136179606153_a_nat @ F @ product_fst_a_nat ) @ ( comp_n7970742859913575170_a_nat @ G @ product_snd_a_nat ) ) @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ ( bNF_vi2654601155092365586_nat_o @ F @ G @ R ) ) ) ) @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ R ) ) ) ).
% convol_image_vimage2p
thf(fact_900_convol__image__vimage2p,axiom,
! [F: option_a > a,G: option_a > nat,R: a > nat > $o] : ( ord_le8666007276011122963_a_nat @ ( image_2274939045924579167_a_nat @ ( bNF_co1502894461497240272_a_nat @ ( comp_o4728027832502833098tion_a @ F @ produc174239612569448963tion_a ) @ ( comp_o636930606119982068tion_a @ G @ produc8952039381521079109tion_a ) ) @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ ( bNF_vi2495386357617530871_nat_o @ F @ G @ R ) ) ) ) @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ R ) ) ) ).
% convol_image_vimage2p
thf(fact_901_convol__image__vimage2p,axiom,
! [F: option_a > option_a,G: option_a > option_a,R: option_a > option_a > $o] : ( ord_le4471550158292877991tion_a @ ( image_6668519322520640453tion_a @ ( bNF_co1154721918818078718tion_a @ ( comp_o511183382729970692tion_a @ F @ produc174239612569448963tion_a ) @ ( comp_o511183382729970692tion_a @ G @ produc8952039381521079109tion_a ) ) @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ ( bNF_vi3938235033518226845on_a_o @ F @ G @ R ) ) ) ) @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ R ) ) ) ).
% convol_image_vimage2p
thf(fact_902_convol__image__vimage2p,axiom,
! [F: option_a > produc3509355604313844263tion_a,G: option_a > nat,R: produc3509355604313844263tion_a > nat > $o] : ( ord_le5649184635953113020_a_nat @ ( image_8907095238652253832_a_nat @ ( bNF_co3083941901121763833_a_nat @ ( comp_o4150946686980370657tion_a @ F @ produc174239612569448963tion_a ) @ ( comp_o636930606119982068tion_a @ G @ produc8952039381521079109tion_a ) ) @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ ( bNF_vi8700166482369260960_nat_o @ F @ G @ R ) ) ) ) @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ R ) ) ) ).
% convol_image_vimage2p
thf(fact_903_convol__image__vimage2p,axiom,
! [F: produc3509355604313844263tion_a > a,G: nat > nat,R: a > nat > $o] : ( ord_le8666007276011122963_a_nat @ ( image_6737578356684938026_a_nat @ ( bNF_co1810985475411362175_a_nat @ ( comp_P7782882863831637408_a_nat @ F @ produc1094642028913761716_a_nat ) @ ( comp_n8250875655535308331_a_nat @ G @ produc8051327305810483698_a_nat ) ) @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ ( bNF_vi5709377802595389348_nat_o @ F @ G @ R ) ) ) ) @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ R ) ) ) ).
% convol_image_vimage2p
thf(fact_904_convol__image__vimage2p,axiom,
! [F: produc3509355604313844263tion_a > option_a,G: nat > option_a,R: option_a > option_a > $o] : ( ord_le4471550158292877991tion_a @ ( image_1391120077856136634tion_a @ ( bNF_co3392937053677906959tion_a @ ( comp_P6361877575466752358_a_nat @ F @ produc1094642028913761716_a_nat ) @ ( comp_n3098180662977613943_a_nat @ G @ produc8051327305810483698_a_nat ) ) @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ ( bNF_vi5538567187793887548on_a_o @ F @ G @ R ) ) ) ) @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ R ) ) ) ).
% convol_image_vimage2p
thf(fact_905_convol__image__vimage2p,axiom,
! [F: produc3509355604313844263tion_a > produc3509355604313844263tion_a,G: nat > nat,R: produc3509355604313844263tion_a > nat > $o] : ( ord_le5649184635953113020_a_nat @ ( image_4137284907621822035_a_nat @ ( bNF_co7748663798163203240_a_nat @ ( comp_P2896547414472712137_a_nat @ F @ produc1094642028913761716_a_nat ) @ ( comp_n8250875655535308331_a_nat @ G @ produc8051327305810483698_a_nat ) ) @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ ( bNF_vi8118414379931976763_nat_o @ F @ G @ R ) ) ) ) @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ R ) ) ) ).
% convol_image_vimage2p
thf(fact_906_iso__tuple__UNIV__I,axiom,
! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_907_iso__tuple__UNIV__I,axiom,
! [X: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X @ top_to3353692345378799459_a_nat ) ).
% iso_tuple_UNIV_I
thf(fact_908_iso__tuple__UNIV__I,axiom,
! [X: produc4760857364844987814_a_nat] : ( member1430561565885860669_a_nat @ X @ top_to7809427810062579212_a_nat ) ).
% iso_tuple_UNIV_I
thf(fact_909_iso__tuple__UNIV__I,axiom,
! [X: produc3509355604313844263tion_a] : ( member5498148017924304208tion_a @ X @ top_to31087889803499607tion_a ) ).
% iso_tuple_UNIV_I
thf(fact_910_UNIV__I,axiom,
! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% UNIV_I
thf(fact_911_UNIV__I,axiom,
! [X: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X @ top_to3353692345378799459_a_nat ) ).
% UNIV_I
thf(fact_912_UNIV__I,axiom,
! [X: produc4760857364844987814_a_nat] : ( member1430561565885860669_a_nat @ X @ top_to7809427810062579212_a_nat ) ).
% UNIV_I
thf(fact_913_UNIV__I,axiom,
! [X: produc3509355604313844263tion_a] : ( member5498148017924304208tion_a @ X @ top_to31087889803499607tion_a ) ).
% UNIV_I
thf(fact_914_Collect__const__case__prod,axiom,
! [P2: $o] :
( ( P2
=> ( ( collec4135126896892755346tion_a
@ ( produc7248784972643495020on_a_o
@ ^ [A6: option_a,B6: option_a] : P2 ) )
= top_to31087889803499607tion_a ) )
& ( ~ P2
=> ( ( collec4135126896892755346tion_a
@ ( produc7248784972643495020on_a_o
@ ^ [A6: option_a,B6: option_a] : P2 ) )
= bot_bo235252021745139059tion_a ) ) ) ).
% Collect_const_case_prod
thf(fact_915_Collect__const__case__prod,axiom,
! [P2: $o] :
( ( P2
=> ( ( collec7438131855006933115_a_nat
@ ( produc7988141324749014923_nat_o
@ ^ [A6: produc3509355604313844263tion_a,B6: nat] : P2 ) )
= top_to7809427810062579212_a_nat ) )
& ( ~ P2
=> ( ( collec7438131855006933115_a_nat
@ ( produc7988141324749014923_nat_o
@ ^ [A6: produc3509355604313844263tion_a,B6: nat] : P2 ) )
= bot_bo8420108762324073968_a_nat ) ) ) ).
% Collect_const_case_prod
thf(fact_916_range__fst,axiom,
( ( image_7272866627766095330tion_a @ produc174239612569448963tion_a @ top_to31087889803499607tion_a )
= top_top_set_option_a ) ).
% range_fst
thf(fact_917_UNIV__eq__I,axiom,
! [A2: set_o] :
( ! [X3: $o] : ( member_o @ X3 @ A2 )
=> ( top_top_set_o = A2 ) ) ).
% UNIV_eq_I
thf(fact_918_UNIV__eq__I,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ! [X3: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X3 @ A2 )
=> ( top_to3353692345378799459_a_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_919_UNIV__eq__I,axiom,
! [A2: set_Pr8660066742269822812_a_nat] :
( ! [X3: produc4760857364844987814_a_nat] : ( member1430561565885860669_a_nat @ X3 @ A2 )
=> ( top_to7809427810062579212_a_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_920_UNIV__eq__I,axiom,
! [A2: set_Pr7585778909603769095tion_a] :
( ! [X3: produc3509355604313844263tion_a] : ( member5498148017924304208tion_a @ X3 @ A2 )
=> ( top_to31087889803499607tion_a = A2 ) ) ).
% UNIV_eq_I
thf(fact_921_UNIV__witness,axiom,
? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_922_UNIV__witness,axiom,
? [X3: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X3 @ top_to3353692345378799459_a_nat ) ).
% UNIV_witness
thf(fact_923_UNIV__witness,axiom,
? [X3: produc4760857364844987814_a_nat] : ( member1430561565885860669_a_nat @ X3 @ top_to7809427810062579212_a_nat ) ).
% UNIV_witness
thf(fact_924_UNIV__witness,axiom,
? [X3: produc3509355604313844263tion_a] : ( member5498148017924304208tion_a @ X3 @ top_to31087889803499607tion_a ) ).
% UNIV_witness
thf(fact_925_top__option__def,axiom,
( top_to4210687906793739827_a_nat
= ( some_s1880800141159441806_a_nat @ top_to3353692345378799459_a_nat ) ) ).
% top_option_def
thf(fact_926_top__option__def,axiom,
( top_to4740459869331219548_a_nat
= ( some_s2314368709036572727_a_nat @ top_to7809427810062579212_a_nat ) ) ).
% top_option_def
thf(fact_927_top__greatest,axiom,
! [A: set_Pr8660066742269822812_a_nat] : ( ord_le5649184635953113020_a_nat @ A @ top_to7809427810062579212_a_nat ) ).
% top_greatest
thf(fact_928_top__greatest,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ top_to3353692345378799459_a_nat ) ).
% top_greatest
thf(fact_929_top__greatest,axiom,
! [A: option5987904771724668803_a_nat] : ( ord_le1496697172603502563_a_nat @ A @ top_to4210687906793739827_a_nat ) ).
% top_greatest
thf(fact_930_top__greatest,axiom,
! [A: option2973356794535198124_a_nat] : ( ord_le8848857416346193420_a_nat @ A @ top_to4740459869331219548_a_nat ) ).
% top_greatest
thf(fact_931_top__greatest,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% top_greatest
thf(fact_932_top_Oextremum__unique,axiom,
! [A: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ top_to7809427810062579212_a_nat @ A )
= ( A = top_to7809427810062579212_a_nat ) ) ).
% top.extremum_unique
thf(fact_933_top_Oextremum__unique,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ top_to3353692345378799459_a_nat @ A )
= ( A = top_to3353692345378799459_a_nat ) ) ).
% top.extremum_unique
thf(fact_934_top_Oextremum__unique,axiom,
! [A: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ top_to4210687906793739827_a_nat @ A )
= ( A = top_to4210687906793739827_a_nat ) ) ).
% top.extremum_unique
thf(fact_935_top_Oextremum__unique,axiom,
! [A: option2973356794535198124_a_nat] :
( ( ord_le8848857416346193420_a_nat @ top_to4740459869331219548_a_nat @ A )
= ( A = top_to4740459869331219548_a_nat ) ) ).
% top.extremum_unique
thf(fact_936_top_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
= ( A = top_top_set_o ) ) ).
% top.extremum_unique
thf(fact_937_top_Oextremum__uniqueI,axiom,
! [A: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ top_to7809427810062579212_a_nat @ A )
=> ( A = top_to7809427810062579212_a_nat ) ) ).
% top.extremum_uniqueI
thf(fact_938_top_Oextremum__uniqueI,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ top_to3353692345378799459_a_nat @ A )
=> ( A = top_to3353692345378799459_a_nat ) ) ).
% top.extremum_uniqueI
thf(fact_939_top_Oextremum__uniqueI,axiom,
! [A: option5987904771724668803_a_nat] :
( ( ord_le1496697172603502563_a_nat @ top_to4210687906793739827_a_nat @ A )
=> ( A = top_to4210687906793739827_a_nat ) ) ).
% top.extremum_uniqueI
thf(fact_940_top_Oextremum__uniqueI,axiom,
! [A: option2973356794535198124_a_nat] :
( ( ord_le8848857416346193420_a_nat @ top_to4740459869331219548_a_nat @ A )
=> ( A = top_to4740459869331219548_a_nat ) ) ).
% top.extremum_uniqueI
thf(fact_941_top_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
=> ( A = top_top_set_o ) ) ).
% top.extremum_uniqueI
thf(fact_942_rangeI,axiom,
! [F: produc3509355604313844263tion_a > $o,X: produc3509355604313844263tion_a] : ( member_o @ ( F @ X ) @ ( image_166006535297930294on_a_o @ F @ top_to31087889803499607tion_a ) ) ).
% rangeI
thf(fact_943_rangeI,axiom,
! [F: produc4760857364844987814_a_nat > $o,X: produc4760857364844987814_a_nat] : ( member_o @ ( F @ X ) @ ( image_6768402879995552939_nat_o @ F @ top_to7809427810062579212_a_nat ) ) ).
% rangeI
thf(fact_944_rangeI,axiom,
! [F: $o > $o,X: $o] : ( member_o @ ( F @ X ) @ ( image_o_o @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_945_rangeI,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,X: product_prod_a_nat] : ( member1430561565885860669_a_nat @ ( F @ X ) @ ( image_3729436797865474090_a_nat @ F @ top_to3353692345378799459_a_nat ) ) ).
% rangeI
thf(fact_946_range__eqI,axiom,
! [B: $o,F: produc3509355604313844263tion_a > $o,X: produc3509355604313844263tion_a] :
( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_166006535297930294on_a_o @ F @ top_to31087889803499607tion_a ) ) ) ).
% range_eqI
thf(fact_947_range__eqI,axiom,
! [B: $o,F: produc4760857364844987814_a_nat > $o,X: produc4760857364844987814_a_nat] :
( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_6768402879995552939_nat_o @ F @ top_to7809427810062579212_a_nat ) ) ) ).
% range_eqI
thf(fact_948_range__eqI,axiom,
! [B: $o,F: $o > $o,X: $o] :
( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_o_o @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_949_range__eqI,axiom,
! [B: produc4760857364844987814_a_nat,F: product_prod_a_nat > produc4760857364844987814_a_nat,X: product_prod_a_nat] :
( ( B
= ( F @ X ) )
=> ( member1430561565885860669_a_nat @ B @ ( image_3729436797865474090_a_nat @ F @ top_to3353692345378799459_a_nat ) ) ) ).
% range_eqI
thf(fact_950_surjD,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,Y: produc4760857364844987814_a_nat] :
( ( ( image_3729436797865474090_a_nat @ F @ top_to3353692345378799459_a_nat )
= top_to7809427810062579212_a_nat )
=> ? [X3: product_prod_a_nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_951_surjD,axiom,
! [F: produc3509355604313844263tion_a > $o,Y: $o] :
( ( ( image_166006535297930294on_a_o @ F @ top_to31087889803499607tion_a )
= top_top_set_o )
=> ? [X3: produc3509355604313844263tion_a] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_952_surjD,axiom,
! [F: produc4760857364844987814_a_nat > $o,Y: $o] :
( ( ( image_6768402879995552939_nat_o @ F @ top_to7809427810062579212_a_nat )
= top_top_set_o )
=> ? [X3: produc4760857364844987814_a_nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_953_surjD,axiom,
! [F: $o > $o,Y: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ? [X3: $o] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_954_surjE,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,Y: produc4760857364844987814_a_nat] :
( ( ( image_3729436797865474090_a_nat @ F @ top_to3353692345378799459_a_nat )
= top_to7809427810062579212_a_nat )
=> ~ ! [X3: product_prod_a_nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_955_surjE,axiom,
! [F: produc3509355604313844263tion_a > $o,Y: $o] :
( ( ( image_166006535297930294on_a_o @ F @ top_to31087889803499607tion_a )
= top_top_set_o )
=> ~ ! [X3: produc3509355604313844263tion_a] :
( Y
= ( ~ ( F @ X3 ) ) ) ) ).
% surjE
thf(fact_956_surjE,axiom,
! [F: produc4760857364844987814_a_nat > $o,Y: $o] :
( ( ( image_6768402879995552939_nat_o @ F @ top_to7809427810062579212_a_nat )
= top_top_set_o )
=> ~ ! [X3: produc4760857364844987814_a_nat] :
( Y
= ( ~ ( F @ X3 ) ) ) ) ).
% surjE
thf(fact_957_surjE,axiom,
! [F: $o > $o,Y: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ~ ! [X3: $o] :
( Y
= ( ~ ( F @ X3 ) ) ) ) ).
% surjE
thf(fact_958_surjI,axiom,
! [G: product_prod_a_nat > produc4760857364844987814_a_nat,F: produc4760857364844987814_a_nat > product_prod_a_nat] :
( ! [X3: produc4760857364844987814_a_nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_3729436797865474090_a_nat @ G @ top_to3353692345378799459_a_nat )
= top_to7809427810062579212_a_nat ) ) ).
% surjI
thf(fact_959_surjI,axiom,
! [G: produc3509355604313844263tion_a > $o,F: $o > produc3509355604313844263tion_a] :
( ! [X3: $o] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_166006535297930294on_a_o @ G @ top_to31087889803499607tion_a )
= top_top_set_o ) ) ).
% surjI
thf(fact_960_surjI,axiom,
! [G: produc4760857364844987814_a_nat > $o,F: $o > produc4760857364844987814_a_nat] :
( ! [X3: $o] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_6768402879995552939_nat_o @ G @ top_to7809427810062579212_a_nat )
= top_top_set_o ) ) ).
% surjI
thf(fact_961_surjI,axiom,
! [G: $o > $o,F: $o > $o] :
( ! [X3: $o] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_o_o @ G @ top_top_set_o )
= top_top_set_o ) ) ).
% surjI
thf(fact_962_surj__def,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat] :
( ( ( image_3729436797865474090_a_nat @ F @ top_to3353692345378799459_a_nat )
= top_to7809427810062579212_a_nat )
= ( ! [Y5: produc4760857364844987814_a_nat] :
? [X2: product_prod_a_nat] :
( Y5
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_963_surj__def,axiom,
! [F: produc3509355604313844263tion_a > $o] :
( ( ( image_166006535297930294on_a_o @ F @ top_to31087889803499607tion_a )
= top_top_set_o )
= ( ! [Y5: $o] :
? [X2: produc3509355604313844263tion_a] :
( Y5
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_964_surj__def,axiom,
! [F: produc4760857364844987814_a_nat > $o] :
( ( ( image_6768402879995552939_nat_o @ F @ top_to7809427810062579212_a_nat )
= top_top_set_o )
= ( ! [Y5: $o] :
? [X2: produc4760857364844987814_a_nat] :
( Y5
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_965_surj__def,axiom,
! [F: $o > $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
= ( ! [Y5: $o] :
? [X2: $o] :
( Y5
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_966_subset__UNIV,axiom,
! [A2: set_Pr8660066742269822812_a_nat] : ( ord_le5649184635953113020_a_nat @ A2 @ top_to7809427810062579212_a_nat ) ).
% subset_UNIV
thf(fact_967_subset__UNIV,axiom,
! [A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A2 @ top_to3353692345378799459_a_nat ) ).
% subset_UNIV
thf(fact_968_subset__UNIV,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ top_top_set_o ) ).
% subset_UNIV
thf(fact_969_rangeE,axiom,
! [B: $o,F: produc3509355604313844263tion_a > $o] :
( ( member_o @ B @ ( image_166006535297930294on_a_o @ F @ top_to31087889803499607tion_a ) )
=> ~ ! [X3: produc3509355604313844263tion_a] :
( B
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_970_rangeE,axiom,
! [B: $o,F: produc4760857364844987814_a_nat > $o] :
( ( member_o @ B @ ( image_6768402879995552939_nat_o @ F @ top_to7809427810062579212_a_nat ) )
=> ~ ! [X3: produc4760857364844987814_a_nat] :
( B
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_971_rangeE,axiom,
! [B: $o,F: $o > $o] :
( ( member_o @ B @ ( image_o_o @ F @ top_top_set_o ) )
=> ~ ! [X3: $o] :
( B
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_972_rangeE,axiom,
! [B: produc4760857364844987814_a_nat,F: product_prod_a_nat > produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ B @ ( image_3729436797865474090_a_nat @ F @ top_to3353692345378799459_a_nat ) )
=> ~ ! [X3: product_prod_a_nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_973_range__composition,axiom,
! [F: $o > $o,G: $o > $o] :
( ( image_o_o
@ ^ [X2: $o] : ( F @ ( G @ X2 ) )
@ top_top_set_o )
= ( image_o_o @ F @ ( image_o_o @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_974_range__composition,axiom,
! [F: $o > $o,G: produc3509355604313844263tion_a > $o] :
( ( image_166006535297930294on_a_o
@ ^ [X2: produc3509355604313844263tion_a] : ( F @ ( G @ X2 ) )
@ top_to31087889803499607tion_a )
= ( image_o_o @ F @ ( image_166006535297930294on_a_o @ G @ top_to31087889803499607tion_a ) ) ) ).
% range_composition
thf(fact_975_range__composition,axiom,
! [F: produc3509355604313844263tion_a > $o,G: $o > produc3509355604313844263tion_a] :
( ( image_o_o
@ ^ [X2: $o] : ( F @ ( G @ X2 ) )
@ top_top_set_o )
= ( image_166006535297930294on_a_o @ F @ ( image_628002824077805460tion_a @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_976_range__composition,axiom,
! [F: $o > $o,G: produc4760857364844987814_a_nat > $o] :
( ( image_6768402879995552939_nat_o
@ ^ [X2: produc4760857364844987814_a_nat] : ( F @ ( G @ X2 ) )
@ top_to7809427810062579212_a_nat )
= ( image_o_o @ F @ ( image_6768402879995552939_nat_o @ G @ top_to7809427810062579212_a_nat ) ) ) ).
% range_composition
thf(fact_977_range__composition,axiom,
! [F: produc4760857364844987814_a_nat > $o,G: $o > produc4760857364844987814_a_nat] :
( ( image_o_o
@ ^ [X2: $o] : ( F @ ( G @ X2 ) )
@ top_top_set_o )
= ( image_6768402879995552939_nat_o @ F @ ( image_2232132205779913721_a_nat @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_978_range__composition,axiom,
! [F: produc4760857364844987814_a_nat > $o,G: product_prod_a_nat > produc4760857364844987814_a_nat] :
( ( image_3076334604395105300_nat_o
@ ^ [X2: product_prod_a_nat] : ( F @ ( G @ X2 ) )
@ top_to3353692345378799459_a_nat )
= ( image_6768402879995552939_nat_o @ F @ ( image_3729436797865474090_a_nat @ G @ top_to3353692345378799459_a_nat ) ) ) ).
% range_composition
thf(fact_979_range__composition,axiom,
! [F: produc3509355604313844263tion_a > $o,G: produc3509355604313844263tion_a > produc3509355604313844263tion_a] :
( ( image_166006535297930294on_a_o
@ ^ [X2: produc3509355604313844263tion_a] : ( F @ ( G @ X2 ) )
@ top_to31087889803499607tion_a )
= ( image_166006535297930294on_a_o @ F @ ( image_6668519322520640453tion_a @ G @ top_to31087889803499607tion_a ) ) ) ).
% range_composition
thf(fact_980_range__composition,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,G: product_prod_a_nat > product_prod_a_nat] :
( ( image_3729436797865474090_a_nat
@ ^ [X2: product_prod_a_nat] : ( F @ ( G @ X2 ) )
@ top_to3353692345378799459_a_nat )
= ( image_3729436797865474090_a_nat @ F @ ( image_8708701149064048897_a_nat @ G @ top_to3353692345378799459_a_nat ) ) ) ).
% range_composition
thf(fact_981_range__composition,axiom,
! [F: produc4760857364844987814_a_nat > $o,G: produc3509355604313844263tion_a > produc4760857364844987814_a_nat] :
( ( image_166006535297930294on_a_o
@ ^ [X2: produc3509355604313844263tion_a] : ( F @ ( G @ X2 ) )
@ top_to31087889803499607tion_a )
= ( image_6768402879995552939_nat_o @ F @ ( image_8907095238652253832_a_nat @ G @ top_to31087889803499607tion_a ) ) ) ).
% range_composition
thf(fact_982_range__composition,axiom,
! [F: produc3509355604313844263tion_a > $o,G: produc4760857364844987814_a_nat > produc3509355604313844263tion_a] :
( ( image_6768402879995552939_nat_o
@ ^ [X2: produc4760857364844987814_a_nat] : ( F @ ( G @ X2 ) )
@ top_to7809427810062579212_a_nat )
= ( image_166006535297930294on_a_o @ F @ ( image_1391120077856136634tion_a @ G @ top_to7809427810062579212_a_nat ) ) ) ).
% range_composition
thf(fact_983_fun_Orel__refl__strong,axiom,
! [X: produc3509355604313844263tion_a > $o,Ra: $o > $o > $o] :
( ! [Z5: $o] :
( ( member_o @ Z5 @ ( image_166006535297930294on_a_o @ X @ top_to31087889803499607tion_a ) )
=> ( Ra @ Z5 @ Z5 ) )
=> ( bNF_re8491395028433271709_a_o_o
@ ^ [Y4: produc3509355604313844263tion_a,Z2: produc3509355604313844263tion_a] : ( Y4 = Z2 )
@ Ra
@ X
@ X ) ) ).
% fun.rel_refl_strong
thf(fact_984_fun_Orel__refl__strong,axiom,
! [X: produc4760857364844987814_a_nat > $o,Ra: $o > $o > $o] :
( ! [Z5: $o] :
( ( member_o @ Z5 @ ( image_6768402879995552939_nat_o @ X @ top_to7809427810062579212_a_nat ) )
=> ( Ra @ Z5 @ Z5 ) )
=> ( bNF_re7959353272085281067at_o_o
@ ^ [Y4: produc4760857364844987814_a_nat,Z2: produc4760857364844987814_a_nat] : ( Y4 = Z2 )
@ Ra
@ X
@ X ) ) ).
% fun.rel_refl_strong
thf(fact_985_fun_Orel__refl__strong,axiom,
! [X: $o > $o,Ra: $o > $o > $o] :
( ! [Z5: $o] :
( ( member_o @ Z5 @ ( image_o_o @ X @ top_top_set_o ) )
=> ( Ra @ Z5 @ Z5 ) )
=> ( bNF_rel_fun_o_o_o_o
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ Ra
@ X
@ X ) ) ).
% fun.rel_refl_strong
thf(fact_986_fun_Orel__refl__strong,axiom,
! [X: product_prod_a_nat > produc4760857364844987814_a_nat,Ra: produc4760857364844987814_a_nat > produc4760857364844987814_a_nat > $o] :
( ! [Z5: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ Z5 @ ( image_3729436797865474090_a_nat @ X @ top_to3353692345378799459_a_nat ) )
=> ( Ra @ Z5 @ Z5 ) )
=> ( bNF_re2369171833462768551_a_nat
@ ^ [Y4: product_prod_a_nat,Z2: product_prod_a_nat] : ( Y4 = Z2 )
@ Ra
@ X
@ X ) ) ).
% fun.rel_refl_strong
thf(fact_987_fun_Orel__mono__strong,axiom,
! [R: $o > $o > $o,X: $o > $o,Y: $o > $o,Ra: $o > $o > $o] :
( ( bNF_rel_fun_o_o_o_o
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ R
@ X
@ Y )
=> ( ! [Z5: $o,Yb: $o] :
( ( member_o @ Z5 @ ( image_o_o @ X @ top_top_set_o ) )
=> ( ( member_o @ Yb @ ( image_o_o @ Y @ top_top_set_o ) )
=> ( ( R @ Z5 @ Yb )
=> ( Ra @ Z5 @ Yb ) ) ) )
=> ( bNF_rel_fun_o_o_o_o
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ Ra
@ X
@ Y ) ) ) ).
% fun.rel_mono_strong
thf(fact_988_fun_Orel__mono__strong,axiom,
! [R: $o > product_prod_a_nat > $o,X: $o > $o,Y: $o > product_prod_a_nat,Ra: $o > product_prod_a_nat > $o] :
( ( bNF_re3038775964670433016_a_nat
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ R
@ X
@ Y )
=> ( ! [Z5: $o,Yb: product_prod_a_nat] :
( ( member_o @ Z5 @ ( image_o_o @ X @ top_top_set_o ) )
=> ( ( member5724188588386418708_a_nat @ Yb @ ( image_7339617019956244688_a_nat @ Y @ top_top_set_o ) )
=> ( ( R @ Z5 @ Yb )
=> ( Ra @ Z5 @ Yb ) ) ) )
=> ( bNF_re3038775964670433016_a_nat
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ Ra
@ X
@ Y ) ) ) ).
% fun.rel_mono_strong
thf(fact_989_fun_Orel__mono__strong,axiom,
! [R: product_prod_a_nat > $o > $o,X: $o > product_prod_a_nat,Y: $o > $o,Ra: product_prod_a_nat > $o > $o] :
( ( bNF_re7998865585964069436_nat_o
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ R
@ X
@ Y )
=> ( ! [Z5: product_prod_a_nat,Yb: $o] :
( ( member5724188588386418708_a_nat @ Z5 @ ( image_7339617019956244688_a_nat @ X @ top_top_set_o ) )
=> ( ( member_o @ Yb @ ( image_o_o @ Y @ top_top_set_o ) )
=> ( ( R @ Z5 @ Yb )
=> ( Ra @ Z5 @ Yb ) ) ) )
=> ( bNF_re7998865585964069436_nat_o
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ Ra
@ X
@ Y ) ) ) ).
% fun.rel_mono_strong
thf(fact_990_fun_Orel__mono__strong,axiom,
! [R: $o > $o > $o,X: produc3509355604313844263tion_a > $o,Y: produc3509355604313844263tion_a > $o,Ra: $o > $o > $o] :
( ( bNF_re8491395028433271709_a_o_o
@ ^ [Y4: produc3509355604313844263tion_a,Z2: produc3509355604313844263tion_a] : ( Y4 = Z2 )
@ R
@ X
@ Y )
=> ( ! [Z5: $o,Yb: $o] :
( ( member_o @ Z5 @ ( image_166006535297930294on_a_o @ X @ top_to31087889803499607tion_a ) )
=> ( ( member_o @ Yb @ ( image_166006535297930294on_a_o @ Y @ top_to31087889803499607tion_a ) )
=> ( ( R @ Z5 @ Yb )
=> ( Ra @ Z5 @ Yb ) ) ) )
=> ( bNF_re8491395028433271709_a_o_o
@ ^ [Y4: produc3509355604313844263tion_a,Z2: produc3509355604313844263tion_a] : ( Y4 = Z2 )
@ Ra
@ X
@ Y ) ) ) ).
% fun.rel_mono_strong
thf(fact_991_fun_Orel__mono__strong,axiom,
! [R: $o > produc3509355604313844263tion_a > $o,X: $o > $o,Y: $o > produc3509355604313844263tion_a,Ra: $o > produc3509355604313844263tion_a > $o] :
( ( bNF_re8060677170909647724tion_a
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ R
@ X
@ Y )
=> ( ! [Z5: $o,Yb: produc3509355604313844263tion_a] :
( ( member_o @ Z5 @ ( image_o_o @ X @ top_top_set_o ) )
=> ( ( member5498148017924304208tion_a @ Yb @ ( image_628002824077805460tion_a @ Y @ top_top_set_o ) )
=> ( ( R @ Z5 @ Yb )
=> ( Ra @ Z5 @ Yb ) ) ) )
=> ( bNF_re8060677170909647724tion_a
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ Ra
@ X
@ Y ) ) ) ).
% fun.rel_mono_strong
thf(fact_992_fun_Orel__mono__strong,axiom,
! [R: produc3509355604313844263tion_a > $o > $o,X: $o > produc3509355604313844263tion_a,Y: $o > $o,Ra: produc3509355604313844263tion_a > $o > $o] :
( ( bNF_re7598680882129772558on_a_o
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ R
@ X
@ Y )
=> ( ! [Z5: produc3509355604313844263tion_a,Yb: $o] :
( ( member5498148017924304208tion_a @ Z5 @ ( image_628002824077805460tion_a @ X @ top_top_set_o ) )
=> ( ( member_o @ Yb @ ( image_o_o @ Y @ top_top_set_o ) )
=> ( ( R @ Z5 @ Yb )
=> ( Ra @ Z5 @ Yb ) ) ) )
=> ( bNF_re7598680882129772558on_a_o
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ Ra
@ X
@ Y ) ) ) ).
% fun.rel_mono_strong
thf(fact_993_fun_Orel__mono__strong,axiom,
! [R: $o > $o > $o,X: produc4760857364844987814_a_nat > $o,Y: produc4760857364844987814_a_nat > $o,Ra: $o > $o > $o] :
( ( bNF_re7959353272085281067at_o_o
@ ^ [Y4: produc4760857364844987814_a_nat,Z2: produc4760857364844987814_a_nat] : ( Y4 = Z2 )
@ R
@ X
@ Y )
=> ( ! [Z5: $o,Yb: $o] :
( ( member_o @ Z5 @ ( image_6768402879995552939_nat_o @ X @ top_to7809427810062579212_a_nat ) )
=> ( ( member_o @ Yb @ ( image_6768402879995552939_nat_o @ Y @ top_to7809427810062579212_a_nat ) )
=> ( ( R @ Z5 @ Yb )
=> ( Ra @ Z5 @ Yb ) ) ) )
=> ( bNF_re7959353272085281067at_o_o
@ ^ [Y4: produc4760857364844987814_a_nat,Z2: produc4760857364844987814_a_nat] : ( Y4 = Z2 )
@ Ra
@ X
@ Y ) ) ) ).
% fun.rel_mono_strong
thf(fact_994_fun_Orel__mono__strong,axiom,
! [R: $o > product_prod_a_nat > $o,X: produc3509355604313844263tion_a > $o,Y: produc3509355604313844263tion_a > product_prod_a_nat,Ra: $o > product_prod_a_nat > $o] :
( ( bNF_re6940856228500498104_a_nat
@ ^ [Y4: produc3509355604313844263tion_a,Z2: produc3509355604313844263tion_a] : ( Y4 = Z2 )
@ R
@ X
@ Y )
=> ( ! [Z5: $o,Yb: product_prod_a_nat] :
( ( member_o @ Z5 @ ( image_166006535297930294on_a_o @ X @ top_to31087889803499607tion_a ) )
=> ( ( member5724188588386418708_a_nat @ Yb @ ( image_2274939045924579167_a_nat @ Y @ top_to31087889803499607tion_a ) )
=> ( ( R @ Z5 @ Yb )
=> ( Ra @ Z5 @ Yb ) ) ) )
=> ( bNF_re6940856228500498104_a_nat
@ ^ [Y4: produc3509355604313844263tion_a,Z2: produc3509355604313844263tion_a] : ( Y4 = Z2 )
@ Ra
@ X
@ Y ) ) ) ).
% fun.rel_mono_strong
thf(fact_995_fun_Orel__mono__strong,axiom,
! [R: $o > produc4760857364844987814_a_nat > $o,X: $o > $o,Y: $o > produc4760857364844987814_a_nat,Ra: $o > produc4760857364844987814_a_nat > $o] :
( ( bNF_re5119610622004122145_a_nat
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ R
@ X
@ Y )
=> ( ! [Z5: $o,Yb: produc4760857364844987814_a_nat] :
( ( member_o @ Z5 @ ( image_o_o @ X @ top_top_set_o ) )
=> ( ( member1430561565885860669_a_nat @ Yb @ ( image_2232132205779913721_a_nat @ Y @ top_top_set_o ) )
=> ( ( R @ Z5 @ Yb )
=> ( Ra @ Z5 @ Yb ) ) ) )
=> ( bNF_re5119610622004122145_a_nat
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ Ra
@ X
@ Y ) ) ) ).
% fun.rel_mono_strong
thf(fact_996_fun_Orel__mono__strong,axiom,
! [R: product_prod_a_nat > $o > $o,X: produc3509355604313844263tion_a > product_prod_a_nat,Y: produc3509355604313844263tion_a > $o,Ra: product_prod_a_nat > $o > $o] :
( ( bNF_re2677573812939358716_nat_o
@ ^ [Y4: produc3509355604313844263tion_a,Z2: produc3509355604313844263tion_a] : ( Y4 = Z2 )
@ R
@ X
@ Y )
=> ( ! [Z5: product_prod_a_nat,Yb: $o] :
( ( member5724188588386418708_a_nat @ Z5 @ ( image_2274939045924579167_a_nat @ X @ top_to31087889803499607tion_a ) )
=> ( ( member_o @ Yb @ ( image_166006535297930294on_a_o @ Y @ top_to31087889803499607tion_a ) )
=> ( ( R @ Z5 @ Yb )
=> ( Ra @ Z5 @ Yb ) ) ) )
=> ( bNF_re2677573812939358716_nat_o
@ ^ [Y4: produc3509355604313844263tion_a,Z2: produc3509355604313844263tion_a] : ( Y4 = Z2 )
@ Ra
@ X
@ Y ) ) ) ).
% fun.rel_mono_strong
thf(fact_997_fun_Orel__cong,axiom,
! [X: $o > $o,Ya: $o > $o,Y: $o > $o,Xa2: $o > $o,R: $o > $o > $o,Ra: $o > $o > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z5: $o,Yb: $o] :
( ( member_o @ Z5 @ ( image_o_o @ Ya @ top_top_set_o ) )
=> ( ( member_o @ Yb @ ( image_o_o @ Xa2 @ top_top_set_o ) )
=> ( ( R @ Z5 @ Yb )
= ( Ra @ Z5 @ Yb ) ) ) )
=> ( ( bNF_rel_fun_o_o_o_o
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ R
@ X
@ Y )
= ( bNF_rel_fun_o_o_o_o
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_998_fun_Orel__cong,axiom,
! [X: $o > $o,Ya: $o > $o,Y: $o > product_prod_a_nat,Xa2: $o > product_prod_a_nat,R: $o > product_prod_a_nat > $o,Ra: $o > product_prod_a_nat > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z5: $o,Yb: product_prod_a_nat] :
( ( member_o @ Z5 @ ( image_o_o @ Ya @ top_top_set_o ) )
=> ( ( member5724188588386418708_a_nat @ Yb @ ( image_7339617019956244688_a_nat @ Xa2 @ top_top_set_o ) )
=> ( ( R @ Z5 @ Yb )
= ( Ra @ Z5 @ Yb ) ) ) )
=> ( ( bNF_re3038775964670433016_a_nat
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ R
@ X
@ Y )
= ( bNF_re3038775964670433016_a_nat
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_999_fun_Orel__cong,axiom,
! [X: $o > product_prod_a_nat,Ya: $o > product_prod_a_nat,Y: $o > $o,Xa2: $o > $o,R: product_prod_a_nat > $o > $o,Ra: product_prod_a_nat > $o > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z5: product_prod_a_nat,Yb: $o] :
( ( member5724188588386418708_a_nat @ Z5 @ ( image_7339617019956244688_a_nat @ Ya @ top_top_set_o ) )
=> ( ( member_o @ Yb @ ( image_o_o @ Xa2 @ top_top_set_o ) )
=> ( ( R @ Z5 @ Yb )
= ( Ra @ Z5 @ Yb ) ) ) )
=> ( ( bNF_re7998865585964069436_nat_o
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ R
@ X
@ Y )
= ( bNF_re7998865585964069436_nat_o
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1000_fun_Orel__cong,axiom,
! [X: produc3509355604313844263tion_a > $o,Ya: produc3509355604313844263tion_a > $o,Y: produc3509355604313844263tion_a > $o,Xa2: produc3509355604313844263tion_a > $o,R: $o > $o > $o,Ra: $o > $o > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z5: $o,Yb: $o] :
( ( member_o @ Z5 @ ( image_166006535297930294on_a_o @ Ya @ top_to31087889803499607tion_a ) )
=> ( ( member_o @ Yb @ ( image_166006535297930294on_a_o @ Xa2 @ top_to31087889803499607tion_a ) )
=> ( ( R @ Z5 @ Yb )
= ( Ra @ Z5 @ Yb ) ) ) )
=> ( ( bNF_re8491395028433271709_a_o_o
@ ^ [Y4: produc3509355604313844263tion_a,Z2: produc3509355604313844263tion_a] : ( Y4 = Z2 )
@ R
@ X
@ Y )
= ( bNF_re8491395028433271709_a_o_o
@ ^ [Y4: produc3509355604313844263tion_a,Z2: produc3509355604313844263tion_a] : ( Y4 = Z2 )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1001_fun_Orel__cong,axiom,
! [X: $o > $o,Ya: $o > $o,Y: $o > produc3509355604313844263tion_a,Xa2: $o > produc3509355604313844263tion_a,R: $o > produc3509355604313844263tion_a > $o,Ra: $o > produc3509355604313844263tion_a > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z5: $o,Yb: produc3509355604313844263tion_a] :
( ( member_o @ Z5 @ ( image_o_o @ Ya @ top_top_set_o ) )
=> ( ( member5498148017924304208tion_a @ Yb @ ( image_628002824077805460tion_a @ Xa2 @ top_top_set_o ) )
=> ( ( R @ Z5 @ Yb )
= ( Ra @ Z5 @ Yb ) ) ) )
=> ( ( bNF_re8060677170909647724tion_a
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ R
@ X
@ Y )
= ( bNF_re8060677170909647724tion_a
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1002_fun_Orel__cong,axiom,
! [X: $o > produc3509355604313844263tion_a,Ya: $o > produc3509355604313844263tion_a,Y: $o > $o,Xa2: $o > $o,R: produc3509355604313844263tion_a > $o > $o,Ra: produc3509355604313844263tion_a > $o > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z5: produc3509355604313844263tion_a,Yb: $o] :
( ( member5498148017924304208tion_a @ Z5 @ ( image_628002824077805460tion_a @ Ya @ top_top_set_o ) )
=> ( ( member_o @ Yb @ ( image_o_o @ Xa2 @ top_top_set_o ) )
=> ( ( R @ Z5 @ Yb )
= ( Ra @ Z5 @ Yb ) ) ) )
=> ( ( bNF_re7598680882129772558on_a_o
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ R
@ X
@ Y )
= ( bNF_re7598680882129772558on_a_o
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1003_fun_Orel__cong,axiom,
! [X: produc4760857364844987814_a_nat > $o,Ya: produc4760857364844987814_a_nat > $o,Y: produc4760857364844987814_a_nat > $o,Xa2: produc4760857364844987814_a_nat > $o,R: $o > $o > $o,Ra: $o > $o > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z5: $o,Yb: $o] :
( ( member_o @ Z5 @ ( image_6768402879995552939_nat_o @ Ya @ top_to7809427810062579212_a_nat ) )
=> ( ( member_o @ Yb @ ( image_6768402879995552939_nat_o @ Xa2 @ top_to7809427810062579212_a_nat ) )
=> ( ( R @ Z5 @ Yb )
= ( Ra @ Z5 @ Yb ) ) ) )
=> ( ( bNF_re7959353272085281067at_o_o
@ ^ [Y4: produc4760857364844987814_a_nat,Z2: produc4760857364844987814_a_nat] : ( Y4 = Z2 )
@ R
@ X
@ Y )
= ( bNF_re7959353272085281067at_o_o
@ ^ [Y4: produc4760857364844987814_a_nat,Z2: produc4760857364844987814_a_nat] : ( Y4 = Z2 )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1004_fun_Orel__cong,axiom,
! [X: produc3509355604313844263tion_a > $o,Ya: produc3509355604313844263tion_a > $o,Y: produc3509355604313844263tion_a > product_prod_a_nat,Xa2: produc3509355604313844263tion_a > product_prod_a_nat,R: $o > product_prod_a_nat > $o,Ra: $o > product_prod_a_nat > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z5: $o,Yb: product_prod_a_nat] :
( ( member_o @ Z5 @ ( image_166006535297930294on_a_o @ Ya @ top_to31087889803499607tion_a ) )
=> ( ( member5724188588386418708_a_nat @ Yb @ ( image_2274939045924579167_a_nat @ Xa2 @ top_to31087889803499607tion_a ) )
=> ( ( R @ Z5 @ Yb )
= ( Ra @ Z5 @ Yb ) ) ) )
=> ( ( bNF_re6940856228500498104_a_nat
@ ^ [Y4: produc3509355604313844263tion_a,Z2: produc3509355604313844263tion_a] : ( Y4 = Z2 )
@ R
@ X
@ Y )
= ( bNF_re6940856228500498104_a_nat
@ ^ [Y4: produc3509355604313844263tion_a,Z2: produc3509355604313844263tion_a] : ( Y4 = Z2 )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1005_fun_Orel__cong,axiom,
! [X: $o > $o,Ya: $o > $o,Y: $o > produc4760857364844987814_a_nat,Xa2: $o > produc4760857364844987814_a_nat,R: $o > produc4760857364844987814_a_nat > $o,Ra: $o > produc4760857364844987814_a_nat > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z5: $o,Yb: produc4760857364844987814_a_nat] :
( ( member_o @ Z5 @ ( image_o_o @ Ya @ top_top_set_o ) )
=> ( ( member1430561565885860669_a_nat @ Yb @ ( image_2232132205779913721_a_nat @ Xa2 @ top_top_set_o ) )
=> ( ( R @ Z5 @ Yb )
= ( Ra @ Z5 @ Yb ) ) ) )
=> ( ( bNF_re5119610622004122145_a_nat
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ R
@ X
@ Y )
= ( bNF_re5119610622004122145_a_nat
@ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1006_fun_Orel__cong,axiom,
! [X: produc3509355604313844263tion_a > product_prod_a_nat,Ya: produc3509355604313844263tion_a > product_prod_a_nat,Y: produc3509355604313844263tion_a > $o,Xa2: produc3509355604313844263tion_a > $o,R: product_prod_a_nat > $o > $o,Ra: product_prod_a_nat > $o > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z5: product_prod_a_nat,Yb: $o] :
( ( member5724188588386418708_a_nat @ Z5 @ ( image_2274939045924579167_a_nat @ Ya @ top_to31087889803499607tion_a ) )
=> ( ( member_o @ Yb @ ( image_166006535297930294on_a_o @ Xa2 @ top_to31087889803499607tion_a ) )
=> ( ( R @ Z5 @ Yb )
= ( Ra @ Z5 @ Yb ) ) ) )
=> ( ( bNF_re2677573812939358716_nat_o
@ ^ [Y4: produc3509355604313844263tion_a,Z2: produc3509355604313844263tion_a] : ( Y4 = Z2 )
@ R
@ X
@ Y )
= ( bNF_re2677573812939358716_nat_o
@ ^ [Y4: produc3509355604313844263tion_a,Z2: produc3509355604313844263tion_a] : ( Y4 = Z2 )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1007_range__subsetD,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,B2: set_Pr8660066742269822812_a_nat,I2: product_prod_a_nat] :
( ( ord_le5649184635953113020_a_nat @ ( image_3729436797865474090_a_nat @ F @ top_to3353692345378799459_a_nat ) @ B2 )
=> ( member1430561565885860669_a_nat @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1008_range__subsetD,axiom,
! [F: produc3509355604313844263tion_a > $o,B2: set_o,I2: produc3509355604313844263tion_a] :
( ( ord_less_eq_set_o @ ( image_166006535297930294on_a_o @ F @ top_to31087889803499607tion_a ) @ B2 )
=> ( member_o @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1009_range__subsetD,axiom,
! [F: produc4760857364844987814_a_nat > $o,B2: set_o,I2: produc4760857364844987814_a_nat] :
( ( ord_less_eq_set_o @ ( image_6768402879995552939_nat_o @ F @ top_to7809427810062579212_a_nat ) @ B2 )
=> ( member_o @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1010_range__subsetD,axiom,
! [F: $o > $o,B2: set_o,I2: $o] :
( ( ord_less_eq_set_o @ ( image_o_o @ F @ top_top_set_o ) @ B2 )
=> ( member_o @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1011_fun_Oset__map,axiom,
! [F: $o > $o,V: $o > $o] :
( ( image_o_o @ ( comp_o_o_o @ F @ V ) @ top_top_set_o )
= ( image_o_o @ F @ ( image_o_o @ V @ top_top_set_o ) ) ) ).
% fun.set_map
thf(fact_1012_fun_Oset__map,axiom,
! [F: $o > $o,V: produc3509355604313844263tion_a > $o] :
( ( image_166006535297930294on_a_o @ ( comp_o8812929617658428164tion_a @ F @ V ) @ top_to31087889803499607tion_a )
= ( image_o_o @ F @ ( image_166006535297930294on_a_o @ V @ top_to31087889803499607tion_a ) ) ) ).
% fun.set_map
thf(fact_1013_fun_Oset__map,axiom,
! [F: produc3509355604313844263tion_a > $o,V: $o > produc3509355604313844263tion_a] :
( ( image_o_o @ ( comp_P4515272243702675076_a_o_o @ F @ V ) @ top_top_set_o )
= ( image_166006535297930294on_a_o @ F @ ( image_628002824077805460tion_a @ V @ top_top_set_o ) ) ) ).
% fun.set_map
thf(fact_1014_fun_Oset__map,axiom,
! [F: $o > $o,V: produc4760857364844987814_a_nat > $o] :
( ( image_6768402879995552939_nat_o @ ( comp_o3516818292788251785_a_nat @ F @ V ) @ top_to7809427810062579212_a_nat )
= ( image_o_o @ F @ ( image_6768402879995552939_nat_o @ V @ top_to7809427810062579212_a_nat ) ) ) ).
% fun.set_map
thf(fact_1015_fun_Oset__map,axiom,
! [F: produc4760857364844987814_a_nat > $o,V: $o > produc4760857364844987814_a_nat] :
( ( image_o_o @ ( comp_P885853961682478089at_o_o @ F @ V ) @ top_top_set_o )
= ( image_6768402879995552939_nat_o @ F @ ( image_2232132205779913721_a_nat @ V @ top_top_set_o ) ) ) ).
% fun.set_map
thf(fact_1016_fun_Oset__map,axiom,
! [F: produc4760857364844987814_a_nat > $o,V: product_prod_a_nat > produc4760857364844987814_a_nat] :
( ( image_3076334604395105300_nat_o @ ( comp_P7346075404395887948_a_nat @ F @ V ) @ top_to3353692345378799459_a_nat )
= ( image_6768402879995552939_nat_o @ F @ ( image_3729436797865474090_a_nat @ V @ top_to3353692345378799459_a_nat ) ) ) ).
% fun.set_map
thf(fact_1017_fun_Oset__map,axiom,
! [F: produc3509355604313844263tion_a > $o,V: produc3509355604313844263tion_a > produc3509355604313844263tion_a] :
( ( image_166006535297930294on_a_o @ ( comp_P3337116118691028755tion_a @ F @ V ) @ top_to31087889803499607tion_a )
= ( image_166006535297930294on_a_o @ F @ ( image_6668519322520640453tion_a @ V @ top_to31087889803499607tion_a ) ) ) ).
% fun.set_map
thf(fact_1018_fun_Oset__map,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,V: product_prod_a_nat > product_prod_a_nat] :
( ( image_3729436797865474090_a_nat @ ( comp_P6608656588004095109_a_nat @ F @ V ) @ top_to3353692345378799459_a_nat )
= ( image_3729436797865474090_a_nat @ F @ ( image_8708701149064048897_a_nat @ V @ top_to3353692345378799459_a_nat ) ) ) ).
% fun.set_map
thf(fact_1019_fun_Oset__map,axiom,
! [F: produc4760857364844987814_a_nat > $o,V: produc3509355604313844263tion_a > produc4760857364844987814_a_nat] :
( ( image_166006535297930294on_a_o @ ( comp_P8503938455319246360tion_a @ F @ V ) @ top_to31087889803499607tion_a )
= ( image_6768402879995552939_nat_o @ F @ ( image_8907095238652253832_a_nat @ V @ top_to31087889803499607tion_a ) ) ) ).
% fun.set_map
thf(fact_1020_fun_Oset__map,axiom,
! [F: produc3509355604313844263tion_a > $o,V: produc4760857364844987814_a_nat > produc3509355604313844263tion_a] :
( ( image_6768402879995552939_nat_o @ ( comp_P4709773760693074362_a_nat @ F @ V ) @ top_to7809427810062579212_a_nat )
= ( image_166006535297930294on_a_o @ F @ ( image_1391120077856136634tion_a @ V @ top_to7809427810062579212_a_nat ) ) ) ).
% fun.set_map
thf(fact_1021_fun_Omap__ident__strong,axiom,
! [T2: produc3509355604313844263tion_a > $o,F: $o > $o] :
( ! [Z5: $o] :
( ( member_o @ Z5 @ ( image_166006535297930294on_a_o @ T2 @ top_to31087889803499607tion_a ) )
=> ( ( F @ Z5 )
= Z5 ) )
=> ( ( comp_o8812929617658428164tion_a @ F @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_1022_fun_Omap__ident__strong,axiom,
! [T2: produc4760857364844987814_a_nat > $o,F: $o > $o] :
( ! [Z5: $o] :
( ( member_o @ Z5 @ ( image_6768402879995552939_nat_o @ T2 @ top_to7809427810062579212_a_nat ) )
=> ( ( F @ Z5 )
= Z5 ) )
=> ( ( comp_o3516818292788251785_a_nat @ F @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_1023_fun_Omap__ident__strong,axiom,
! [T2: $o > $o,F: $o > $o] :
( ! [Z5: $o] :
( ( member_o @ Z5 @ ( image_o_o @ T2 @ top_top_set_o ) )
=> ( ( F @ Z5 )
= Z5 ) )
=> ( ( comp_o_o_o @ F @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_1024_fun_Omap__ident__strong,axiom,
! [T2: product_prod_a_nat > produc4760857364844987814_a_nat,F: produc4760857364844987814_a_nat > produc4760857364844987814_a_nat] :
( ! [Z5: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ Z5 @ ( image_3729436797865474090_a_nat @ T2 @ top_to3353692345378799459_a_nat ) )
=> ( ( F @ Z5 )
= Z5 ) )
=> ( ( comp_P6732065935450665390_a_nat @ F @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_1025_comp__surj,axiom,
! [F: $o > $o,G: $o > $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ( ( ( image_o_o @ G @ top_top_set_o )
= top_top_set_o )
=> ( ( image_o_o @ ( comp_o_o_o @ G @ F ) @ top_top_set_o )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_1026_comp__surj,axiom,
! [F: $o > produc3509355604313844263tion_a,G: produc3509355604313844263tion_a > $o] :
( ( ( image_628002824077805460tion_a @ F @ top_top_set_o )
= top_to31087889803499607tion_a )
=> ( ( ( image_166006535297930294on_a_o @ G @ top_to31087889803499607tion_a )
= top_top_set_o )
=> ( ( image_o_o @ ( comp_P4515272243702675076_a_o_o @ G @ F ) @ top_top_set_o )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_1027_comp__surj,axiom,
! [F: produc3509355604313844263tion_a > $o,G: $o > $o] :
( ( ( image_166006535297930294on_a_o @ F @ top_to31087889803499607tion_a )
= top_top_set_o )
=> ( ( ( image_o_o @ G @ top_top_set_o )
= top_top_set_o )
=> ( ( image_166006535297930294on_a_o @ ( comp_o8812929617658428164tion_a @ G @ F ) @ top_to31087889803499607tion_a )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_1028_comp__surj,axiom,
! [F: $o > produc4760857364844987814_a_nat,G: produc4760857364844987814_a_nat > $o] :
( ( ( image_2232132205779913721_a_nat @ F @ top_top_set_o )
= top_to7809427810062579212_a_nat )
=> ( ( ( image_6768402879995552939_nat_o @ G @ top_to7809427810062579212_a_nat )
= top_top_set_o )
=> ( ( image_o_o @ ( comp_P885853961682478089at_o_o @ G @ F ) @ top_top_set_o )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_1029_comp__surj,axiom,
! [F: produc4760857364844987814_a_nat > $o,G: $o > $o] :
( ( ( image_6768402879995552939_nat_o @ F @ top_to7809427810062579212_a_nat )
= top_top_set_o )
=> ( ( ( image_o_o @ G @ top_top_set_o )
= top_top_set_o )
=> ( ( image_6768402879995552939_nat_o @ ( comp_o3516818292788251785_a_nat @ G @ F ) @ top_to7809427810062579212_a_nat )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_1030_comp__surj,axiom,
! [F: produc3509355604313844263tion_a > produc3509355604313844263tion_a,G: produc3509355604313844263tion_a > $o] :
( ( ( image_6668519322520640453tion_a @ F @ top_to31087889803499607tion_a )
= top_to31087889803499607tion_a )
=> ( ( ( image_166006535297930294on_a_o @ G @ top_to31087889803499607tion_a )
= top_top_set_o )
=> ( ( image_166006535297930294on_a_o @ ( comp_P3337116118691028755tion_a @ G @ F ) @ top_to31087889803499607tion_a )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_1031_comp__surj,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,G: produc4760857364844987814_a_nat > $o] :
( ( ( image_3729436797865474090_a_nat @ F @ top_to3353692345378799459_a_nat )
= top_to7809427810062579212_a_nat )
=> ( ( ( image_6768402879995552939_nat_o @ G @ top_to7809427810062579212_a_nat )
= top_top_set_o )
=> ( ( image_3076334604395105300_nat_o @ ( comp_P7346075404395887948_a_nat @ G @ F ) @ top_to3353692345378799459_a_nat )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_1032_comp__surj,axiom,
! [F: product_prod_a_nat > product_prod_a_nat,G: product_prod_a_nat > produc4760857364844987814_a_nat] :
( ( ( image_8708701149064048897_a_nat @ F @ top_to3353692345378799459_a_nat )
= top_to3353692345378799459_a_nat )
=> ( ( ( image_3729436797865474090_a_nat @ G @ top_to3353692345378799459_a_nat )
= top_to7809427810062579212_a_nat )
=> ( ( image_3729436797865474090_a_nat @ ( comp_P6608656588004095109_a_nat @ G @ F ) @ top_to3353692345378799459_a_nat )
= top_to7809427810062579212_a_nat ) ) ) ).
% comp_surj
thf(fact_1033_comp__surj,axiom,
! [F: produc4760857364844987814_a_nat > produc3509355604313844263tion_a,G: produc3509355604313844263tion_a > $o] :
( ( ( image_1391120077856136634tion_a @ F @ top_to7809427810062579212_a_nat )
= top_to31087889803499607tion_a )
=> ( ( ( image_166006535297930294on_a_o @ G @ top_to31087889803499607tion_a )
= top_top_set_o )
=> ( ( image_6768402879995552939_nat_o @ ( comp_P4709773760693074362_a_nat @ G @ F ) @ top_to7809427810062579212_a_nat )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_1034_comp__surj,axiom,
! [F: produc3509355604313844263tion_a > produc4760857364844987814_a_nat,G: produc4760857364844987814_a_nat > $o] :
( ( ( image_8907095238652253832_a_nat @ F @ top_to31087889803499607tion_a )
= top_to7809427810062579212_a_nat )
=> ( ( ( image_6768402879995552939_nat_o @ G @ top_to7809427810062579212_a_nat )
= top_top_set_o )
=> ( ( image_166006535297930294on_a_o @ ( comp_P8503938455319246360tion_a @ G @ F ) @ top_to31087889803499607tion_a )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_1035_Collect__case__prod__Grp__eqD,axiom,
! [Z: product_prod_a_nat,A2: set_a,F: a > nat] :
( ( member5724188588386418708_a_nat @ Z @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ ( bNF_Grp_a_nat @ A2 @ F ) ) ) )
=> ( ( comp_a328195826791242666_a_nat @ F @ product_fst_a_nat @ Z )
= ( product_snd_a_nat @ Z ) ) ) ).
% Collect_case_prod_Grp_eqD
thf(fact_1036_Collect__case__prod__Grp__eqD,axiom,
! [Z: produc3509355604313844263tion_a,A2: set_option_a,F: option_a > option_a] :
( ( member5498148017924304208tion_a @ Z @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ ( bNF_Gr5654676046521400910tion_a @ A2 @ F ) ) ) )
=> ( ( comp_o511183382729970692tion_a @ F @ produc174239612569448963tion_a @ Z )
= ( produc8952039381521079109tion_a @ Z ) ) ) ).
% Collect_case_prod_Grp_eqD
thf(fact_1037_Collect__case__prod__Grp__eqD,axiom,
! [Z: produc4760857364844987814_a_nat,A2: set_Pr7585778909603769095tion_a,F: produc3509355604313844263tion_a > nat] :
( ( member1430561565885860669_a_nat @ Z @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ ( bNF_Gr2566800059905468329_a_nat @ A2 @ F ) ) ) )
=> ( ( comp_P712212119343490300_a_nat @ F @ produc1094642028913761716_a_nat @ Z )
= ( produc8051327305810483698_a_nat @ Z ) ) ) ).
% Collect_case_prod_Grp_eqD
thf(fact_1038_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_1039_top__empty__eq,axiom,
( top_to1985590801843794146_nat_o
= ( ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ top_to3353692345378799459_a_nat ) ) ) ).
% top_empty_eq
thf(fact_1040_top__empty__eq,axiom,
( top_to4981642658365807993_nat_o
= ( ^ [X2: produc4760857364844987814_a_nat] : ( member1430561565885860669_a_nat @ X2 @ top_to7809427810062579212_a_nat ) ) ) ).
% top_empty_eq
thf(fact_1041_top__empty__eq,axiom,
( top_to7271036243499817798on_a_o
= ( ^ [X2: produc3509355604313844263tion_a] : ( member5498148017924304208tion_a @ X2 @ top_to31087889803499607tion_a ) ) ) ).
% top_empty_eq
thf(fact_1042_top__empty__eq2,axiom,
( top_to7883107037694708781_nat_o
= ( ^ [X2: produc3509355604313844263tion_a,Y5: nat] : ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ X2 @ Y5 ) @ top_to7809427810062579212_a_nat ) ) ) ).
% top_empty_eq2
thf(fact_1043_top__empty__eq2,axiom,
( top_to8344230362804011838on_a_o
= ( ^ [X2: option_a,Y5: option_a] : ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y5 ) @ top_to31087889803499607tion_a ) ) ) ).
% top_empty_eq2
thf(fact_1044_top__empty__eq2,axiom,
( top_top_a_nat_o
= ( ^ [X2: a,Y5: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y5 ) @ top_to3353692345378799459_a_nat ) ) ) ).
% top_empty_eq2
thf(fact_1045_Collect__case__prod__Grp__in,axiom,
! [Z: product_prod_a_nat,A2: set_a,F: a > nat] :
( ( member5724188588386418708_a_nat @ Z @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ ( bNF_Grp_a_nat @ A2 @ F ) ) ) )
=> ( member_a @ ( product_fst_a_nat @ Z ) @ A2 ) ) ).
% Collect_case_prod_Grp_in
thf(fact_1046_Collect__case__prod__Grp__in,axiom,
! [Z: produc3509355604313844263tion_a,A2: set_option_a,F: option_a > option_a] :
( ( member5498148017924304208tion_a @ Z @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ ( bNF_Gr5654676046521400910tion_a @ A2 @ F ) ) ) )
=> ( member_option_a @ ( produc174239612569448963tion_a @ Z ) @ A2 ) ) ).
% Collect_case_prod_Grp_in
thf(fact_1047_Collect__case__prod__Grp__in,axiom,
! [Z: produc4760857364844987814_a_nat,A2: set_Pr7585778909603769095tion_a,F: produc3509355604313844263tion_a > nat] :
( ( member1430561565885860669_a_nat @ Z @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ ( bNF_Gr2566800059905468329_a_nat @ A2 @ F ) ) ) )
=> ( member5498148017924304208tion_a @ ( produc1094642028913761716_a_nat @ Z ) @ A2 ) ) ).
% Collect_case_prod_Grp_in
thf(fact_1048_fun_Opred__cong,axiom,
! [X: produc3509355604313844263tion_a > $o,Ya: produc3509355604313844263tion_a > $o,P2: $o > $o,Pa: $o > $o] :
( ( X = Ya )
=> ( ! [Z5: $o] :
( ( member_o @ Z5 @ ( image_166006535297930294on_a_o @ Ya @ top_to31087889803499607tion_a ) )
=> ( ( P2 @ Z5 )
= ( Pa @ Z5 ) ) )
=> ( ( basic_8531274769467203284on_a_o
@ ^ [Uu: produc3509355604313844263tion_a] : $true
@ P2
@ X )
= ( basic_8531274769467203284on_a_o
@ ^ [Uu: produc3509355604313844263tion_a] : $true
@ Pa
@ Ya ) ) ) ) ).
% fun.pred_cong
thf(fact_1049_fun_Opred__cong,axiom,
! [X: produc4760857364844987814_a_nat > $o,Ya: produc4760857364844987814_a_nat > $o,P2: $o > $o,Pa: $o > $o] :
( ( X = Ya )
=> ( ! [Z5: $o] :
( ( member_o @ Z5 @ ( image_6768402879995552939_nat_o @ Ya @ top_to7809427810062579212_a_nat ) )
=> ( ( P2 @ Z5 )
= ( Pa @ Z5 ) ) )
=> ( ( basic_6539822665531120781_nat_o
@ ^ [Uu: produc4760857364844987814_a_nat] : $true
@ P2
@ X )
= ( basic_6539822665531120781_nat_o
@ ^ [Uu: produc4760857364844987814_a_nat] : $true
@ Pa
@ Ya ) ) ) ) ).
% fun.pred_cong
thf(fact_1050_fun_Opred__cong,axiom,
! [X: $o > $o,Ya: $o > $o,P2: $o > $o,Pa: $o > $o] :
( ( X = Ya )
=> ( ! [Z5: $o] :
( ( member_o @ Z5 @ ( image_o_o @ Ya @ top_top_set_o ) )
=> ( ( P2 @ Z5 )
= ( Pa @ Z5 ) ) )
=> ( ( basic_pred_fun_o_o
@ ^ [Uu: $o] : $true
@ P2
@ X )
= ( basic_pred_fun_o_o
@ ^ [Uu: $o] : $true
@ Pa
@ Ya ) ) ) ) ).
% fun.pred_cong
thf(fact_1051_fun_Opred__cong,axiom,
! [X: product_prod_a_nat > produc4760857364844987814_a_nat,Ya: product_prod_a_nat > produc4760857364844987814_a_nat,P2: produc4760857364844987814_a_nat > $o,Pa: produc4760857364844987814_a_nat > $o] :
( ( X = Ya )
=> ( ! [Z5: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ Z5 @ ( image_3729436797865474090_a_nat @ Ya @ top_to3353692345378799459_a_nat ) )
=> ( ( P2 @ Z5 )
= ( Pa @ Z5 ) ) )
=> ( ( basic_1871686548785688264_a_nat
@ ^ [Uu: product_prod_a_nat] : $true
@ P2
@ X )
= ( basic_1871686548785688264_a_nat
@ ^ [Uu: product_prod_a_nat] : $true
@ Pa
@ Ya ) ) ) ) ).
% fun.pred_cong
thf(fact_1052_fun_Opred__mono__strong,axiom,
! [P2: $o > $o,X: produc3509355604313844263tion_a > $o,Pa: $o > $o] :
( ( basic_8531274769467203284on_a_o
@ ^ [Uu: produc3509355604313844263tion_a] : $true
@ P2
@ X )
=> ( ! [Z5: $o] :
( ( member_o @ Z5 @ ( image_166006535297930294on_a_o @ X @ top_to31087889803499607tion_a ) )
=> ( ( P2 @ Z5 )
=> ( Pa @ Z5 ) ) )
=> ( basic_8531274769467203284on_a_o
@ ^ [Uu: produc3509355604313844263tion_a] : $true
@ Pa
@ X ) ) ) ).
% fun.pred_mono_strong
thf(fact_1053_fun_Opred__mono__strong,axiom,
! [P2: $o > $o,X: produc4760857364844987814_a_nat > $o,Pa: $o > $o] :
( ( basic_6539822665531120781_nat_o
@ ^ [Uu: produc4760857364844987814_a_nat] : $true
@ P2
@ X )
=> ( ! [Z5: $o] :
( ( member_o @ Z5 @ ( image_6768402879995552939_nat_o @ X @ top_to7809427810062579212_a_nat ) )
=> ( ( P2 @ Z5 )
=> ( Pa @ Z5 ) ) )
=> ( basic_6539822665531120781_nat_o
@ ^ [Uu: produc4760857364844987814_a_nat] : $true
@ Pa
@ X ) ) ) ).
% fun.pred_mono_strong
thf(fact_1054_fun_Opred__mono__strong,axiom,
! [P2: $o > $o,X: $o > $o,Pa: $o > $o] :
( ( basic_pred_fun_o_o
@ ^ [Uu: $o] : $true
@ P2
@ X )
=> ( ! [Z5: $o] :
( ( member_o @ Z5 @ ( image_o_o @ X @ top_top_set_o ) )
=> ( ( P2 @ Z5 )
=> ( Pa @ Z5 ) ) )
=> ( basic_pred_fun_o_o
@ ^ [Uu: $o] : $true
@ Pa
@ X ) ) ) ).
% fun.pred_mono_strong
thf(fact_1055_fun_Opred__mono__strong,axiom,
! [P2: produc4760857364844987814_a_nat > $o,X: product_prod_a_nat > produc4760857364844987814_a_nat,Pa: produc4760857364844987814_a_nat > $o] :
( ( basic_1871686548785688264_a_nat
@ ^ [Uu: product_prod_a_nat] : $true
@ P2
@ X )
=> ( ! [Z5: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ Z5 @ ( image_3729436797865474090_a_nat @ X @ top_to3353692345378799459_a_nat ) )
=> ( ( P2 @ Z5 )
=> ( Pa @ Z5 ) ) )
=> ( basic_1871686548785688264_a_nat
@ ^ [Uu: product_prod_a_nat] : $true
@ Pa
@ X ) ) ) ).
% fun.pred_mono_strong
thf(fact_1056_type__definition_ORep__range,axiom,
! [Rep: product_prod_a_nat > produc4760857364844987814_a_nat,Abs: produc4760857364844987814_a_nat > product_prod_a_nat,A2: set_Pr8660066742269822812_a_nat] :
( ( type_d3363870972157910353_a_nat @ Rep @ Abs @ A2 )
=> ( ( image_3729436797865474090_a_nat @ Rep @ top_to3353692345378799459_a_nat )
= A2 ) ) ).
% type_definition.Rep_range
thf(fact_1057_type__definition_ORep__range,axiom,
! [Rep: produc3509355604313844263tion_a > $o,Abs: $o > produc3509355604313844263tion_a,A2: set_o] :
( ( type_d9194230788701541213on_a_o @ Rep @ Abs @ A2 )
=> ( ( image_166006535297930294on_a_o @ Rep @ top_to31087889803499607tion_a )
= A2 ) ) ).
% type_definition.Rep_range
thf(fact_1058_type__definition_ORep__range,axiom,
! [Rep: produc4760857364844987814_a_nat > $o,Abs: $o > produc4760857364844987814_a_nat,A2: set_o] :
( ( type_d7953064638307121348_nat_o @ Rep @ Abs @ A2 )
=> ( ( image_6768402879995552939_nat_o @ Rep @ top_to7809427810062579212_a_nat )
= A2 ) ) ).
% type_definition.Rep_range
thf(fact_1059_type__definition_ORep__range,axiom,
! [Rep: $o > $o,Abs: $o > $o,A2: set_o] :
( ( type_definition_o_o @ Rep @ Abs @ A2 )
=> ( ( image_o_o @ Rep @ top_top_set_o )
= A2 ) ) ).
% type_definition.Rep_range
thf(fact_1060_type__definition_OAbs__image,axiom,
! [Rep: produc4760857364844987814_a_nat > product_prod_a_nat,Abs: product_prod_a_nat > produc4760857364844987814_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( type_d6372012530977374289_a_nat @ Rep @ Abs @ A2 )
=> ( ( image_3729436797865474090_a_nat @ Abs @ A2 )
= top_to7809427810062579212_a_nat ) ) ).
% type_definition.Abs_image
thf(fact_1061_type__definition_OAbs__image,axiom,
! [Rep: $o > produc3509355604313844263tion_a,Abs: produc3509355604313844263tion_a > $o,A2: set_Pr7585778909603769095tion_a] :
( ( type_d432855040626640571tion_a @ Rep @ Abs @ A2 )
=> ( ( image_166006535297930294on_a_o @ Abs @ A2 )
= top_top_set_o ) ) ).
% type_definition.Abs_image
thf(fact_1062_type__definition_OAbs__image,axiom,
! [Rep: $o > produc4760857364844987814_a_nat,Abs: produc4760857364844987814_a_nat > $o,A2: set_Pr8660066742269822812_a_nat] :
( ( type_d3416793964091482130_a_nat @ Rep @ Abs @ A2 )
=> ( ( image_6768402879995552939_nat_o @ Abs @ A2 )
= top_top_set_o ) ) ).
% type_definition.Abs_image
thf(fact_1063_type__definition_OAbs__image,axiom,
! [Rep: $o > $o,Abs: $o > $o,A2: set_o] :
( ( type_definition_o_o @ Rep @ Abs @ A2 )
=> ( ( image_o_o @ Abs @ A2 )
= top_top_set_o ) ) ).
% type_definition.Abs_image
thf(fact_1064_fun_Orel__compp__Grp,axiom,
! [R: produc3509355604313844263tion_a > nat > $o] :
( ( bNF_re5878663065082895966_a_nat
@ ^ [Y4: product_prod_a_nat,Z2: product_prod_a_nat] : ( Y4 = Z2 )
@ R )
= ( relcom5939993930347576672at_nat
@ ( conver2095598742721174056tion_a
@ ( bNF_Gr6306292117806463247tion_a
@ ( collec4943286744929717527_a_nat
@ ^ [X2: product_prod_a_nat > produc4760857364844987814_a_nat] : ( ord_le5649184635953113020_a_nat @ ( image_3729436797865474090_a_nat @ X2 @ top_to3353692345378799459_a_nat ) @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ R ) ) ) )
@ ( comp_P6873071508423116251_a_nat @ produc1094642028913761716_a_nat ) ) )
@ ( bNF_Gr2637526933174132606at_nat
@ ( collec4943286744929717527_a_nat
@ ^ [X2: product_prod_a_nat > produc4760857364844987814_a_nat] : ( ord_le5649184635953113020_a_nat @ ( image_3729436797865474090_a_nat @ X2 @ top_to3353692345378799459_a_nat ) @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ R ) ) ) )
@ ( comp_P6605823348199399384_a_nat @ produc8051327305810483698_a_nat ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_1065_fstOp__in,axiom,
! [Ac2: product_prod_a_nat,P2: a > nat > $o,Q: nat > nat > $o] :
( ( member5724188588386418708_a_nat @ Ac2 @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ ( relcompp_a_nat_nat @ P2 @ Q ) ) ) )
=> ( member5724188588386418708_a_nat @ ( bNF_fstOp_a_nat_nat @ P2 @ Q @ Ac2 ) @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ P2 ) ) ) ) ).
% fstOp_in
thf(fact_1066_fstOp__in,axiom,
! [Ac2: produc3509355604313844263tion_a,P2: option_a > option_a > $o,Q: option_a > option_a > $o] :
( ( member5498148017924304208tion_a @ Ac2 @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ ( relcom1089236864311994323tion_a @ P2 @ Q ) ) ) )
=> ( member5498148017924304208tion_a @ ( bNF_fs7976035448042954642tion_a @ P2 @ Q @ Ac2 ) @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ P2 ) ) ) ) ).
% fstOp_in
thf(fact_1067_fstOp__in,axiom,
! [Ac2: produc4760857364844987814_a_nat,P2: produc3509355604313844263tion_a > nat > $o,Q: nat > nat > $o] :
( ( member1430561565885860669_a_nat @ Ac2 @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ ( relcom5357432296622887768at_nat @ P2 @ Q ) ) ) )
=> ( member1430561565885860669_a_nat @ ( bNF_fs4236391834175287831at_nat @ P2 @ Q @ Ac2 ) @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ P2 ) ) ) ) ).
% fstOp_in
thf(fact_1068_sndOp__in,axiom,
! [Ac2: product_prod_a_nat,P2: a > a > $o,Q: a > nat > $o] :
( ( member5724188588386418708_a_nat @ Ac2 @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ ( relcompp_a_a_nat @ P2 @ Q ) ) ) )
=> ( member5724188588386418708_a_nat @ ( bNF_sndOp_a_a_nat @ P2 @ Q @ Ac2 ) @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ Q ) ) ) ) ).
% sndOp_in
thf(fact_1069_sndOp__in,axiom,
! [Ac2: product_prod_a_nat,P2: a > produc3509355604313844263tion_a > $o,Q: produc3509355604313844263tion_a > nat > $o] :
( ( member5724188588386418708_a_nat @ Ac2 @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ ( relcom2185508823153564202_a_nat @ P2 @ Q ) ) ) )
=> ( member1430561565885860669_a_nat @ ( bNF_sn8225408376946506605_a_nat @ P2 @ Q @ Ac2 ) @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ Q ) ) ) ) ).
% sndOp_in
thf(fact_1070_sndOp__in,axiom,
! [Ac2: produc3509355604313844263tion_a,P2: option_a > option_a > $o,Q: option_a > option_a > $o] :
( ( member5498148017924304208tion_a @ Ac2 @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ ( relcom1089236864311994323tion_a @ P2 @ Q ) ) ) )
=> ( member5498148017924304208tion_a @ ( bNF_sn4238875959383316176tion_a @ P2 @ Q @ Ac2 ) @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ Q ) ) ) ) ).
% sndOp_in
thf(fact_1071_sndOp__in,axiom,
! [Ac2: produc4760857364844987814_a_nat,P2: produc3509355604313844263tion_a > a > $o,Q: a > nat > $o] :
( ( member1430561565885860669_a_nat @ Ac2 @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ ( relcom6486514688412681240_a_nat @ P2 @ Q ) ) ) )
=> ( member5724188588386418708_a_nat @ ( bNF_sn3303042205350847835_a_nat @ P2 @ Q @ Ac2 ) @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ Q ) ) ) ) ).
% sndOp_in
thf(fact_1072_sndOp__in,axiom,
! [Ac2: produc4760857364844987814_a_nat,P2: produc3509355604313844263tion_a > produc3509355604313844263tion_a > $o,Q: produc3509355604313844263tion_a > nat > $o] :
( ( member1430561565885860669_a_nat @ Ac2 @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ ( relcom4954024199764030273_a_nat @ P2 @ Q ) ) ) )
=> ( member1430561565885860669_a_nat @ ( bNF_sn437989323118565508_a_nat @ P2 @ Q @ Ac2 ) @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ Q ) ) ) ) ).
% sndOp_in
thf(fact_1073_range__constant,axiom,
! [X: produc4760857364844987814_a_nat] :
( ( image_3729436797865474090_a_nat
@ ^ [Uu: product_prod_a_nat] : X
@ top_to3353692345378799459_a_nat )
= ( insert8345643852111475670_a_nat @ X @ bot_bo8420108762324073968_a_nat ) ) ).
% range_constant
thf(fact_1074_range__constant,axiom,
! [X: $o] :
( ( image_166006535297930294on_a_o
@ ^ [Uu: produc3509355604313844263tion_a] : X
@ top_to31087889803499607tion_a )
= ( insert_o @ X @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_1075_range__constant,axiom,
! [X: $o] :
( ( image_6768402879995552939_nat_o
@ ^ [Uu: produc4760857364844987814_a_nat] : X
@ top_to7809427810062579212_a_nat )
= ( insert_o @ X @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_1076_range__constant,axiom,
! [X: $o] :
( ( image_o_o
@ ^ [Uu: $o] : X
@ top_top_set_o )
= ( insert_o @ X @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_1077_insert__iff,axiom,
! [A: $o,B: $o,A2: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A2 ) )
= ( ( A = B )
| ( member_o @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1078_insert__iff,axiom,
! [A: product_prod_a_nat,B: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A @ ( insert8054603423593749677_a_nat @ B @ A2 ) )
= ( ( A = B )
| ( member5724188588386418708_a_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1079_insert__iff,axiom,
! [A: produc4760857364844987814_a_nat,B: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat] :
( ( member1430561565885860669_a_nat @ A @ ( insert8345643852111475670_a_nat @ B @ A2 ) )
= ( ( A = B )
| ( member1430561565885860669_a_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1080_insert__iff,axiom,
! [A: produc3509355604313844263tion_a,B: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a] :
( ( member5498148017924304208tion_a @ A @ ( insert1246254401036548087tion_a @ B @ A2 ) )
= ( ( A = B )
| ( member5498148017924304208tion_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1081_insertCI,axiom,
! [A: $o,B2: set_o,B: $o] :
( ( ~ ( member_o @ A @ B2 )
=> ( A = B ) )
=> ( member_o @ A @ ( insert_o @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1082_insertCI,axiom,
! [A: product_prod_a_nat,B2: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat] :
( ( ~ ( member5724188588386418708_a_nat @ A @ B2 )
=> ( A = B ) )
=> ( member5724188588386418708_a_nat @ A @ ( insert8054603423593749677_a_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1083_insertCI,axiom,
! [A: produc4760857364844987814_a_nat,B2: set_Pr8660066742269822812_a_nat,B: produc4760857364844987814_a_nat] :
( ( ~ ( member1430561565885860669_a_nat @ A @ B2 )
=> ( A = B ) )
=> ( member1430561565885860669_a_nat @ A @ ( insert8345643852111475670_a_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1084_insertCI,axiom,
! [A: produc3509355604313844263tion_a,B2: set_Pr7585778909603769095tion_a,B: produc3509355604313844263tion_a] :
( ( ~ ( member5498148017924304208tion_a @ A @ B2 )
=> ( A = B ) )
=> ( member5498148017924304208tion_a @ A @ ( insert1246254401036548087tion_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1085_insert__image,axiom,
! [X: $o,A2: set_o,F: $o > $o] :
( ( member_o @ X @ A2 )
=> ( ( insert_o @ ( F @ X ) @ ( image_o_o @ F @ A2 ) )
= ( image_o_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1086_insert__image,axiom,
! [X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,F: product_prod_a_nat > produc4760857364844987814_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A2 )
=> ( ( insert8345643852111475670_a_nat @ ( F @ X ) @ ( image_3729436797865474090_a_nat @ F @ A2 ) )
= ( image_3729436797865474090_a_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1087_insert__image,axiom,
! [X: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat,F: produc4760857364844987814_a_nat > $o] :
( ( member1430561565885860669_a_nat @ X @ A2 )
=> ( ( insert_o @ ( F @ X ) @ ( image_6768402879995552939_nat_o @ F @ A2 ) )
= ( image_6768402879995552939_nat_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1088_insert__image,axiom,
! [X: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a,F: produc3509355604313844263tion_a > $o] :
( ( member5498148017924304208tion_a @ X @ A2 )
=> ( ( insert_o @ ( F @ X ) @ ( image_166006535297930294on_a_o @ F @ A2 ) )
= ( image_166006535297930294on_a_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1089_image__insert,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,A: product_prod_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( image_3729436797865474090_a_nat @ F @ ( insert8054603423593749677_a_nat @ A @ B2 ) )
= ( insert8345643852111475670_a_nat @ ( F @ A ) @ ( image_3729436797865474090_a_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1090_image__insert,axiom,
! [F: produc3509355604313844263tion_a > $o,A: produc3509355604313844263tion_a,B2: set_Pr7585778909603769095tion_a] :
( ( image_166006535297930294on_a_o @ F @ ( insert1246254401036548087tion_a @ A @ B2 ) )
= ( insert_o @ ( F @ A ) @ ( image_166006535297930294on_a_o @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1091_image__insert,axiom,
! [F: produc4760857364844987814_a_nat > $o,A: produc4760857364844987814_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ( image_6768402879995552939_nat_o @ F @ ( insert8345643852111475670_a_nat @ A @ B2 ) )
= ( insert_o @ ( F @ A ) @ ( image_6768402879995552939_nat_o @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1092_image__insert,axiom,
! [F: $o > $o,A: $o,B2: set_o] :
( ( image_o_o @ F @ ( insert_o @ A @ B2 ) )
= ( insert_o @ ( F @ A ) @ ( image_o_o @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1093_singletonI,axiom,
! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_1094_singletonI,axiom,
! [A: product_prod_a_nat] : ( member5724188588386418708_a_nat @ A @ ( insert8054603423593749677_a_nat @ A @ bot_bo9049108969261143879_a_nat ) ) ).
% singletonI
thf(fact_1095_singletonI,axiom,
! [A: produc4760857364844987814_a_nat] : ( member1430561565885860669_a_nat @ A @ ( insert8345643852111475670_a_nat @ A @ bot_bo8420108762324073968_a_nat ) ) ).
% singletonI
thf(fact_1096_singletonI,axiom,
! [A: produc3509355604313844263tion_a] : ( member5498148017924304208tion_a @ A @ ( insert1246254401036548087tion_a @ A @ bot_bo235252021745139059tion_a ) ) ).
% singletonI
thf(fact_1097_insert__subset,axiom,
! [X: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a,B2: set_Pr7585778909603769095tion_a] :
( ( ord_le4471550158292877991tion_a @ ( insert1246254401036548087tion_a @ X @ A2 ) @ B2 )
= ( ( member5498148017924304208tion_a @ X @ B2 )
& ( ord_le4471550158292877991tion_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1098_insert__subset,axiom,
! [X: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ ( insert8345643852111475670_a_nat @ X @ A2 ) @ B2 )
= ( ( member1430561565885860669_a_nat @ X @ B2 )
& ( ord_le5649184635953113020_a_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1099_insert__subset,axiom,
! [X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ X @ A2 ) @ B2 )
= ( ( member5724188588386418708_a_nat @ X @ B2 )
& ( ord_le8666007276011122963_a_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1100_insert__subset,axiom,
! [X: $o,A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X @ A2 ) @ B2 )
= ( ( member_o @ X @ B2 )
& ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1101_swap__simp,axiom,
! [X: nat,Y: produc3509355604313844263tion_a] :
( ( produc2088345664931682910tion_a @ ( produc3238187050400694270tion_a @ X @ Y ) )
= ( produc5457283509376734880_a_nat @ Y @ X ) ) ).
% swap_simp
thf(fact_1102_swap__simp,axiom,
! [X: nat,Y: a] :
( ( product_swap_nat_a @ ( product_Pair_nat_a @ X @ Y ) )
= ( product_Pair_a_nat @ Y @ X ) ) ).
% swap_simp
thf(fact_1103_swap__simp,axiom,
! [X: produc3509355604313844263tion_a,Y: nat] :
( ( produc4307442123907723520_a_nat @ ( produc5457283509376734880_a_nat @ X @ Y ) )
= ( produc3238187050400694270tion_a @ Y @ X ) ) ).
% swap_simp
thf(fact_1104_swap__simp,axiom,
! [X: option_a,Y: option_a] :
( ( produc1669947526446001591tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) )
= ( produc9011544418120257559tion_a @ Y @ X ) ) ).
% swap_simp
thf(fact_1105_swap__simp,axiom,
! [X: a,Y: nat] :
( ( product_swap_a_nat @ ( product_Pair_a_nat @ X @ Y ) )
= ( product_Pair_nat_a @ Y @ X ) ) ).
% swap_simp
thf(fact_1106_case__swap,axiom,
! [F: a > nat > produc4760857364844987814_a_nat,P: product_prod_a_nat] :
( ( produc8358520178856414376_a_nat
@ ^ [Y5: nat,X2: a] : ( F @ X2 @ Y5 )
@ ( product_swap_a_nat @ P ) )
= ( produc4004610663780792650_a_nat @ F @ P ) ) ).
% case_swap
thf(fact_1107_case__swap,axiom,
! [F: produc3509355604313844263tion_a > nat > $o,P: produc4760857364844987814_a_nat] :
( ( produc8689481078867993261on_a_o
@ ^ [Y5: nat,X2: produc3509355604313844263tion_a] : ( F @ X2 @ Y5 )
@ ( produc4307442123907723520_a_nat @ P ) )
= ( produc7988141324749014923_nat_o @ F @ P ) ) ).
% case_swap
thf(fact_1108_case__swap,axiom,
! [F: produc3509355604313844263tion_a > nat > produc4760857364844987814_a_nat,P: produc4760857364844987814_a_nat] :
( ( produc9185384946573455057_a_nat
@ ^ [Y5: nat,X2: produc3509355604313844263tion_a] : ( F @ X2 @ Y5 )
@ ( produc4307442123907723520_a_nat @ P ) )
= ( produc5088488592959153011_a_nat @ F @ P ) ) ).
% case_swap
thf(fact_1109_case__swap,axiom,
! [F: nat > a > produc4760857364844987814_a_nat,P: product_prod_nat_a] :
( ( produc4004610663780792650_a_nat
@ ^ [Y5: a,X2: nat] : ( F @ X2 @ Y5 )
@ ( product_swap_nat_a @ P ) )
= ( produc8358520178856414376_a_nat @ F @ P ) ) ).
% case_swap
thf(fact_1110_case__swap,axiom,
! [F: option_a > option_a > $o,P: produc3509355604313844263tion_a] :
( ( produc7248784972643495020on_a_o
@ ^ [Y5: option_a,X2: option_a] : ( F @ X2 @ Y5 )
@ ( produc1669947526446001591tion_a @ P ) )
= ( produc7248784972643495020on_a_o @ F @ P ) ) ).
% case_swap
thf(fact_1111_case__swap,axiom,
! [F: nat > produc3509355604313844263tion_a > $o,P: produc5129331709516570188tion_a] :
( ( produc7988141324749014923_nat_o
@ ^ [Y5: produc3509355604313844263tion_a,X2: nat] : ( F @ X2 @ Y5 )
@ ( produc2088345664931682910tion_a @ P ) )
= ( produc8689481078867993261on_a_o @ F @ P ) ) ).
% case_swap
thf(fact_1112_case__swap,axiom,
! [F: option_a > option_a > produc3509355604313844263tion_a,P: produc3509355604313844263tion_a] :
( ( produc2781663484249587963tion_a
@ ^ [Y5: option_a,X2: option_a] : ( F @ X2 @ Y5 )
@ ( produc1669947526446001591tion_a @ P ) )
= ( produc2781663484249587963tion_a @ F @ P ) ) ).
% case_swap
thf(fact_1113_case__swap,axiom,
! [F: nat > produc3509355604313844263tion_a > produc4760857364844987814_a_nat,P: produc5129331709516570188tion_a] :
( ( produc5088488592959153011_a_nat
@ ^ [Y5: produc3509355604313844263tion_a,X2: nat] : ( F @ X2 @ Y5 )
@ ( produc2088345664931682910tion_a @ P ) )
= ( produc9185384946573455057_a_nat @ F @ P ) ) ).
% case_swap
thf(fact_1114_case__swap,axiom,
! [F: option_a > option_a > option_a,P: produc3509355604313844263tion_a] :
( ( produc6953443519211250200tion_a
@ ^ [Y5: option_a,X2: option_a] : ( F @ X2 @ Y5 )
@ ( produc1669947526446001591tion_a @ P ) )
= ( produc6953443519211250200tion_a @ F @ P ) ) ).
% case_swap
thf(fact_1115_singleton__insert__inj__eq,axiom,
! [B: produc4760857364844987814_a_nat,A: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat] :
( ( ( insert8345643852111475670_a_nat @ B @ bot_bo8420108762324073968_a_nat )
= ( insert8345643852111475670_a_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_le5649184635953113020_a_nat @ A2 @ ( insert8345643852111475670_a_nat @ B @ bot_bo8420108762324073968_a_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1116_singleton__insert__inj__eq,axiom,
! [B: product_prod_a_nat,A: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ( insert8054603423593749677_a_nat @ B @ bot_bo9049108969261143879_a_nat )
= ( insert8054603423593749677_a_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_le8666007276011122963_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1117_singleton__insert__inj__eq,axiom,
! [B: $o,A: $o,A2: set_o] :
( ( ( insert_o @ B @ bot_bot_set_o )
= ( insert_o @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1118_singleton__insert__inj__eq_H,axiom,
! [A: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat,B: produc4760857364844987814_a_nat] :
( ( ( insert8345643852111475670_a_nat @ A @ A2 )
= ( insert8345643852111475670_a_nat @ B @ bot_bo8420108762324073968_a_nat ) )
= ( ( A = B )
& ( ord_le5649184635953113020_a_nat @ A2 @ ( insert8345643852111475670_a_nat @ B @ bot_bo8420108762324073968_a_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1119_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat] :
( ( ( insert8054603423593749677_a_nat @ A @ A2 )
= ( insert8054603423593749677_a_nat @ B @ bot_bo9049108969261143879_a_nat ) )
= ( ( A = B )
& ( ord_le8666007276011122963_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1120_singleton__insert__inj__eq_H,axiom,
! [A: $o,A2: set_o,B: $o] :
( ( ( insert_o @ A @ A2 )
= ( insert_o @ B @ bot_bot_set_o ) )
= ( ( A = B )
& ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1121_snd__swap,axiom,
! [X: produc3509355604313844263tion_a] :
( ( produc8952039381521079109tion_a @ ( produc1669947526446001591tion_a @ X ) )
= ( produc174239612569448963tion_a @ X ) ) ).
% snd_swap
thf(fact_1122_fst__swap,axiom,
! [X: produc3509355604313844263tion_a] :
( ( produc174239612569448963tion_a @ ( produc1669947526446001591tion_a @ X ) )
= ( produc8952039381521079109tion_a @ X ) ) ).
% fst_swap
thf(fact_1123_pair__in__swap__image,axiom,
! [Y: nat,X: produc3509355604313844263tion_a,A2: set_Pr8660066742269822812_a_nat] :
( ( member1799035910557443043tion_a @ ( produc3238187050400694270tion_a @ Y @ X ) @ ( image_4505759252293404409tion_a @ produc4307442123907723520_a_nat @ A2 ) )
= ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ X @ Y ) @ A2 ) ) ).
% pair_in_swap_image
thf(fact_1124_pair__in__swap__image,axiom,
! [Y: nat,X: a,A2: set_Pr4934435412358123699_a_nat] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ Y @ X ) @ ( image_2723492575932950055_nat_a @ product_swap_a_nat @ A2 ) )
= ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ A2 ) ) ).
% pair_in_swap_image
thf(fact_1125_pair__in__swap__image,axiom,
! [Y: produc3509355604313844263tion_a,X: nat,A2: set_Pr2852166431588183298tion_a] :
( ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ Y @ X ) @ ( image_9075947210971128057_a_nat @ produc2088345664931682910tion_a @ A2 ) )
= ( member1799035910557443043tion_a @ ( produc3238187050400694270tion_a @ X @ Y ) @ A2 ) ) ).
% pair_in_swap_image
thf(fact_1126_pair__in__swap__image,axiom,
! [Y: option_a,X: option_a,A2: set_Pr7585778909603769095tion_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ X ) @ ( image_6668519322520640453tion_a @ produc1669947526446001591tion_a @ A2 ) )
= ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ A2 ) ) ).
% pair_in_swap_image
thf(fact_1127_pair__in__swap__image,axiom,
! [Y: a,X: nat,A2: set_Pr4193341848836149977_nat_a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y @ X ) @ ( image_5148427718660454439_a_nat @ product_swap_nat_a @ A2 ) )
= ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X @ Y ) @ A2 ) ) ).
% pair_in_swap_image
thf(fact_1128_singletonD,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1129_singletonD,axiom,
! [B: product_prod_a_nat,A: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ B @ ( insert8054603423593749677_a_nat @ A @ bot_bo9049108969261143879_a_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1130_singletonD,axiom,
! [B: produc4760857364844987814_a_nat,A: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ B @ ( insert8345643852111475670_a_nat @ A @ bot_bo8420108762324073968_a_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1131_singletonD,axiom,
! [B: produc3509355604313844263tion_a,A: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ B @ ( insert1246254401036548087tion_a @ A @ bot_bo235252021745139059tion_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1132_singleton__iff,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1133_singleton__iff,axiom,
! [B: product_prod_a_nat,A: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ B @ ( insert8054603423593749677_a_nat @ A @ bot_bo9049108969261143879_a_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1134_singleton__iff,axiom,
! [B: produc4760857364844987814_a_nat,A: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ B @ ( insert8345643852111475670_a_nat @ A @ bot_bo8420108762324073968_a_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1135_singleton__iff,axiom,
! [B: produc3509355604313844263tion_a,A: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ B @ ( insert1246254401036548087tion_a @ A @ bot_bo235252021745139059tion_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1136_insert__subsetI,axiom,
! [X: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a,X5: set_Pr7585778909603769095tion_a] :
( ( member5498148017924304208tion_a @ X @ A2 )
=> ( ( ord_le4471550158292877991tion_a @ X5 @ A2 )
=> ( ord_le4471550158292877991tion_a @ ( insert1246254401036548087tion_a @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1137_insert__subsetI,axiom,
! [X: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat,X5: set_Pr8660066742269822812_a_nat] :
( ( member1430561565885860669_a_nat @ X @ A2 )
=> ( ( ord_le5649184635953113020_a_nat @ X5 @ A2 )
=> ( ord_le5649184635953113020_a_nat @ ( insert8345643852111475670_a_nat @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1138_insert__subsetI,axiom,
! [X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,X5: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ X5 @ A2 )
=> ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1139_insert__subsetI,axiom,
! [X: $o,A2: set_o,X5: set_o] :
( ( member_o @ X @ A2 )
=> ( ( ord_less_eq_set_o @ X5 @ A2 )
=> ( ord_less_eq_set_o @ ( insert_o @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1140_insert__mono,axiom,
! [C2: set_Pr8660066742269822812_a_nat,D: set_Pr8660066742269822812_a_nat,A: produc4760857364844987814_a_nat] :
( ( ord_le5649184635953113020_a_nat @ C2 @ D )
=> ( ord_le5649184635953113020_a_nat @ ( insert8345643852111475670_a_nat @ A @ C2 ) @ ( insert8345643852111475670_a_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_1141_insert__mono,axiom,
! [C2: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ D )
=> ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ A @ C2 ) @ ( insert8054603423593749677_a_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_1142_insert__mono,axiom,
! [C2: set_o,D: set_o,A: $o] :
( ( ord_less_eq_set_o @ C2 @ D )
=> ( ord_less_eq_set_o @ ( insert_o @ A @ C2 ) @ ( insert_o @ A @ D ) ) ) ).
% insert_mono
thf(fact_1143_subset__insert,axiom,
! [X: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a,B2: set_Pr7585778909603769095tion_a] :
( ~ ( member5498148017924304208tion_a @ X @ A2 )
=> ( ( ord_le4471550158292877991tion_a @ A2 @ ( insert1246254401036548087tion_a @ X @ B2 ) )
= ( ord_le4471550158292877991tion_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1144_subset__insert,axiom,
! [X: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ~ ( member1430561565885860669_a_nat @ X @ A2 )
=> ( ( ord_le5649184635953113020_a_nat @ A2 @ ( insert8345643852111475670_a_nat @ X @ B2 ) )
= ( ord_le5649184635953113020_a_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1145_subset__insert,axiom,
! [X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ X @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ X @ B2 ) )
= ( ord_le8666007276011122963_a_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1146_subset__insert,axiom,
! [X: $o,A2: set_o,B2: set_o] :
( ~ ( member_o @ X @ A2 )
=> ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) )
= ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1147_subset__insertI,axiom,
! [B2: set_Pr8660066742269822812_a_nat,A: produc4760857364844987814_a_nat] : ( ord_le5649184635953113020_a_nat @ B2 @ ( insert8345643852111475670_a_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_1148_subset__insertI,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat] : ( ord_le8666007276011122963_a_nat @ B2 @ ( insert8054603423593749677_a_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_1149_subset__insertI,axiom,
! [B2: set_o,A: $o] : ( ord_less_eq_set_o @ B2 @ ( insert_o @ A @ B2 ) ) ).
% subset_insertI
thf(fact_1150_subset__insertI2,axiom,
! [A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat,B: produc4760857364844987814_a_nat] :
( ( ord_le5649184635953113020_a_nat @ A2 @ B2 )
=> ( ord_le5649184635953113020_a_nat @ A2 @ ( insert8345643852111475670_a_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_1151_subset__insertI2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_1152_subset__insertI2,axiom,
! [A2: set_o,B2: set_o,B: $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_1153_mk__disjoint__insert,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ? [B7: set_o] :
( ( A2
= ( insert_o @ A @ B7 ) )
& ~ ( member_o @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_1154_mk__disjoint__insert,axiom,
! [A: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A @ A2 )
=> ? [B7: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( insert8054603423593749677_a_nat @ A @ B7 ) )
& ~ ( member5724188588386418708_a_nat @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_1155_mk__disjoint__insert,axiom,
! [A: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat] :
( ( member1430561565885860669_a_nat @ A @ A2 )
=> ? [B7: set_Pr8660066742269822812_a_nat] :
( ( A2
= ( insert8345643852111475670_a_nat @ A @ B7 ) )
& ~ ( member1430561565885860669_a_nat @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_1156_mk__disjoint__insert,axiom,
! [A: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a] :
( ( member5498148017924304208tion_a @ A @ A2 )
=> ? [B7: set_Pr7585778909603769095tion_a] :
( ( A2
= ( insert1246254401036548087tion_a @ A @ B7 ) )
& ~ ( member5498148017924304208tion_a @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_1157_insert__eq__iff,axiom,
! [A: $o,A2: set_o,B: $o,B2: set_o] :
( ~ ( member_o @ A @ A2 )
=> ( ~ ( member_o @ B @ B2 )
=> ( ( ( insert_o @ A @ A2 )
= ( insert_o @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A = ~ B )
=> ? [C5: set_o] :
( ( A2
= ( insert_o @ B @ C5 ) )
& ~ ( member_o @ B @ C5 )
& ( B2
= ( insert_o @ A @ C5 ) )
& ~ ( member_o @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1158_insert__eq__iff,axiom,
! [A: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ A @ A2 )
=> ( ~ ( member5724188588386418708_a_nat @ B @ B2 )
=> ( ( ( insert8054603423593749677_a_nat @ A @ A2 )
= ( insert8054603423593749677_a_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C5: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( insert8054603423593749677_a_nat @ B @ C5 ) )
& ~ ( member5724188588386418708_a_nat @ B @ C5 )
& ( B2
= ( insert8054603423593749677_a_nat @ A @ C5 ) )
& ~ ( member5724188588386418708_a_nat @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1159_insert__eq__iff,axiom,
! [A: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat,B: produc4760857364844987814_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ~ ( member1430561565885860669_a_nat @ A @ A2 )
=> ( ~ ( member1430561565885860669_a_nat @ B @ B2 )
=> ( ( ( insert8345643852111475670_a_nat @ A @ A2 )
= ( insert8345643852111475670_a_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C5: set_Pr8660066742269822812_a_nat] :
( ( A2
= ( insert8345643852111475670_a_nat @ B @ C5 ) )
& ~ ( member1430561565885860669_a_nat @ B @ C5 )
& ( B2
= ( insert8345643852111475670_a_nat @ A @ C5 ) )
& ~ ( member1430561565885860669_a_nat @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1160_insert__eq__iff,axiom,
! [A: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a,B: produc3509355604313844263tion_a,B2: set_Pr7585778909603769095tion_a] :
( ~ ( member5498148017924304208tion_a @ A @ A2 )
=> ( ~ ( member5498148017924304208tion_a @ B @ B2 )
=> ( ( ( insert1246254401036548087tion_a @ A @ A2 )
= ( insert1246254401036548087tion_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C5: set_Pr7585778909603769095tion_a] :
( ( A2
= ( insert1246254401036548087tion_a @ B @ C5 ) )
& ~ ( member5498148017924304208tion_a @ B @ C5 )
& ( B2
= ( insert1246254401036548087tion_a @ A @ C5 ) )
& ~ ( member5498148017924304208tion_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1161_insert__absorb,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ( ( insert_o @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1162_insert__absorb,axiom,
! [A: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A @ A2 )
=> ( ( insert8054603423593749677_a_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1163_insert__absorb,axiom,
! [A: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat] :
( ( member1430561565885860669_a_nat @ A @ A2 )
=> ( ( insert8345643852111475670_a_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1164_insert__absorb,axiom,
! [A: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a] :
( ( member5498148017924304208tion_a @ A @ A2 )
=> ( ( insert1246254401036548087tion_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1165_insert__ident,axiom,
! [X: $o,A2: set_o,B2: set_o] :
( ~ ( member_o @ X @ A2 )
=> ( ~ ( member_o @ X @ B2 )
=> ( ( ( insert_o @ X @ A2 )
= ( insert_o @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_1166_insert__ident,axiom,
! [X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ X @ A2 )
=> ( ~ ( member5724188588386418708_a_nat @ X @ B2 )
=> ( ( ( insert8054603423593749677_a_nat @ X @ A2 )
= ( insert8054603423593749677_a_nat @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_1167_insert__ident,axiom,
! [X: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat,B2: set_Pr8660066742269822812_a_nat] :
( ~ ( member1430561565885860669_a_nat @ X @ A2 )
=> ( ~ ( member1430561565885860669_a_nat @ X @ B2 )
=> ( ( ( insert8345643852111475670_a_nat @ X @ A2 )
= ( insert8345643852111475670_a_nat @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_1168_insert__ident,axiom,
! [X: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a,B2: set_Pr7585778909603769095tion_a] :
( ~ ( member5498148017924304208tion_a @ X @ A2 )
=> ( ~ ( member5498148017924304208tion_a @ X @ B2 )
=> ( ( ( insert1246254401036548087tion_a @ X @ A2 )
= ( insert1246254401036548087tion_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_1169_Set_Oset__insert,axiom,
! [X: $o,A2: set_o] :
( ( member_o @ X @ A2 )
=> ~ ! [B7: set_o] :
( ( A2
= ( insert_o @ X @ B7 ) )
=> ( member_o @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_1170_Set_Oset__insert,axiom,
! [X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A2 )
=> ~ ! [B7: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( insert8054603423593749677_a_nat @ X @ B7 ) )
=> ( member5724188588386418708_a_nat @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_1171_Set_Oset__insert,axiom,
! [X: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat] :
( ( member1430561565885860669_a_nat @ X @ A2 )
=> ~ ! [B7: set_Pr8660066742269822812_a_nat] :
( ( A2
= ( insert8345643852111475670_a_nat @ X @ B7 ) )
=> ( member1430561565885860669_a_nat @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_1172_Set_Oset__insert,axiom,
! [X: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a] :
( ( member5498148017924304208tion_a @ X @ A2 )
=> ~ ! [B7: set_Pr7585778909603769095tion_a] :
( ( A2
= ( insert1246254401036548087tion_a @ X @ B7 ) )
=> ( member5498148017924304208tion_a @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_1173_insertI2,axiom,
! [A: $o,B2: set_o,B: $o] :
( ( member_o @ A @ B2 )
=> ( member_o @ A @ ( insert_o @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1174_insertI2,axiom,
! [A: product_prod_a_nat,B2: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A @ B2 )
=> ( member5724188588386418708_a_nat @ A @ ( insert8054603423593749677_a_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1175_insertI2,axiom,
! [A: produc4760857364844987814_a_nat,B2: set_Pr8660066742269822812_a_nat,B: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ A @ B2 )
=> ( member1430561565885860669_a_nat @ A @ ( insert8345643852111475670_a_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1176_insertI2,axiom,
! [A: produc3509355604313844263tion_a,B2: set_Pr7585778909603769095tion_a,B: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ A @ B2 )
=> ( member5498148017924304208tion_a @ A @ ( insert1246254401036548087tion_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1177_insertI1,axiom,
! [A: $o,B2: set_o] : ( member_o @ A @ ( insert_o @ A @ B2 ) ) ).
% insertI1
thf(fact_1178_insertI1,axiom,
! [A: product_prod_a_nat,B2: set_Pr4934435412358123699_a_nat] : ( member5724188588386418708_a_nat @ A @ ( insert8054603423593749677_a_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_1179_insertI1,axiom,
! [A: produc4760857364844987814_a_nat,B2: set_Pr8660066742269822812_a_nat] : ( member1430561565885860669_a_nat @ A @ ( insert8345643852111475670_a_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_1180_insertI1,axiom,
! [A: produc3509355604313844263tion_a,B2: set_Pr7585778909603769095tion_a] : ( member5498148017924304208tion_a @ A @ ( insert1246254401036548087tion_a @ A @ B2 ) ) ).
% insertI1
thf(fact_1181_insertE,axiom,
! [A: $o,B: $o,A2: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A2 ) )
=> ( ( A = ~ B )
=> ( member_o @ A @ A2 ) ) ) ).
% insertE
thf(fact_1182_insertE,axiom,
! [A: product_prod_a_nat,B: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A @ ( insert8054603423593749677_a_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member5724188588386418708_a_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_1183_insertE,axiom,
! [A: produc4760857364844987814_a_nat,B: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat] :
( ( member1430561565885860669_a_nat @ A @ ( insert8345643852111475670_a_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member1430561565885860669_a_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_1184_insertE,axiom,
! [A: produc3509355604313844263tion_a,B: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a] :
( ( member5498148017924304208tion_a @ A @ ( insert1246254401036548087tion_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member5498148017924304208tion_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_1185_insert__compr,axiom,
( insert_o
= ( ^ [A6: $o,B5: set_o] :
( collect_o
@ ^ [X2: $o] :
( ( X2 = A6 )
| ( member_o @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_1186_insert__compr,axiom,
( insert8054603423593749677_a_nat
= ( ^ [A6: product_prod_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( X2 = A6 )
| ( member5724188588386418708_a_nat @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_1187_insert__compr,axiom,
( insert8345643852111475670_a_nat
= ( ^ [A6: produc4760857364844987814_a_nat,B5: set_Pr8660066742269822812_a_nat] :
( collec7438131855006933115_a_nat
@ ^ [X2: produc4760857364844987814_a_nat] :
( ( X2 = A6 )
| ( member1430561565885860669_a_nat @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_1188_insert__compr,axiom,
( insert1246254401036548087tion_a
= ( ^ [A6: produc3509355604313844263tion_a,B5: set_Pr7585778909603769095tion_a] :
( collec4135126896892755346tion_a
@ ^ [X2: produc3509355604313844263tion_a] :
( ( X2 = A6 )
| ( member5498148017924304208tion_a @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_1189_in__rel__def,axiom,
( fun_in2859719455692874677_a_nat
= ( ^ [R3: set_Pr8660066742269822812_a_nat,X2: produc3509355604313844263tion_a,Y5: nat] : ( member1430561565885860669_a_nat @ ( produc5457283509376734880_a_nat @ X2 @ Y5 ) @ R3 ) ) ) ).
% in_rel_def
thf(fact_1190_in__rel__def,axiom,
( fun_in406819195844002114tion_a
= ( ^ [R3: set_Pr7585778909603769095tion_a,X2: option_a,Y5: option_a] : ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y5 ) @ R3 ) ) ) ).
% in_rel_def
thf(fact_1191_in__rel__def,axiom,
( fun_in_rel_a_nat
= ( ^ [R3: set_Pr4934435412358123699_a_nat,X2: a,Y5: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y5 ) @ R3 ) ) ) ).
% in_rel_def
thf(fact_1192_subset__singleton__iff,axiom,
! [X5: set_Pr8660066742269822812_a_nat,A: produc4760857364844987814_a_nat] :
( ( ord_le5649184635953113020_a_nat @ X5 @ ( insert8345643852111475670_a_nat @ A @ bot_bo8420108762324073968_a_nat ) )
= ( ( X5 = bot_bo8420108762324073968_a_nat )
| ( X5
= ( insert8345643852111475670_a_nat @ A @ bot_bo8420108762324073968_a_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1193_subset__singleton__iff,axiom,
! [X5: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X5 @ ( insert8054603423593749677_a_nat @ A @ bot_bo9049108969261143879_a_nat ) )
= ( ( X5 = bot_bo9049108969261143879_a_nat )
| ( X5
= ( insert8054603423593749677_a_nat @ A @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1194_subset__singleton__iff,axiom,
! [X5: set_o,A: $o] :
( ( ord_less_eq_set_o @ X5 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( ( X5 = bot_bot_set_o )
| ( X5
= ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_1195_subset__singletonD,axiom,
! [A2: set_Pr8660066742269822812_a_nat,X: produc4760857364844987814_a_nat] :
( ( ord_le5649184635953113020_a_nat @ A2 @ ( insert8345643852111475670_a_nat @ X @ bot_bo8420108762324073968_a_nat ) )
=> ( ( A2 = bot_bo8420108762324073968_a_nat )
| ( A2
= ( insert8345643852111475670_a_nat @ X @ bot_bo8420108762324073968_a_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1196_subset__singletonD,axiom,
! [A2: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) )
=> ( ( A2 = bot_bo9049108969261143879_a_nat )
| ( A2
= ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1197_subset__singletonD,axiom,
! [A2: set_o,X: $o] :
( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) )
=> ( ( A2 = bot_bot_set_o )
| ( A2
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% subset_singletonD
thf(fact_1198_in__rel__Collect__case__prod__eq,axiom,
! [X5: option_a > option_a > $o] :
( ( fun_in406819195844002114tion_a @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ X5 ) ) )
= X5 ) ).
% in_rel_Collect_case_prod_eq
thf(fact_1199_in__rel__Collect__case__prod__eq,axiom,
! [X5: produc3509355604313844263tion_a > nat > $o] :
( ( fun_in2859719455692874677_a_nat @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ X5 ) ) )
= X5 ) ).
% in_rel_Collect_case_prod_eq
thf(fact_1200_image__constant__conv,axiom,
! [A2: set_Pr4934435412358123699_a_nat,C: produc4760857364844987814_a_nat] :
( ( ( A2 = bot_bo9049108969261143879_a_nat )
=> ( ( image_3729436797865474090_a_nat
@ ^ [X2: product_prod_a_nat] : C
@ A2 )
= bot_bo8420108762324073968_a_nat ) )
& ( ( A2 != bot_bo9049108969261143879_a_nat )
=> ( ( image_3729436797865474090_a_nat
@ ^ [X2: product_prod_a_nat] : C
@ A2 )
= ( insert8345643852111475670_a_nat @ C @ bot_bo8420108762324073968_a_nat ) ) ) ) ).
% image_constant_conv
thf(fact_1201_image__constant__conv,axiom,
! [A2: set_Pr7585778909603769095tion_a,C: $o] :
( ( ( A2 = bot_bo235252021745139059tion_a )
=> ( ( image_166006535297930294on_a_o
@ ^ [X2: produc3509355604313844263tion_a] : C
@ A2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bo235252021745139059tion_a )
=> ( ( image_166006535297930294on_a_o
@ ^ [X2: produc3509355604313844263tion_a] : C
@ A2 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_1202_image__constant__conv,axiom,
! [A2: set_Pr8660066742269822812_a_nat,C: $o] :
( ( ( A2 = bot_bo8420108762324073968_a_nat )
=> ( ( image_6768402879995552939_nat_o
@ ^ [X2: produc4760857364844987814_a_nat] : C
@ A2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bo8420108762324073968_a_nat )
=> ( ( image_6768402879995552939_nat_o
@ ^ [X2: produc4760857364844987814_a_nat] : C
@ A2 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_1203_image__constant__conv,axiom,
! [A2: set_o,C: $o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X2: $o] : C
@ A2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X2: $o] : C
@ A2 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_1204_image__constant,axiom,
! [X: $o,A2: set_o,C: $o] :
( ( member_o @ X @ A2 )
=> ( ( image_o_o
@ ^ [X2: $o] : C
@ A2 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_1205_image__constant,axiom,
! [X: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,C: produc4760857364844987814_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A2 )
=> ( ( image_3729436797865474090_a_nat
@ ^ [X2: product_prod_a_nat] : C
@ A2 )
= ( insert8345643852111475670_a_nat @ C @ bot_bo8420108762324073968_a_nat ) ) ) ).
% image_constant
thf(fact_1206_image__constant,axiom,
! [X: produc4760857364844987814_a_nat,A2: set_Pr8660066742269822812_a_nat,C: $o] :
( ( member1430561565885860669_a_nat @ X @ A2 )
=> ( ( image_6768402879995552939_nat_o
@ ^ [X2: produc4760857364844987814_a_nat] : C
@ A2 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_1207_image__constant,axiom,
! [X: produc3509355604313844263tion_a,A2: set_Pr7585778909603769095tion_a,C: $o] :
( ( member5498148017924304208tion_a @ X @ A2 )
=> ( ( image_166006535297930294on_a_o
@ ^ [X2: produc3509355604313844263tion_a] : C
@ A2 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_1208_Grp__fst__snd,axiom,
! [R: option_a > option_a > $o] :
( ( relcom6243778819037660208tion_a @ ( conver6767049270244447698tion_a @ ( bNF_Gr2851282060293067563tion_a @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ R ) ) @ produc174239612569448963tion_a ) ) @ ( bNF_Gr2851282060293067563tion_a @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ R ) ) @ produc8952039381521079109tion_a ) )
= R ) ).
% Grp_fst_snd
thf(fact_1209_Grp__fst__snd,axiom,
! [R: produc3509355604313844263tion_a > nat > $o] :
( ( relcom9166779600205804078at_nat @ ( conver3169331626652451274tion_a @ ( bNF_Gr2569508567399099313tion_a @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ R ) ) @ produc1094642028913761716_a_nat ) ) @ ( bNF_Gr8625867392921115590at_nat @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ R ) ) @ produc8051327305810483698_a_nat ) )
= R ) ).
% Grp_fst_snd
thf(fact_1210_range__eq__singletonD,axiom,
! [F: product_prod_a_nat > produc4760857364844987814_a_nat,A: produc4760857364844987814_a_nat,X: product_prod_a_nat] :
( ( ( image_3729436797865474090_a_nat @ F @ top_to3353692345378799459_a_nat )
= ( insert8345643852111475670_a_nat @ A @ bot_bo8420108762324073968_a_nat ) )
=> ( ( F @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_1211_range__eq__singletonD,axiom,
! [F: produc3509355604313844263tion_a > $o,A: $o,X: produc3509355604313844263tion_a] :
( ( ( image_166006535297930294on_a_o @ F @ top_to31087889803499607tion_a )
= ( insert_o @ A @ bot_bot_set_o ) )
=> ( ( F @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_1212_range__eq__singletonD,axiom,
! [F: produc4760857364844987814_a_nat > $o,A: $o,X: produc4760857364844987814_a_nat] :
( ( ( image_6768402879995552939_nat_o @ F @ top_to7809427810062579212_a_nat )
= ( insert_o @ A @ bot_bot_set_o ) )
=> ( ( F @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_1213_range__eq__singletonD,axiom,
! [F: $o > $o,A: $o,X: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= ( insert_o @ A @ bot_bot_set_o ) )
=> ( ( F @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_1214_Collect__case__prod__in__rel__leE,axiom,
! [X5: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X5 @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ ( fun_in_rel_a_nat @ Y2 ) ) ) )
=> ( ord_le8666007276011122963_a_nat @ X5 @ Y2 ) ) ).
% Collect_case_prod_in_rel_leE
thf(fact_1215_Collect__case__prod__in__rel__leE,axiom,
! [X5: set_Pr7585778909603769095tion_a,Y2: set_Pr7585778909603769095tion_a] :
( ( ord_le4471550158292877991tion_a @ X5 @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ ( fun_in406819195844002114tion_a @ Y2 ) ) ) )
=> ( ord_le4471550158292877991tion_a @ X5 @ Y2 ) ) ).
% Collect_case_prod_in_rel_leE
thf(fact_1216_Collect__case__prod__in__rel__leE,axiom,
! [X5: set_Pr8660066742269822812_a_nat,Y2: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ X5 @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ ( fun_in2859719455692874677_a_nat @ Y2 ) ) ) )
=> ( ord_le5649184635953113020_a_nat @ X5 @ Y2 ) ) ).
% Collect_case_prod_in_rel_leE
thf(fact_1217_Collect__case__prod__in__rel__leI,axiom,
! [X5: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X5 @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ X5 @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ ( fun_in_rel_a_nat @ Y2 ) ) ) ) ) ).
% Collect_case_prod_in_rel_leI
thf(fact_1218_Collect__case__prod__in__rel__leI,axiom,
! [X5: set_Pr7585778909603769095tion_a,Y2: set_Pr7585778909603769095tion_a] :
( ( ord_le4471550158292877991tion_a @ X5 @ Y2 )
=> ( ord_le4471550158292877991tion_a @ X5 @ ( collec4135126896892755346tion_a @ ( produc7248784972643495020on_a_o @ ( fun_in406819195844002114tion_a @ Y2 ) ) ) ) ) ).
% Collect_case_prod_in_rel_leI
thf(fact_1219_Collect__case__prod__in__rel__leI,axiom,
! [X5: set_Pr8660066742269822812_a_nat,Y2: set_Pr8660066742269822812_a_nat] :
( ( ord_le5649184635953113020_a_nat @ X5 @ Y2 )
=> ( ord_le5649184635953113020_a_nat @ X5 @ ( collec7438131855006933115_a_nat @ ( produc7988141324749014923_nat_o @ ( fun_in2859719455692874677_a_nat @ Y2 ) ) ) ) ) ).
% Collect_case_prod_in_rel_leI
thf(fact_1220_prod_Oswap__def,axiom,
( produc2088345664931682910tion_a
= ( ^ [P3: produc5129331709516570188tion_a] : ( produc5457283509376734880_a_nat @ ( produc5832230846834443088tion_a @ P3 ) @ ( produc8098917606792496914tion_a @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_1221_prod_Oswap__def,axiom,
( product_swap_nat_a
= ( ^ [P3: product_prod_nat_a] : ( product_Pair_a_nat @ ( product_snd_nat_a @ P3 ) @ ( product_fst_nat_a @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_1222_prod_Oswap__def,axiom,
( produc1669947526446001591tion_a
= ( ^ [P3: produc3509355604313844263tion_a] : ( produc9011544418120257559tion_a @ ( produc8952039381521079109tion_a @ P3 ) @ ( produc174239612569448963tion_a @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_1223_subset__singleton__iff__Uniq,axiom,
! [A2: set_Pr7585778909603769095tion_a] :
( ( ? [A6: produc3509355604313844263tion_a] : ( ord_le4471550158292877991tion_a @ A2 @ ( insert1246254401036548087tion_a @ A6 @ bot_bo235252021745139059tion_a ) ) )
= ( uniq_P5457002341418850476tion_a
@ ^ [X2: produc3509355604313844263tion_a] : ( member5498148017924304208tion_a @ X2 @ A2 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1224_subset__singleton__iff__Uniq,axiom,
! [A2: set_Pr8660066742269822812_a_nat] :
( ( ? [A6: produc4760857364844987814_a_nat] : ( ord_le5649184635953113020_a_nat @ A2 @ ( insert8345643852111475670_a_nat @ A6 @ bot_bo8420108762324073968_a_nat ) ) )
= ( uniq_P7604065738380836065_a_nat
@ ^ [X2: produc4760857364844987814_a_nat] : ( member1430561565885860669_a_nat @ X2 @ A2 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1225_subset__singleton__iff__Uniq,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( ? [A6: product_prod_a_nat] : ( ord_le8666007276011122963_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ A6 @ bot_bo9049108969261143879_a_nat ) ) )
= ( uniq_P6155568909923467704_a_nat
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ A2 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1226_subset__singleton__iff__Uniq,axiom,
! [A2: set_o] :
( ( ? [A6: $o] : ( ord_less_eq_set_o @ A2 @ ( insert_o @ A6 @ bot_bot_set_o ) ) )
= ( uniq_o
@ ^ [X2: $o] : ( member_o @ X2 @ A2 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1227_these__insert__Some,axiom,
! [X: a,A2: set_option_a] :
( ( these_a @ ( insert_option_a @ ( some_a @ X ) @ A2 ) )
= ( insert_a @ X @ ( these_a @ A2 ) ) ) ).
% these_insert_Some
thf(fact_1228_these__insert__Some,axiom,
! [X: set_Pr4934435412358123699_a_nat,A2: set_op4138567020912711225_a_nat] :
( ( these_1236029516712174902_a_nat @ ( insert8208376198844224947_a_nat @ ( some_s1880800141159441806_a_nat @ X ) @ A2 ) )
= ( insert6577045944563236579_a_nat @ X @ ( these_1236029516712174902_a_nat @ A2 ) ) ) ).
% these_insert_Some
thf(fact_1229_these__insert__Some,axiom,
! [X: set_Pr8660066742269822812_a_nat,A2: set_op4623351964121498594_a_nat] :
( ( these_3555539573381603295_a_nat @ ( insert8722057353638028252_a_nat @ ( some_s2314368709036572727_a_nat @ X ) @ A2 ) )
= ( insert7606643247661561228_a_nat @ X @ ( these_3555539573381603295_a_nat @ A2 ) ) ) ).
% these_insert_Some
thf(fact_1230_these__image__Some__eq,axiom,
! [A2: set_a] :
( ( these_a @ ( image_a_option_a @ some_a @ A2 ) )
= A2 ) ).
% these_image_Some_eq
thf(fact_1231_these__image__Some__eq,axiom,
! [A2: set_se6121441497158405097_a_nat] :
( ( these_1236029516712174902_a_nat @ ( image_4776182516663252797_a_nat @ some_s1880800141159441806_a_nat @ A2 ) )
= A2 ) ).
% these_image_Some_eq
thf(fact_1232_these__image__Some__eq,axiom,
! [A2: set_se7986904383068512786_a_nat] :
( ( these_3555539573381603295_a_nat @ ( image_4143212015729212815_a_nat @ some_s2314368709036572727_a_nat @ A2 ) )
= A2 ) ).
% these_image_Some_eq
thf(fact_1233_in__these__eq,axiom,
! [X: $o,A2: set_option_o] :
( ( member_o @ X @ ( these_o @ A2 ) )
= ( member_option_o @ ( some_o @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_1234_in__these__eq,axiom,
! [X: product_prod_a_nat,A2: set_op2732609354225412611_a_nat] :
( ( member5724188588386418708_a_nat @ X @ ( these_4186761423770586880_a_nat @ A2 ) )
= ( member1239542624773947236_a_nat @ ( some_P6251353102471802712_a_nat @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_1235_in__these__eq,axiom,
! [X: produc4760857364844987814_a_nat,A2: set_op8842393985335015468_a_nat] :
( ( member1430561565885860669_a_nat @ X @ ( these_8720864711012592169_a_nat @ A2 ) )
= ( member7104896805454144013_a_nat @ ( some_P8082868956047141505_a_nat @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_1236_in__these__eq,axiom,
! [X: produc3509355604313844263tion_a,A2: set_op6528554486239853133tion_a] :
( ( member5498148017924304208tion_a @ X @ ( these_8388281722090972516tion_a @ A2 ) )
= ( member7330119243174117782tion_a @ ( some_P7306582813780734732tion_a @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_1237_in__these__eq,axiom,
! [X: a,A2: set_option_a] :
( ( member_a @ X @ ( these_a @ A2 ) )
= ( member_option_a @ ( some_a @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_1238_in__these__eq,axiom,
! [X: set_Pr4934435412358123699_a_nat,A2: set_op4138567020912711225_a_nat] :
( ( member8034581882086557258_a_nat @ X @ ( these_1236029516712174902_a_nat @ A2 ) )
= ( member4183983893482796698_a_nat @ ( some_s1880800141159441806_a_nat @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_1239_in__these__eq,axiom,
! [X: set_Pr8660066742269822812_a_nat,A2: set_op4623351964121498594_a_nat] :
( ( member904957861751271667_a_nat @ X @ ( these_3555539573381603295_a_nat @ A2 ) )
= ( member1062715794267306179_a_nat @ ( some_s2314368709036572727_a_nat @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_1240_is__singletonI_H,axiom,
! [A2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X3: $o,Y3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( member_o @ Y3 @ A2 )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_o @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1241_is__singletonI_H,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( A2 != bot_bo9049108969261143879_a_nat )
=> ( ! [X3: product_prod_a_nat,Y3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A2 )
=> ( ( member5724188588386418708_a_nat @ Y3 @ A2 )
=> ( X3 = Y3 ) ) )
=> ( is_sin7638514652509285001_a_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1242_is__singletonI_H,axiom,
! [A2: set_Pr8660066742269822812_a_nat] :
( ( A2 != bot_bo8420108762324073968_a_nat )
=> ( ! [X3: produc4760857364844987814_a_nat,Y3: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ X3 @ A2 )
=> ( ( member1430561565885860669_a_nat @ Y3 @ A2 )
=> ( X3 = Y3 ) ) )
=> ( is_sin523092863759094194_a_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1243_is__singletonI_H,axiom,
! [A2: set_Pr7585778909603769095tion_a] :
( ( A2 != bot_bo235252021745139059tion_a )
=> ( ! [X3: produc3509355604313844263tion_a,Y3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ X3 @ A2 )
=> ( ( member5498148017924304208tion_a @ Y3 @ A2 )
=> ( X3 = Y3 ) ) )
=> ( is_sin6176258158573935899tion_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1244_the__elem__image__unique,axiom,
! [A2: set_o,F: $o > $o,X: $o] :
( ( A2 != bot_bot_set_o )
=> ( ! [Y3: $o] :
( ( member_o @ Y3 @ A2 )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_elem_o @ ( image_o_o @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1245_the__elem__image__unique,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F: product_prod_a_nat > produc4760857364844987814_a_nat,X: product_prod_a_nat] :
( ( A2 != bot_bo9049108969261143879_a_nat )
=> ( ! [Y3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Y3 @ A2 )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_el944771114868072817_a_nat @ ( image_3729436797865474090_a_nat @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1246_the__elem__image__unique,axiom,
! [A2: set_Pr8660066742269822812_a_nat,F: produc4760857364844987814_a_nat > $o,X: produc4760857364844987814_a_nat] :
( ( A2 != bot_bo8420108762324073968_a_nat )
=> ( ! [Y3: produc4760857364844987814_a_nat] :
( ( member1430561565885860669_a_nat @ Y3 @ A2 )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_elem_o @ ( image_6768402879995552939_nat_o @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1247_the__elem__image__unique,axiom,
! [A2: set_Pr7585778909603769095tion_a,F: produc3509355604313844263tion_a > $o,X: produc3509355604313844263tion_a] :
( ( A2 != bot_bo235252021745139059tion_a )
=> ( ! [Y3: produc3509355604313844263tion_a] :
( ( member5498148017924304208tion_a @ Y3 @ A2 )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_elem_o @ ( image_166006535297930294on_a_o @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1248_prod__set__simps_I1_J,axiom,
! [X: produc3509355604313844263tion_a,Y: nat] :
( ( basic_3724282984221625776_a_nat @ ( produc5457283509376734880_a_nat @ X @ Y ) )
= ( insert1246254401036548087tion_a @ X @ bot_bo235252021745139059tion_a ) ) ).
% prod_set_simps(1)
thf(fact_1249_prod__set__simps_I1_J,axiom,
! [X: option_a,Y: option_a] :
( ( basic_3033240920312245511tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) )
= ( insert_option_a @ X @ bot_bot_set_option_a ) ) ).
% prod_set_simps(1)
thf(fact_1250_prod__set__simps_I1_J,axiom,
! [X: a,Y: nat] :
( ( basic_fsts_a_nat @ ( product_Pair_a_nat @ X @ Y ) )
= ( insert_a @ X @ bot_bot_set_a ) ) ).
% prod_set_simps(1)
thf(fact_1251_prod__set__simps_I2_J,axiom,
! [X: produc3509355604313844263tion_a,Y: nat] :
( ( basic_1320122631689494770_a_nat @ ( produc5457283509376734880_a_nat @ X @ Y ) )
= ( insert_nat @ Y @ bot_bot_set_nat ) ) ).
% prod_set_simps(2)
thf(fact_1252_prod__set__simps_I2_J,axiom,
! [X: option_a,Y: option_a] :
( ( basic_2804023534644499525tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) )
= ( insert_option_a @ Y @ bot_bot_set_option_a ) ) ).
% prod_set_simps(2)
thf(fact_1253_prod__set__simps_I2_J,axiom,
! [X: a,Y: nat] :
( ( basic_snds_a_nat @ ( product_Pair_a_nat @ X @ Y ) )
= ( insert_nat @ Y @ bot_bot_set_nat ) ) ).
% prod_set_simps(2)
thf(fact_1254_fsts_Ocases,axiom,
! [A: option_a,P: produc3509355604313844263tion_a] :
( ( member_option_a @ A @ ( basic_3033240920312245511tion_a @ P ) )
=> ( A
= ( produc174239612569448963tion_a @ P ) ) ) ).
% fsts.cases
thf(fact_1255_fsts_Osimps,axiom,
! [A: option_a,P: produc3509355604313844263tion_a] :
( ( member_option_a @ A @ ( basic_3033240920312245511tion_a @ P ) )
= ( A
= ( produc174239612569448963tion_a @ P ) ) ) ).
% fsts.simps
thf(fact_1256_fsts_Ointros,axiom,
! [P: produc3509355604313844263tion_a] : ( member_option_a @ ( produc174239612569448963tion_a @ P ) @ ( basic_3033240920312245511tion_a @ P ) ) ).
% fsts.intros
thf(fact_1257_prod__set__defs_I1_J,axiom,
( basic_3033240920312245511tion_a
= ( ^ [P3: produc3509355604313844263tion_a] : ( insert_option_a @ ( produc174239612569448963tion_a @ P3 ) @ bot_bot_set_option_a ) ) ) ).
% prod_set_defs(1)
thf(fact_1258_UNIV__option__conv,axiom,
( top_top_set_option_a
= ( insert_option_a @ none_a @ ( image_a_option_a @ some_a @ top_top_set_a ) ) ) ).
% UNIV_option_conv
thf(fact_1259_UNIV__option__conv,axiom,
( top_to4965271386993507561_a_nat
= ( insert8208376198844224947_a_nat @ none_s3625552246908112402_a_nat @ ( image_4776182516663252797_a_nat @ some_s1880800141159441806_a_nat @ top_to4675147776276609689_a_nat ) ) ) ).
% UNIV_option_conv
thf(fact_1260_UNIV__option__conv,axiom,
( top_to525796794233645714_a_nat
= ( insert8722057353638028252_a_nat @ none_s1042040178515302587_a_nat @ ( image_4143212015729212815_a_nat @ some_s2314368709036572727_a_nat @ top_to8120023364063973570_a_nat ) ) ) ).
% UNIV_option_conv
thf(fact_1261_bind__singleton__conv__image,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F: product_prod_a_nat > produc4760857364844987814_a_nat] :
( ( bind_P2472235213782352632_a_nat @ A2
@ ^ [X2: product_prod_a_nat] : ( insert8345643852111475670_a_nat @ ( F @ X2 ) @ bot_bo8420108762324073968_a_nat ) )
= ( image_3729436797865474090_a_nat @ F @ A2 ) ) ).
% bind_singleton_conv_image
thf(fact_1262_bind__singleton__conv__image,axiom,
! [A2: set_Pr7585778909603769095tion_a,F: produc3509355604313844263tion_a > $o] :
( ( bind_P734035699143971588on_a_o @ A2
@ ^ [X2: produc3509355604313844263tion_a] : ( insert_o @ ( F @ X2 ) @ bot_bot_set_o ) )
= ( image_166006535297930294on_a_o @ F @ A2 ) ) ).
% bind_singleton_conv_image
thf(fact_1263_bind__singleton__conv__image,axiom,
! [A2: set_Pr8660066742269822812_a_nat,F: produc4760857364844987814_a_nat > $o] :
( ( bind_P3633667746293460061_nat_o @ A2
@ ^ [X2: produc4760857364844987814_a_nat] : ( insert_o @ ( F @ X2 ) @ bot_bot_set_o ) )
= ( image_6768402879995552939_nat_o @ F @ A2 ) ) ).
% bind_singleton_conv_image
thf(fact_1264_bind__singleton__conv__image,axiom,
! [A2: set_o,F: $o > $o] :
( ( bind_o_o @ A2
@ ^ [X2: $o] : ( insert_o @ ( F @ X2 ) @ bot_bot_set_o ) )
= ( image_o_o @ F @ A2 ) ) ).
% bind_singleton_conv_image
thf(fact_1265_not__Some__eq,axiom,
! [X: option_a] :
( ( ! [Y5: a] :
( X
!= ( some_a @ Y5 ) ) )
= ( X = none_a ) ) ).
% not_Some_eq
thf(fact_1266_not__Some__eq,axiom,
! [X: option5987904771724668803_a_nat] :
( ( ! [Y5: set_Pr4934435412358123699_a_nat] :
( X
!= ( some_s1880800141159441806_a_nat @ Y5 ) ) )
= ( X = none_s3625552246908112402_a_nat ) ) ).
% not_Some_eq
thf(fact_1267_not__Some__eq,axiom,
! [X: option2973356794535198124_a_nat] :
( ( ! [Y5: set_Pr8660066742269822812_a_nat] :
( X
!= ( some_s2314368709036572727_a_nat @ Y5 ) ) )
= ( X = none_s1042040178515302587_a_nat ) ) ).
% not_Some_eq
thf(fact_1268_not__None__eq,axiom,
! [X: option_a] :
( ( X != none_a )
= ( ? [Y5: a] :
( X
= ( some_a @ Y5 ) ) ) ) ).
% not_None_eq
thf(fact_1269_not__None__eq,axiom,
! [X: option5987904771724668803_a_nat] :
( ( X != none_s3625552246908112402_a_nat )
= ( ? [Y5: set_Pr4934435412358123699_a_nat] :
( X
= ( some_s1880800141159441806_a_nat @ Y5 ) ) ) ) ).
% not_None_eq
thf(fact_1270_not__None__eq,axiom,
! [X: option2973356794535198124_a_nat] :
( ( X != none_s1042040178515302587_a_nat )
= ( ? [Y5: set_Pr8660066742269822812_a_nat] :
( X
= ( some_s2314368709036572727_a_nat @ Y5 ) ) ) ) ).
% not_None_eq
thf(fact_1271_less__eq__option__None__code,axiom,
! [X: option5987904771724668803_a_nat] : ( ord_le1496697172603502563_a_nat @ none_s3625552246908112402_a_nat @ X ) ).
% less_eq_option_None_code
thf(fact_1272_less__eq__option__None__code,axiom,
! [X: option2973356794535198124_a_nat] : ( ord_le8848857416346193420_a_nat @ none_s1042040178515302587_a_nat @ X ) ).
% less_eq_option_None_code
thf(fact_1273_option_Ocollapse,axiom,
! [Option: option2973356794535198124_a_nat] :
( ( Option != none_s1042040178515302587_a_nat )
=> ( ( some_s2314368709036572727_a_nat @ ( the_se5449783460274439624_a_nat @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_1274_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1275_Inf__bool__def,axiom,
( complete_Inf_Inf_o
= ( ^ [A5: set_o] :
~ ( member_o @ $false @ A5 ) ) ) ).
% Inf_bool_def
% Conjectures (2)
thf(conj_0,hypothesis,
( ord_le5649184635953113020_a_nat @ ( ground3966248510942147843tion_a @ c )
@ ( image_3729436797865474090_a_nat
@ ( produc4004610663780792650_a_nat
@ ^ [F3: a] : ( produc5457283509376734880_a_nat @ ( produc9011544418120257559tion_a @ ( some_a @ F3 ) @ ( some_a @ F3 ) ) ) )
@ f ) ) ).
thf(conj_1,conjecture,
( ord_le8666007276011122963_a_nat
@ ( ground_funas_gctxt_a
@ ( ground8556684339173901107on_a_a
@ ^ [F3: produc3509355604313844263tion_a] : ( the_a @ ( produc174239612569448963tion_a @ F3 ) )
@ c ) )
@ f ) ).
%------------------------------------------------------------------------------