%------------------------------------------------------------------------------
% 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    : Eval_FO/0004_Cluster/prob_00059_002359__15184344_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1899 ( 554 unt; 639 typ;   0 def)
%            Number of atoms       : 4451 (2047 equ;   0 cnn)
%            Maximal formula atoms :   15 (   3 avg)
%            Number of connectives : 14909 ( 923   ~;  74   |; 404   &;11823   @)
%                                         (   0 <=>;1685  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   24 (   7 avg)
%            Number of types       :  149 ( 148 usr)
%            Number of type conns  : 2409 (2409   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  494 ( 491 usr;  50 con; 0-7 aty)
%            Number of variables   : 4234 (  80   ^;3927   !; 227   ?;4234   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 13:05:24.974
%------------------------------------------------------------------------------
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thf(sy_c_List_Oextract_001tf__a,type,
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thf(sy_c_RBT__Impl_Oord_Orbt__union__with_001tf__a_001tf__b,type,
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thf(sy_c_RBT__Impl_Oord_Orbt__union__with__key_001tf__a_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J,type,
    rBT_rb8558014646470190020list_a: ( a > a > $o ) > ( a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a ) > rBT_rb484615002580549772list_a > rBT_rb484615002580549772list_a > rBT_rb484615002580549772list_a ).

thf(sy_c_RBT__Impl_Oord_Orbt__union__with__key_001tf__a_001tf__b,type,
    rBT_rb1234474489191313366ey_a_b: ( a > a > $o ) > ( a > b > b > b ) > rBT_rbt_a_b > rBT_rbt_a_b > rBT_rbt_a_b ).

thf(sy_c_RBT__Impl_Oord_Osinter__with_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    rBT_si4727287429423564033list_a: ( ( a > a > $o ) > ( a > a > $o ) > $o ) > ( ( a > a > $o ) > produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a ) > list_P1472896444292205030list_a > list_P1472896444292205030list_a > list_P1472896444292205030list_a ).

thf(sy_c_RBT__Impl_Oord_Osinter__with_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    rBT_si8725346231877325896list_a: ( list_a > list_a > $o ) > ( list_a > list_a > list_a > list_a ) > list_P321204300973800749list_a > list_P321204300973800749list_a > list_P321204300973800749list_a ).

thf(sy_c_RBT__Impl_Oord_Osinter__with_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J_J,type,
    rBT_si5295904238279282010list_a: ( list_a > list_a > $o ) > ( list_a > produc2547433716817190550list_a > produc2547433716817190550list_a > produc2547433716817190550list_a ) > list_P1945613012453903807list_a > list_P1945613012453903807list_a > list_P1945613012453903807list_a ).

thf(sy_c_RBT__Impl_Oord_Osinter__with_001tf__a_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J,type,
    rBT_si6934442379753512503list_a: ( a > a > $o ) > ( a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a ) > list_P5792129111969709340list_a > list_P5792129111969709340list_a > list_P5792129111969709340list_a ).

thf(sy_c_RBT__Impl_Oord_Osinter__with_001tf__a_001tf__b,type,
    rBT_sinter_with_a_b: ( a > a > $o ) > ( a > b > b > b ) > list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b ).

thf(sy_c_RBT__Impl_Oord_Osunion__with_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    rBT_su4068828071963083764list_a: ( ( a > a > $o ) > ( a > a > $o ) > $o ) > ( ( a > a > $o ) > produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a ) > list_P1472896444292205030list_a > list_P1472896444292205030list_a > list_P1472896444292205030list_a ).

thf(sy_c_RBT__Impl_Oord_Osunion__with_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    rBT_su5562828613431444283list_a: ( list_a > list_a > $o ) > ( list_a > list_a > list_a > list_a ) > list_P321204300973800749list_a > list_P321204300973800749list_a > list_P321204300973800749list_a ).

thf(sy_c_RBT__Impl_Oord_Osunion__with_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J_J,type,
    rBT_su605796190718627789list_a: ( list_a > list_a > $o ) > ( list_a > produc2547433716817190550list_a > produc2547433716817190550list_a > produc2547433716817190550list_a ) > list_P1945613012453903807list_a > list_P1945613012453903807list_a > list_P1945613012453903807list_a ).

thf(sy_c_RBT__Impl_Oord_Osunion__with_001tf__a_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J,type,
    rBT_su2724219885554377002list_a: ( a > a > $o ) > ( a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a ) > list_P5792129111969709340list_a > list_P5792129111969709340list_a > list_P5792129111969709340list_a ).

thf(sy_c_RBT__Impl_Oord_Osunion__with_001tf__a_001tf__b,type,
    rBT_sunion_with_a_b: ( a > a > $o ) > ( a > b > b > b ) > list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b ).

thf(sy_c_RBT__Impl_Opaint_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    rBT_pa804577256866744895list_a: rBT_color > rBT_rb9014005217275227734list_a > rBT_rb9014005217275227734list_a ).

thf(sy_c_RBT__Impl_Opaint_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    rBT_pa6702398345666083206list_a: rBT_color > rBT_rb146355364677272861list_a > rBT_rb146355364677272861list_a ).

thf(sy_c_RBT__Impl_Opaint_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J_J,type,
    rBT_pa6697467494130406808list_a: rBT_color > rBT_rb6721489163287533103list_a > rBT_rb6721489163287533103list_a ).

thf(sy_c_RBT__Impl_Opaint_001tf__a_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J,type,
    rBT_pa7206040881279535989list_a: rBT_color > rBT_rb484615002580549772list_a > rBT_rb484615002580549772list_a ).

thf(sy_c_RBT__Impl_Opaint_001tf__a_001tf__b,type,
    rBT_paint_a_b: rBT_color > rBT_rbt_a_b > rBT_rbt_a_b ).

thf(sy_c_RBT__Impl_Orbt_OBranch_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
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thf(sy_c_RBT__Impl_Orbt_OBranch_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_RBT__Impl_Orbt_OBranch_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J_J,type,
    rBT_Br1204270329063416912list_a: rBT_color > rBT_rb6721489163287533103list_a > list_a > produc2547433716817190550list_a > rBT_rb6721489163287533103list_a > rBT_rb6721489163287533103list_a ).

thf(sy_c_RBT__Impl_Orbt_OBranch_001tf__a_001t__List__Olist_Itf__a_J,type,
    rBT_Branch_a_list_a: rBT_color > rBT_rbt_a_list_a > a > list_a > rBT_rbt_a_list_a > rBT_rbt_a_list_a ).

thf(sy_c_RBT__Impl_Orbt_OBranch_001tf__a_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
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thf(sy_c_RBT__Impl_Orbt_OBranch_001tf__a_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J,type,
    rBT_Br706236987277315629list_a: rBT_color > rBT_rb484615002580549772list_a > a > produc271001111946445293list_a > rBT_rb484615002580549772list_a > rBT_rb484615002580549772list_a ).

thf(sy_c_RBT__Impl_Orbt_OBranch_001tf__a_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
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thf(sy_c_RBT__Impl_Orbt_OBranch_001tf__a_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J_J,type,
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thf(sy_c_RBT__Impl_Orbt_OBranch_001tf__a_001tf__b,type,
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thf(sy_c_RBT__Impl_Orbt_OEmpty_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
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thf(sy_c_RBT__Impl_Orbt_OEmpty_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_RBT__Impl_Orbt_OEmpty_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J_J,type,
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thf(sy_c_RBT__Impl_Orbt_OEmpty_001tf__a_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_RBT__Impl_Orbt_OEmpty_001tf__a_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_J_J,type,
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thf(sy_c_RBT__Impl_Orbt_OEmpty_001tf__a_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
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thf(sy_c_RBT__Impl_Orbt_OEmpty_001tf__a_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J,type,
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thf(sy_c_RBT__Impl_Orbt_OEmpty_001tf__a_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
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thf(sy_c_RBT__Impl_Orbt_OEmpty_001tf__a_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J_J,type,
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thf(sy_c_RBT__Impl_Orbt_OEmpty_001tf__a_001tf__b,type,
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thf(sy_c_RBT__Impl_Orbt__baliL_001tf__a_001tf__b,type,
    rBT_rbt_baliL_a_b: rBT_rbt_a_b > a > b > rBT_rbt_a_b > rBT_rbt_a_b ).

thf(sy_c_RBT__Impl_Orbt__baliR_001tf__a_001tf__b,type,
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thf(sy_c_RBT__Impl_Orbt__joinL_001tf__a_001tf__b,type,
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thf(sy_c_RBT__Impl_Orbt__joinR_001tf__a_001tf__b,type,
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thf(sy_c_RBT__Impl_Orbt__join_001tf__a_001tf__b,type,
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thf(sy_c_RBT__Impl_Orbt__split__min_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
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thf(sy_c_RBT__Impl_Orbt__split__min_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_RBT__Impl_Orbt__split__min_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J_J,type,
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thf(sy_c_RBT__Impl_Orbt__split__min_001tf__a_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_RBT__Impl_Orbt__split__min_001tf__a_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_J_J,type,
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thf(sy_c_RBT__Impl_Orbt__split__min_001tf__a_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
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thf(sy_c_RBT__Impl_Orbt__split__min_001tf__a_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
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thf(sy_c_RBT__Impl_Orbt__split__min_001tf__a_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J_J,type,
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thf(sy_c_RBT__Impl_Orbt__split__min_001tf__a_001tf__b,type,
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thf(sy_c_RBT__ext_Orbt__entries__next_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
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thf(sy_c_RBT__ext_Orbt__entries__next_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J_J_J,type,
    member5573719349037835490list_a: produc1828243372476419385list_a > set_Pr6785862820624809881list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J_J_J_J,type,
    member2402001272945804526list_a: produc7879897333239164229list_a > set_Pr5822853913412458917list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_J,type,
    member8745141763671436032od_a_b: produc2639161875728825943od_a_b > set_Pr7142564897319301815od_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
    member4889668945541975382list_a: produc8685980395799941037list_a > set_Pr2070066670564046349list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
    member55517302436591865list_a: produc6446781272394708176list_a > set_Pr4912978608699121200list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J_J,type,
    member852939521359900607list_a: produc2547433716817190550list_a > set_Pr7222558492679949814list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J_J,type,
    member8909059974025578786list_a: produc4030704393689519609list_a > set_Pr2723921704722382297list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J_J_J,type,
    member8728132036948691176list_a: produc6957332542911023295list_a > set_Pr3437823854051686047list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__RBT____Impl__Orbt_Itf__a_Mtf__b_J_J_J,type,
    member6846102102718116240bt_a_b: produc1750692505412360441bt_a_b > set_Pr5333203134345213999bt_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    member1426531481828664017od_a_b: product_prod_a_b > set_Product_prod_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__RBT____Impl__Orbt_Itf__a_Mtf__b_J_J,type,
    member4322261901662495203bt_a_b: produc3797860095223585356bt_a_b > set_Pr3657771483386757890bt_a_b > $o ).

thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

thf(sy_v_k,type,
    k: a ).

thf(sy_v_less,type,
    less: a > a > $o ).

thf(sy_v_less__eq,type,
    less_eq: a > a > $o ).

thf(sy_v_t,type,
    t: rBT_rbt_a_b ).

thf(sy_v_v,type,
    v: b ).

thf(sy_v_v_H,type,
    v2: b ).

% Relevant facts (1254)
thf(fact_0_local_Odual__order_Oasym,axiom,
    ! [B: a,A: a] :
      ( ( less @ B @ A )
     => ~ ( less @ A @ B ) ) ).

% local.dual_order.asym
thf(fact_1_local_Odual__order_Ostrict__implies__not__eq,axiom,
    ! [B: a,A: a] :
      ( ( less @ B @ A )
     => ( A != B ) ) ).

% local.dual_order.strict_implies_not_eq
thf(fact_2_local_Odual__order_Ostrict__trans,axiom,
    ! [B: a,A: a,C: a] :
      ( ( less @ B @ A )
     => ( ( less @ C @ B )
       => ( less @ C @ A ) ) ) ).

% local.dual_order.strict_trans
thf(fact_3_local_Oorder_Oasym,axiom,
    ! [A: a,B: a] :
      ( ( less @ A @ B )
     => ~ ( less @ B @ A ) ) ).

% local.order.asym
thf(fact_4_local_Oorder_Oirrefl,axiom,
    ! [A: a] :
      ~ ( less @ A @ A ) ).

% local.order.irrefl
thf(fact_5_local_Oorder_Ostrict__implies__not__eq,axiom,
    ! [A: a,B: a] :
      ( ( less @ A @ B )
     => ( A != B ) ) ).

% local.order.strict_implies_not_eq
thf(fact_6_local_Oorder_Ostrict__trans,axiom,
    ! [A: a,B: a,C: a] :
      ( ( less @ A @ B )
     => ( ( less @ B @ C )
       => ( less @ A @ C ) ) ) ).

% local.order.strict_trans
thf(fact_7_local_Oantisym__conv3,axiom,
    ! [Y: a,X: a] :
      ( ~ ( less @ Y @ X )
     => ( ( ~ ( less @ X @ Y ) )
        = ( X = Y ) ) ) ).

% local.antisym_conv3
thf(fact_8_local_Oless__asym,axiom,
    ! [X: a,Y: a] :
      ( ( less @ X @ Y )
     => ~ ( less @ Y @ X ) ) ).

% local.less_asym
thf(fact_9_local_Oless__asym_H,axiom,
    ! [A: a,B: a] :
      ( ( less @ A @ B )
     => ~ ( less @ B @ A ) ) ).

% local.less_asym'
thf(fact_10_local_Oless__imp__neq,axiom,
    ! [X: a,Y: a] :
      ( ( less @ X @ Y )
     => ( X != Y ) ) ).

% local.less_imp_neq
thf(fact_11_local_Oless__imp__not__eq,axiom,
    ! [X: a,Y: a] :
      ( ( less @ X @ Y )
     => ( X != Y ) ) ).

% local.less_imp_not_eq
thf(fact_12_local_Oless__imp__not__eq2,axiom,
    ! [X: a,Y: a] :
      ( ( less @ X @ Y )
     => ( Y != X ) ) ).

% local.less_imp_not_eq2
thf(fact_13_local_Oless__imp__not__less,axiom,
    ! [X: a,Y: a] :
      ( ( less @ X @ Y )
     => ~ ( less @ Y @ X ) ) ).

% local.less_imp_not_less
thf(fact_14_local_Oless__imp__triv,axiom,
    ! [X: a,Y: a,P: $o] :
      ( ( less @ X @ Y )
     => ( ( less @ Y @ X )
       => P ) ) ).

% local.less_imp_triv
thf(fact_15_local_Oless__irrefl,axiom,
    ! [X: a] :
      ~ ( less @ X @ X ) ).

% local.less_irrefl
thf(fact_16_local_Oless__linear,axiom,
    ! [X: a,Y: a] :
      ( ( less @ X @ Y )
      | ( X = Y )
      | ( less @ Y @ X ) ) ).

% local.less_linear
thf(fact_17_local_Oless__not__sym,axiom,
    ! [X: a,Y: a] :
      ( ( less @ X @ Y )
     => ~ ( less @ Y @ X ) ) ).

% local.less_not_sym
thf(fact_18_local_Oless__trans,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( less @ X @ Y )
     => ( ( less @ Y @ Z )
       => ( less @ X @ Z ) ) ) ).

% local.less_trans
thf(fact_19_local_Olinorder__cases,axiom,
    ! [X: a,Y: a] :
      ( ~ ( less @ X @ Y )
     => ( ( X != Y )
       => ( less @ Y @ X ) ) ) ).

% local.linorder_cases
thf(fact_20_local_Olinorder__less__wlog,axiom,
    ! [P: a > a > $o,A: a,B: a] :
      ( ! [A2: a,B2: a] :
          ( ( less @ A2 @ B2 )
         => ( P @ A2 @ B2 ) )
     => ( ! [A2: a] : ( P @ A2 @ A2 )
       => ( ! [A2: a,B2: a] :
              ( ( P @ B2 @ A2 )
             => ( P @ A2 @ B2 ) )
         => ( P @ A @ B ) ) ) ) ).

% local.linorder_less_wlog
thf(fact_21_local_OneqE,axiom,
    ! [X: a,Y: a] :
      ( ( X != Y )
     => ( ~ ( less @ X @ Y )
       => ( less @ Y @ X ) ) ) ).

% local.neqE
thf(fact_22_local_Oneq__iff,axiom,
    ! [X: a,Y: a] :
      ( ( X != Y )
      = ( ( less @ X @ Y )
        | ( less @ Y @ X ) ) ) ).

% local.neq_iff
thf(fact_23_local_Onot__less__iff__gr__or__eq,axiom,
    ! [X: a,Y: a] :
      ( ( ~ ( less @ X @ Y ) )
      = ( ( less @ Y @ X )
        | ( X = Y ) ) ) ).

% local.not_less_iff_gr_or_eq
thf(fact_24_local_Oord__eq__less__trans,axiom,
    ! [A: a,B: a,C: a] :
      ( ( A = B )
     => ( ( less @ B @ C )
       => ( less @ A @ C ) ) ) ).

% local.ord_eq_less_trans
thf(fact_25_local_Oord__less__eq__trans,axiom,
    ! [A: a,B: a,C: a] :
      ( ( less @ A @ B )
     => ( ( B = C )
       => ( less @ A @ C ) ) ) ).

% local.ord_less_eq_trans
thf(fact_26_local_Orbt__insert__is__rbt,axiom,
    ! [T: rBT_rb484615002580549772list_a,K: a,V: produc271001111946445293list_a] :
      ( ( rBT_is3736607230663913552list_a @ less @ T )
     => ( rBT_is3736607230663913552list_a @ less @ ( rBT_rb4834844912256151443list_a @ less @ K @ V @ T ) ) ) ).

% local.rbt_insert_is_rbt
thf(fact_27_local_Orbt__insert__is__rbt,axiom,
    ! [T: rBT_rbt_a_b,K: a,V: b] :
      ( ( rBT_is_rbt_a_b @ less @ T )
     => ( rBT_is_rbt_a_b @ less @ ( rBT_rbt_insert_a_b @ less @ K @ V @ T ) ) ) ).

% local.rbt_insert_is_rbt
thf(fact_28_local_Orbt__interw__is__rbt,axiom,
    ! [T1: rBT_rb484615002580549772list_a,T2: rBT_rb484615002580549772list_a,F: produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a] :
      ( ( rBT_is3736607230663913552list_a @ less @ T1 )
     => ( ( rBT_is3736607230663913552list_a @ less @ T2 )
       => ( rBT_is3736607230663913552list_a @ less @ ( rBT_rb1815534573182949414list_a @ less @ F @ T1 @ T2 ) ) ) ) ).

% local.rbt_interw_is_rbt
thf(fact_29_local_Orbt__interw__is__rbt,axiom,
    ! [T1: rBT_rbt_a_b,T2: rBT_rbt_a_b,F: b > b > b] :
      ( ( rBT_is_rbt_a_b @ less @ T1 )
     => ( ( rBT_is_rbt_a_b @ less @ T2 )
       => ( rBT_is_rbt_a_b @ less @ ( rBT_rb6197565798871309880th_a_b @ less @ F @ T1 @ T2 ) ) ) ) ).

% local.rbt_interw_is_rbt
thf(fact_30_local_Orbt__unionw__is__rbt,axiom,
    ! [Lt: rBT_rb484615002580549772list_a,Rt: rBT_rb484615002580549772list_a,F: produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a] :
      ( ( rBT_is3736607230663913552list_a @ less @ Lt )
     => ( ( rBT_is3736607230663913552list_a @ less @ Rt )
       => ( rBT_is3736607230663913552list_a @ less @ ( rBT_rb6828684115838589721list_a @ less @ F @ Lt @ Rt ) ) ) ) ).

% local.rbt_unionw_is_rbt
thf(fact_31_local_Orbt__unionw__is__rbt,axiom,
    ! [Lt: rBT_rbt_a_b,Rt: rBT_rbt_a_b,F: b > b > b] :
      ( ( rBT_is_rbt_a_b @ less @ Lt )
     => ( ( rBT_is_rbt_a_b @ less @ Rt )
       => ( rBT_is_rbt_a_b @ less @ ( rBT_rb7608173054702221355th_a_b @ less @ F @ Lt @ Rt ) ) ) ) ).

% local.rbt_unionw_is_rbt
thf(fact_32_local_Orbt__inter__is__rbt,axiom,
    ! [T1: rBT_rb484615002580549772list_a,T2: rBT_rb484615002580549772list_a] :
      ( ( rBT_is3736607230663913552list_a @ less @ T1 )
     => ( ( rBT_is3736607230663913552list_a @ less @ T2 )
       => ( rBT_is3736607230663913552list_a @ less @ ( rBT_rb664717106984517952list_a @ less @ T1 @ T2 ) ) ) ) ).

% local.rbt_inter_is_rbt
thf(fact_33_local_Orbt__inter__is__rbt,axiom,
    ! [T1: rBT_rbt_a_b,T2: rBT_rbt_a_b] :
      ( ( rBT_is_rbt_a_b @ less @ T1 )
     => ( ( rBT_is_rbt_a_b @ less @ T2 )
       => ( rBT_is_rbt_a_b @ less @ ( rBT_rbt_inter_a_b @ less @ T1 @ T2 ) ) ) ) ).

% local.rbt_inter_is_rbt
thf(fact_34_local_Orbt__union__is__rbt,axiom,
    ! [Lt: rBT_rb484615002580549772list_a,Rt: rBT_rb484615002580549772list_a] :
      ( ( rBT_is3736607230663913552list_a @ less @ Lt )
     => ( ( rBT_is3736607230663913552list_a @ less @ Rt )
       => ( rBT_is3736607230663913552list_a @ less @ ( rBT_rb1703184032556916915list_a @ less @ Lt @ Rt ) ) ) ) ).

% local.rbt_union_is_rbt
thf(fact_35_local_Orbt__union__is__rbt,axiom,
    ! [Lt: rBT_rbt_a_b,Rt: rBT_rbt_a_b] :
      ( ( rBT_is_rbt_a_b @ less @ Lt )
     => ( ( rBT_is_rbt_a_b @ less @ Rt )
       => ( rBT_is_rbt_a_b @ less @ ( rBT_rbt_union_a_b @ less @ Lt @ Rt ) ) ) ) ).

% local.rbt_union_is_rbt
thf(fact_36_local_Orbt__insertw__is__rbt,axiom,
    ! [T: rBT_rb484615002580549772list_a,F: produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a,K: a,V: produc271001111946445293list_a] :
      ( ( rBT_is3736607230663913552list_a @ less @ T )
     => ( rBT_is3736607230663913552list_a @ less @ ( rBT_rb7563067580523141625list_a @ less @ F @ K @ V @ T ) ) ) ).

% local.rbt_insertw_is_rbt
thf(fact_37_local_Orbt__insertw__is__rbt,axiom,
    ! [T: rBT_rbt_a_b,F: b > b > b,K: a,V: b] :
      ( ( rBT_is_rbt_a_b @ less @ T )
     => ( rBT_is_rbt_a_b @ less @ ( rBT_rb5519417620787648267th_a_b @ less @ F @ K @ V @ T ) ) ) ).

% local.rbt_insertw_is_rbt
thf(fact_38_local_Orbt__minus__is__rbt,axiom,
    ! [T1: rBT_rb484615002580549772list_a,T2: rBT_rb484615002580549772list_a] :
      ( ( rBT_is3736607230663913552list_a @ less @ T1 )
     => ( ( rBT_is3736607230663913552list_a @ less @ T2 )
       => ( rBT_is3736607230663913552list_a @ less @ ( rBT_rb863753708588712564list_a @ less @ T1 @ T2 ) ) ) ) ).

% local.rbt_minus_is_rbt
thf(fact_39_local_Orbt__minus__is__rbt,axiom,
    ! [T1: rBT_rbt_a_b,T2: rBT_rbt_a_b] :
      ( ( rBT_is_rbt_a_b @ less @ T1 )
     => ( ( rBT_is_rbt_a_b @ less @ T2 )
       => ( rBT_is_rbt_a_b @ less @ ( rBT_rbt_minus_a_b @ less @ T1 @ T2 ) ) ) ) ).

% local.rbt_minus_is_rbt
thf(fact_40_local_Ois__rbt__rbt__unionwk,axiom,
    ! [T1: rBT_rb484615002580549772list_a,T2: rBT_rb484615002580549772list_a,F: a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a] :
      ( ( rBT_is3736607230663913552list_a @ less @ T1 )
     => ( ( rBT_is3736607230663913552list_a @ less @ T2 )
       => ( rBT_is3736607230663913552list_a @ less @ ( rBT_rb8558014646470190020list_a @ less @ F @ T1 @ T2 ) ) ) ) ).

% local.is_rbt_rbt_unionwk
thf(fact_41_local_Ois__rbt__rbt__unionwk,axiom,
    ! [T1: rBT_rbt_a_b,T2: rBT_rbt_a_b,F: a > b > b > b] :
      ( ( rBT_is_rbt_a_b @ less @ T1 )
     => ( ( rBT_is_rbt_a_b @ less @ T2 )
       => ( rBT_is_rbt_a_b @ less @ ( rBT_rb1234474489191313366ey_a_b @ less @ F @ T1 @ T2 ) ) ) ) ).

% local.is_rbt_rbt_unionwk
thf(fact_42_local_Orbt__interwk__is__rbt,axiom,
    ! [T1: rBT_rb484615002580549772list_a,T2: rBT_rb484615002580549772list_a,F: a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a] :
      ( ( rBT_is3736607230663913552list_a @ less @ T1 )
     => ( ( rBT_is3736607230663913552list_a @ less @ T2 )
       => ( rBT_is3736607230663913552list_a @ less @ ( rBT_rb3443340721405695735list_a @ less @ F @ T1 @ T2 ) ) ) ) ).

% local.rbt_interwk_is_rbt
thf(fact_43_local_Orbt__interwk__is__rbt,axiom,
    ! [T1: rBT_rbt_a_b,T2: rBT_rbt_a_b,F: a > b > b > b] :
      ( ( rBT_is_rbt_a_b @ less @ T1 )
     => ( ( rBT_is_rbt_a_b @ less @ T2 )
       => ( rBT_is_rbt_a_b @ less @ ( rBT_rb935285606175209993ey_a_b @ less @ F @ T1 @ T2 ) ) ) ) ).

% local.rbt_interwk_is_rbt
thf(fact_44_local_Orbt__delete__in__tree,axiom,
    ! [T: rBT_rb6525319661942305155bt_a_b,K: a,V: produc3797860095223585356bt_a_b,X: a] :
      ( ( rBT_is4165910115021524777bt_a_b @ less @ T )
     => ( ( member6846102102718116240bt_a_b @ ( produc3015636708179276019bt_a_b @ K @ V ) @ ( set_Pr1558196515228517412bt_a_b @ ( rBT_en699320873491845330bt_a_b @ ( rBT_rb9108385973946467252bt_a_b @ less @ X @ T ) ) ) )
        = ( ( X != K )
          & ( member6846102102718116240bt_a_b @ ( produc3015636708179276019bt_a_b @ K @ V ) @ ( set_Pr1558196515228517412bt_a_b @ ( rBT_en699320873491845330bt_a_b @ T ) ) ) ) ) ) ).

% local.rbt_delete_in_tree
thf(fact_45_local_Orbt__delete__in__tree,axiom,
    ! [T: rBT_rb3549771087195502139list_a,K: a,V: list_P5792129111969709340list_a,X: a] :
      ( ( rBT_is6567381017377407743list_a @ less @ T )
     => ( ( member2402001272945804526list_a @ ( produc4159846287338058421list_a @ K @ V ) @ ( set_Pr3960553176861824346list_a @ ( rBT_en7750752156332278038list_a @ ( rBT_rb2086199898300189108list_a @ less @ X @ T ) ) ) )
        = ( ( X != K )
          & ( member2402001272945804526list_a @ ( produc4159846287338058421list_a @ K @ V ) @ ( set_Pr3960553176861824346list_a @ ( rBT_en7750752156332278038list_a @ T ) ) ) ) ) ) ).

% local.rbt_delete_in_tree
thf(fact_46_local_Orbt__delete__in__tree,axiom,
    ! [T: rBT_rb4976296014333920589od_a_b,K: a,V: list_P1467974523212632366od_a_b,X: a] :
      ( ( rBT_is1764093772723789329od_a_b @ less @ T )
     => ( ( member8745141763671436032od_a_b @ ( produc6643264353547226055od_a_b @ K @ V ) @ ( set_Pr187010687427835756od_a_b @ ( rBT_en1619617535027037992od_a_b @ ( rBT_rb3560263105592137158od_a_b @ less @ X @ T ) ) ) )
        = ( ( X != K )
          & ( member8745141763671436032od_a_b @ ( produc6643264353547226055od_a_b @ K @ V ) @ ( set_Pr187010687427835756od_a_b @ ( rBT_en1619617535027037992od_a_b @ T ) ) ) ) ) ) ).

% local.rbt_delete_in_tree
thf(fact_47_local_Orbt__delete__in__tree,axiom,
    ! [T: rBT_rbt_a_list_a,K: a,V: list_a,X: a] :
      ( ( rBT_is_rbt_a_list_a @ less @ T )
     => ( ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ K @ V ) @ ( set_Pr3039917418810828226list_a @ ( rBT_entries_a_list_a @ ( rBT_rb3109870203101518876list_a @ less @ X @ T ) ) ) )
        = ( ( X != K )
          & ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ K @ V ) @ ( set_Pr3039917418810828226list_a @ ( rBT_entries_a_list_a @ T ) ) ) ) ) ) ).

% local.rbt_delete_in_tree
thf(fact_48_local_Orbt__delete__in__tree,axiom,
    ! [T: rBT_rb484615002580549772list_a,K: a,V: produc271001111946445293list_a,X: a] :
      ( ( rBT_is3736607230663913552list_a @ less @ T )
     => ( ( member852939521359900607list_a @ ( produc4562151247771834630list_a @ K @ V ) @ ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ ( rBT_rb9212334439554585221list_a @ less @ X @ T ) ) ) )
        = ( ( X != K )
          & ( member852939521359900607list_a @ ( produc4562151247771834630list_a @ K @ V ) @ ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ T ) ) ) ) ) ) ).

% local.rbt_delete_in_tree
thf(fact_49_local_Orbt__delete__in__tree,axiom,
    ! [T: rBT_rbt_a_b,K: a,V: b,X: a] :
      ( ( rBT_is_rbt_a_b @ less @ T )
     => ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K @ V ) @ ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ ( rBT_rbt_delete_a_b @ less @ X @ T ) ) ) )
        = ( ( X != K )
          & ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K @ V ) @ ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ T ) ) ) ) ) ) ).

% local.rbt_delete_in_tree
thf(fact_50_rbt__insert__entries__None,axiom,
    ! [T: rBT_rb6525319661942305155bt_a_b,K: a,V: produc3797860095223585356bt_a_b] :
      ( ( rBT_is4165910115021524777bt_a_b @ less @ T )
     => ( ( ( rBT_rb1295924438074454757bt_a_b @ less @ T @ K )
          = none_P6847224758046513579bt_a_b )
       => ( ( set_Pr1558196515228517412bt_a_b @ ( rBT_en699320873491845330bt_a_b @ ( rBT_rb4857517976831245094bt_a_b @ less @ K @ V @ T ) ) )
          = ( insert5022450467486791465bt_a_b @ ( produc3015636708179276019bt_a_b @ K @ V ) @ ( set_Pr1558196515228517412bt_a_b @ ( rBT_en699320873491845330bt_a_b @ T ) ) ) ) ) ) ).

% rbt_insert_entries_None
thf(fact_51_rbt__insert__entries__None,axiom,
    ! [T: rBT_rb3549771087195502139list_a,K: a,V: list_P5792129111969709340list_a] :
      ( ( rBT_is6567381017377407743list_a @ less @ T )
     => ( ( ( rBT_rb7528920427811653699list_a @ less @ T @ K )
          = none_l1589864400833858557list_a )
       => ( ( set_Pr3960553176861824346list_a @ ( rBT_en7750752156332278038list_a @ ( rBT_rb8368079579106888130list_a @ less @ K @ V @ T ) ) )
          = ( insert3480438106477388565list_a @ ( produc4159846287338058421list_a @ K @ V ) @ ( set_Pr3960553176861824346list_a @ ( rBT_en7750752156332278038list_a @ T ) ) ) ) ) ) ).

% rbt_insert_entries_None
thf(fact_52_rbt__insert__entries__None,axiom,
    ! [T: rBT_rb4976296014333920589od_a_b,K: a,V: list_P1467974523212632366od_a_b] :
      ( ( rBT_is1764093772723789329od_a_b @ less @ T )
     => ( ( ( rBT_rb4890583358032336725od_a_b @ less @ T @ K )
          = none_l1809674250386656527od_a_b )
       => ( ( set_Pr187010687427835756od_a_b @ ( rBT_en1619617535027037992od_a_b @ ( rBT_rb5924729889950403540od_a_b @ less @ K @ V @ T ) ) )
          = ( insert1231944051548111911od_a_b @ ( produc6643264353547226055od_a_b @ K @ V ) @ ( set_Pr187010687427835756od_a_b @ ( rBT_en1619617535027037992od_a_b @ T ) ) ) ) ) ) ).

% rbt_insert_entries_None
thf(fact_53_rbt__insert__entries__None,axiom,
    ! [T: rBT_rbt_a_list_a,K: a,V: list_a] :
      ( ( rBT_is_rbt_a_list_a @ less @ T )
     => ( ( ( rBT_rb8775438530296890539list_a @ less @ T @ K )
          = none_list_a )
       => ( ( set_Pr3039917418810828226list_a @ ( rBT_entries_a_list_a @ ( rBT_rb3324229616542404650list_a @ less @ K @ V @ T ) ) )
          = ( insert4787181130391233917list_a @ ( produc6670463072477821725list_a @ K @ V ) @ ( set_Pr3039917418810828226list_a @ ( rBT_entries_a_list_a @ T ) ) ) ) ) ) ).

% rbt_insert_entries_None
thf(fact_54_rbt__insert__entries__None,axiom,
    ! [T: rBT_rb3198946194961646575list_a,K: a,V: produc3854044251064639184list_a] :
      ( ( rBT_is6464524471971283763list_a @ less @ T )
     => ( ( ( rBT_rb5625365584452904567list_a @ less @ T @ K )
          = none_P5893993846586699057list_a )
       => ( ( set_Pr7668085575154557838list_a @ ( rBT_en1395339733974451658list_a @ ( rBT_rb2370039884616530038list_a @ less @ K @ V @ T ) ) )
          = ( insert9219734766891377609list_a @ ( produc5594513834575919081list_a @ K @ V ) @ ( set_Pr7668085575154557838list_a @ ( rBT_en1395339733974451658list_a @ T ) ) ) ) ) ) ).

% rbt_insert_entries_None
thf(fact_55_rbt__insert__entries__None,axiom,
    ! [T: rBT_rb3712131685166902597od_a_b,K: a,V: produc1700058555794149158od_a_b] :
      ( ( rBT_is5832149757158040201od_a_b @ less @ T )
     => ( ( ( rBT_rb5455008113374026189od_a_b @ less @ T @ K )
          = none_P912253767061701255od_a_b )
       => ( ( set_Pr6417284609035544548od_a_b @ ( rBT_en6144929471051798560od_a_b @ ( rBT_rb1080756271447205580od_a_b @ less @ K @ V @ T ) ) )
          = ( insert1097306472773324575od_a_b @ ( produc2692732248440677695od_a_b @ K @ V ) @ ( set_Pr6417284609035544548od_a_b @ ( rBT_en6144929471051798560od_a_b @ T ) ) ) ) ) ) ).

% rbt_insert_entries_None
thf(fact_56_rbt__insert__entries__None,axiom,
    ! [T: rBT_rb484615002580549772list_a,K: a,V: produc271001111946445293list_a] :
      ( ( rBT_is3736607230663913552list_a @ less @ T )
     => ( ( ( rBT_rb4780624521278262164list_a @ less @ T @ K )
          = none_P4075134247637483086list_a )
       => ( ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ ( rBT_rb4834844912256151443list_a @ less @ K @ V @ T ) ) )
          = ( insert6203622070194670694list_a @ ( produc4562151247771834630list_a @ K @ V ) @ ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ T ) ) ) ) ) ) ).

% rbt_insert_entries_None
thf(fact_57_rbt__insert__entries__None,axiom,
    ! [T: rBT_rbt_a_b,K: a,V: b] :
      ( ( rBT_is_rbt_a_b @ less @ T )
     => ( ( ( rBT_rbt_lookup_a_b @ less @ T @ K )
          = none_b )
       => ( ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ ( rBT_rbt_insert_a_b @ less @ K @ V @ T ) ) )
          = ( insert4534936386344385144od_a_b @ ( product_Pair_a_b @ K @ V ) @ ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ T ) ) ) ) ) ) ).

% rbt_insert_entries_None
thf(fact_58_insert__Diff__single,axiom,
    ! [A: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a] :
      ( ( insert6203622070194670694list_a @ A @ ( minus_4036318849104940381list_a @ A3 @ ( insert6203622070194670694list_a @ A @ bot_bo119984060639512674list_a ) ) )
      = ( insert6203622070194670694list_a @ A @ A3 ) ) ).

% insert_Diff_single
thf(fact_59_insert__Diff__single,axiom,
    ! [A: product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ( insert4534936386344385144od_a_b @ A @ ( minus_6888070959853530735od_a_b @ A3 @ ( insert4534936386344385144od_a_b @ A @ bot_bo3428410327501040244od_a_b ) ) )
      = ( insert4534936386344385144od_a_b @ A @ A3 ) ) ).

% insert_Diff_single
thf(fact_60_insert__Diff__single,axiom,
    ! [A: a,A3: set_a] :
      ( ( insert_a @ A @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) )
      = ( insert_a @ A @ A3 ) ) ).

% insert_Diff_single
thf(fact_61_Diff__insert0,axiom,
    ! [X: produc1392733710753813600list_a,A3: set_Pr1696764298067387072list_a,B3: set_Pr1696764298067387072list_a] :
      ( ~ ( member8740110733080369545list_a @ X @ A3 )
     => ( ( minus_1324476095901101607list_a @ A3 @ ( insert5583651178174595632list_a @ X @ B3 ) )
        = ( minus_1324476095901101607list_a @ A3 @ B3 ) ) ) ).

% Diff_insert0
thf(fact_62_Diff__insert0,axiom,
    ! [X: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a,B3: set_Pr7222558492679949814list_a] :
      ( ~ ( member852939521359900607list_a @ X @ A3 )
     => ( ( minus_4036318849104940381list_a @ A3 @ ( insert6203622070194670694list_a @ X @ B3 ) )
        = ( minus_4036318849104940381list_a @ A3 @ B3 ) ) ) ).

% Diff_insert0
thf(fact_63_Diff__insert0,axiom,
    ! [X: list_a,A3: set_list_a,B3: set_list_a] :
      ( ~ ( member_list_a @ X @ A3 )
     => ( ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ X @ B3 ) )
        = ( minus_646659088055828811list_a @ A3 @ B3 ) ) ) ).

% Diff_insert0
thf(fact_64_Diff__insert0,axiom,
    ! [X: a > a > $o,A3: set_a_a_o,B3: set_a_a_o] :
      ( ~ ( member_a_a_o @ X @ A3 )
     => ( ( minus_399184413438596917_a_a_o @ A3 @ ( insert_a_a_o @ X @ B3 ) )
        = ( minus_399184413438596917_a_a_o @ A3 @ B3 ) ) ) ).

% Diff_insert0
thf(fact_65_Diff__insert0,axiom,
    ! [X: product_prod_a_b,A3: set_Product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ X @ A3 )
     => ( ( minus_6888070959853530735od_a_b @ A3 @ ( insert4534936386344385144od_a_b @ X @ B3 ) )
        = ( minus_6888070959853530735od_a_b @ A3 @ B3 ) ) ) ).

% Diff_insert0
thf(fact_66_Diff__insert0,axiom,
    ! [X: a,A3: set_a,B3: set_a] :
      ( ~ ( member_a @ X @ A3 )
     => ( ( minus_minus_set_a @ A3 @ ( insert_a @ X @ B3 ) )
        = ( minus_minus_set_a @ A3 @ B3 ) ) ) ).

% Diff_insert0
thf(fact_67_insert__Diff1,axiom,
    ! [X: produc1392733710753813600list_a,B3: set_Pr1696764298067387072list_a,A3: set_Pr1696764298067387072list_a] :
      ( ( member8740110733080369545list_a @ X @ B3 )
     => ( ( minus_1324476095901101607list_a @ ( insert5583651178174595632list_a @ X @ A3 ) @ B3 )
        = ( minus_1324476095901101607list_a @ A3 @ B3 ) ) ) ).

% insert_Diff1
thf(fact_68_insert__Diff1,axiom,
    ! [X: produc2547433716817190550list_a,B3: set_Pr7222558492679949814list_a,A3: set_Pr7222558492679949814list_a] :
      ( ( member852939521359900607list_a @ X @ B3 )
     => ( ( minus_4036318849104940381list_a @ ( insert6203622070194670694list_a @ X @ A3 ) @ B3 )
        = ( minus_4036318849104940381list_a @ A3 @ B3 ) ) ) ).

% insert_Diff1
thf(fact_69_insert__Diff1,axiom,
    ! [X: list_a,B3: set_list_a,A3: set_list_a] :
      ( ( member_list_a @ X @ B3 )
     => ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A3 ) @ B3 )
        = ( minus_646659088055828811list_a @ A3 @ B3 ) ) ) ).

% insert_Diff1
thf(fact_70_insert__Diff1,axiom,
    ! [X: a > a > $o,B3: set_a_a_o,A3: set_a_a_o] :
      ( ( member_a_a_o @ X @ B3 )
     => ( ( minus_399184413438596917_a_a_o @ ( insert_a_a_o @ X @ A3 ) @ B3 )
        = ( minus_399184413438596917_a_a_o @ A3 @ B3 ) ) ) ).

% insert_Diff1
thf(fact_71_insert__Diff1,axiom,
    ! [X: product_prod_a_b,B3: set_Product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ B3 )
     => ( ( minus_6888070959853530735od_a_b @ ( insert4534936386344385144od_a_b @ X @ A3 ) @ B3 )
        = ( minus_6888070959853530735od_a_b @ A3 @ B3 ) ) ) ).

% insert_Diff1
thf(fact_72_insert__Diff1,axiom,
    ! [X: a,B3: set_a,A3: set_a] :
      ( ( member_a @ X @ B3 )
     => ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B3 )
        = ( minus_minus_set_a @ A3 @ B3 ) ) ) ).

% insert_Diff1
thf(fact_73_Diff__empty,axiom,
    ! [A3: set_Product_prod_a_b] :
      ( ( minus_6888070959853530735od_a_b @ A3 @ bot_bo3428410327501040244od_a_b )
      = A3 ) ).

% Diff_empty
thf(fact_74_Diff__empty,axiom,
    ! [A3: set_a] :
      ( ( minus_minus_set_a @ A3 @ bot_bot_set_a )
      = A3 ) ).

% Diff_empty
thf(fact_75_empty__Diff,axiom,
    ! [A3: set_Product_prod_a_b] :
      ( ( minus_6888070959853530735od_a_b @ bot_bo3428410327501040244od_a_b @ A3 )
      = bot_bo3428410327501040244od_a_b ) ).

% empty_Diff
thf(fact_76_empty__Diff,axiom,
    ! [A3: set_a] :
      ( ( minus_minus_set_a @ bot_bot_set_a @ A3 )
      = bot_bot_set_a ) ).

% empty_Diff
thf(fact_77_Diff__cancel,axiom,
    ! [A3: set_Product_prod_a_b] :
      ( ( minus_6888070959853530735od_a_b @ A3 @ A3 )
      = bot_bo3428410327501040244od_a_b ) ).

% Diff_cancel
thf(fact_78_Diff__cancel,axiom,
    ! [A3: set_a] :
      ( ( minus_minus_set_a @ A3 @ A3 )
      = bot_bot_set_a ) ).

% Diff_cancel
thf(fact_79_mem__Collect__eq,axiom,
    ! [A: produc1392733710753813600list_a,P: produc1392733710753813600list_a > $o] :
      ( ( member8740110733080369545list_a @ A @ ( collec7621445833558445515list_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
    ! [A: produc2547433716817190550list_a,P: produc2547433716817190550list_a > $o] :
      ( ( member852939521359900607list_a @ A @ ( collec1680782243818011649list_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_81_mem__Collect__eq,axiom,
    ! [A: list_a,P: list_a > $o] :
      ( ( member_list_a @ A @ ( collect_list_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_82_mem__Collect__eq,axiom,
    ! [A: a > a > $o,P: ( a > a > $o ) > $o] :
      ( ( member_a_a_o @ A @ ( collect_a_a_o @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_83_mem__Collect__eq,axiom,
    ! [A: product_prod_a_b,P: product_prod_a_b > $o] :
      ( ( member1426531481828664017od_a_b @ A @ ( collec3336397801687681299od_a_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_84_mem__Collect__eq,axiom,
    ! [A: a,P: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_85_Collect__mem__eq,axiom,
    ! [A3: set_Pr1696764298067387072list_a] :
      ( ( collec7621445833558445515list_a
        @ ^ [X2: produc1392733710753813600list_a] : ( member8740110733080369545list_a @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_86_Collect__mem__eq,axiom,
    ! [A3: set_Pr7222558492679949814list_a] :
      ( ( collec1680782243818011649list_a
        @ ^ [X2: produc2547433716817190550list_a] : ( member852939521359900607list_a @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_87_Collect__mem__eq,axiom,
    ! [A3: set_list_a] :
      ( ( collect_list_a
        @ ^ [X2: list_a] : ( member_list_a @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_88_Collect__mem__eq,axiom,
    ! [A3: set_a_a_o] :
      ( ( collect_a_a_o
        @ ^ [X2: a > a > $o] : ( member_a_a_o @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_89_Collect__mem__eq,axiom,
    ! [A3: set_Product_prod_a_b] :
      ( ( collec3336397801687681299od_a_b
        @ ^ [X2: product_prod_a_b] : ( member1426531481828664017od_a_b @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_90_Collect__mem__eq,axiom,
    ! [A3: set_a] :
      ( ( collect_a
        @ ^ [X2: a] : ( member_a @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_91_Collect__cong,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ! [X3: a] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_a @ P )
        = ( collect_a @ Q ) ) ) ).

% Collect_cong
thf(fact_92_Collect__cong,axiom,
    ! [P: product_prod_a_b > $o,Q: product_prod_a_b > $o] :
      ( ! [X3: product_prod_a_b] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec3336397801687681299od_a_b @ P )
        = ( collec3336397801687681299od_a_b @ Q ) ) ) ).

% Collect_cong
thf(fact_93_singletonI,axiom,
    ! [A: produc1392733710753813600list_a] : ( member8740110733080369545list_a @ A @ ( insert5583651178174595632list_a @ A @ bot_bo7554134525451881772list_a ) ) ).

% singletonI
thf(fact_94_singletonI,axiom,
    ! [A: produc2547433716817190550list_a] : ( member852939521359900607list_a @ A @ ( insert6203622070194670694list_a @ A @ bot_bo119984060639512674list_a ) ) ).

% singletonI
thf(fact_95_singletonI,axiom,
    ! [A: list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).

% singletonI
thf(fact_96_singletonI,axiom,
    ! [A: a > a > $o] : ( member_a_a_o @ A @ ( insert_a_a_o @ A @ bot_bot_set_a_a_o ) ) ).

% singletonI
thf(fact_97_singletonI,axiom,
    ! [A: product_prod_a_b] : ( member1426531481828664017od_a_b @ A @ ( insert4534936386344385144od_a_b @ A @ bot_bo3428410327501040244od_a_b ) ) ).

% singletonI
thf(fact_98_singletonI,axiom,
    ! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).

% singletonI
thf(fact_99_local_OacyclicI__order,axiom,
    ! [R: set_Pr4048851178543822343list_a,F: list_a > a] :
      ( ! [A2: list_a,B2: list_a] :
          ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A2 @ B2 ) @ R )
         => ( less @ ( F @ B2 ) @ ( F @ A2 ) ) )
     => ( transi3405315890175678931list_a @ R ) ) ).

% local.acyclicI_order
thf(fact_100_local_Orbt__delete__is__rbt,axiom,
    ! [T: rBT_rb484615002580549772list_a,K: a] :
      ( ( rBT_is3736607230663913552list_a @ less @ T )
     => ( rBT_is3736607230663913552list_a @ less @ ( rBT_rb9212334439554585221list_a @ less @ K @ T ) ) ) ).

% local.rbt_delete_is_rbt
thf(fact_101_local_Orbt__delete__is__rbt,axiom,
    ! [T: rBT_rbt_a_b,K: a] :
      ( ( rBT_is_rbt_a_b @ less @ T )
     => ( rBT_is_rbt_a_b @ less @ ( rBT_rbt_delete_a_b @ less @ K @ T ) ) ) ).

% local.rbt_delete_is_rbt
thf(fact_102_empty__Collect__eq,axiom,
    ! [P: product_prod_a_b > $o] :
      ( ( bot_bo3428410327501040244od_a_b
        = ( collec3336397801687681299od_a_b @ P ) )
      = ( ! [X2: product_prod_a_b] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_103_empty__Collect__eq,axiom,
    ! [P: a > $o] :
      ( ( bot_bot_set_a
        = ( collect_a @ P ) )
      = ( ! [X2: a] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_104_Collect__empty__eq,axiom,
    ! [P: product_prod_a_b > $o] :
      ( ( ( collec3336397801687681299od_a_b @ P )
        = bot_bo3428410327501040244od_a_b )
      = ( ! [X2: product_prod_a_b] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_105_Collect__empty__eq,axiom,
    ! [P: a > $o] :
      ( ( ( collect_a @ P )
        = bot_bot_set_a )
      = ( ! [X2: a] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_106_all__not__in__conv,axiom,
    ! [A3: set_Pr1696764298067387072list_a] :
      ( ( ! [X2: produc1392733710753813600list_a] :
            ~ ( member8740110733080369545list_a @ X2 @ A3 ) )
      = ( A3 = bot_bo7554134525451881772list_a ) ) ).

% all_not_in_conv
thf(fact_107_all__not__in__conv,axiom,
    ! [A3: set_Pr7222558492679949814list_a] :
      ( ( ! [X2: produc2547433716817190550list_a] :
            ~ ( member852939521359900607list_a @ X2 @ A3 ) )
      = ( A3 = bot_bo119984060639512674list_a ) ) ).

% all_not_in_conv
thf(fact_108_all__not__in__conv,axiom,
    ! [A3: set_list_a] :
      ( ( ! [X2: list_a] :
            ~ ( member_list_a @ X2 @ A3 ) )
      = ( A3 = bot_bot_set_list_a ) ) ).

% all_not_in_conv
thf(fact_109_all__not__in__conv,axiom,
    ! [A3: set_a_a_o] :
      ( ( ! [X2: a > a > $o] :
            ~ ( member_a_a_o @ X2 @ A3 ) )
      = ( A3 = bot_bot_set_a_a_o ) ) ).

% all_not_in_conv
thf(fact_110_all__not__in__conv,axiom,
    ! [A3: set_Product_prod_a_b] :
      ( ( ! [X2: product_prod_a_b] :
            ~ ( member1426531481828664017od_a_b @ X2 @ A3 ) )
      = ( A3 = bot_bo3428410327501040244od_a_b ) ) ).

% all_not_in_conv
thf(fact_111_all__not__in__conv,axiom,
    ! [A3: set_a] :
      ( ( ! [X2: a] :
            ~ ( member_a @ X2 @ A3 ) )
      = ( A3 = bot_bot_set_a ) ) ).

% all_not_in_conv
thf(fact_112_empty__iff,axiom,
    ! [C: produc1392733710753813600list_a] :
      ~ ( member8740110733080369545list_a @ C @ bot_bo7554134525451881772list_a ) ).

% empty_iff
thf(fact_113_empty__iff,axiom,
    ! [C: produc2547433716817190550list_a] :
      ~ ( member852939521359900607list_a @ C @ bot_bo119984060639512674list_a ) ).

% empty_iff
thf(fact_114_empty__iff,axiom,
    ! [C: list_a] :
      ~ ( member_list_a @ C @ bot_bot_set_list_a ) ).

% empty_iff
thf(fact_115_empty__iff,axiom,
    ! [C: a > a > $o] :
      ~ ( member_a_a_o @ C @ bot_bot_set_a_a_o ) ).

% empty_iff
thf(fact_116_empty__iff,axiom,
    ! [C: product_prod_a_b] :
      ~ ( member1426531481828664017od_a_b @ C @ bot_bo3428410327501040244od_a_b ) ).

% empty_iff
thf(fact_117_empty__iff,axiom,
    ! [C: a] :
      ~ ( member_a @ C @ bot_bot_set_a ) ).

% empty_iff
thf(fact_118_insert__absorb2,axiom,
    ! [X: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a] :
      ( ( insert6203622070194670694list_a @ X @ ( insert6203622070194670694list_a @ X @ A3 ) )
      = ( insert6203622070194670694list_a @ X @ A3 ) ) ).

% insert_absorb2
thf(fact_119_insert__absorb2,axiom,
    ! [X: product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ( insert4534936386344385144od_a_b @ X @ ( insert4534936386344385144od_a_b @ X @ A3 ) )
      = ( insert4534936386344385144od_a_b @ X @ A3 ) ) ).

% insert_absorb2
thf(fact_120_insert__absorb2,axiom,
    ! [X: a,A3: set_a] :
      ( ( insert_a @ X @ ( insert_a @ X @ A3 ) )
      = ( insert_a @ X @ A3 ) ) ).

% insert_absorb2
thf(fact_121_insert__iff,axiom,
    ! [A: produc1392733710753813600list_a,B: produc1392733710753813600list_a,A3: set_Pr1696764298067387072list_a] :
      ( ( member8740110733080369545list_a @ A @ ( insert5583651178174595632list_a @ B @ A3 ) )
      = ( ( A = B )
        | ( member8740110733080369545list_a @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_122_insert__iff,axiom,
    ! [A: produc2547433716817190550list_a,B: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a] :
      ( ( member852939521359900607list_a @ A @ ( insert6203622070194670694list_a @ B @ A3 ) )
      = ( ( A = B )
        | ( member852939521359900607list_a @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_123_insert__iff,axiom,
    ! [A: list_a,B: list_a,A3: set_list_a] :
      ( ( member_list_a @ A @ ( insert_list_a @ B @ A3 ) )
      = ( ( A = B )
        | ( member_list_a @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_124_insert__iff,axiom,
    ! [A: a > a > $o,B: a > a > $o,A3: set_a_a_o] :
      ( ( member_a_a_o @ A @ ( insert_a_a_o @ B @ A3 ) )
      = ( ( A = B )
        | ( member_a_a_o @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_125_insert__iff,axiom,
    ! [A: product_prod_a_b,B: product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ A @ ( insert4534936386344385144od_a_b @ B @ A3 ) )
      = ( ( A = B )
        | ( member1426531481828664017od_a_b @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_126_insert__iff,axiom,
    ! [A: a,B: a,A3: set_a] :
      ( ( member_a @ A @ ( insert_a @ B @ A3 ) )
      = ( ( A = B )
        | ( member_a @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_127_insertCI,axiom,
    ! [A: produc1392733710753813600list_a,B3: set_Pr1696764298067387072list_a,B: produc1392733710753813600list_a] :
      ( ( ~ ( member8740110733080369545list_a @ A @ B3 )
       => ( A = B ) )
     => ( member8740110733080369545list_a @ A @ ( insert5583651178174595632list_a @ B @ B3 ) ) ) ).

% insertCI
thf(fact_128_insertCI,axiom,
    ! [A: produc2547433716817190550list_a,B3: set_Pr7222558492679949814list_a,B: produc2547433716817190550list_a] :
      ( ( ~ ( member852939521359900607list_a @ A @ B3 )
       => ( A = B ) )
     => ( member852939521359900607list_a @ A @ ( insert6203622070194670694list_a @ B @ B3 ) ) ) ).

% insertCI
thf(fact_129_insertCI,axiom,
    ! [A: list_a,B3: set_list_a,B: list_a] :
      ( ( ~ ( member_list_a @ A @ B3 )
       => ( A = B ) )
     => ( member_list_a @ A @ ( insert_list_a @ B @ B3 ) ) ) ).

% insertCI
thf(fact_130_insertCI,axiom,
    ! [A: a > a > $o,B3: set_a_a_o,B: a > a > $o] :
      ( ( ~ ( member_a_a_o @ A @ B3 )
       => ( A = B ) )
     => ( member_a_a_o @ A @ ( insert_a_a_o @ B @ B3 ) ) ) ).

% insertCI
thf(fact_131_insertCI,axiom,
    ! [A: product_prod_a_b,B3: set_Product_prod_a_b,B: product_prod_a_b] :
      ( ( ~ ( member1426531481828664017od_a_b @ A @ B3 )
       => ( A = B ) )
     => ( member1426531481828664017od_a_b @ A @ ( insert4534936386344385144od_a_b @ B @ B3 ) ) ) ).

% insertCI
thf(fact_132_insertCI,axiom,
    ! [A: a,B3: set_a,B: a] :
      ( ( ~ ( member_a @ A @ B3 )
       => ( A = B ) )
     => ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).

% insertCI
thf(fact_133_Diff__idemp,axiom,
    ! [A3: set_Product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ( minus_6888070959853530735od_a_b @ ( minus_6888070959853530735od_a_b @ A3 @ B3 ) @ B3 )
      = ( minus_6888070959853530735od_a_b @ A3 @ B3 ) ) ).

% Diff_idemp
thf(fact_134_Diff__idemp,axiom,
    ! [A3: set_a,B3: set_a] :
      ( ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ B3 )
      = ( minus_minus_set_a @ A3 @ B3 ) ) ).

% Diff_idemp
thf(fact_135_Diff__iff,axiom,
    ! [C: produc1392733710753813600list_a,A3: set_Pr1696764298067387072list_a,B3: set_Pr1696764298067387072list_a] :
      ( ( member8740110733080369545list_a @ C @ ( minus_1324476095901101607list_a @ A3 @ B3 ) )
      = ( ( member8740110733080369545list_a @ C @ A3 )
        & ~ ( member8740110733080369545list_a @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_136_Diff__iff,axiom,
    ! [C: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a,B3: set_Pr7222558492679949814list_a] :
      ( ( member852939521359900607list_a @ C @ ( minus_4036318849104940381list_a @ A3 @ B3 ) )
      = ( ( member852939521359900607list_a @ C @ A3 )
        & ~ ( member852939521359900607list_a @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_137_Diff__iff,axiom,
    ! [C: list_a,A3: set_list_a,B3: set_list_a] :
      ( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A3 @ B3 ) )
      = ( ( member_list_a @ C @ A3 )
        & ~ ( member_list_a @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_138_Diff__iff,axiom,
    ! [C: a > a > $o,A3: set_a_a_o,B3: set_a_a_o] :
      ( ( member_a_a_o @ C @ ( minus_399184413438596917_a_a_o @ A3 @ B3 ) )
      = ( ( member_a_a_o @ C @ A3 )
        & ~ ( member_a_a_o @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_139_Diff__iff,axiom,
    ! [C: product_prod_a_b,A3: set_Product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ ( minus_6888070959853530735od_a_b @ A3 @ B3 ) )
      = ( ( member1426531481828664017od_a_b @ C @ A3 )
        & ~ ( member1426531481828664017od_a_b @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_140_Diff__iff,axiom,
    ! [C: a,A3: set_a,B3: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) )
      = ( ( member_a @ C @ A3 )
        & ~ ( member_a @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_141_DiffI,axiom,
    ! [C: produc1392733710753813600list_a,A3: set_Pr1696764298067387072list_a,B3: set_Pr1696764298067387072list_a] :
      ( ( member8740110733080369545list_a @ C @ A3 )
     => ( ~ ( member8740110733080369545list_a @ C @ B3 )
       => ( member8740110733080369545list_a @ C @ ( minus_1324476095901101607list_a @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_142_DiffI,axiom,
    ! [C: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a,B3: set_Pr7222558492679949814list_a] :
      ( ( member852939521359900607list_a @ C @ A3 )
     => ( ~ ( member852939521359900607list_a @ C @ B3 )
       => ( member852939521359900607list_a @ C @ ( minus_4036318849104940381list_a @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_143_DiffI,axiom,
    ! [C: list_a,A3: set_list_a,B3: set_list_a] :
      ( ( member_list_a @ C @ A3 )
     => ( ~ ( member_list_a @ C @ B3 )
       => ( member_list_a @ C @ ( minus_646659088055828811list_a @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_144_DiffI,axiom,
    ! [C: a > a > $o,A3: set_a_a_o,B3: set_a_a_o] :
      ( ( member_a_a_o @ C @ A3 )
     => ( ~ ( member_a_a_o @ C @ B3 )
       => ( member_a_a_o @ C @ ( minus_399184413438596917_a_a_o @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_145_DiffI,axiom,
    ! [C: product_prod_a_b,A3: set_Product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ A3 )
     => ( ~ ( member1426531481828664017od_a_b @ C @ B3 )
       => ( member1426531481828664017od_a_b @ C @ ( minus_6888070959853530735od_a_b @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_146_DiffI,axiom,
    ! [C: a,A3: set_a,B3: set_a] :
      ( ( member_a @ C @ A3 )
     => ( ~ ( member_a @ C @ B3 )
       => ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_147_ex__in__conv,axiom,
    ! [A3: set_Pr1696764298067387072list_a] :
      ( ( ? [X2: produc1392733710753813600list_a] : ( member8740110733080369545list_a @ X2 @ A3 ) )
      = ( A3 != bot_bo7554134525451881772list_a ) ) ).

% ex_in_conv
thf(fact_148_ex__in__conv,axiom,
    ! [A3: set_Pr7222558492679949814list_a] :
      ( ( ? [X2: produc2547433716817190550list_a] : ( member852939521359900607list_a @ X2 @ A3 ) )
      = ( A3 != bot_bo119984060639512674list_a ) ) ).

% ex_in_conv
thf(fact_149_ex__in__conv,axiom,
    ! [A3: set_list_a] :
      ( ( ? [X2: list_a] : ( member_list_a @ X2 @ A3 ) )
      = ( A3 != bot_bot_set_list_a ) ) ).

% ex_in_conv
thf(fact_150_ex__in__conv,axiom,
    ! [A3: set_a_a_o] :
      ( ( ? [X2: a > a > $o] : ( member_a_a_o @ X2 @ A3 ) )
      = ( A3 != bot_bot_set_a_a_o ) ) ).

% ex_in_conv
thf(fact_151_ex__in__conv,axiom,
    ! [A3: set_Product_prod_a_b] :
      ( ( ? [X2: product_prod_a_b] : ( member1426531481828664017od_a_b @ X2 @ A3 ) )
      = ( A3 != bot_bo3428410327501040244od_a_b ) ) ).

% ex_in_conv
thf(fact_152_ex__in__conv,axiom,
    ! [A3: set_a] :
      ( ( ? [X2: a] : ( member_a @ X2 @ A3 ) )
      = ( A3 != bot_bot_set_a ) ) ).

% ex_in_conv
thf(fact_153_equals0I,axiom,
    ! [A3: set_Pr1696764298067387072list_a] :
      ( ! [Y2: produc1392733710753813600list_a] :
          ~ ( member8740110733080369545list_a @ Y2 @ A3 )
     => ( A3 = bot_bo7554134525451881772list_a ) ) ).

% equals0I
thf(fact_154_equals0I,axiom,
    ! [A3: set_Pr7222558492679949814list_a] :
      ( ! [Y2: produc2547433716817190550list_a] :
          ~ ( member852939521359900607list_a @ Y2 @ A3 )
     => ( A3 = bot_bo119984060639512674list_a ) ) ).

% equals0I
thf(fact_155_equals0I,axiom,
    ! [A3: set_list_a] :
      ( ! [Y2: list_a] :
          ~ ( member_list_a @ Y2 @ A3 )
     => ( A3 = bot_bot_set_list_a ) ) ).

% equals0I
thf(fact_156_equals0I,axiom,
    ! [A3: set_a_a_o] :
      ( ! [Y2: a > a > $o] :
          ~ ( member_a_a_o @ Y2 @ A3 )
     => ( A3 = bot_bot_set_a_a_o ) ) ).

% equals0I
thf(fact_157_equals0I,axiom,
    ! [A3: set_Product_prod_a_b] :
      ( ! [Y2: product_prod_a_b] :
          ~ ( member1426531481828664017od_a_b @ Y2 @ A3 )
     => ( A3 = bot_bo3428410327501040244od_a_b ) ) ).

% equals0I
thf(fact_158_equals0I,axiom,
    ! [A3: set_a] :
      ( ! [Y2: a] :
          ~ ( member_a @ Y2 @ A3 )
     => ( A3 = bot_bot_set_a ) ) ).

% equals0I
thf(fact_159_equals0D,axiom,
    ! [A3: set_Pr1696764298067387072list_a,A: produc1392733710753813600list_a] :
      ( ( A3 = bot_bo7554134525451881772list_a )
     => ~ ( member8740110733080369545list_a @ A @ A3 ) ) ).

% equals0D
thf(fact_160_equals0D,axiom,
    ! [A3: set_Pr7222558492679949814list_a,A: produc2547433716817190550list_a] :
      ( ( A3 = bot_bo119984060639512674list_a )
     => ~ ( member852939521359900607list_a @ A @ A3 ) ) ).

% equals0D
thf(fact_161_equals0D,axiom,
    ! [A3: set_list_a,A: list_a] :
      ( ( A3 = bot_bot_set_list_a )
     => ~ ( member_list_a @ A @ A3 ) ) ).

% equals0D
thf(fact_162_equals0D,axiom,
    ! [A3: set_a_a_o,A: a > a > $o] :
      ( ( A3 = bot_bot_set_a_a_o )
     => ~ ( member_a_a_o @ A @ A3 ) ) ).

% equals0D
thf(fact_163_equals0D,axiom,
    ! [A3: set_Product_prod_a_b,A: product_prod_a_b] :
      ( ( A3 = bot_bo3428410327501040244od_a_b )
     => ~ ( member1426531481828664017od_a_b @ A @ A3 ) ) ).

% equals0D
thf(fact_164_equals0D,axiom,
    ! [A3: set_a,A: a] :
      ( ( A3 = bot_bot_set_a )
     => ~ ( member_a @ A @ A3 ) ) ).

% equals0D
thf(fact_165_emptyE,axiom,
    ! [A: produc1392733710753813600list_a] :
      ~ ( member8740110733080369545list_a @ A @ bot_bo7554134525451881772list_a ) ).

% emptyE
thf(fact_166_emptyE,axiom,
    ! [A: produc2547433716817190550list_a] :
      ~ ( member852939521359900607list_a @ A @ bot_bo119984060639512674list_a ) ).

% emptyE
thf(fact_167_emptyE,axiom,
    ! [A: list_a] :
      ~ ( member_list_a @ A @ bot_bot_set_list_a ) ).

% emptyE
thf(fact_168_emptyE,axiom,
    ! [A: a > a > $o] :
      ~ ( member_a_a_o @ A @ bot_bot_set_a_a_o ) ).

% emptyE
thf(fact_169_emptyE,axiom,
    ! [A: product_prod_a_b] :
      ~ ( member1426531481828664017od_a_b @ A @ bot_bo3428410327501040244od_a_b ) ).

% emptyE
thf(fact_170_emptyE,axiom,
    ! [A: a] :
      ~ ( member_a @ A @ bot_bot_set_a ) ).

% emptyE
thf(fact_171_mk__disjoint__insert,axiom,
    ! [A: produc1392733710753813600list_a,A3: set_Pr1696764298067387072list_a] :
      ( ( member8740110733080369545list_a @ A @ A3 )
     => ? [B4: set_Pr1696764298067387072list_a] :
          ( ( A3
            = ( insert5583651178174595632list_a @ A @ B4 ) )
          & ~ ( member8740110733080369545list_a @ A @ B4 ) ) ) ).

% mk_disjoint_insert
thf(fact_172_mk__disjoint__insert,axiom,
    ! [A: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a] :
      ( ( member852939521359900607list_a @ A @ A3 )
     => ? [B4: set_Pr7222558492679949814list_a] :
          ( ( A3
            = ( insert6203622070194670694list_a @ A @ B4 ) )
          & ~ ( member852939521359900607list_a @ A @ B4 ) ) ) ).

% mk_disjoint_insert
thf(fact_173_mk__disjoint__insert,axiom,
    ! [A: list_a,A3: set_list_a] :
      ( ( member_list_a @ A @ A3 )
     => ? [B4: set_list_a] :
          ( ( A3
            = ( insert_list_a @ A @ B4 ) )
          & ~ ( member_list_a @ A @ B4 ) ) ) ).

% mk_disjoint_insert
thf(fact_174_mk__disjoint__insert,axiom,
    ! [A: a > a > $o,A3: set_a_a_o] :
      ( ( member_a_a_o @ A @ A3 )
     => ? [B4: set_a_a_o] :
          ( ( A3
            = ( insert_a_a_o @ A @ B4 ) )
          & ~ ( member_a_a_o @ A @ B4 ) ) ) ).

% mk_disjoint_insert
thf(fact_175_mk__disjoint__insert,axiom,
    ! [A: product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ A @ A3 )
     => ? [B4: set_Product_prod_a_b] :
          ( ( A3
            = ( insert4534936386344385144od_a_b @ A @ B4 ) )
          & ~ ( member1426531481828664017od_a_b @ A @ B4 ) ) ) ).

% mk_disjoint_insert
thf(fact_176_mk__disjoint__insert,axiom,
    ! [A: a,A3: set_a] :
      ( ( member_a @ A @ A3 )
     => ? [B4: set_a] :
          ( ( A3
            = ( insert_a @ A @ B4 ) )
          & ~ ( member_a @ A @ B4 ) ) ) ).

% mk_disjoint_insert
thf(fact_177_insert__commute,axiom,
    ! [X: produc2547433716817190550list_a,Y: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a] :
      ( ( insert6203622070194670694list_a @ X @ ( insert6203622070194670694list_a @ Y @ A3 ) )
      = ( insert6203622070194670694list_a @ Y @ ( insert6203622070194670694list_a @ X @ A3 ) ) ) ).

% insert_commute
thf(fact_178_insert__commute,axiom,
    ! [X: product_prod_a_b,Y: product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ( insert4534936386344385144od_a_b @ X @ ( insert4534936386344385144od_a_b @ Y @ A3 ) )
      = ( insert4534936386344385144od_a_b @ Y @ ( insert4534936386344385144od_a_b @ X @ A3 ) ) ) ).

% insert_commute
thf(fact_179_insert__commute,axiom,
    ! [X: a,Y: a,A3: set_a] :
      ( ( insert_a @ X @ ( insert_a @ Y @ A3 ) )
      = ( insert_a @ Y @ ( insert_a @ X @ A3 ) ) ) ).

% insert_commute
thf(fact_180_insert__eq__iff,axiom,
    ! [A: produc1392733710753813600list_a,A3: set_Pr1696764298067387072list_a,B: produc1392733710753813600list_a,B3: set_Pr1696764298067387072list_a] :
      ( ~ ( member8740110733080369545list_a @ A @ A3 )
     => ( ~ ( member8740110733080369545list_a @ B @ B3 )
       => ( ( ( insert5583651178174595632list_a @ A @ A3 )
            = ( insert5583651178174595632list_a @ B @ B3 ) )
          = ( ( ( A = B )
             => ( A3 = B3 ) )
            & ( ( A != B )
             => ? [C2: set_Pr1696764298067387072list_a] :
                  ( ( A3
                    = ( insert5583651178174595632list_a @ B @ C2 ) )
                  & ~ ( member8740110733080369545list_a @ B @ C2 )
                  & ( B3
                    = ( insert5583651178174595632list_a @ A @ C2 ) )
                  & ~ ( member8740110733080369545list_a @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_181_insert__eq__iff,axiom,
    ! [A: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a,B: produc2547433716817190550list_a,B3: set_Pr7222558492679949814list_a] :
      ( ~ ( member852939521359900607list_a @ A @ A3 )
     => ( ~ ( member852939521359900607list_a @ B @ B3 )
       => ( ( ( insert6203622070194670694list_a @ A @ A3 )
            = ( insert6203622070194670694list_a @ B @ B3 ) )
          = ( ( ( A = B )
             => ( A3 = B3 ) )
            & ( ( A != B )
             => ? [C2: set_Pr7222558492679949814list_a] :
                  ( ( A3
                    = ( insert6203622070194670694list_a @ B @ C2 ) )
                  & ~ ( member852939521359900607list_a @ B @ C2 )
                  & ( B3
                    = ( insert6203622070194670694list_a @ A @ C2 ) )
                  & ~ ( member852939521359900607list_a @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_182_insert__eq__iff,axiom,
    ! [A: list_a,A3: set_list_a,B: list_a,B3: set_list_a] :
      ( ~ ( member_list_a @ A @ A3 )
     => ( ~ ( member_list_a @ B @ B3 )
       => ( ( ( insert_list_a @ A @ A3 )
            = ( insert_list_a @ B @ B3 ) )
          = ( ( ( A = B )
             => ( A3 = B3 ) )
            & ( ( A != B )
             => ? [C2: set_list_a] :
                  ( ( A3
                    = ( insert_list_a @ B @ C2 ) )
                  & ~ ( member_list_a @ B @ C2 )
                  & ( B3
                    = ( insert_list_a @ A @ C2 ) )
                  & ~ ( member_list_a @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_183_insert__eq__iff,axiom,
    ! [A: a > a > $o,A3: set_a_a_o,B: a > a > $o,B3: set_a_a_o] :
      ( ~ ( member_a_a_o @ A @ A3 )
     => ( ~ ( member_a_a_o @ B @ B3 )
       => ( ( ( insert_a_a_o @ A @ A3 )
            = ( insert_a_a_o @ B @ B3 ) )
          = ( ( ( A = B )
             => ( A3 = B3 ) )
            & ( ( A != B )
             => ? [C2: set_a_a_o] :
                  ( ( A3
                    = ( insert_a_a_o @ B @ C2 ) )
                  & ~ ( member_a_a_o @ B @ C2 )
                  & ( B3
                    = ( insert_a_a_o @ A @ C2 ) )
                  & ~ ( member_a_a_o @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_184_insert__eq__iff,axiom,
    ! [A: product_prod_a_b,A3: set_Product_prod_a_b,B: product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ A @ A3 )
     => ( ~ ( member1426531481828664017od_a_b @ B @ B3 )
       => ( ( ( insert4534936386344385144od_a_b @ A @ A3 )
            = ( insert4534936386344385144od_a_b @ B @ B3 ) )
          = ( ( ( A = B )
             => ( A3 = B3 ) )
            & ( ( A != B )
             => ? [C2: set_Product_prod_a_b] :
                  ( ( A3
                    = ( insert4534936386344385144od_a_b @ B @ C2 ) )
                  & ~ ( member1426531481828664017od_a_b @ B @ C2 )
                  & ( B3
                    = ( insert4534936386344385144od_a_b @ A @ C2 ) )
                  & ~ ( member1426531481828664017od_a_b @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_185_insert__eq__iff,axiom,
    ! [A: a,A3: set_a,B: a,B3: set_a] :
      ( ~ ( member_a @ A @ A3 )
     => ( ~ ( member_a @ B @ B3 )
       => ( ( ( insert_a @ A @ A3 )
            = ( insert_a @ B @ B3 ) )
          = ( ( ( A = B )
             => ( A3 = B3 ) )
            & ( ( A != B )
             => ? [C2: set_a] :
                  ( ( A3
                    = ( insert_a @ B @ C2 ) )
                  & ~ ( member_a @ B @ C2 )
                  & ( B3
                    = ( insert_a @ A @ C2 ) )
                  & ~ ( member_a @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_186_insert__absorb,axiom,
    ! [A: produc1392733710753813600list_a,A3: set_Pr1696764298067387072list_a] :
      ( ( member8740110733080369545list_a @ A @ A3 )
     => ( ( insert5583651178174595632list_a @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_187_insert__absorb,axiom,
    ! [A: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a] :
      ( ( member852939521359900607list_a @ A @ A3 )
     => ( ( insert6203622070194670694list_a @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_188_insert__absorb,axiom,
    ! [A: list_a,A3: set_list_a] :
      ( ( member_list_a @ A @ A3 )
     => ( ( insert_list_a @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_189_insert__absorb,axiom,
    ! [A: a > a > $o,A3: set_a_a_o] :
      ( ( member_a_a_o @ A @ A3 )
     => ( ( insert_a_a_o @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_190_insert__absorb,axiom,
    ! [A: product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ A @ A3 )
     => ( ( insert4534936386344385144od_a_b @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_191_insert__absorb,axiom,
    ! [A: a,A3: set_a] :
      ( ( member_a @ A @ A3 )
     => ( ( insert_a @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_192_insert__ident,axiom,
    ! [X: produc1392733710753813600list_a,A3: set_Pr1696764298067387072list_a,B3: set_Pr1696764298067387072list_a] :
      ( ~ ( member8740110733080369545list_a @ X @ A3 )
     => ( ~ ( member8740110733080369545list_a @ X @ B3 )
       => ( ( ( insert5583651178174595632list_a @ X @ A3 )
            = ( insert5583651178174595632list_a @ X @ B3 ) )
          = ( A3 = B3 ) ) ) ) ).

% insert_ident
thf(fact_193_insert__ident,axiom,
    ! [X: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a,B3: set_Pr7222558492679949814list_a] :
      ( ~ ( member852939521359900607list_a @ X @ A3 )
     => ( ~ ( member852939521359900607list_a @ X @ B3 )
       => ( ( ( insert6203622070194670694list_a @ X @ A3 )
            = ( insert6203622070194670694list_a @ X @ B3 ) )
          = ( A3 = B3 ) ) ) ) ).

% insert_ident
thf(fact_194_insert__ident,axiom,
    ! [X: list_a,A3: set_list_a,B3: set_list_a] :
      ( ~ ( member_list_a @ X @ A3 )
     => ( ~ ( member_list_a @ X @ B3 )
       => ( ( ( insert_list_a @ X @ A3 )
            = ( insert_list_a @ X @ B3 ) )
          = ( A3 = B3 ) ) ) ) ).

% insert_ident
thf(fact_195_insert__ident,axiom,
    ! [X: a > a > $o,A3: set_a_a_o,B3: set_a_a_o] :
      ( ~ ( member_a_a_o @ X @ A3 )
     => ( ~ ( member_a_a_o @ X @ B3 )
       => ( ( ( insert_a_a_o @ X @ A3 )
            = ( insert_a_a_o @ X @ B3 ) )
          = ( A3 = B3 ) ) ) ) ).

% insert_ident
thf(fact_196_insert__ident,axiom,
    ! [X: product_prod_a_b,A3: set_Product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ X @ A3 )
     => ( ~ ( member1426531481828664017od_a_b @ X @ B3 )
       => ( ( ( insert4534936386344385144od_a_b @ X @ A3 )
            = ( insert4534936386344385144od_a_b @ X @ B3 ) )
          = ( A3 = B3 ) ) ) ) ).

% insert_ident
thf(fact_197_insert__ident,axiom,
    ! [X: a,A3: set_a,B3: set_a] :
      ( ~ ( member_a @ X @ A3 )
     => ( ~ ( member_a @ X @ B3 )
       => ( ( ( insert_a @ X @ A3 )
            = ( insert_a @ X @ B3 ) )
          = ( A3 = B3 ) ) ) ) ).

% insert_ident
thf(fact_198_Set_Oset__insert,axiom,
    ! [X: produc1392733710753813600list_a,A3: set_Pr1696764298067387072list_a] :
      ( ( member8740110733080369545list_a @ X @ A3 )
     => ~ ! [B4: set_Pr1696764298067387072list_a] :
            ( ( A3
              = ( insert5583651178174595632list_a @ X @ B4 ) )
           => ( member8740110733080369545list_a @ X @ B4 ) ) ) ).

% Set.set_insert
thf(fact_199_Set_Oset__insert,axiom,
    ! [X: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a] :
      ( ( member852939521359900607list_a @ X @ A3 )
     => ~ ! [B4: set_Pr7222558492679949814list_a] :
            ( ( A3
              = ( insert6203622070194670694list_a @ X @ B4 ) )
           => ( member852939521359900607list_a @ X @ B4 ) ) ) ).

% Set.set_insert
thf(fact_200_Set_Oset__insert,axiom,
    ! [X: list_a,A3: set_list_a] :
      ( ( member_list_a @ X @ A3 )
     => ~ ! [B4: set_list_a] :
            ( ( A3
              = ( insert_list_a @ X @ B4 ) )
           => ( member_list_a @ X @ B4 ) ) ) ).

% Set.set_insert
thf(fact_201_Set_Oset__insert,axiom,
    ! [X: a > a > $o,A3: set_a_a_o] :
      ( ( member_a_a_o @ X @ A3 )
     => ~ ! [B4: set_a_a_o] :
            ( ( A3
              = ( insert_a_a_o @ X @ B4 ) )
           => ( member_a_a_o @ X @ B4 ) ) ) ).

% Set.set_insert
thf(fact_202_Set_Oset__insert,axiom,
    ! [X: product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ A3 )
     => ~ ! [B4: set_Product_prod_a_b] :
            ( ( A3
              = ( insert4534936386344385144od_a_b @ X @ B4 ) )
           => ( member1426531481828664017od_a_b @ X @ B4 ) ) ) ).

% Set.set_insert
thf(fact_203_Set_Oset__insert,axiom,
    ! [X: a,A3: set_a] :
      ( ( member_a @ X @ A3 )
     => ~ ! [B4: set_a] :
            ( ( A3
              = ( insert_a @ X @ B4 ) )
           => ( member_a @ X @ B4 ) ) ) ).

% Set.set_insert
thf(fact_204_insertI2,axiom,
    ! [A: produc1392733710753813600list_a,B3: set_Pr1696764298067387072list_a,B: produc1392733710753813600list_a] :
      ( ( member8740110733080369545list_a @ A @ B3 )
     => ( member8740110733080369545list_a @ A @ ( insert5583651178174595632list_a @ B @ B3 ) ) ) ).

% insertI2
thf(fact_205_insertI2,axiom,
    ! [A: produc2547433716817190550list_a,B3: set_Pr7222558492679949814list_a,B: produc2547433716817190550list_a] :
      ( ( member852939521359900607list_a @ A @ B3 )
     => ( member852939521359900607list_a @ A @ ( insert6203622070194670694list_a @ B @ B3 ) ) ) ).

% insertI2
thf(fact_206_insertI2,axiom,
    ! [A: list_a,B3: set_list_a,B: list_a] :
      ( ( member_list_a @ A @ B3 )
     => ( member_list_a @ A @ ( insert_list_a @ B @ B3 ) ) ) ).

% insertI2
thf(fact_207_insertI2,axiom,
    ! [A: a > a > $o,B3: set_a_a_o,B: a > a > $o] :
      ( ( member_a_a_o @ A @ B3 )
     => ( member_a_a_o @ A @ ( insert_a_a_o @ B @ B3 ) ) ) ).

% insertI2
thf(fact_208_insertI2,axiom,
    ! [A: product_prod_a_b,B3: set_Product_prod_a_b,B: product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ A @ B3 )
     => ( member1426531481828664017od_a_b @ A @ ( insert4534936386344385144od_a_b @ B @ B3 ) ) ) ).

% insertI2
thf(fact_209_insertI2,axiom,
    ! [A: a,B3: set_a,B: a] :
      ( ( member_a @ A @ B3 )
     => ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).

% insertI2
thf(fact_210_insertI1,axiom,
    ! [A: produc1392733710753813600list_a,B3: set_Pr1696764298067387072list_a] : ( member8740110733080369545list_a @ A @ ( insert5583651178174595632list_a @ A @ B3 ) ) ).

% insertI1
thf(fact_211_insertI1,axiom,
    ! [A: produc2547433716817190550list_a,B3: set_Pr7222558492679949814list_a] : ( member852939521359900607list_a @ A @ ( insert6203622070194670694list_a @ A @ B3 ) ) ).

% insertI1
thf(fact_212_insertI1,axiom,
    ! [A: list_a,B3: set_list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ B3 ) ) ).

% insertI1
thf(fact_213_insertI1,axiom,
    ! [A: a > a > $o,B3: set_a_a_o] : ( member_a_a_o @ A @ ( insert_a_a_o @ A @ B3 ) ) ).

% insertI1
thf(fact_214_insertI1,axiom,
    ! [A: product_prod_a_b,B3: set_Product_prod_a_b] : ( member1426531481828664017od_a_b @ A @ ( insert4534936386344385144od_a_b @ A @ B3 ) ) ).

% insertI1
thf(fact_215_insertI1,axiom,
    ! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).

% insertI1
thf(fact_216_insertE,axiom,
    ! [A: produc1392733710753813600list_a,B: produc1392733710753813600list_a,A3: set_Pr1696764298067387072list_a] :
      ( ( member8740110733080369545list_a @ A @ ( insert5583651178174595632list_a @ B @ A3 ) )
     => ( ( A != B )
       => ( member8740110733080369545list_a @ A @ A3 ) ) ) ).

% insertE
thf(fact_217_insertE,axiom,
    ! [A: produc2547433716817190550list_a,B: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a] :
      ( ( member852939521359900607list_a @ A @ ( insert6203622070194670694list_a @ B @ A3 ) )
     => ( ( A != B )
       => ( member852939521359900607list_a @ A @ A3 ) ) ) ).

% insertE
thf(fact_218_insertE,axiom,
    ! [A: list_a,B: list_a,A3: set_list_a] :
      ( ( member_list_a @ A @ ( insert_list_a @ B @ A3 ) )
     => ( ( A != B )
       => ( member_list_a @ A @ A3 ) ) ) ).

% insertE
thf(fact_219_insertE,axiom,
    ! [A: a > a > $o,B: a > a > $o,A3: set_a_a_o] :
      ( ( member_a_a_o @ A @ ( insert_a_a_o @ B @ A3 ) )
     => ( ( A != B )
       => ( member_a_a_o @ A @ A3 ) ) ) ).

% insertE
thf(fact_220_insertE,axiom,
    ! [A: product_prod_a_b,B: product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ A @ ( insert4534936386344385144od_a_b @ B @ A3 ) )
     => ( ( A != B )
       => ( member1426531481828664017od_a_b @ A @ A3 ) ) ) ).

% insertE
thf(fact_221_insertE,axiom,
    ! [A: a,B: a,A3: set_a] :
      ( ( member_a @ A @ ( insert_a @ B @ A3 ) )
     => ( ( A != B )
       => ( member_a @ A @ A3 ) ) ) ).

% insertE
thf(fact_222_DiffD2,axiom,
    ! [C: produc1392733710753813600list_a,A3: set_Pr1696764298067387072list_a,B3: set_Pr1696764298067387072list_a] :
      ( ( member8740110733080369545list_a @ C @ ( minus_1324476095901101607list_a @ A3 @ B3 ) )
     => ~ ( member8740110733080369545list_a @ C @ B3 ) ) ).

% DiffD2
thf(fact_223_DiffD2,axiom,
    ! [C: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a,B3: set_Pr7222558492679949814list_a] :
      ( ( member852939521359900607list_a @ C @ ( minus_4036318849104940381list_a @ A3 @ B3 ) )
     => ~ ( member852939521359900607list_a @ C @ B3 ) ) ).

% DiffD2
thf(fact_224_DiffD2,axiom,
    ! [C: list_a,A3: set_list_a,B3: set_list_a] :
      ( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A3 @ B3 ) )
     => ~ ( member_list_a @ C @ B3 ) ) ).

% DiffD2
thf(fact_225_DiffD2,axiom,
    ! [C: a > a > $o,A3: set_a_a_o,B3: set_a_a_o] :
      ( ( member_a_a_o @ C @ ( minus_399184413438596917_a_a_o @ A3 @ B3 ) )
     => ~ ( member_a_a_o @ C @ B3 ) ) ).

% DiffD2
thf(fact_226_DiffD2,axiom,
    ! [C: product_prod_a_b,A3: set_Product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ ( minus_6888070959853530735od_a_b @ A3 @ B3 ) )
     => ~ ( member1426531481828664017od_a_b @ C @ B3 ) ) ).

% DiffD2
thf(fact_227_DiffD2,axiom,
    ! [C: a,A3: set_a,B3: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) )
     => ~ ( member_a @ C @ B3 ) ) ).

% DiffD2
thf(fact_228_DiffD1,axiom,
    ! [C: produc1392733710753813600list_a,A3: set_Pr1696764298067387072list_a,B3: set_Pr1696764298067387072list_a] :
      ( ( member8740110733080369545list_a @ C @ ( minus_1324476095901101607list_a @ A3 @ B3 ) )
     => ( member8740110733080369545list_a @ C @ A3 ) ) ).

% DiffD1
thf(fact_229_DiffD1,axiom,
    ! [C: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a,B3: set_Pr7222558492679949814list_a] :
      ( ( member852939521359900607list_a @ C @ ( minus_4036318849104940381list_a @ A3 @ B3 ) )
     => ( member852939521359900607list_a @ C @ A3 ) ) ).

% DiffD1
thf(fact_230_DiffD1,axiom,
    ! [C: list_a,A3: set_list_a,B3: set_list_a] :
      ( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A3 @ B3 ) )
     => ( member_list_a @ C @ A3 ) ) ).

% DiffD1
thf(fact_231_DiffD1,axiom,
    ! [C: a > a > $o,A3: set_a_a_o,B3: set_a_a_o] :
      ( ( member_a_a_o @ C @ ( minus_399184413438596917_a_a_o @ A3 @ B3 ) )
     => ( member_a_a_o @ C @ A3 ) ) ).

% DiffD1
thf(fact_232_DiffD1,axiom,
    ! [C: product_prod_a_b,A3: set_Product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ ( minus_6888070959853530735od_a_b @ A3 @ B3 ) )
     => ( member1426531481828664017od_a_b @ C @ A3 ) ) ).

% DiffD1
thf(fact_233_DiffD1,axiom,
    ! [C: a,A3: set_a,B3: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) )
     => ( member_a @ C @ A3 ) ) ).

% DiffD1
thf(fact_234_DiffE,axiom,
    ! [C: produc1392733710753813600list_a,A3: set_Pr1696764298067387072list_a,B3: set_Pr1696764298067387072list_a] :
      ( ( member8740110733080369545list_a @ C @ ( minus_1324476095901101607list_a @ A3 @ B3 ) )
     => ~ ( ( member8740110733080369545list_a @ C @ A3 )
         => ( member8740110733080369545list_a @ C @ B3 ) ) ) ).

% DiffE
thf(fact_235_DiffE,axiom,
    ! [C: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a,B3: set_Pr7222558492679949814list_a] :
      ( ( member852939521359900607list_a @ C @ ( minus_4036318849104940381list_a @ A3 @ B3 ) )
     => ~ ( ( member852939521359900607list_a @ C @ A3 )
         => ( member852939521359900607list_a @ C @ B3 ) ) ) ).

% DiffE
thf(fact_236_DiffE,axiom,
    ! [C: list_a,A3: set_list_a,B3: set_list_a] :
      ( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A3 @ B3 ) )
     => ~ ( ( member_list_a @ C @ A3 )
         => ( member_list_a @ C @ B3 ) ) ) ).

% DiffE
thf(fact_237_DiffE,axiom,
    ! [C: a > a > $o,A3: set_a_a_o,B3: set_a_a_o] :
      ( ( member_a_a_o @ C @ ( minus_399184413438596917_a_a_o @ A3 @ B3 ) )
     => ~ ( ( member_a_a_o @ C @ A3 )
         => ( member_a_a_o @ C @ B3 ) ) ) ).

% DiffE
thf(fact_238_DiffE,axiom,
    ! [C: product_prod_a_b,A3: set_Product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ ( minus_6888070959853530735od_a_b @ A3 @ B3 ) )
     => ~ ( ( member1426531481828664017od_a_b @ C @ A3 )
         => ( member1426531481828664017od_a_b @ C @ B3 ) ) ) ).

% DiffE
thf(fact_239_DiffE,axiom,
    ! [C: a,A3: set_a,B3: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) )
     => ~ ( ( member_a @ C @ A3 )
         => ( member_a @ C @ B3 ) ) ) ).

% DiffE
thf(fact_240_singleton__inject,axiom,
    ! [A: produc2547433716817190550list_a,B: produc2547433716817190550list_a] :
      ( ( ( insert6203622070194670694list_a @ A @ bot_bo119984060639512674list_a )
        = ( insert6203622070194670694list_a @ B @ bot_bo119984060639512674list_a ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_241_singleton__inject,axiom,
    ! [A: product_prod_a_b,B: product_prod_a_b] :
      ( ( ( insert4534936386344385144od_a_b @ A @ bot_bo3428410327501040244od_a_b )
        = ( insert4534936386344385144od_a_b @ B @ bot_bo3428410327501040244od_a_b ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_242_singleton__inject,axiom,
    ! [A: a,B: a] :
      ( ( ( insert_a @ A @ bot_bot_set_a )
        = ( insert_a @ B @ bot_bot_set_a ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_243_insert__not__empty,axiom,
    ! [A: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a] :
      ( ( insert6203622070194670694list_a @ A @ A3 )
     != bot_bo119984060639512674list_a ) ).

% insert_not_empty
thf(fact_244_insert__not__empty,axiom,
    ! [A: product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ( insert4534936386344385144od_a_b @ A @ A3 )
     != bot_bo3428410327501040244od_a_b ) ).

% insert_not_empty
thf(fact_245_insert__not__empty,axiom,
    ! [A: a,A3: set_a] :
      ( ( insert_a @ A @ A3 )
     != bot_bot_set_a ) ).

% insert_not_empty
thf(fact_246_doubleton__eq__iff,axiom,
    ! [A: produc2547433716817190550list_a,B: produc2547433716817190550list_a,C: produc2547433716817190550list_a,D: produc2547433716817190550list_a] :
      ( ( ( insert6203622070194670694list_a @ A @ ( insert6203622070194670694list_a @ B @ bot_bo119984060639512674list_a ) )
        = ( insert6203622070194670694list_a @ C @ ( insert6203622070194670694list_a @ D @ bot_bo119984060639512674list_a ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_247_doubleton__eq__iff,axiom,
    ! [A: product_prod_a_b,B: product_prod_a_b,C: product_prod_a_b,D: product_prod_a_b] :
      ( ( ( insert4534936386344385144od_a_b @ A @ ( insert4534936386344385144od_a_b @ B @ bot_bo3428410327501040244od_a_b ) )
        = ( insert4534936386344385144od_a_b @ C @ ( insert4534936386344385144od_a_b @ D @ bot_bo3428410327501040244od_a_b ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_248_doubleton__eq__iff,axiom,
    ! [A: a,B: a,C: a,D: a] :
      ( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
        = ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_249_singleton__iff,axiom,
    ! [B: produc1392733710753813600list_a,A: produc1392733710753813600list_a] :
      ( ( member8740110733080369545list_a @ B @ ( insert5583651178174595632list_a @ A @ bot_bo7554134525451881772list_a ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_250_singleton__iff,axiom,
    ! [B: produc2547433716817190550list_a,A: produc2547433716817190550list_a] :
      ( ( member852939521359900607list_a @ B @ ( insert6203622070194670694list_a @ A @ bot_bo119984060639512674list_a ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_251_singleton__iff,axiom,
    ! [B: list_a,A: list_a] :
      ( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_252_singleton__iff,axiom,
    ! [B: a > a > $o,A: a > a > $o] :
      ( ( member_a_a_o @ B @ ( insert_a_a_o @ A @ bot_bot_set_a_a_o ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_253_singleton__iff,axiom,
    ! [B: product_prod_a_b,A: product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ B @ ( insert4534936386344385144od_a_b @ A @ bot_bo3428410327501040244od_a_b ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_254_singleton__iff,axiom,
    ! [B: a,A: a] :
      ( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_255_singletonD,axiom,
    ! [B: produc1392733710753813600list_a,A: produc1392733710753813600list_a] :
      ( ( member8740110733080369545list_a @ B @ ( insert5583651178174595632list_a @ A @ bot_bo7554134525451881772list_a ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_256_singletonD,axiom,
    ! [B: produc2547433716817190550list_a,A: produc2547433716817190550list_a] :
      ( ( member852939521359900607list_a @ B @ ( insert6203622070194670694list_a @ A @ bot_bo119984060639512674list_a ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_257_singletonD,axiom,
    ! [B: list_a,A: list_a] :
      ( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_258_singletonD,axiom,
    ! [B: a > a > $o,A: a > a > $o] :
      ( ( member_a_a_o @ B @ ( insert_a_a_o @ A @ bot_bot_set_a_a_o ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_259_singletonD,axiom,
    ! [B: product_prod_a_b,A: product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ B @ ( insert4534936386344385144od_a_b @ A @ bot_bo3428410327501040244od_a_b ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_260_singletonD,axiom,
    ! [B: a,A: a] :
      ( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_261_insert__Diff__if,axiom,
    ! [X: produc1392733710753813600list_a,B3: set_Pr1696764298067387072list_a,A3: set_Pr1696764298067387072list_a] :
      ( ( ( member8740110733080369545list_a @ X @ B3 )
       => ( ( minus_1324476095901101607list_a @ ( insert5583651178174595632list_a @ X @ A3 ) @ B3 )
          = ( minus_1324476095901101607list_a @ A3 @ B3 ) ) )
      & ( ~ ( member8740110733080369545list_a @ X @ B3 )
       => ( ( minus_1324476095901101607list_a @ ( insert5583651178174595632list_a @ X @ A3 ) @ B3 )
          = ( insert5583651178174595632list_a @ X @ ( minus_1324476095901101607list_a @ A3 @ B3 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_262_insert__Diff__if,axiom,
    ! [X: produc2547433716817190550list_a,B3: set_Pr7222558492679949814list_a,A3: set_Pr7222558492679949814list_a] :
      ( ( ( member852939521359900607list_a @ X @ B3 )
       => ( ( minus_4036318849104940381list_a @ ( insert6203622070194670694list_a @ X @ A3 ) @ B3 )
          = ( minus_4036318849104940381list_a @ A3 @ B3 ) ) )
      & ( ~ ( member852939521359900607list_a @ X @ B3 )
       => ( ( minus_4036318849104940381list_a @ ( insert6203622070194670694list_a @ X @ A3 ) @ B3 )
          = ( insert6203622070194670694list_a @ X @ ( minus_4036318849104940381list_a @ A3 @ B3 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_263_insert__Diff__if,axiom,
    ! [X: list_a,B3: set_list_a,A3: set_list_a] :
      ( ( ( member_list_a @ X @ B3 )
       => ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A3 ) @ B3 )
          = ( minus_646659088055828811list_a @ A3 @ B3 ) ) )
      & ( ~ ( member_list_a @ X @ B3 )
       => ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A3 ) @ B3 )
          = ( insert_list_a @ X @ ( minus_646659088055828811list_a @ A3 @ B3 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_264_insert__Diff__if,axiom,
    ! [X: a > a > $o,B3: set_a_a_o,A3: set_a_a_o] :
      ( ( ( member_a_a_o @ X @ B3 )
       => ( ( minus_399184413438596917_a_a_o @ ( insert_a_a_o @ X @ A3 ) @ B3 )
          = ( minus_399184413438596917_a_a_o @ A3 @ B3 ) ) )
      & ( ~ ( member_a_a_o @ X @ B3 )
       => ( ( minus_399184413438596917_a_a_o @ ( insert_a_a_o @ X @ A3 ) @ B3 )
          = ( insert_a_a_o @ X @ ( minus_399184413438596917_a_a_o @ A3 @ B3 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_265_insert__Diff__if,axiom,
    ! [X: product_prod_a_b,B3: set_Product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ( ( member1426531481828664017od_a_b @ X @ B3 )
       => ( ( minus_6888070959853530735od_a_b @ ( insert4534936386344385144od_a_b @ X @ A3 ) @ B3 )
          = ( minus_6888070959853530735od_a_b @ A3 @ B3 ) ) )
      & ( ~ ( member1426531481828664017od_a_b @ X @ B3 )
       => ( ( minus_6888070959853530735od_a_b @ ( insert4534936386344385144od_a_b @ X @ A3 ) @ B3 )
          = ( insert4534936386344385144od_a_b @ X @ ( minus_6888070959853530735od_a_b @ A3 @ B3 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_266_insert__Diff__if,axiom,
    ! [X: a,B3: set_a,A3: set_a] :
      ( ( ( member_a @ X @ B3 )
       => ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B3 )
          = ( minus_minus_set_a @ A3 @ B3 ) ) )
      & ( ~ ( member_a @ X @ B3 )
       => ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B3 )
          = ( insert_a @ X @ ( minus_minus_set_a @ A3 @ B3 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_267_Diff__insert__absorb,axiom,
    ! [X: produc1392733710753813600list_a,A3: set_Pr1696764298067387072list_a] :
      ( ~ ( member8740110733080369545list_a @ X @ A3 )
     => ( ( minus_1324476095901101607list_a @ ( insert5583651178174595632list_a @ X @ A3 ) @ ( insert5583651178174595632list_a @ X @ bot_bo7554134525451881772list_a ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_268_Diff__insert__absorb,axiom,
    ! [X: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a] :
      ( ~ ( member852939521359900607list_a @ X @ A3 )
     => ( ( minus_4036318849104940381list_a @ ( insert6203622070194670694list_a @ X @ A3 ) @ ( insert6203622070194670694list_a @ X @ bot_bo119984060639512674list_a ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_269_Diff__insert__absorb,axiom,
    ! [X: list_a,A3: set_list_a] :
      ( ~ ( member_list_a @ X @ A3 )
     => ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A3 ) @ ( insert_list_a @ X @ bot_bot_set_list_a ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_270_Diff__insert__absorb,axiom,
    ! [X: a > a > $o,A3: set_a_a_o] :
      ( ~ ( member_a_a_o @ X @ A3 )
     => ( ( minus_399184413438596917_a_a_o @ ( insert_a_a_o @ X @ A3 ) @ ( insert_a_a_o @ X @ bot_bot_set_a_a_o ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_271_Diff__insert__absorb,axiom,
    ! [X: product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ X @ A3 )
     => ( ( minus_6888070959853530735od_a_b @ ( insert4534936386344385144od_a_b @ X @ A3 ) @ ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_272_Diff__insert__absorb,axiom,
    ! [X: a,A3: set_a] :
      ( ~ ( member_a @ X @ A3 )
     => ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ ( insert_a @ X @ bot_bot_set_a ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_273_Diff__insert2,axiom,
    ! [A3: set_Pr7222558492679949814list_a,A: produc2547433716817190550list_a,B3: set_Pr7222558492679949814list_a] :
      ( ( minus_4036318849104940381list_a @ A3 @ ( insert6203622070194670694list_a @ A @ B3 ) )
      = ( minus_4036318849104940381list_a @ ( minus_4036318849104940381list_a @ A3 @ ( insert6203622070194670694list_a @ A @ bot_bo119984060639512674list_a ) ) @ B3 ) ) ).

% Diff_insert2
thf(fact_274_Diff__insert2,axiom,
    ! [A3: set_Product_prod_a_b,A: product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ( minus_6888070959853530735od_a_b @ A3 @ ( insert4534936386344385144od_a_b @ A @ B3 ) )
      = ( minus_6888070959853530735od_a_b @ ( minus_6888070959853530735od_a_b @ A3 @ ( insert4534936386344385144od_a_b @ A @ bot_bo3428410327501040244od_a_b ) ) @ B3 ) ) ).

% Diff_insert2
thf(fact_275_Diff__insert2,axiom,
    ! [A3: set_a,A: a,B3: set_a] :
      ( ( minus_minus_set_a @ A3 @ ( insert_a @ A @ B3 ) )
      = ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B3 ) ) ).

% Diff_insert2
thf(fact_276_insert__Diff,axiom,
    ! [A: produc1392733710753813600list_a,A3: set_Pr1696764298067387072list_a] :
      ( ( member8740110733080369545list_a @ A @ A3 )
     => ( ( insert5583651178174595632list_a @ A @ ( minus_1324476095901101607list_a @ A3 @ ( insert5583651178174595632list_a @ A @ bot_bo7554134525451881772list_a ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_277_insert__Diff,axiom,
    ! [A: produc2547433716817190550list_a,A3: set_Pr7222558492679949814list_a] :
      ( ( member852939521359900607list_a @ A @ A3 )
     => ( ( insert6203622070194670694list_a @ A @ ( minus_4036318849104940381list_a @ A3 @ ( insert6203622070194670694list_a @ A @ bot_bo119984060639512674list_a ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_278_insert__Diff,axiom,
    ! [A: list_a,A3: set_list_a] :
      ( ( member_list_a @ A @ A3 )
     => ( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_279_insert__Diff,axiom,
    ! [A: a > a > $o,A3: set_a_a_o] :
      ( ( member_a_a_o @ A @ A3 )
     => ( ( insert_a_a_o @ A @ ( minus_399184413438596917_a_a_o @ A3 @ ( insert_a_a_o @ A @ bot_bot_set_a_a_o ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_280_insert__Diff,axiom,
    ! [A: product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ A @ A3 )
     => ( ( insert4534936386344385144od_a_b @ A @ ( minus_6888070959853530735od_a_b @ A3 @ ( insert4534936386344385144od_a_b @ A @ bot_bo3428410327501040244od_a_b ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_281_insert__Diff,axiom,
    ! [A: a,A3: set_a] :
      ( ( member_a @ A @ A3 )
     => ( ( insert_a @ A @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_282_Diff__insert,axiom,
    ! [A3: set_Pr7222558492679949814list_a,A: produc2547433716817190550list_a,B3: set_Pr7222558492679949814list_a] :
      ( ( minus_4036318849104940381list_a @ A3 @ ( insert6203622070194670694list_a @ A @ B3 ) )
      = ( minus_4036318849104940381list_a @ ( minus_4036318849104940381list_a @ A3 @ B3 ) @ ( insert6203622070194670694list_a @ A @ bot_bo119984060639512674list_a ) ) ) ).

% Diff_insert
thf(fact_283_Diff__insert,axiom,
    ! [A3: set_Product_prod_a_b,A: product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ( minus_6888070959853530735od_a_b @ A3 @ ( insert4534936386344385144od_a_b @ A @ B3 ) )
      = ( minus_6888070959853530735od_a_b @ ( minus_6888070959853530735od_a_b @ A3 @ B3 ) @ ( insert4534936386344385144od_a_b @ A @ bot_bo3428410327501040244od_a_b ) ) ) ).

% Diff_insert
thf(fact_284_Diff__insert,axiom,
    ! [A3: set_a,A: a,B3: set_a] :
      ( ( minus_minus_set_a @ A3 @ ( insert_a @ A @ B3 ) )
      = ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).

% Diff_insert
thf(fact_285_not__Some__eq,axiom,
    ! [X: option2417831327529529196od_a_b] :
      ( ( ! [Y3: produc1700058555794149158od_a_b] :
            ( X
           != ( some_P2412010246944353803od_a_b @ Y3 ) ) )
      = ( X = none_P912253767061701255od_a_b ) ) ).

% not_Some_eq
thf(fact_286_not__Some__eq,axiom,
    ! [X: option110588449099563443list_a] :
      ( ( ! [Y3: produc271001111946445293list_a] :
            ( X
           != ( some_P1402039549733047762list_a @ Y3 ) ) )
      = ( X = none_P4075134247637483086list_a ) ) ).

% not_Some_eq
thf(fact_287_not__Some__eq,axiom,
    ! [X: option788170273582809878list_a] :
      ( ( ! [Y3: produc3854044251064639184list_a] :
            ( X
           != ( some_P5354654743593010357list_a @ Y3 ) ) )
      = ( X = none_P5893993846586699057list_a ) ) ).

% not_Some_eq
thf(fact_288_not__Some__eq,axiom,
    ! [X: option4277935495599621869list_a] :
      ( ( ! [Y3: produc9164743771328383783list_a] :
            ( X
           != ( some_P5335617695965928972list_a @ Y3 ) ) )
      = ( X = none_P3970268120610169480list_a ) ) ).

% not_Some_eq
thf(fact_289_not__Some__eq,axiom,
    ! [X: option4732895764270368476list_a] :
      ( ( ! [Y3: produc2547433716817190550list_a] :
            ( X
           != ( some_P3259385407967395195list_a @ Y3 ) ) )
      = ( X = none_P1759628928084742647list_a ) ) ).

% not_Some_eq
thf(fact_290_not__Some__eq,axiom,
    ! [X: option_list_a] :
      ( ( ! [Y3: list_a] :
            ( X
           != ( some_list_a @ Y3 ) ) )
      = ( X = none_list_a ) ) ).

% not_Some_eq
thf(fact_291_not__Some__eq,axiom,
    ! [X: option_b] :
      ( ( ! [Y3: b] :
            ( X
           != ( some_b @ Y3 ) ) )
      = ( X = none_b ) ) ).

% not_Some_eq
thf(fact_292_not__None__eq,axiom,
    ! [X: option2417831327529529196od_a_b] :
      ( ( X != none_P912253767061701255od_a_b )
      = ( ? [Y3: produc1700058555794149158od_a_b] :
            ( X
            = ( some_P2412010246944353803od_a_b @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_293_not__None__eq,axiom,
    ! [X: option110588449099563443list_a] :
      ( ( X != none_P4075134247637483086list_a )
      = ( ? [Y3: produc271001111946445293list_a] :
            ( X
            = ( some_P1402039549733047762list_a @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_294_not__None__eq,axiom,
    ! [X: option788170273582809878list_a] :
      ( ( X != none_P5893993846586699057list_a )
      = ( ? [Y3: produc3854044251064639184list_a] :
            ( X
            = ( some_P5354654743593010357list_a @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_295_not__None__eq,axiom,
    ! [X: option4277935495599621869list_a] :
      ( ( X != none_P3970268120610169480list_a )
      = ( ? [Y3: produc9164743771328383783list_a] :
            ( X
            = ( some_P5335617695965928972list_a @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_296_not__None__eq,axiom,
    ! [X: option4732895764270368476list_a] :
      ( ( X != none_P1759628928084742647list_a )
      = ( ? [Y3: produc2547433716817190550list_a] :
            ( X
            = ( some_P3259385407967395195list_a @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_297_not__None__eq,axiom,
    ! [X: option_list_a] :
      ( ( X != none_list_a )
      = ( ? [Y3: list_a] :
            ( X
            = ( some_list_a @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_298_not__None__eq,axiom,
    ! [X: option_b] :
      ( ( X != none_b )
      = ( ? [Y3: b] :
            ( X
            = ( some_b @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_299_local_Orbt__lookup__in__tree,axiom,
    ! [T: rBT_rb6525319661942305155bt_a_b,K: a,V: produc3797860095223585356bt_a_b] :
      ( ( rBT_rb4191689393857777698bt_a_b @ less @ T )
     => ( ( ( rBT_rb1295924438074454757bt_a_b @ less @ T @ K )
          = ( some_P7253062255424944423bt_a_b @ V ) )
        = ( member6846102102718116240bt_a_b @ ( produc3015636708179276019bt_a_b @ K @ V ) @ ( set_Pr1558196515228517412bt_a_b @ ( rBT_en699320873491845330bt_a_b @ T ) ) ) ) ) ).

% local.rbt_lookup_in_tree
thf(fact_300_local_Orbt__lookup__in__tree,axiom,
    ! [T: rBT_rb3549771087195502139list_a,K: a,V: list_P5792129111969709340list_a] :
      ( ( rBT_rb1805918226279206342list_a @ less @ T )
     => ( ( ( rBT_rb7528920427811653699list_a @ less @ T @ K )
          = ( some_l6838989086144800129list_a @ V ) )
        = ( member2402001272945804526list_a @ ( produc4159846287338058421list_a @ K @ V ) @ ( set_Pr3960553176861824346list_a @ ( rBT_en7750752156332278038list_a @ T ) ) ) ) ) ).

% local.rbt_lookup_in_tree
thf(fact_301_local_Orbt__lookup__in__tree,axiom,
    ! [T: rBT_rb4976296014333920589od_a_b,K: a,V: list_P1467974523212632366od_a_b] :
      ( ( rBT_rb3998897555662963672od_a_b @ less @ T )
     => ( ( ( rBT_rb4890583358032336725od_a_b @ less @ T @ K )
          = ( some_l1253897223994727059od_a_b @ V ) )
        = ( member8745141763671436032od_a_b @ ( produc6643264353547226055od_a_b @ K @ V ) @ ( set_Pr187010687427835756od_a_b @ ( rBT_en1619617535027037992od_a_b @ T ) ) ) ) ) ).

% local.rbt_lookup_in_tree
thf(fact_302_local_Orbt__lookup__in__tree,axiom,
    ! [T: rBT_rb3198946194961646575list_a,K: a,V: produc3854044251064639184list_a] :
      ( ( rBT_rb6007450547276413562list_a @ less @ T )
     => ( ( ( rBT_rb5625365584452904567list_a @ less @ T @ K )
          = ( some_P5354654743593010357list_a @ V ) )
        = ( member8909059974025578786list_a @ ( produc5594513834575919081list_a @ K @ V ) @ ( set_Pr7668085575154557838list_a @ ( rBT_en1395339733974451658list_a @ T ) ) ) ) ) ).

% local.rbt_lookup_in_tree
thf(fact_303_local_Orbt__lookup__in__tree,axiom,
    ! [T: rBT_rb6886338251485743302list_a,K: a,V: produc9164743771328383783list_a] :
      ( ( rBT_rb3676718616835825873list_a @ less @ T )
     => ( ( ( rBT_rb9209483341951570382list_a @ less @ T @ K )
          = ( some_P5335617695965928972list_a @ V ) )
        = ( member55517302436591865list_a @ ( produc3570814007050167616list_a @ K @ V ) @ ( set_Pr218118668326575973list_a @ ( rBT_en5472518308710218785list_a @ T ) ) ) ) ) ).

% local.rbt_lookup_in_tree
thf(fact_304_local_Orbt__lookup__in__tree,axiom,
    ! [T: rBT_rb6027196121907741877list_a,K: a,V: produc2547433716817190550list_a] :
      ( ( rBT_rb3455411056480160832list_a @ less @ T )
     => ( ( ( rBT_rb6302383274397067581list_a @ less @ T @ K )
          = ( some_P3259385407967395195list_a @ V ) )
        = ( member8728132036948691176list_a @ ( produc3540107409463719087list_a @ K @ V ) @ ( set_Pr8732349045776383828list_a @ ( rBT_en6992304632074839952list_a @ T ) ) ) ) ) ).

% local.rbt_lookup_in_tree
thf(fact_305_local_Orbt__lookup__in__tree,axiom,
    ! [T: rBT_rbt_a_list_a,K: a,V: list_a] :
      ( ( rBT_rb7395583532625325102list_a @ less @ T )
     => ( ( ( rBT_rb8775438530296890539list_a @ less @ T @ K )
          = ( some_list_a @ V ) )
        = ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ K @ V ) @ ( set_Pr3039917418810828226list_a @ ( rBT_entries_a_list_a @ T ) ) ) ) ) ).

% local.rbt_lookup_in_tree
thf(fact_306_local_Orbt__lookup__in__tree,axiom,
    ! [T: rBT_rb484615002580549772list_a,K: a,V: produc271001111946445293list_a] :
      ( ( rBT_rb6864293667938249111list_a @ less @ T )
     => ( ( ( rBT_rb4780624521278262164list_a @ less @ T @ K )
          = ( some_P1402039549733047762list_a @ V ) )
        = ( member852939521359900607list_a @ ( produc4562151247771834630list_a @ K @ V ) @ ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ T ) ) ) ) ) ).

% local.rbt_lookup_in_tree
thf(fact_307_local_Orbt__lookup__in__tree,axiom,
    ! [T: rBT_rbt_a_b,K: a,V: b] :
      ( ( rBT_rbt_sorted_a_b @ less @ T )
     => ( ( ( rBT_rbt_lookup_a_b @ less @ T @ K )
          = ( some_b @ V ) )
        = ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K @ V ) @ ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ T ) ) ) ) ) ).

% local.rbt_lookup_in_tree
thf(fact_308_local_Oirreflp__on__less,axiom,
    ! [A3: set_a] : ( irreflp_on_a @ A3 @ less ) ).

% local.irreflp_on_less
thf(fact_309_local_Oasymp__on__less,axiom,
    ! [A3: set_a] : ( asymp_on_a @ A3 @ less ) ).

% local.asymp_on_less
thf(fact_310_local_Orbt__lookup__from__in__tree,axiom,
    ! [T1: rBT_rb6525319661942305155bt_a_b,T2: rBT_rb6525319661942305155bt_a_b,K: a] :
      ( ( rBT_rb4191689393857777698bt_a_b @ less @ T1 )
     => ( ( rBT_rb4191689393857777698bt_a_b @ less @ T2 )
       => ( ! [V2: produc3797860095223585356bt_a_b] :
              ( ( member6846102102718116240bt_a_b @ ( produc3015636708179276019bt_a_b @ K @ V2 ) @ ( set_Pr1558196515228517412bt_a_b @ ( rBT_en699320873491845330bt_a_b @ T1 ) ) )
              = ( member6846102102718116240bt_a_b @ ( produc3015636708179276019bt_a_b @ K @ V2 ) @ ( set_Pr1558196515228517412bt_a_b @ ( rBT_en699320873491845330bt_a_b @ T2 ) ) ) )
         => ( ( rBT_rb1295924438074454757bt_a_b @ less @ T1 @ K )
            = ( rBT_rb1295924438074454757bt_a_b @ less @ T2 @ K ) ) ) ) ) ).

% local.rbt_lookup_from_in_tree
thf(fact_311_local_Orbt__lookup__from__in__tree,axiom,
    ! [T1: rBT_rb3549771087195502139list_a,T2: rBT_rb3549771087195502139list_a,K: a] :
      ( ( rBT_rb1805918226279206342list_a @ less @ T1 )
     => ( ( rBT_rb1805918226279206342list_a @ less @ T2 )
       => ( ! [V2: list_P5792129111969709340list_a] :
              ( ( member2402001272945804526list_a @ ( produc4159846287338058421list_a @ K @ V2 ) @ ( set_Pr3960553176861824346list_a @ ( rBT_en7750752156332278038list_a @ T1 ) ) )
              = ( member2402001272945804526list_a @ ( produc4159846287338058421list_a @ K @ V2 ) @ ( set_Pr3960553176861824346list_a @ ( rBT_en7750752156332278038list_a @ T2 ) ) ) )
         => ( ( rBT_rb7528920427811653699list_a @ less @ T1 @ K )
            = ( rBT_rb7528920427811653699list_a @ less @ T2 @ K ) ) ) ) ) ).

% local.rbt_lookup_from_in_tree
thf(fact_312_local_Orbt__lookup__from__in__tree,axiom,
    ! [T1: rBT_rb4976296014333920589od_a_b,T2: rBT_rb4976296014333920589od_a_b,K: a] :
      ( ( rBT_rb3998897555662963672od_a_b @ less @ T1 )
     => ( ( rBT_rb3998897555662963672od_a_b @ less @ T2 )
       => ( ! [V2: list_P1467974523212632366od_a_b] :
              ( ( member8745141763671436032od_a_b @ ( produc6643264353547226055od_a_b @ K @ V2 ) @ ( set_Pr187010687427835756od_a_b @ ( rBT_en1619617535027037992od_a_b @ T1 ) ) )
              = ( member8745141763671436032od_a_b @ ( produc6643264353547226055od_a_b @ K @ V2 ) @ ( set_Pr187010687427835756od_a_b @ ( rBT_en1619617535027037992od_a_b @ T2 ) ) ) )
         => ( ( rBT_rb4890583358032336725od_a_b @ less @ T1 @ K )
            = ( rBT_rb4890583358032336725od_a_b @ less @ T2 @ K ) ) ) ) ) ).

% local.rbt_lookup_from_in_tree
thf(fact_313_local_Orbt__lookup__from__in__tree,axiom,
    ! [T1: rBT_rbt_a_list_a,T2: rBT_rbt_a_list_a,K: a] :
      ( ( rBT_rb7395583532625325102list_a @ less @ T1 )
     => ( ( rBT_rb7395583532625325102list_a @ less @ T2 )
       => ( ! [V2: list_a] :
              ( ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ K @ V2 ) @ ( set_Pr3039917418810828226list_a @ ( rBT_entries_a_list_a @ T1 ) ) )
              = ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ K @ V2 ) @ ( set_Pr3039917418810828226list_a @ ( rBT_entries_a_list_a @ T2 ) ) ) )
         => ( ( rBT_rb8775438530296890539list_a @ less @ T1 @ K )
            = ( rBT_rb8775438530296890539list_a @ less @ T2 @ K ) ) ) ) ) ).

% local.rbt_lookup_from_in_tree
thf(fact_314_local_Orbt__lookup__from__in__tree,axiom,
    ! [T1: rBT_rb484615002580549772list_a,T2: rBT_rb484615002580549772list_a,K: a] :
      ( ( rBT_rb6864293667938249111list_a @ less @ T1 )
     => ( ( rBT_rb6864293667938249111list_a @ less @ T2 )
       => ( ! [V2: produc271001111946445293list_a] :
              ( ( member852939521359900607list_a @ ( produc4562151247771834630list_a @ K @ V2 ) @ ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ T1 ) ) )
              = ( member852939521359900607list_a @ ( produc4562151247771834630list_a @ K @ V2 ) @ ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ T2 ) ) ) )
         => ( ( rBT_rb4780624521278262164list_a @ less @ T1 @ K )
            = ( rBT_rb4780624521278262164list_a @ less @ T2 @ K ) ) ) ) ) ).

% local.rbt_lookup_from_in_tree
thf(fact_315_local_Orbt__lookup__from__in__tree,axiom,
    ! [T1: rBT_rbt_a_b,T2: rBT_rbt_a_b,K: a] :
      ( ( rBT_rbt_sorted_a_b @ less @ T1 )
     => ( ( rBT_rbt_sorted_a_b @ less @ T2 )
       => ( ! [V2: b] :
              ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K @ V2 ) @ ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ T1 ) ) )
              = ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K @ V2 ) @ ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ T2 ) ) ) )
         => ( ( rBT_rbt_lookup_a_b @ less @ T1 @ K )
            = ( rBT_rbt_lookup_a_b @ less @ T2 @ K ) ) ) ) ) ).

% local.rbt_lookup_from_in_tree
thf(fact_316_local_Orbt__lookup__rbt__insert,axiom,
    ! [T: rBT_rb3198946194961646575list_a,K: a,V: produc3854044251064639184list_a] :
      ( ( rBT_is6464524471971283763list_a @ less @ T )
     => ( ( rBT_rb5625365584452904567list_a @ less @ ( rBT_rb2370039884616530038list_a @ less @ K @ V @ T ) )
        = ( fun_up513072731191583893list_a @ ( rBT_rb5625365584452904567list_a @ less @ T ) @ K @ ( some_P5354654743593010357list_a @ V ) ) ) ) ).

% local.rbt_lookup_rbt_insert
thf(fact_317_local_Orbt__lookup__rbt__insert,axiom,
    ! [T: rBT_rb6886338251485743302list_a,K: a,V: produc9164743771328383783list_a] :
      ( ( rBT_is5094447046737250954list_a @ less @ T )
     => ( ( rBT_rb9209483341951570382list_a @ less @ ( rBT_rb5899567038564170445list_a @ less @ K @ V @ T ) )
        = ( fun_up2516339909567971180list_a @ ( rBT_rb9209483341951570382list_a @ less @ T ) @ K @ ( some_P5335617695965928972list_a @ V ) ) ) ) ).

% local.rbt_lookup_rbt_insert
thf(fact_318_local_Orbt__lookup__rbt__insert,axiom,
    ! [T: rBT_rb6027196121907741877list_a,K: a,V: produc2547433716817190550list_a] :
      ( ( rBT_is6679524918181081593list_a @ less @ T )
     => ( ( rBT_rb6302383274397067581list_a @ less @ ( rBT_rb1928131432470246972list_a @ less @ K @ V @ T ) )
        = ( fun_up7085013570333959003list_a @ ( rBT_rb6302383274397067581list_a @ less @ T ) @ K @ ( some_P3259385407967395195list_a @ V ) ) ) ) ).

% local.rbt_lookup_rbt_insert
thf(fact_319_local_Orbt__lookup__rbt__insert,axiom,
    ! [T: rBT_rbt_a_list_a,K: a,V: list_a] :
      ( ( rBT_is_rbt_a_list_a @ less @ T )
     => ( ( rBT_rb8775438530296890539list_a @ less @ ( rBT_rb3324229616542404650list_a @ less @ K @ V @ T ) )
        = ( fun_up4547247863559967817list_a @ ( rBT_rb8775438530296890539list_a @ less @ T ) @ K @ ( some_list_a @ V ) ) ) ) ).

% local.rbt_lookup_rbt_insert
thf(fact_320_local_Orbt__lookup__rbt__insert,axiom,
    ! [T: rBT_rb484615002580549772list_a,K: a,V: produc271001111946445293list_a] :
      ( ( rBT_is3736607230663913552list_a @ less @ T )
     => ( ( rBT_rb4780624521278262164list_a @ less @ ( rBT_rb4834844912256151443list_a @ less @ K @ V @ T ) )
        = ( fun_up8065330075003041074list_a @ ( rBT_rb4780624521278262164list_a @ less @ T ) @ K @ ( some_P1402039549733047762list_a @ V ) ) ) ) ).

% local.rbt_lookup_rbt_insert
thf(fact_321_local_Orbt__lookup__rbt__insert,axiom,
    ! [T: rBT_rbt_a_b,K: a,V: b] :
      ( ( rBT_is_rbt_a_b @ less @ T )
     => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_insert_a_b @ less @ K @ V @ T ) )
        = ( fun_upd_a_option_b @ ( rBT_rbt_lookup_a_b @ less @ T ) @ K @ ( some_b @ V ) ) ) ) ).

% local.rbt_lookup_rbt_insert
thf(fact_322_local_Orbt__lookup__Empty,axiom,
    ( ( rBT_rb5625365584452904567list_a @ less @ rBT_Em1305389575258925283list_a )
    = ( ^ [X2: a] : none_P5893993846586699057list_a ) ) ).

% local.rbt_lookup_Empty
thf(fact_323_local_Orbt__lookup__Empty,axiom,
    ( ( rBT_rb5455008113374026189od_a_b @ less @ rBT_Em8678960569159055417od_a_b )
    = ( ^ [X2: a] : none_P912253767061701255od_a_b ) ) ).

% local.rbt_lookup_Empty
thf(fact_324_local_Orbt__lookup__Empty,axiom,
    ( ( rBT_rb4780624521278262164list_a @ less @ rBT_Em8565251403037144576list_a )
    = ( ^ [X2: a] : none_P4075134247637483086list_a ) ) ).

% local.rbt_lookup_Empty
thf(fact_325_local_Orbt__lookup__Empty,axiom,
    ( ( rBT_rbt_lookup_a_b @ less @ rBT_Empty_a_b )
    = ( ^ [X2: a] : none_b ) ) ).

% local.rbt_lookup_Empty
thf(fact_326_local_Orbt__lookup_Osimps_I1_J,axiom,
    ! [K: a] :
      ( ( rBT_rb5625365584452904567list_a @ less @ rBT_Em1305389575258925283list_a @ K )
      = none_P5893993846586699057list_a ) ).

% local.rbt_lookup.simps(1)
thf(fact_327_local_Orbt__lookup_Osimps_I1_J,axiom,
    ! [K: a] :
      ( ( rBT_rb5455008113374026189od_a_b @ less @ rBT_Em8678960569159055417od_a_b @ K )
      = none_P912253767061701255od_a_b ) ).

% local.rbt_lookup.simps(1)
thf(fact_328_local_Orbt__lookup_Osimps_I1_J,axiom,
    ! [K: a] :
      ( ( rBT_rb4780624521278262164list_a @ less @ rBT_Em8565251403037144576list_a @ K )
      = none_P4075134247637483086list_a ) ).

% local.rbt_lookup.simps(1)
thf(fact_329_local_Orbt__lookup_Osimps_I1_J,axiom,
    ! [K: a] :
      ( ( rBT_rbt_lookup_a_b @ less @ rBT_Empty_a_b @ K )
      = none_b ) ).

% local.rbt_lookup.simps(1)
thf(fact_330_local_Orbt__lookup_Osimps_I2_J,axiom,
    ! [K: a,X: a,Uu: rBT_color,L: rBT_rb3198946194961646575list_a,Y: produc3854044251064639184list_a,R: rBT_rb3198946194961646575list_a] :
      ( ( ( less @ K @ X )
       => ( ( rBT_rb5625365584452904567list_a @ less @ ( rBT_Br137153042119060240list_a @ Uu @ L @ X @ Y @ R ) @ K )
          = ( rBT_rb5625365584452904567list_a @ less @ L @ K ) ) )
      & ( ~ ( less @ K @ X )
       => ( ( ( less @ X @ K )
           => ( ( rBT_rb5625365584452904567list_a @ less @ ( rBT_Br137153042119060240list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( rBT_rb5625365584452904567list_a @ less @ R @ K ) ) )
          & ( ~ ( less @ X @ K )
           => ( ( rBT_rb5625365584452904567list_a @ less @ ( rBT_Br137153042119060240list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( some_P5354654743593010357list_a @ Y ) ) ) ) ) ) ).

% local.rbt_lookup.simps(2)
thf(fact_331_local_Orbt__lookup_Osimps_I2_J,axiom,
    ! [K: a,X: a,Uu: rBT_color,L: rBT_rb6886338251485743302list_a,Y: produc9164743771328383783list_a,R: rBT_rb6886338251485743302list_a] :
      ( ( ( less @ K @ X )
       => ( ( rBT_rb9209483341951570382list_a @ less @ ( rBT_Br4059219879412995943list_a @ Uu @ L @ X @ Y @ R ) @ K )
          = ( rBT_rb9209483341951570382list_a @ less @ L @ K ) ) )
      & ( ~ ( less @ K @ X )
       => ( ( ( less @ X @ K )
           => ( ( rBT_rb9209483341951570382list_a @ less @ ( rBT_Br4059219879412995943list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( rBT_rb9209483341951570382list_a @ less @ R @ K ) ) )
          & ( ~ ( less @ X @ K )
           => ( ( rBT_rb9209483341951570382list_a @ less @ ( rBT_Br4059219879412995943list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( some_P5335617695965928972list_a @ Y ) ) ) ) ) ) ).

% local.rbt_lookup.simps(2)
thf(fact_332_local_Orbt__lookup_Osimps_I2_J,axiom,
    ! [K: a,X: a,Uu: rBT_color,L: rBT_rb6027196121907741877list_a,Y: produc2547433716817190550list_a,R: rBT_rb6027196121907741877list_a] :
      ( ( ( less @ K @ X )
       => ( ( rBT_rb6302383274397067581list_a @ less @ ( rBT_Br5621580029570968278list_a @ Uu @ L @ X @ Y @ R ) @ K )
          = ( rBT_rb6302383274397067581list_a @ less @ L @ K ) ) )
      & ( ~ ( less @ K @ X )
       => ( ( ( less @ X @ K )
           => ( ( rBT_rb6302383274397067581list_a @ less @ ( rBT_Br5621580029570968278list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( rBT_rb6302383274397067581list_a @ less @ R @ K ) ) )
          & ( ~ ( less @ X @ K )
           => ( ( rBT_rb6302383274397067581list_a @ less @ ( rBT_Br5621580029570968278list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( some_P3259385407967395195list_a @ Y ) ) ) ) ) ) ).

% local.rbt_lookup.simps(2)
thf(fact_333_local_Orbt__lookup_Osimps_I2_J,axiom,
    ! [K: a,X: a,Uu: rBT_color,L: rBT_rbt_a_list_a,Y: list_a,R: rBT_rbt_a_list_a] :
      ( ( ( less @ K @ X )
       => ( ( rBT_rb8775438530296890539list_a @ less @ ( rBT_Branch_a_list_a @ Uu @ L @ X @ Y @ R ) @ K )
          = ( rBT_rb8775438530296890539list_a @ less @ L @ K ) ) )
      & ( ~ ( less @ K @ X )
       => ( ( ( less @ X @ K )
           => ( ( rBT_rb8775438530296890539list_a @ less @ ( rBT_Branch_a_list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( rBT_rb8775438530296890539list_a @ less @ R @ K ) ) )
          & ( ~ ( less @ X @ K )
           => ( ( rBT_rb8775438530296890539list_a @ less @ ( rBT_Branch_a_list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( some_list_a @ Y ) ) ) ) ) ) ).

% local.rbt_lookup.simps(2)
thf(fact_334_local_Orbt__lookup_Osimps_I2_J,axiom,
    ! [K: a,X: a,Uu: rBT_color,L: rBT_rb484615002580549772list_a,Y: produc271001111946445293list_a,R: rBT_rb484615002580549772list_a] :
      ( ( ( less @ K @ X )
       => ( ( rBT_rb4780624521278262164list_a @ less @ ( rBT_Br706236987277315629list_a @ Uu @ L @ X @ Y @ R ) @ K )
          = ( rBT_rb4780624521278262164list_a @ less @ L @ K ) ) )
      & ( ~ ( less @ K @ X )
       => ( ( ( less @ X @ K )
           => ( ( rBT_rb4780624521278262164list_a @ less @ ( rBT_Br706236987277315629list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( rBT_rb4780624521278262164list_a @ less @ R @ K ) ) )
          & ( ~ ( less @ X @ K )
           => ( ( rBT_rb4780624521278262164list_a @ less @ ( rBT_Br706236987277315629list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( some_P1402039549733047762list_a @ Y ) ) ) ) ) ) ).

% local.rbt_lookup.simps(2)
thf(fact_335_local_Orbt__lookup_Osimps_I2_J,axiom,
    ! [K: a,X: a,Uu: rBT_color,L: rBT_rbt_a_b,Y: b,R: rBT_rbt_a_b] :
      ( ( ( less @ K @ X )
       => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_Branch_a_b @ Uu @ L @ X @ Y @ R ) @ K )
          = ( rBT_rbt_lookup_a_b @ less @ L @ K ) ) )
      & ( ~ ( less @ K @ X )
       => ( ( ( less @ X @ K )
           => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_Branch_a_b @ Uu @ L @ X @ Y @ R ) @ K )
              = ( rBT_rbt_lookup_a_b @ less @ R @ K ) ) )
          & ( ~ ( less @ X @ K )
           => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_Branch_a_b @ Uu @ L @ X @ Y @ R ) @ K )
              = ( some_b @ Y ) ) ) ) ) ) ).

% local.rbt_lookup.simps(2)
thf(fact_336_local_Orbt__sorted__entries__right__unique,axiom,
    ! [K: a,V: produc3797860095223585356bt_a_b,T: rBT_rb6525319661942305155bt_a_b,V3: produc3797860095223585356bt_a_b] :
      ( ( member6846102102718116240bt_a_b @ ( produc3015636708179276019bt_a_b @ K @ V ) @ ( set_Pr1558196515228517412bt_a_b @ ( rBT_en699320873491845330bt_a_b @ T ) ) )
     => ( ( member6846102102718116240bt_a_b @ ( produc3015636708179276019bt_a_b @ K @ V3 ) @ ( set_Pr1558196515228517412bt_a_b @ ( rBT_en699320873491845330bt_a_b @ T ) ) )
       => ( ( rBT_rb4191689393857777698bt_a_b @ less @ T )
         => ( V = V3 ) ) ) ) ).

% local.rbt_sorted_entries_right_unique
thf(fact_337_local_Orbt__sorted__entries__right__unique,axiom,
    ! [K: a,V: list_P5792129111969709340list_a,T: rBT_rb3549771087195502139list_a,V3: list_P5792129111969709340list_a] :
      ( ( member2402001272945804526list_a @ ( produc4159846287338058421list_a @ K @ V ) @ ( set_Pr3960553176861824346list_a @ ( rBT_en7750752156332278038list_a @ T ) ) )
     => ( ( member2402001272945804526list_a @ ( produc4159846287338058421list_a @ K @ V3 ) @ ( set_Pr3960553176861824346list_a @ ( rBT_en7750752156332278038list_a @ T ) ) )
       => ( ( rBT_rb1805918226279206342list_a @ less @ T )
         => ( V = V3 ) ) ) ) ).

% local.rbt_sorted_entries_right_unique
thf(fact_338_local_Orbt__sorted__entries__right__unique,axiom,
    ! [K: a,V: list_P1467974523212632366od_a_b,T: rBT_rb4976296014333920589od_a_b,V3: list_P1467974523212632366od_a_b] :
      ( ( member8745141763671436032od_a_b @ ( produc6643264353547226055od_a_b @ K @ V ) @ ( set_Pr187010687427835756od_a_b @ ( rBT_en1619617535027037992od_a_b @ T ) ) )
     => ( ( member8745141763671436032od_a_b @ ( produc6643264353547226055od_a_b @ K @ V3 ) @ ( set_Pr187010687427835756od_a_b @ ( rBT_en1619617535027037992od_a_b @ T ) ) )
       => ( ( rBT_rb3998897555662963672od_a_b @ less @ T )
         => ( V = V3 ) ) ) ) ).

% local.rbt_sorted_entries_right_unique
thf(fact_339_local_Orbt__sorted__entries__right__unique,axiom,
    ! [K: a,V: list_a,T: rBT_rbt_a_list_a,V3: list_a] :
      ( ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ K @ V ) @ ( set_Pr3039917418810828226list_a @ ( rBT_entries_a_list_a @ T ) ) )
     => ( ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ K @ V3 ) @ ( set_Pr3039917418810828226list_a @ ( rBT_entries_a_list_a @ T ) ) )
       => ( ( rBT_rb7395583532625325102list_a @ less @ T )
         => ( V = V3 ) ) ) ) ).

% local.rbt_sorted_entries_right_unique
thf(fact_340_local_Orbt__sorted__entries__right__unique,axiom,
    ! [K: a,V: produc271001111946445293list_a,T: rBT_rb484615002580549772list_a,V3: produc271001111946445293list_a] :
      ( ( member852939521359900607list_a @ ( produc4562151247771834630list_a @ K @ V ) @ ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ T ) ) )
     => ( ( member852939521359900607list_a @ ( produc4562151247771834630list_a @ K @ V3 ) @ ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ T ) ) )
       => ( ( rBT_rb6864293667938249111list_a @ less @ T )
         => ( V = V3 ) ) ) ) ).

% local.rbt_sorted_entries_right_unique
thf(fact_341_local_Orbt__sorted__entries__right__unique,axiom,
    ! [K: a,V: b,T: rBT_rbt_a_b,V3: b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K @ V ) @ ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ T ) ) )
     => ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K @ V3 ) @ ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ T ) ) )
       => ( ( rBT_rbt_sorted_a_b @ less @ T )
         => ( V = V3 ) ) ) ) ).

% local.rbt_sorted_entries_right_unique
thf(fact_342_local_Orbt__del_Ocases,axiom,
    ! [X: produc4880734740204102731bt_a_b] :
      ( ! [X3: a] :
          ( X
         != ( produc324221335107646021bt_a_b @ X3 @ rBT_Empty_a_b ) )
     => ~ ! [X3: a,C3: rBT_color,A2: rBT_rbt_a_b,Y2: a,S: b,B2: rBT_rbt_a_b] :
            ( X
           != ( produc324221335107646021bt_a_b @ X3 @ ( rBT_Branch_a_b @ C3 @ A2 @ Y2 @ S @ B2 ) ) ) ) ).

% local.rbt_del.cases
thf(fact_343_local_Orbt__del_Ocases,axiom,
    ! [X: produc7648788782784911161list_a] :
      ( ! [X3: a] :
          ( X
         != ( produc2705466823935335219list_a @ X3 @ rBT_Em8565251403037144576list_a ) )
     => ~ ! [X3: a,C3: rBT_color,A2: rBT_rb484615002580549772list_a,Y2: a,S: produc271001111946445293list_a,B2: rBT_rb484615002580549772list_a] :
            ( X
           != ( produc2705466823935335219list_a @ X3 @ ( rBT_Br706236987277315629list_a @ C3 @ A2 @ Y2 @ S @ B2 ) ) ) ) ).

% local.rbt_del.cases
thf(fact_344_local_Orbt__split_Ocases,axiom,
    ! [X: produc727794225953898581_a_b_a] :
      ( ! [K2: a] :
          ( X
         != ( produc2410990786219400071_a_b_a @ rBT_Empty_a_b @ K2 ) )
     => ~ ! [Uu2: rBT_color,L2: rBT_rbt_a_b,A2: a,B2: b,R2: rBT_rbt_a_b,X3: a] :
            ( X
           != ( produc2410990786219400071_a_b_a @ ( rBT_Branch_a_b @ Uu2 @ L2 @ A2 @ B2 @ R2 ) @ X3 ) ) ) ).

% local.rbt_split.cases
thf(fact_345_local_Orbt__split_Ocases,axiom,
    ! [X: produc4172418568614467815st_a_a] :
      ( ! [K2: a] :
          ( X
         != ( produc5537483632639485721st_a_a @ rBT_Em8565251403037144576list_a @ K2 ) )
     => ~ ! [Uu2: rBT_color,L2: rBT_rb484615002580549772list_a,A2: a,B2: produc271001111946445293list_a,R2: rBT_rb484615002580549772list_a,X3: a] :
            ( X
           != ( produc5537483632639485721st_a_a @ ( rBT_Br706236987277315629list_a @ Uu2 @ L2 @ A2 @ B2 @ R2 ) @ X3 ) ) ) ).

% local.rbt_split.cases
thf(fact_346_local_Orbt__sorted_Osimps_I1_J,axiom,
    rBT_rbt_sorted_a_b @ less @ rBT_Empty_a_b ).

% local.rbt_sorted.simps(1)
thf(fact_347_local_Orbt__sorted_Osimps_I1_J,axiom,
    rBT_rb6864293667938249111list_a @ less @ rBT_Em8565251403037144576list_a ).

% local.rbt_sorted.simps(1)
thf(fact_348_local_Ois__rbt__rbt__sorted,axiom,
    ! [T: rBT_rb484615002580549772list_a] :
      ( ( rBT_is3736607230663913552list_a @ less @ T )
     => ( rBT_rb6864293667938249111list_a @ less @ T ) ) ).

% local.is_rbt_rbt_sorted
thf(fact_349_local_Ois__rbt__rbt__sorted,axiom,
    ! [T: rBT_rbt_a_b] :
      ( ( rBT_is_rbt_a_b @ less @ T )
     => ( rBT_rbt_sorted_a_b @ less @ T ) ) ).

% local.is_rbt_rbt_sorted
thf(fact_350_local_OEmpty__is__rbt,axiom,
    rBT_is3736607230663913552list_a @ less @ rBT_Em8565251403037144576list_a ).

% local.Empty_is_rbt
thf(fact_351_local_OEmpty__is__rbt,axiom,
    rBT_is_rbt_a_b @ less @ rBT_Empty_a_b ).

% local.Empty_is_rbt
thf(fact_352_local_Orbt__insert__rbt__sorted,axiom,
    ! [T: rBT_rb484615002580549772list_a,K: a,V: produc271001111946445293list_a] :
      ( ( rBT_rb6864293667938249111list_a @ less @ T )
     => ( rBT_rb6864293667938249111list_a @ less @ ( rBT_rb4834844912256151443list_a @ less @ K @ V @ T ) ) ) ).

% local.rbt_insert_rbt_sorted
thf(fact_353_local_Orbt__insert__rbt__sorted,axiom,
    ! [T: rBT_rbt_a_b,K: a,V: b] :
      ( ( rBT_rbt_sorted_a_b @ less @ T )
     => ( rBT_rbt_sorted_a_b @ less @ ( rBT_rbt_insert_a_b @ less @ K @ V @ T ) ) ) ).

% local.rbt_insert_rbt_sorted
thf(fact_354_local_Orbt__sorted__delete,axiom,
    ! [T: rBT_rbt_a_b,X: a] :
      ( ( rBT_rbt_sorted_a_b @ less @ T )
     => ( rBT_rbt_sorted_a_b @ less @ ( rBT_rbt_delete_a_b @ less @ X @ T ) ) ) ).

% local.rbt_sorted_delete
thf(fact_355_local_Orbt__sorted__delete,axiom,
    ! [T: rBT_rb484615002580549772list_a,X: a] :
      ( ( rBT_rb6864293667938249111list_a @ less @ T )
     => ( rBT_rb6864293667938249111list_a @ less @ ( rBT_rb9212334439554585221list_a @ less @ X @ T ) ) ) ).

% local.rbt_sorted_delete
thf(fact_356_local_Orbt__insertw__rbt__sorted,axiom,
    ! [T: rBT_rb484615002580549772list_a,F: produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a,K: a,V: produc271001111946445293list_a] :
      ( ( rBT_rb6864293667938249111list_a @ less @ T )
     => ( rBT_rb6864293667938249111list_a @ less @ ( rBT_rb7563067580523141625list_a @ less @ F @ K @ V @ T ) ) ) ).

% local.rbt_insertw_rbt_sorted
thf(fact_357_local_Orbt__insertw__rbt__sorted,axiom,
    ! [T: rBT_rbt_a_b,F: b > b > b,K: a,V: b] :
      ( ( rBT_rbt_sorted_a_b @ less @ T )
     => ( rBT_rbt_sorted_a_b @ less @ ( rBT_rb5519417620787648267th_a_b @ less @ F @ K @ V @ T ) ) ) ).

% local.rbt_insertw_rbt_sorted
thf(fact_358_option_Oinject,axiom,
    ! [X22: produc271001111946445293list_a,Y22: produc271001111946445293list_a] :
      ( ( ( some_P1402039549733047762list_a @ X22 )
        = ( some_P1402039549733047762list_a @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_359_option_Oinject,axiom,
    ! [X22: produc3854044251064639184list_a,Y22: produc3854044251064639184list_a] :
      ( ( ( some_P5354654743593010357list_a @ X22 )
        = ( some_P5354654743593010357list_a @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_360_option_Oinject,axiom,
    ! [X22: produc9164743771328383783list_a,Y22: produc9164743771328383783list_a] :
      ( ( ( some_P5335617695965928972list_a @ X22 )
        = ( some_P5335617695965928972list_a @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_361_option_Oinject,axiom,
    ! [X22: produc2547433716817190550list_a,Y22: produc2547433716817190550list_a] :
      ( ( ( some_P3259385407967395195list_a @ X22 )
        = ( some_P3259385407967395195list_a @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_362_option_Oinject,axiom,
    ! [X22: list_a,Y22: list_a] :
      ( ( ( some_list_a @ X22 )
        = ( some_list_a @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_363_option_Oinject,axiom,
    ! [X22: b,Y22: b] :
      ( ( ( some_b @ X22 )
        = ( some_b @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_364_local_Oentries__eqI,axiom,
    ! [T1: rBT_rb484615002580549772list_a,T2: rBT_rb484615002580549772list_a] :
      ( ( rBT_rb6864293667938249111list_a @ less @ T1 )
     => ( ( rBT_rb6864293667938249111list_a @ less @ T2 )
       => ( ( ( rBT_rb4780624521278262164list_a @ less @ T1 )
            = ( rBT_rb4780624521278262164list_a @ less @ T2 ) )
         => ( ( rBT_en3875316870356833511list_a @ T1 )
            = ( rBT_en3875316870356833511list_a @ T2 ) ) ) ) ) ).

% local.entries_eqI
thf(fact_365_local_Oentries__eqI,axiom,
    ! [T1: rBT_rbt_a_b,T2: rBT_rbt_a_b] :
      ( ( rBT_rbt_sorted_a_b @ less @ T1 )
     => ( ( rBT_rbt_sorted_a_b @ less @ T2 )
       => ( ( ( rBT_rbt_lookup_a_b @ less @ T1 )
            = ( rBT_rbt_lookup_a_b @ less @ T2 ) )
         => ( ( rBT_entries_a_b @ T1 )
            = ( rBT_entries_a_b @ T2 ) ) ) ) ) ).

% local.entries_eqI
thf(fact_366_local_Oentries__rbt__lookup,axiom,
    ! [T1: rBT_rb484615002580549772list_a,T2: rBT_rb484615002580549772list_a] :
      ( ( rBT_rb6864293667938249111list_a @ less @ T1 )
     => ( ( rBT_rb6864293667938249111list_a @ less @ T2 )
       => ( ( ( rBT_en3875316870356833511list_a @ T1 )
            = ( rBT_en3875316870356833511list_a @ T2 ) )
          = ( ( rBT_rb4780624521278262164list_a @ less @ T1 )
            = ( rBT_rb4780624521278262164list_a @ less @ T2 ) ) ) ) ) ).

% local.entries_rbt_lookup
thf(fact_367_local_Oentries__rbt__lookup,axiom,
    ! [T1: rBT_rbt_a_b,T2: rBT_rbt_a_b] :
      ( ( rBT_rbt_sorted_a_b @ less @ T1 )
     => ( ( rBT_rbt_sorted_a_b @ less @ T2 )
       => ( ( ( rBT_entries_a_b @ T1 )
            = ( rBT_entries_a_b @ T2 ) )
          = ( ( rBT_rbt_lookup_a_b @ less @ T1 )
            = ( rBT_rbt_lookup_a_b @ less @ T2 ) ) ) ) ) ).

% local.entries_rbt_lookup
thf(fact_368_local_Oset__entries__inject,axiom,
    ! [T1: rBT_rb484615002580549772list_a,T2: rBT_rb484615002580549772list_a] :
      ( ( rBT_rb6864293667938249111list_a @ less @ T1 )
     => ( ( rBT_rb6864293667938249111list_a @ less @ T2 )
       => ( ( ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ T1 ) )
            = ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ T2 ) ) )
          = ( ( rBT_en3875316870356833511list_a @ T1 )
            = ( rBT_en3875316870356833511list_a @ T2 ) ) ) ) ) ).

% local.set_entries_inject
thf(fact_369_local_Oset__entries__inject,axiom,
    ! [T1: rBT_rbt_a_b,T2: rBT_rbt_a_b] :
      ( ( rBT_rbt_sorted_a_b @ less @ T1 )
     => ( ( rBT_rbt_sorted_a_b @ less @ T2 )
       => ( ( ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ T1 ) )
            = ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ T2 ) ) )
          = ( ( rBT_entries_a_b @ T1 )
            = ( rBT_entries_a_b @ T2 ) ) ) ) ) ).

% local.set_entries_inject
thf(fact_370_local_Orbt__lookup__rbt__insert_H,axiom,
    ! [T: rBT_rb3198946194961646575list_a,K: a,V: produc3854044251064639184list_a] :
      ( ( rBT_rb6007450547276413562list_a @ less @ T )
     => ( ( rBT_rb5625365584452904567list_a @ less @ ( rBT_rb2370039884616530038list_a @ less @ K @ V @ T ) )
        = ( fun_up513072731191583893list_a @ ( rBT_rb5625365584452904567list_a @ less @ T ) @ K @ ( some_P5354654743593010357list_a @ V ) ) ) ) ).

% local.rbt_lookup_rbt_insert'
thf(fact_371_local_Orbt__lookup__rbt__insert_H,axiom,
    ! [T: rBT_rb6886338251485743302list_a,K: a,V: produc9164743771328383783list_a] :
      ( ( rBT_rb3676718616835825873list_a @ less @ T )
     => ( ( rBT_rb9209483341951570382list_a @ less @ ( rBT_rb5899567038564170445list_a @ less @ K @ V @ T ) )
        = ( fun_up2516339909567971180list_a @ ( rBT_rb9209483341951570382list_a @ less @ T ) @ K @ ( some_P5335617695965928972list_a @ V ) ) ) ) ).

% local.rbt_lookup_rbt_insert'
thf(fact_372_local_Orbt__lookup__rbt__insert_H,axiom,
    ! [T: rBT_rb6027196121907741877list_a,K: a,V: produc2547433716817190550list_a] :
      ( ( rBT_rb3455411056480160832list_a @ less @ T )
     => ( ( rBT_rb6302383274397067581list_a @ less @ ( rBT_rb1928131432470246972list_a @ less @ K @ V @ T ) )
        = ( fun_up7085013570333959003list_a @ ( rBT_rb6302383274397067581list_a @ less @ T ) @ K @ ( some_P3259385407967395195list_a @ V ) ) ) ) ).

% local.rbt_lookup_rbt_insert'
thf(fact_373_local_Orbt__lookup__rbt__insert_H,axiom,
    ! [T: rBT_rbt_a_list_a,K: a,V: list_a] :
      ( ( rBT_rb7395583532625325102list_a @ less @ T )
     => ( ( rBT_rb8775438530296890539list_a @ less @ ( rBT_rb3324229616542404650list_a @ less @ K @ V @ T ) )
        = ( fun_up4547247863559967817list_a @ ( rBT_rb8775438530296890539list_a @ less @ T ) @ K @ ( some_list_a @ V ) ) ) ) ).

% local.rbt_lookup_rbt_insert'
thf(fact_374_local_Orbt__lookup__rbt__insert_H,axiom,
    ! [T: rBT_rb484615002580549772list_a,K: a,V: produc271001111946445293list_a] :
      ( ( rBT_rb6864293667938249111list_a @ less @ T )
     => ( ( rBT_rb4780624521278262164list_a @ less @ ( rBT_rb4834844912256151443list_a @ less @ K @ V @ T ) )
        = ( fun_up8065330075003041074list_a @ ( rBT_rb4780624521278262164list_a @ less @ T ) @ K @ ( some_P1402039549733047762list_a @ V ) ) ) ) ).

% local.rbt_lookup_rbt_insert'
thf(fact_375_local_Orbt__lookup__rbt__insert_H,axiom,
    ! [T: rBT_rbt_a_b,K: a,V: b] :
      ( ( rBT_rbt_sorted_a_b @ less @ T )
     => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_insert_a_b @ less @ K @ V @ T ) )
        = ( fun_upd_a_option_b @ ( rBT_rbt_lookup_a_b @ less @ T ) @ K @ ( some_b @ V ) ) ) ) ).

% local.rbt_lookup_rbt_insert'
thf(fact_376_option_Odistinct_I1_J,axiom,
    ! [X22: produc1700058555794149158od_a_b] :
      ( none_P912253767061701255od_a_b
     != ( some_P2412010246944353803od_a_b @ X22 ) ) ).

% option.distinct(1)
thf(fact_377_option_Odistinct_I1_J,axiom,
    ! [X22: produc271001111946445293list_a] :
      ( none_P4075134247637483086list_a
     != ( some_P1402039549733047762list_a @ X22 ) ) ).

% option.distinct(1)
thf(fact_378_option_Odistinct_I1_J,axiom,
    ! [X22: produc3854044251064639184list_a] :
      ( none_P5893993846586699057list_a
     != ( some_P5354654743593010357list_a @ X22 ) ) ).

% option.distinct(1)
thf(fact_379_option_Odistinct_I1_J,axiom,
    ! [X22: produc9164743771328383783list_a] :
      ( none_P3970268120610169480list_a
     != ( some_P5335617695965928972list_a @ X22 ) ) ).

% option.distinct(1)
thf(fact_380_option_Odistinct_I1_J,axiom,
    ! [X22: produc2547433716817190550list_a] :
      ( none_P1759628928084742647list_a
     != ( some_P3259385407967395195list_a @ X22 ) ) ).

% option.distinct(1)
thf(fact_381_option_Odistinct_I1_J,axiom,
    ! [X22: list_a] :
      ( none_list_a
     != ( some_list_a @ X22 ) ) ).

% option.distinct(1)
thf(fact_382_option_Odistinct_I1_J,axiom,
    ! [X22: b] :
      ( none_b
     != ( some_b @ X22 ) ) ).

% option.distinct(1)
thf(fact_383_option_OdiscI,axiom,
    ! [Option: option2417831327529529196od_a_b,X22: produc1700058555794149158od_a_b] :
      ( ( Option
        = ( some_P2412010246944353803od_a_b @ X22 ) )
     => ( Option != none_P912253767061701255od_a_b ) ) ).

% option.discI
thf(fact_384_option_OdiscI,axiom,
    ! [Option: option110588449099563443list_a,X22: produc271001111946445293list_a] :
      ( ( Option
        = ( some_P1402039549733047762list_a @ X22 ) )
     => ( Option != none_P4075134247637483086list_a ) ) ).

% option.discI
thf(fact_385_option_OdiscI,axiom,
    ! [Option: option788170273582809878list_a,X22: produc3854044251064639184list_a] :
      ( ( Option
        = ( some_P5354654743593010357list_a @ X22 ) )
     => ( Option != none_P5893993846586699057list_a ) ) ).

% option.discI
thf(fact_386_option_OdiscI,axiom,
    ! [Option: option4277935495599621869list_a,X22: produc9164743771328383783list_a] :
      ( ( Option
        = ( some_P5335617695965928972list_a @ X22 ) )
     => ( Option != none_P3970268120610169480list_a ) ) ).

% option.discI
thf(fact_387_option_OdiscI,axiom,
    ! [Option: option4732895764270368476list_a,X22: produc2547433716817190550list_a] :
      ( ( Option
        = ( some_P3259385407967395195list_a @ X22 ) )
     => ( Option != none_P1759628928084742647list_a ) ) ).

% option.discI
thf(fact_388_option_OdiscI,axiom,
    ! [Option: option_list_a,X22: list_a] :
      ( ( Option
        = ( some_list_a @ X22 ) )
     => ( Option != none_list_a ) ) ).

% option.discI
thf(fact_389_option_OdiscI,axiom,
    ! [Option: option_b,X22: b] :
      ( ( Option
        = ( some_b @ X22 ) )
     => ( Option != none_b ) ) ).

% option.discI
thf(fact_390_option_Oexhaust,axiom,
    ! [Y: option2417831327529529196od_a_b] :
      ( ( Y != none_P912253767061701255od_a_b )
     => ~ ! [X23: produc1700058555794149158od_a_b] :
            ( Y
           != ( some_P2412010246944353803od_a_b @ X23 ) ) ) ).

% option.exhaust
thf(fact_391_option_Oexhaust,axiom,
    ! [Y: option110588449099563443list_a] :
      ( ( Y != none_P4075134247637483086list_a )
     => ~ ! [X23: produc271001111946445293list_a] :
            ( Y
           != ( some_P1402039549733047762list_a @ X23 ) ) ) ).

% option.exhaust
thf(fact_392_option_Oexhaust,axiom,
    ! [Y: option788170273582809878list_a] :
      ( ( Y != none_P5893993846586699057list_a )
     => ~ ! [X23: produc3854044251064639184list_a] :
            ( Y
           != ( some_P5354654743593010357list_a @ X23 ) ) ) ).

% option.exhaust
thf(fact_393_option_Oexhaust,axiom,
    ! [Y: option4277935495599621869list_a] :
      ( ( Y != none_P3970268120610169480list_a )
     => ~ ! [X23: produc9164743771328383783list_a] :
            ( Y
           != ( some_P5335617695965928972list_a @ X23 ) ) ) ).

% option.exhaust
thf(fact_394_option_Oexhaust,axiom,
    ! [Y: option4732895764270368476list_a] :
      ( ( Y != none_P1759628928084742647list_a )
     => ~ ! [X23: produc2547433716817190550list_a] :
            ( Y
           != ( some_P3259385407967395195list_a @ X23 ) ) ) ).

% option.exhaust
thf(fact_395_option_Oexhaust,axiom,
    ! [Y: option_list_a] :
      ( ( Y != none_list_a )
     => ~ ! [X23: list_a] :
            ( Y
           != ( some_list_a @ X23 ) ) ) ).

% option.exhaust
thf(fact_396_option_Oexhaust,axiom,
    ! [Y: option_b] :
      ( ( Y != none_b )
     => ~ ! [X23: b] :
            ( Y
           != ( some_b @ X23 ) ) ) ).

% option.exhaust
thf(fact_397_split__option__ex,axiom,
    ( ( ^ [P2: option2417831327529529196od_a_b > $o] :
        ? [X4: option2417831327529529196od_a_b] : ( P2 @ X4 ) )
    = ( ^ [P3: option2417831327529529196od_a_b > $o] :
          ( ( P3 @ none_P912253767061701255od_a_b )
          | ? [X2: produc1700058555794149158od_a_b] : ( P3 @ ( some_P2412010246944353803od_a_b @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_398_split__option__ex,axiom,
    ( ( ^ [P2: option110588449099563443list_a > $o] :
        ? [X4: option110588449099563443list_a] : ( P2 @ X4 ) )
    = ( ^ [P3: option110588449099563443list_a > $o] :
          ( ( P3 @ none_P4075134247637483086list_a )
          | ? [X2: produc271001111946445293list_a] : ( P3 @ ( some_P1402039549733047762list_a @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_399_split__option__ex,axiom,
    ( ( ^ [P2: option788170273582809878list_a > $o] :
        ? [X4: option788170273582809878list_a] : ( P2 @ X4 ) )
    = ( ^ [P3: option788170273582809878list_a > $o] :
          ( ( P3 @ none_P5893993846586699057list_a )
          | ? [X2: produc3854044251064639184list_a] : ( P3 @ ( some_P5354654743593010357list_a @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_400_split__option__ex,axiom,
    ( ( ^ [P2: option4277935495599621869list_a > $o] :
        ? [X4: option4277935495599621869list_a] : ( P2 @ X4 ) )
    = ( ^ [P3: option4277935495599621869list_a > $o] :
          ( ( P3 @ none_P3970268120610169480list_a )
          | ? [X2: produc9164743771328383783list_a] : ( P3 @ ( some_P5335617695965928972list_a @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_401_split__option__ex,axiom,
    ( ( ^ [P2: option4732895764270368476list_a > $o] :
        ? [X4: option4732895764270368476list_a] : ( P2 @ X4 ) )
    = ( ^ [P3: option4732895764270368476list_a > $o] :
          ( ( P3 @ none_P1759628928084742647list_a )
          | ? [X2: produc2547433716817190550list_a] : ( P3 @ ( some_P3259385407967395195list_a @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_402_split__option__ex,axiom,
    ( ( ^ [P2: option_list_a > $o] :
        ? [X4: option_list_a] : ( P2 @ X4 ) )
    = ( ^ [P3: option_list_a > $o] :
          ( ( P3 @ none_list_a )
          | ? [X2: list_a] : ( P3 @ ( some_list_a @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_403_split__option__ex,axiom,
    ( ( ^ [P2: option_b > $o] :
        ? [X4: option_b] : ( P2 @ X4 ) )
    = ( ^ [P3: option_b > $o] :
          ( ( P3 @ none_b )
          | ? [X2: b] : ( P3 @ ( some_b @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_404_split__option__all,axiom,
    ( ( ^ [P2: option2417831327529529196od_a_b > $o] :
        ! [X4: option2417831327529529196od_a_b] : ( P2 @ X4 ) )
    = ( ^ [P3: option2417831327529529196od_a_b > $o] :
          ( ( P3 @ none_P912253767061701255od_a_b )
          & ! [X2: produc1700058555794149158od_a_b] : ( P3 @ ( some_P2412010246944353803od_a_b @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_405_split__option__all,axiom,
    ( ( ^ [P2: option110588449099563443list_a > $o] :
        ! [X4: option110588449099563443list_a] : ( P2 @ X4 ) )
    = ( ^ [P3: option110588449099563443list_a > $o] :
          ( ( P3 @ none_P4075134247637483086list_a )
          & ! [X2: produc271001111946445293list_a] : ( P3 @ ( some_P1402039549733047762list_a @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_406_split__option__all,axiom,
    ( ( ^ [P2: option788170273582809878list_a > $o] :
        ! [X4: option788170273582809878list_a] : ( P2 @ X4 ) )
    = ( ^ [P3: option788170273582809878list_a > $o] :
          ( ( P3 @ none_P5893993846586699057list_a )
          & ! [X2: produc3854044251064639184list_a] : ( P3 @ ( some_P5354654743593010357list_a @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_407_split__option__all,axiom,
    ( ( ^ [P2: option4277935495599621869list_a > $o] :
        ! [X4: option4277935495599621869list_a] : ( P2 @ X4 ) )
    = ( ^ [P3: option4277935495599621869list_a > $o] :
          ( ( P3 @ none_P3970268120610169480list_a )
          & ! [X2: produc9164743771328383783list_a] : ( P3 @ ( some_P5335617695965928972list_a @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_408_split__option__all,axiom,
    ( ( ^ [P2: option4732895764270368476list_a > $o] :
        ! [X4: option4732895764270368476list_a] : ( P2 @ X4 ) )
    = ( ^ [P3: option4732895764270368476list_a > $o] :
          ( ( P3 @ none_P1759628928084742647list_a )
          & ! [X2: produc2547433716817190550list_a] : ( P3 @ ( some_P3259385407967395195list_a @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_409_split__option__all,axiom,
    ( ( ^ [P2: option_list_a > $o] :
        ! [X4: option_list_a] : ( P2 @ X4 ) )
    = ( ^ [P3: option_list_a > $o] :
          ( ( P3 @ none_list_a )
          & ! [X2: list_a] : ( P3 @ ( some_list_a @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_410_split__option__all,axiom,
    ( ( ^ [P2: option_b > $o] :
        ! [X4: option_b] : ( P2 @ X4 ) )
    = ( ^ [P3: option_b > $o] :
          ( ( P3 @ none_b )
          & ! [X2: b] : ( P3 @ ( some_b @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_411_combine__options__cases,axiom,
    ! [X: option_b,P: option_b > option_b > $o,Y: option_b] :
      ( ( ( X = none_b )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_b )
         => ( P @ X @ Y ) )
       => ( ! [A2: b,B2: b] :
              ( ( X
                = ( some_b @ A2 ) )
             => ( ( Y
                  = ( some_b @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_412_combine__options__cases,axiom,
    ! [X: option_b,P: option_b > option_list_a > $o,Y: option_list_a] :
      ( ( ( X = none_b )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_list_a )
         => ( P @ X @ Y ) )
       => ( ! [A2: b,B2: list_a] :
              ( ( X
                = ( some_b @ A2 ) )
             => ( ( Y
                  = ( some_list_a @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_413_combine__options__cases,axiom,
    ! [X: option_list_a,P: option_list_a > option_b > $o,Y: option_b] :
      ( ( ( X = none_list_a )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_b )
         => ( P @ X @ Y ) )
       => ( ! [A2: list_a,B2: b] :
              ( ( X
                = ( some_list_a @ A2 ) )
             => ( ( Y
                  = ( some_b @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_414_combine__options__cases,axiom,
    ! [X: option_list_a,P: option_list_a > option_list_a > $o,Y: option_list_a] :
      ( ( ( X = none_list_a )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_list_a )
         => ( P @ X @ Y ) )
       => ( ! [A2: list_a,B2: list_a] :
              ( ( X
                = ( some_list_a @ A2 ) )
             => ( ( Y
                  = ( some_list_a @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_415_combine__options__cases,axiom,
    ! [X: option_b,P: option_b > option4277935495599621869list_a > $o,Y: option4277935495599621869list_a] :
      ( ( ( X = none_b )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P3970268120610169480list_a )
         => ( P @ X @ Y ) )
       => ( ! [A2: b,B2: produc9164743771328383783list_a] :
              ( ( X
                = ( some_b @ A2 ) )
             => ( ( Y
                  = ( some_P5335617695965928972list_a @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_416_combine__options__cases,axiom,
    ! [X: option4277935495599621869list_a,P: option4277935495599621869list_a > option_b > $o,Y: option_b] :
      ( ( ( X = none_P3970268120610169480list_a )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_b )
         => ( P @ X @ Y ) )
       => ( ! [A2: produc9164743771328383783list_a,B2: b] :
              ( ( X
                = ( some_P5335617695965928972list_a @ A2 ) )
             => ( ( Y
                  = ( some_b @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_417_combine__options__cases,axiom,
    ! [X: option4277935495599621869list_a,P: option4277935495599621869list_a > option_list_a > $o,Y: option_list_a] :
      ( ( ( X = none_P3970268120610169480list_a )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_list_a )
         => ( P @ X @ Y ) )
       => ( ! [A2: produc9164743771328383783list_a,B2: list_a] :
              ( ( X
                = ( some_P5335617695965928972list_a @ A2 ) )
             => ( ( Y
                  = ( some_list_a @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_418_combine__options__cases,axiom,
    ! [X: option_list_a,P: option_list_a > option4277935495599621869list_a > $o,Y: option4277935495599621869list_a] :
      ( ( ( X = none_list_a )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P3970268120610169480list_a )
         => ( P @ X @ Y ) )
       => ( ! [A2: list_a,B2: produc9164743771328383783list_a] :
              ( ( X
                = ( some_list_a @ A2 ) )
             => ( ( Y
                  = ( some_P5335617695965928972list_a @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_419_combine__options__cases,axiom,
    ! [X: option_b,P: option_b > option788170273582809878list_a > $o,Y: option788170273582809878list_a] :
      ( ( ( X = none_b )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P5893993846586699057list_a )
         => ( P @ X @ Y ) )
       => ( ! [A2: b,B2: produc3854044251064639184list_a] :
              ( ( X
                = ( some_b @ A2 ) )
             => ( ( Y
                  = ( some_P5354654743593010357list_a @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_420_combine__options__cases,axiom,
    ! [X: option788170273582809878list_a,P: option788170273582809878list_a > option_b > $o,Y: option_b] :
      ( ( ( X = none_P5893993846586699057list_a )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_b )
         => ( P @ X @ Y ) )
       => ( ! [A2: produc3854044251064639184list_a,B2: b] :
              ( ( X
                = ( some_P5354654743593010357list_a @ A2 ) )
             => ( ( Y
                  = ( some_b @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_421_local_Orbt__split__min__rbt__lookup,axiom,
    ! [T: rBT_rb3712131685166902597od_a_b,A: a,B: produc1700058555794149158od_a_b,T3: rBT_rb3712131685166902597od_a_b,K: a] :
      ( ( ( rBT_rb7919835307695533164od_a_b @ T )
        = ( produc8241341905585961537od_a_b @ A @ ( produc5452134863492728084od_a_b @ B @ T3 ) ) )
     => ( ( rBT_rb2608035895457119440od_a_b @ less @ T )
       => ( ( T != rBT_Em8678960569159055417od_a_b )
         => ( ( ( less @ K @ A )
             => ( ( rBT_rb5455008113374026189od_a_b @ less @ T @ K )
                = none_P912253767061701255od_a_b ) )
            & ( ~ ( less @ K @ A )
             => ( ( ( K = A )
                 => ( ( rBT_rb5455008113374026189od_a_b @ less @ T @ K )
                    = ( some_P2412010246944353803od_a_b @ B ) ) )
                & ( ( K != A )
                 => ( ( rBT_rb5455008113374026189od_a_b @ less @ T @ K )
                    = ( rBT_rb5455008113374026189od_a_b @ less @ T3 @ K ) ) ) ) ) ) ) ) ) ).

% local.rbt_split_min_rbt_lookup
thf(fact_422_local_Orbt__split__min__rbt__lookup,axiom,
    ! [T: rBT_rb3198946194961646575list_a,A: a,B: produc3854044251064639184list_a,T3: rBT_rb3198946194961646575list_a,K: a] :
      ( ( ( rBT_rb7789113769008717334list_a @ T )
        = ( produc6417676956431212181list_a @ A @ ( produc3494809192345822824list_a @ B @ T3 ) ) )
     => ( ( rBT_rb6007450547276413562list_a @ less @ T )
       => ( ( T != rBT_Em1305389575258925283list_a )
         => ( ( ( less @ K @ A )
             => ( ( rBT_rb5625365584452904567list_a @ less @ T @ K )
                = none_P5893993846586699057list_a ) )
            & ( ~ ( less @ K @ A )
             => ( ( ( K = A )
                 => ( ( rBT_rb5625365584452904567list_a @ less @ T @ K )
                    = ( some_P5354654743593010357list_a @ B ) ) )
                & ( ( K != A )
                 => ( ( rBT_rb5625365584452904567list_a @ less @ T @ K )
                    = ( rBT_rb5625365584452904567list_a @ less @ T3 @ K ) ) ) ) ) ) ) ) ) ).

% local.rbt_split_min_rbt_lookup
thf(fact_423_local_Orbt__split__min__rbt__lookup,axiom,
    ! [T: rBT_rb6886338251485743302list_a,A: a,B: produc9164743771328383783list_a,T3: rBT_rb6886338251485743302list_a,K: a] :
      ( ( ( rBT_rb8581871447322525805list_a @ T )
        = ( produc1707771300154040643list_a @ A @ ( produc4857060868021337238list_a @ B @ T3 ) ) )
     => ( ( rBT_rb3676718616835825873list_a @ less @ T )
       => ( ( T != rBT_Em7568626898658680890list_a )
         => ( ( ( less @ K @ A )
             => ( ( rBT_rb9209483341951570382list_a @ less @ T @ K )
                = none_P3970268120610169480list_a ) )
            & ( ~ ( less @ K @ A )
             => ( ( ( K = A )
                 => ( ( rBT_rb9209483341951570382list_a @ less @ T @ K )
                    = ( some_P5335617695965928972list_a @ B ) ) )
                & ( ( K != A )
                 => ( ( rBT_rb9209483341951570382list_a @ less @ T @ K )
                    = ( rBT_rb9209483341951570382list_a @ less @ T3 @ K ) ) ) ) ) ) ) ) ) ).

% local.rbt_split_min_rbt_lookup
thf(fact_424_local_Orbt__split__min__rbt__lookup,axiom,
    ! [T: rBT_rb6027196121907741877list_a,A: a,B: produc2547433716817190550list_a,T3: rBT_rb6027196121907741877list_a,K: a] :
      ( ( ( rBT_rb8767210468718574556list_a @ T )
        = ( produc1389796149088856353list_a @ A @ ( produc2517009662524402676list_a @ B @ T3 ) ) )
     => ( ( rBT_rb3455411056480160832list_a @ less @ T )
       => ( ( T != rBT_Em302963693327321001list_a )
         => ( ( ( less @ K @ A )
             => ( ( rBT_rb6302383274397067581list_a @ less @ T @ K )
                = none_P1759628928084742647list_a ) )
            & ( ~ ( less @ K @ A )
             => ( ( ( K = A )
                 => ( ( rBT_rb6302383274397067581list_a @ less @ T @ K )
                    = ( some_P3259385407967395195list_a @ B ) ) )
                & ( ( K != A )
                 => ( ( rBT_rb6302383274397067581list_a @ less @ T @ K )
                    = ( rBT_rb6302383274397067581list_a @ less @ T3 @ K ) ) ) ) ) ) ) ) ) ).

% local.rbt_split_min_rbt_lookup
thf(fact_425_local_Orbt__split__min__rbt__lookup,axiom,
    ! [T: rBT_rbt_a_list_a,A: a,B: list_a,T3: rBT_rbt_a_list_a,K: a] :
      ( ( ( rBT_rb8633090166358492618list_a @ T )
        = ( produc8008000883041212029list_a @ A @ ( produc8650245363082908752list_a @ B @ T3 ) ) )
     => ( ( rBT_rb7395583532625325102list_a @ less @ T )
       => ( ( T != rBT_Empty_a_list_a )
         => ( ( ( less @ K @ A )
             => ( ( rBT_rb8775438530296890539list_a @ less @ T @ K )
                = none_list_a ) )
            & ( ~ ( less @ K @ A )
             => ( ( ( K = A )
                 => ( ( rBT_rb8775438530296890539list_a @ less @ T @ K )
                    = ( some_list_a @ B ) ) )
                & ( ( K != A )
                 => ( ( rBT_rb8775438530296890539list_a @ less @ T @ K )
                    = ( rBT_rb8775438530296890539list_a @ less @ T3 @ K ) ) ) ) ) ) ) ) ) ).

% local.rbt_split_min_rbt_lookup
thf(fact_426_local_Orbt__split__min__rbt__lookup,axiom,
    ! [T: rBT_rb484615002580549772list_a,A: a,B: produc271001111946445293list_a,T3: rBT_rb484615002580549772list_a,K: a] :
      ( ( ( rBT_rb7388485119689856307list_a @ T )
        = ( produc9186377531069891279list_a @ A @ ( produc5162862577585215778list_a @ B @ T3 ) ) )
     => ( ( rBT_rb6864293667938249111list_a @ less @ T )
       => ( ( T != rBT_Em8565251403037144576list_a )
         => ( ( ( less @ K @ A )
             => ( ( rBT_rb4780624521278262164list_a @ less @ T @ K )
                = none_P4075134247637483086list_a ) )
            & ( ~ ( less @ K @ A )
             => ( ( ( K = A )
                 => ( ( rBT_rb4780624521278262164list_a @ less @ T @ K )
                    = ( some_P1402039549733047762list_a @ B ) ) )
                & ( ( K != A )
                 => ( ( rBT_rb4780624521278262164list_a @ less @ T @ K )
                    = ( rBT_rb4780624521278262164list_a @ less @ T3 @ K ) ) ) ) ) ) ) ) ) ).

% local.rbt_split_min_rbt_lookup
thf(fact_427_local_Orbt__split__min__rbt__lookup,axiom,
    ! [T: rBT_rbt_a_b,A: a,B: b,T3: rBT_rbt_a_b,K: a] :
      ( ( ( rBT_rb5461770324497761349in_a_b @ T )
        = ( produc3015636708179276019bt_a_b @ A @ ( produc8464718726981904454bt_a_b @ B @ T3 ) ) )
     => ( ( rBT_rbt_sorted_a_b @ less @ T )
       => ( ( T != rBT_Empty_a_b )
         => ( ( ( less @ K @ A )
             => ( ( rBT_rbt_lookup_a_b @ less @ T @ K )
                = none_b ) )
            & ( ~ ( less @ K @ A )
             => ( ( ( K = A )
                 => ( ( rBT_rbt_lookup_a_b @ less @ T @ K )
                    = ( some_b @ B ) ) )
                & ( ( K != A )
                 => ( ( rBT_rbt_lookup_a_b @ less @ T @ K )
                    = ( rBT_rbt_lookup_a_b @ less @ T3 @ K ) ) ) ) ) ) ) ) ) ).

% local.rbt_split_min_rbt_lookup
thf(fact_428_local_Orbt__lookup__rbt__union,axiom,
    ! [S2: rBT_rb484615002580549772list_a,T: rBT_rb484615002580549772list_a] :
      ( ( rBT_rb6864293667938249111list_a @ less @ S2 )
     => ( ( rBT_rb6864293667938249111list_a @ less @ T )
       => ( ( rBT_rb4780624521278262164list_a @ less @ ( rBT_rb1703184032556916915list_a @ less @ S2 @ T ) )
          = ( map_ad2112687585880850084list_a @ ( rBT_rb4780624521278262164list_a @ less @ S2 ) @ ( rBT_rb4780624521278262164list_a @ less @ T ) ) ) ) ) ).

% local.rbt_lookup_rbt_union
thf(fact_429_local_Orbt__lookup__rbt__union,axiom,
    ! [S2: rBT_rbt_a_b,T: rBT_rbt_a_b] :
      ( ( rBT_rbt_sorted_a_b @ less @ S2 )
     => ( ( rBT_rbt_sorted_a_b @ less @ T )
       => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_union_a_b @ less @ S2 @ T ) )
          = ( map_add_a_b @ ( rBT_rbt_lookup_a_b @ less @ S2 ) @ ( rBT_rbt_lookup_a_b @ less @ T ) ) ) ) ) ).

% local.rbt_lookup_rbt_union
thf(fact_430_local_Orbt__lookup__delete,axiom,
    ! [T: rBT_rb3198946194961646575list_a,X: a,K: a] :
      ( ( rBT_in8151905902596479116list_a @ T )
     => ( ( rBT_rb6007450547276413562list_a @ less @ T )
       => ( ( ( X = K )
           => ( ( rBT_rb5625365584452904567list_a @ less @ ( rBT_rb4214032327585210216list_a @ less @ X @ T ) @ K )
              = none_P5893993846586699057list_a ) )
          & ( ( X != K )
           => ( ( rBT_rb5625365584452904567list_a @ less @ ( rBT_rb4214032327585210216list_a @ less @ X @ T ) @ K )
              = ( rBT_rb5625365584452904567list_a @ less @ T @ K ) ) ) ) ) ) ).

% local.rbt_lookup_delete
thf(fact_431_local_Orbt__lookup__delete,axiom,
    ! [T: rBT_rb3712131685166902597od_a_b,X: a,K: a] :
      ( ( rBT_in4934967524375278818od_a_b @ T )
     => ( ( rBT_rb2608035895457119440od_a_b @ less @ T )
       => ( ( ( X = K )
           => ( ( rBT_rb5455008113374026189od_a_b @ less @ ( rBT_rb5169571330711275454od_a_b @ less @ X @ T ) @ K )
              = none_P912253767061701255od_a_b ) )
          & ( ( X != K )
           => ( ( rBT_rb5455008113374026189od_a_b @ less @ ( rBT_rb5169571330711275454od_a_b @ less @ X @ T ) @ K )
              = ( rBT_rb5455008113374026189od_a_b @ less @ T @ K ) ) ) ) ) ) ).

% local.rbt_lookup_delete
thf(fact_432_local_Orbt__lookup__delete,axiom,
    ! [T: rBT_rb484615002580549772list_a,X: a,K: a] :
      ( ( rBT_in516323401141568425list_a @ T )
     => ( ( rBT_rb6864293667938249111list_a @ less @ T )
       => ( ( ( X = K )
           => ( ( rBT_rb4780624521278262164list_a @ less @ ( rBT_rb9212334439554585221list_a @ less @ X @ T ) @ K )
              = none_P4075134247637483086list_a ) )
          & ( ( X != K )
           => ( ( rBT_rb4780624521278262164list_a @ less @ ( rBT_rb9212334439554585221list_a @ less @ X @ T ) @ K )
              = ( rBT_rb4780624521278262164list_a @ less @ T @ K ) ) ) ) ) ) ).

% local.rbt_lookup_delete
thf(fact_433_local_Orbt__lookup__delete,axiom,
    ! [T: rBT_rbt_a_b,X: a,K: a] :
      ( ( rBT_inv_12_a_b @ T )
     => ( ( rBT_rbt_sorted_a_b @ less @ T )
       => ( ( ( X = K )
           => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_delete_a_b @ less @ X @ T ) @ K )
              = none_b ) )
          & ( ( X != K )
           => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_delete_a_b @ less @ X @ T ) @ K )
              = ( rBT_rbt_lookup_a_b @ less @ T @ K ) ) ) ) ) ) ).

% local.rbt_lookup_delete
thf(fact_434_local_Omap__of__entries,axiom,
    ! [T: rBT_rb484615002580549772list_a] :
      ( ( rBT_rb6864293667938249111list_a @ less @ T )
     => ( ( map_of4901303744667269458list_a @ ( rBT_en3875316870356833511list_a @ T ) )
        = ( rBT_rb4780624521278262164list_a @ less @ T ) ) ) ).

% local.map_of_entries
thf(fact_435_local_Omap__of__entries,axiom,
    ! [T: rBT_rbt_a_b] :
      ( ( rBT_rbt_sorted_a_b @ less @ T )
     => ( ( map_of_a_b @ ( rBT_entries_a_b @ T ) )
        = ( rBT_rbt_lookup_a_b @ less @ T ) ) ) ).

% local.map_of_entries
thf(fact_436_asymp__onI,axiom,
    ! [A3: set_Pr1696764298067387072list_a,R3: produc1392733710753813600list_a > produc1392733710753813600list_a > $o] :
      ( ! [X3: produc1392733710753813600list_a,Y2: produc1392733710753813600list_a] :
          ( ( member8740110733080369545list_a @ X3 @ A3 )
         => ( ( member8740110733080369545list_a @ Y2 @ A3 )
           => ( ( R3 @ X3 @ Y2 )
             => ~ ( R3 @ Y2 @ X3 ) ) ) )
     => ( asymp_3850313391995159806list_a @ A3 @ R3 ) ) ).

% asymp_onI
thf(fact_437_asymp__onI,axiom,
    ! [A3: set_Pr7222558492679949814list_a,R3: produc2547433716817190550list_a > produc2547433716817190550list_a > $o] :
      ( ! [X3: produc2547433716817190550list_a,Y2: produc2547433716817190550list_a] :
          ( ( member852939521359900607list_a @ X3 @ A3 )
         => ( ( member852939521359900607list_a @ Y2 @ A3 )
           => ( ( R3 @ X3 @ Y2 )
             => ~ ( R3 @ Y2 @ X3 ) ) ) )
     => ( asymp_7041679217503909684list_a @ A3 @ R3 ) ) ).

% asymp_onI
thf(fact_438_asymp__onI,axiom,
    ! [A3: set_Product_prod_a_b,R3: product_prod_a_b > product_prod_a_b > $o] :
      ( ! [X3: product_prod_a_b,Y2: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X3 @ A3 )
         => ( ( member1426531481828664017od_a_b @ Y2 @ A3 )
           => ( ( R3 @ X3 @ Y2 )
             => ~ ( R3 @ Y2 @ X3 ) ) ) )
     => ( asymp_1385369521986542406od_a_b @ A3 @ R3 ) ) ).

% asymp_onI
thf(fact_439_asymp__onI,axiom,
    ! [A3: set_list_a,R3: list_a > list_a > $o] :
      ( ! [X3: list_a,Y2: list_a] :
          ( ( member_list_a @ X3 @ A3 )
         => ( ( member_list_a @ Y2 @ A3 )
           => ( ( R3 @ X3 @ Y2 )
             => ~ ( R3 @ Y2 @ X3 ) ) ) )
     => ( asymp_on_list_a @ A3 @ R3 ) ) ).

% asymp_onI
thf(fact_440_asymp__onI,axiom,
    ! [A3: set_a_a_o,R3: ( a > a > $o ) > ( a > a > $o ) > $o] :
      ( ! [X3: a > a > $o,Y2: a > a > $o] :
          ( ( member_a_a_o @ X3 @ A3 )
         => ( ( member_a_a_o @ Y2 @ A3 )
           => ( ( R3 @ X3 @ Y2 )
             => ~ ( R3 @ Y2 @ X3 ) ) ) )
     => ( asymp_on_a_a_o @ A3 @ R3 ) ) ).

% asymp_onI
thf(fact_441_asymp__onI,axiom,
    ! [A3: set_a,R3: a > a > $o] :
      ( ! [X3: a,Y2: a] :
          ( ( member_a @ X3 @ A3 )
         => ( ( member_a @ Y2 @ A3 )
           => ( ( R3 @ X3 @ Y2 )
             => ~ ( R3 @ Y2 @ X3 ) ) ) )
     => ( asymp_on_a @ A3 @ R3 ) ) ).

% asymp_onI
thf(fact_442_local_Orbt__map__entry_Osimps_I1_J,axiom,
    ! [K: a,F: produc271001111946445293list_a > produc271001111946445293list_a] :
      ( ( rBT_rb9060310988641997584list_a @ less @ K @ F @ rBT_Em8565251403037144576list_a )
      = rBT_Em8565251403037144576list_a ) ).

% local.rbt_map_entry.simps(1)
thf(fact_443_local_Orbt__map__entry_Osimps_I1_J,axiom,
    ! [K: a,F: b > b] :
      ( ( rBT_rb4700473395652492066ry_a_b @ less @ K @ F @ rBT_Empty_a_b )
      = rBT_Empty_a_b ) ).

% local.rbt_map_entry.simps(1)
thf(fact_444_local_Orbt__map__entry__rbt__sorted,axiom,
    ! [K: a,F: produc271001111946445293list_a > produc271001111946445293list_a,T: rBT_rb484615002580549772list_a] :
      ( ( rBT_rb6864293667938249111list_a @ less @ ( rBT_rb9060310988641997584list_a @ less @ K @ F @ T ) )
      = ( rBT_rb6864293667938249111list_a @ less @ T ) ) ).

% local.rbt_map_entry_rbt_sorted
thf(fact_445_local_Orbt__map__entry__rbt__sorted,axiom,
    ! [K: a,F: b > b,T: rBT_rbt_a_b] :
      ( ( rBT_rbt_sorted_a_b @ less @ ( rBT_rb4700473395652492066ry_a_b @ less @ K @ F @ T ) )
      = ( rBT_rbt_sorted_a_b @ less @ T ) ) ).

% local.rbt_map_entry_rbt_sorted
thf(fact_446_local_Orbt__map__entry_Osimps_I2_J,axiom,
    ! [K: a,X: a,F: produc271001111946445293list_a > produc271001111946445293list_a,C: rBT_color,Lt: rBT_rb484615002580549772list_a,V: produc271001111946445293list_a,Rt: rBT_rb484615002580549772list_a] :
      ( ( ( less @ K @ X )
       => ( ( rBT_rb9060310988641997584list_a @ less @ K @ F @ ( rBT_Br706236987277315629list_a @ C @ Lt @ X @ V @ Rt ) )
          = ( rBT_Br706236987277315629list_a @ C @ ( rBT_rb9060310988641997584list_a @ less @ K @ F @ Lt ) @ X @ V @ Rt ) ) )
      & ( ~ ( less @ K @ X )
       => ( ( ( less @ X @ K )
           => ( ( rBT_rb9060310988641997584list_a @ less @ K @ F @ ( rBT_Br706236987277315629list_a @ C @ Lt @ X @ V @ Rt ) )
              = ( rBT_Br706236987277315629list_a @ C @ Lt @ X @ V @ ( rBT_rb9060310988641997584list_a @ less @ K @ F @ Rt ) ) ) )
          & ( ~ ( less @ X @ K )
           => ( ( rBT_rb9060310988641997584list_a @ less @ K @ F @ ( rBT_Br706236987277315629list_a @ C @ Lt @ X @ V @ Rt ) )
              = ( rBT_Br706236987277315629list_a @ C @ Lt @ X @ ( F @ V ) @ Rt ) ) ) ) ) ) ).

% local.rbt_map_entry.simps(2)
thf(fact_447_local_Orbt__map__entry_Osimps_I2_J,axiom,
    ! [K: a,X: a,F: b > b,C: rBT_color,Lt: rBT_rbt_a_b,V: b,Rt: rBT_rbt_a_b] :
      ( ( ( less @ K @ X )
       => ( ( rBT_rb4700473395652492066ry_a_b @ less @ K @ F @ ( rBT_Branch_a_b @ C @ Lt @ X @ V @ Rt ) )
          = ( rBT_Branch_a_b @ C @ ( rBT_rb4700473395652492066ry_a_b @ less @ K @ F @ Lt ) @ X @ V @ Rt ) ) )
      & ( ~ ( less @ K @ X )
       => ( ( ( less @ X @ K )
           => ( ( rBT_rb4700473395652492066ry_a_b @ less @ K @ F @ ( rBT_Branch_a_b @ C @ Lt @ X @ V @ Rt ) )
              = ( rBT_Branch_a_b @ C @ Lt @ X @ V @ ( rBT_rb4700473395652492066ry_a_b @ less @ K @ F @ Rt ) ) ) )
          & ( ~ ( less @ X @ K )
           => ( ( rBT_rb4700473395652492066ry_a_b @ less @ K @ F @ ( rBT_Branch_a_b @ C @ Lt @ X @ V @ Rt ) )
              = ( rBT_Branch_a_b @ C @ Lt @ X @ ( F @ V ) @ Rt ) ) ) ) ) ) ).

% local.rbt_map_entry.simps(2)
thf(fact_448_local_Omap__rbt__sorted,axiom,
    ! [F: a > b > b,T: rBT_rbt_a_b] :
      ( ( rBT_rbt_sorted_a_b @ less @ ( rBT_map_a_b_b @ F @ T ) )
      = ( rBT_rbt_sorted_a_b @ less @ T ) ) ).

% local.map_rbt_sorted
thf(fact_449_local_Omap__rbt__sorted,axiom,
    ! [F: a > produc271001111946445293list_a > b,T: rBT_rb484615002580549772list_a] :
      ( ( rBT_rbt_sorted_a_b @ less @ ( rBT_ma2596417598595193579st_a_b @ F @ T ) )
      = ( rBT_rb6864293667938249111list_a @ less @ T ) ) ).

% local.map_rbt_sorted
thf(fact_450_local_Omap__rbt__sorted,axiom,
    ! [F: a > b > produc271001111946445293list_a,T: rBT_rbt_a_b] :
      ( ( rBT_rb6864293667938249111list_a @ less @ ( rBT_ma8342713975248396615list_a @ F @ T ) )
      = ( rBT_rbt_sorted_a_b @ less @ T ) ) ).

% local.map_rbt_sorted
thf(fact_451_local_Omap__rbt__sorted,axiom,
    ! [F: a > produc271001111946445293list_a > produc271001111946445293list_a,T: rBT_rb484615002580549772list_a] :
      ( ( rBT_rb6864293667938249111list_a @ less @ ( rBT_ma6168333958713114073list_a @ F @ T ) )
      = ( rBT_rb6864293667938249111list_a @ less @ T ) ) ).

% local.map_rbt_sorted
thf(fact_452_map__upd__nonempty,axiom,
    ! [T: a > option_b,K: a,X: b] :
      ( ( fun_upd_a_option_b @ T @ K @ ( some_b @ X ) )
     != ( ^ [X2: a] : none_b ) ) ).

% map_upd_nonempty
thf(fact_453_map__upd__nonempty,axiom,
    ! [T: product_prod_a_b > option_b,K: product_prod_a_b,X: b] :
      ( ( fun_up3276301394330262810tion_b @ T @ K @ ( some_b @ X ) )
     != ( ^ [X2: product_prod_a_b] : none_b ) ) ).

% map_upd_nonempty
thf(fact_454_local_Orbt__delete,axiom,
    ! [T: rBT_rbt_a_b,X: a] :
      ( ( rBT_inv_12_a_b @ T )
     => ( rBT_inv_12_a_b @ ( rBT_rbt_delete_a_b @ less @ X @ T ) ) ) ).

% local.rbt_delete
thf(fact_455_local_Orbt__delete,axiom,
    ! [T: rBT_rb484615002580549772list_a,X: a] :
      ( ( rBT_in516323401141568425list_a @ T )
     => ( rBT_in516323401141568425list_a @ ( rBT_rb9212334439554585221list_a @ less @ X @ T ) ) ) ).

% local.rbt_delete
thf(fact_456_map__add__assoc,axiom,
    ! [M1: a > option_b,M2: a > option_b,M3: a > option_b] :
      ( ( map_add_a_b @ M1 @ ( map_add_a_b @ M2 @ M3 ) )
      = ( map_add_a_b @ ( map_add_a_b @ M1 @ M2 ) @ M3 ) ) ).

% map_add_assoc
thf(fact_457_map__add__find__right,axiom,
    ! [N: a > option_b,K: a,Xx: b,M: a > option_b] :
      ( ( ( N @ K )
        = ( some_b @ Xx ) )
     => ( ( map_add_a_b @ M @ N @ K )
        = ( some_b @ Xx ) ) ) ).

% map_add_find_right
thf(fact_458_map__add__None,axiom,
    ! [M: a > option_b,N: a > option_b,K: a] :
      ( ( ( map_add_a_b @ M @ N @ K )
        = none_b )
      = ( ( ( N @ K )
          = none_b )
        & ( ( M @ K )
          = none_b ) ) ) ).

% map_add_None
thf(fact_459_map__add__eq__empty__iff,axiom,
    ! [F: a > option_b,G: a > option_b] :
      ( ( ( map_add_a_b @ F @ G )
        = ( ^ [X2: a] : none_b ) )
      = ( ( F
          = ( ^ [X2: a] : none_b ) )
        & ( G
          = ( ^ [X2: a] : none_b ) ) ) ) ).

% map_add_eq_empty_iff
thf(fact_460_map__add__upd,axiom,
    ! [F: product_prod_a_b > option_b,G: product_prod_a_b > option_b,X: product_prod_a_b,Y: b] :
      ( ( map_ad5145250906507364748_a_b_b @ F @ ( fun_up3276301394330262810tion_b @ G @ X @ ( some_b @ Y ) ) )
      = ( fun_up3276301394330262810tion_b @ ( map_ad5145250906507364748_a_b_b @ F @ G ) @ X @ ( some_b @ Y ) ) ) ).

% map_add_upd
thf(fact_461_map__add__upd,axiom,
    ! [F: a > option_b,G: a > option_b,X: a,Y: b] :
      ( ( map_add_a_b @ F @ ( fun_upd_a_option_b @ G @ X @ ( some_b @ Y ) ) )
      = ( fun_upd_a_option_b @ ( map_add_a_b @ F @ G ) @ X @ ( some_b @ Y ) ) ) ).

% map_add_upd
thf(fact_462_local_Orbt__map__entry__is__rbt,axiom,
    ! [K: a,F: produc271001111946445293list_a > produc271001111946445293list_a,T: rBT_rb484615002580549772list_a] :
      ( ( rBT_is3736607230663913552list_a @ less @ ( rBT_rb9060310988641997584list_a @ less @ K @ F @ T ) )
      = ( rBT_is3736607230663913552list_a @ less @ T ) ) ).

% local.rbt_map_entry_is_rbt
thf(fact_463_local_Orbt__map__entry__is__rbt,axiom,
    ! [K: a,F: b > b,T: rBT_rbt_a_b] :
      ( ( rBT_is_rbt_a_b @ less @ ( rBT_rb4700473395652492066ry_a_b @ less @ K @ F @ T ) )
      = ( rBT_is_rbt_a_b @ less @ T ) ) ).

% local.rbt_map_entry_is_rbt
thf(fact_464_local_Omap__is__rbt,axiom,
    ! [F: a > produc271001111946445293list_a > b,T: rBT_rb484615002580549772list_a] :
      ( ( rBT_is_rbt_a_b @ less @ ( rBT_ma2596417598595193579st_a_b @ F @ T ) )
      = ( rBT_is3736607230663913552list_a @ less @ T ) ) ).

% local.map_is_rbt
thf(fact_465_local_Omap__is__rbt,axiom,
    ! [F: a > b > produc271001111946445293list_a,T: rBT_rbt_a_b] :
      ( ( rBT_is3736607230663913552list_a @ less @ ( rBT_ma8342713975248396615list_a @ F @ T ) )
      = ( rBT_is_rbt_a_b @ less @ T ) ) ).

% local.map_is_rbt
thf(fact_466_local_Omap__is__rbt,axiom,
    ! [F: a > produc271001111946445293list_a > produc271001111946445293list_a,T: rBT_rb484615002580549772list_a] :
      ( ( rBT_is3736607230663913552list_a @ less @ ( rBT_ma6168333958713114073list_a @ F @ T ) )
      = ( rBT_is3736607230663913552list_a @ less @ T ) ) ).

% local.map_is_rbt
thf(fact_467_local_Omap__is__rbt,axiom,
    ! [F: a > b > b,T: rBT_rbt_a_b] :
      ( ( rBT_is_rbt_a_b @ less @ ( rBT_map_a_b_b @ F @ T ) )
      = ( rBT_is_rbt_a_b @ less @ T ) ) ).

% local.map_is_rbt
thf(fact_468_local_Orbt__lookup__rbt__bulkload,axiom,
    ! [Xs: list_P5792129111969709340list_a] :
      ( ( rBT_rb4780624521278262164list_a @ less @ ( rBT_rb6861261436685619186list_a @ less @ Xs ) )
      = ( map_of4901303744667269458list_a @ Xs ) ) ).

% local.rbt_lookup_rbt_bulkload
thf(fact_469_local_Orbt__lookup__rbt__bulkload,axiom,
    ! [Xs: list_P1467974523212632366od_a_b] :
      ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_bulkload_a_b @ less @ Xs ) )
      = ( map_of_a_b @ Xs ) ) ).

% local.rbt_lookup_rbt_bulkload
thf(fact_470_map__add__SomeD,axiom,
    ! [M: a > option_b,N: a > option_b,K: a,X: b] :
      ( ( ( map_add_a_b @ M @ N @ K )
        = ( some_b @ X ) )
     => ( ( ( N @ K )
          = ( some_b @ X ) )
        | ( ( ( N @ K )
            = none_b )
          & ( ( M @ K )
            = ( some_b @ X ) ) ) ) ) ).

% map_add_SomeD
thf(fact_471_map__add__Some__iff,axiom,
    ! [M: a > option_b,N: a > option_b,K: a,X: b] :
      ( ( ( map_add_a_b @ M @ N @ K )
        = ( some_b @ X ) )
      = ( ( ( N @ K )
          = ( some_b @ X ) )
        | ( ( ( N @ K )
            = none_b )
          & ( ( M @ K )
            = ( some_b @ X ) ) ) ) ) ).

% map_add_Some_iff
thf(fact_472_weak__map__of__SomeI,axiom,
    ! [K: b,X: rBT_rbt_a_b,L: list_P3138634381613558876bt_a_b] :
      ( ( member4322261901662495203bt_a_b @ ( produc8464718726981904454bt_a_b @ K @ X ) @ ( set_Pr1756970846036224119bt_a_b @ L ) )
     => ? [X3: rBT_rbt_a_b] :
          ( ( map_of_b_RBT_rbt_a_b @ L @ K )
          = ( some_RBT_rbt_a_b @ X3 ) ) ) ).

% weak_map_of_SomeI
thf(fact_473_weak__map__of__SomeI,axiom,
    ! [K: a,X: produc3797860095223585356bt_a_b,L: list_P6241832400464525193bt_a_b] :
      ( ( member6846102102718116240bt_a_b @ ( produc3015636708179276019bt_a_b @ K @ X ) @ ( set_Pr1558196515228517412bt_a_b @ L ) )
     => ? [X3: produc3797860095223585356bt_a_b] :
          ( ( map_of7816238735838981543bt_a_b @ L @ K )
          = ( some_P7253062255424944423bt_a_b @ X3 ) ) ) ).

% weak_map_of_SomeI
thf(fact_474_weak__map__of__SomeI,axiom,
    ! [K: a,X: list_P5792129111969709340list_a,L: list_P982604105241552843list_a] :
      ( ( member2402001272945804526list_a @ ( produc4159846287338058421list_a @ K @ X ) @ ( set_Pr3960553176861824346list_a @ L ) )
     => ? [X3: list_P5792129111969709340list_a] :
          ( ( map_of8882757051520389377list_a @ L @ K )
          = ( some_l6838989086144800129list_a @ X3 ) ) ) ).

% weak_map_of_SomeI
thf(fact_475_weak__map__of__SomeI,axiom,
    ! [K: a,X: list_P1467974523212632366od_a_b,L: list_P5700792096213675741od_a_b] :
      ( ( member8745141763671436032od_a_b @ ( produc6643264353547226055od_a_b @ K @ X ) @ ( set_Pr187010687427835756od_a_b @ L ) )
     => ? [X3: list_P1467974523212632366od_a_b] :
          ( ( map_of2529231726428571155od_a_b @ L @ K )
          = ( some_l1253897223994727059od_a_b @ X3 ) ) ) ).

% weak_map_of_SomeI
thf(fact_476_weak__map__of__SomeI,axiom,
    ! [K: a,X: list_a,L: list_P4541805568828049459list_a] :
      ( ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ K @ X ) @ ( set_Pr3039917418810828226list_a @ L ) )
     => ? [X3: list_a] :
          ( ( map_of_a_list_a @ L @ K )
          = ( some_list_a @ X3 ) ) ) ).

% weak_map_of_SomeI
thf(fact_477_weak__map__of__SomeI,axiom,
    ! [K: list_a,X: produc2547433716817190550list_a,L: list_P1945613012453903807list_a] :
      ( ( member5573719349037835490list_a @ ( produc8961369442823515561list_a @ K @ X ) @ ( set_Pr7132271252953855310list_a @ L ) )
     => ? [X3: produc2547433716817190550list_a] :
          ( ( map_of4460908170151070709list_a @ L @ K )
          = ( some_P3259385407967395195list_a @ X3 ) ) ) ).

% weak_map_of_SomeI
thf(fact_478_weak__map__of__SomeI,axiom,
    ! [K: list_a,X: list_a,L: list_P321204300973800749list_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ K @ X ) @ ( set_Pr2906193453920572092list_a @ L ) )
     => ? [X3: list_a] :
          ( ( map_of_list_a_list_a @ L @ K )
          = ( some_list_a @ X3 ) ) ) ).

% weak_map_of_SomeI
thf(fact_479_weak__map__of__SomeI,axiom,
    ! [K: a > a > $o,X: produc9164743771328383783list_a,L: list_P1472896444292205030list_a] :
      ( ( member8740110733080369545list_a @ ( produc8684083420967988176list_a @ K @ X ) @ ( set_Pr6051718963112299509list_a @ L ) )
     => ? [X3: produc9164743771328383783list_a] :
          ( ( map_of1656090001276894236list_a @ L @ K )
          = ( some_P5335617695965928972list_a @ X3 ) ) ) ).

% weak_map_of_SomeI
thf(fact_480_weak__map__of__SomeI,axiom,
    ! [K: a,X: produc271001111946445293list_a,L: list_P5792129111969709340list_a] :
      ( ( member852939521359900607list_a @ ( produc4562151247771834630list_a @ K @ X ) @ ( set_Pr4782379691873284651list_a @ L ) )
     => ? [X3: produc271001111946445293list_a] :
          ( ( map_of4901303744667269458list_a @ L @ K )
          = ( some_P1402039549733047762list_a @ X3 ) ) ) ).

% weak_map_of_SomeI
thf(fact_481_weak__map__of__SomeI,axiom,
    ! [K: a,X: b,L: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K @ X ) @ ( set_Product_prod_a_b2 @ L ) )
     => ? [X3: b] :
          ( ( map_of_a_b @ L @ K )
          = ( some_b @ X3 ) ) ) ).

% weak_map_of_SomeI
thf(fact_482_map__of__SomeD,axiom,
    ! [Xs: list_P3138634381613558876bt_a_b,K: b,Y: rBT_rbt_a_b] :
      ( ( ( map_of_b_RBT_rbt_a_b @ Xs @ K )
        = ( some_RBT_rbt_a_b @ Y ) )
     => ( member4322261901662495203bt_a_b @ ( produc8464718726981904454bt_a_b @ K @ Y ) @ ( set_Pr1756970846036224119bt_a_b @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_483_map__of__SomeD,axiom,
    ! [Xs: list_P6241832400464525193bt_a_b,K: a,Y: produc3797860095223585356bt_a_b] :
      ( ( ( map_of7816238735838981543bt_a_b @ Xs @ K )
        = ( some_P7253062255424944423bt_a_b @ Y ) )
     => ( member6846102102718116240bt_a_b @ ( produc3015636708179276019bt_a_b @ K @ Y ) @ ( set_Pr1558196515228517412bt_a_b @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_484_map__of__SomeD,axiom,
    ! [Xs: list_P982604105241552843list_a,K: a,Y: list_P5792129111969709340list_a] :
      ( ( ( map_of8882757051520389377list_a @ Xs @ K )
        = ( some_l6838989086144800129list_a @ Y ) )
     => ( member2402001272945804526list_a @ ( produc4159846287338058421list_a @ K @ Y ) @ ( set_Pr3960553176861824346list_a @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_485_map__of__SomeD,axiom,
    ! [Xs: list_P5700792096213675741od_a_b,K: a,Y: list_P1467974523212632366od_a_b] :
      ( ( ( map_of2529231726428571155od_a_b @ Xs @ K )
        = ( some_l1253897223994727059od_a_b @ Y ) )
     => ( member8745141763671436032od_a_b @ ( produc6643264353547226055od_a_b @ K @ Y ) @ ( set_Pr187010687427835756od_a_b @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_486_map__of__SomeD,axiom,
    ! [Xs: list_P4541805568828049459list_a,K: a,Y: list_a] :
      ( ( ( map_of_a_list_a @ Xs @ K )
        = ( some_list_a @ Y ) )
     => ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ K @ Y ) @ ( set_Pr3039917418810828226list_a @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_487_map__of__SomeD,axiom,
    ! [Xs: list_P1945613012453903807list_a,K: list_a,Y: produc2547433716817190550list_a] :
      ( ( ( map_of4460908170151070709list_a @ Xs @ K )
        = ( some_P3259385407967395195list_a @ Y ) )
     => ( member5573719349037835490list_a @ ( produc8961369442823515561list_a @ K @ Y ) @ ( set_Pr7132271252953855310list_a @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_488_map__of__SomeD,axiom,
    ! [Xs: list_P321204300973800749list_a,K: list_a,Y: list_a] :
      ( ( ( map_of_list_a_list_a @ Xs @ K )
        = ( some_list_a @ Y ) )
     => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ K @ Y ) @ ( set_Pr2906193453920572092list_a @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_489_map__of__SomeD,axiom,
    ! [Xs: list_P1472896444292205030list_a,K: a > a > $o,Y: produc9164743771328383783list_a] :
      ( ( ( map_of1656090001276894236list_a @ Xs @ K )
        = ( some_P5335617695965928972list_a @ Y ) )
     => ( member8740110733080369545list_a @ ( produc8684083420967988176list_a @ K @ Y ) @ ( set_Pr6051718963112299509list_a @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_490_map__of__SomeD,axiom,
    ! [Xs: list_P5792129111969709340list_a,K: a,Y: produc271001111946445293list_a] :
      ( ( ( map_of4901303744667269458list_a @ Xs @ K )
        = ( some_P1402039549733047762list_a @ Y ) )
     => ( member852939521359900607list_a @ ( produc4562151247771834630list_a @ K @ Y ) @ ( set_Pr4782379691873284651list_a @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_491_map__of__SomeD,axiom,
    ! [Xs: list_P1467974523212632366od_a_b,K: a,Y: b] :
      ( ( ( map_of_a_b @ Xs @ K )
        = ( some_b @ Y ) )
     => ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K @ Y ) @ ( set_Product_prod_a_b2 @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_492_asymp__onD,axiom,
    ! [A3: set_Pr1696764298067387072list_a,R3: produc1392733710753813600list_a > produc1392733710753813600list_a > $o,X: produc1392733710753813600list_a,Y: produc1392733710753813600list_a] :
      ( ( asymp_3850313391995159806list_a @ A3 @ R3 )
     => ( ( member8740110733080369545list_a @ X @ A3 )
       => ( ( member8740110733080369545list_a @ Y @ A3 )
         => ( ( R3 @ X @ Y )
           => ~ ( R3 @ Y @ X ) ) ) ) ) ).

% asymp_onD
thf(fact_493_asymp__onD,axiom,
    ! [A3: set_Pr7222558492679949814list_a,R3: produc2547433716817190550list_a > produc2547433716817190550list_a > $o,X: produc2547433716817190550list_a,Y: produc2547433716817190550list_a] :
      ( ( asymp_7041679217503909684list_a @ A3 @ R3 )
     => ( ( member852939521359900607list_a @ X @ A3 )
       => ( ( member852939521359900607list_a @ Y @ A3 )
         => ( ( R3 @ X @ Y )
           => ~ ( R3 @ Y @ X ) ) ) ) ) ).

% asymp_onD
thf(fact_494_asymp__onD,axiom,
    ! [A3: set_Product_prod_a_b,R3: product_prod_a_b > product_prod_a_b > $o,X: product_prod_a_b,Y: product_prod_a_b] :
      ( ( asymp_1385369521986542406od_a_b @ A3 @ R3 )
     => ( ( member1426531481828664017od_a_b @ X @ A3 )
       => ( ( member1426531481828664017od_a_b @ Y @ A3 )
         => ( ( R3 @ X @ Y )
           => ~ ( R3 @ Y @ X ) ) ) ) ) ).

% asymp_onD
thf(fact_495_asymp__onD,axiom,
    ! [A3: set_list_a,R3: list_a > list_a > $o,X: list_a,Y: list_a] :
      ( ( asymp_on_list_a @ A3 @ R3 )
     => ( ( member_list_a @ X @ A3 )
       => ( ( member_list_a @ Y @ A3 )
         => ( ( R3 @ X @ Y )
           => ~ ( R3 @ Y @ X ) ) ) ) ) ).

% asymp_onD
thf(fact_496_asymp__onD,axiom,
    ! [A3: set_a_a_o,R3: ( a > a > $o ) > ( a > a > $o ) > $o,X: a > a > $o,Y: a > a > $o] :
      ( ( asymp_on_a_a_o @ A3 @ R3 )
     => ( ( member_a_a_o @ X @ A3 )
       => ( ( member_a_a_o @ Y @ A3 )
         => ( ( R3 @ X @ Y )
           => ~ ( R3 @ Y @ X ) ) ) ) ) ).

% asymp_onD
thf(fact_497_asymp__onD,axiom,
    ! [A3: set_a,R3: a > a > $o,X: a,Y: a] :
      ( ( asymp_on_a @ A3 @ R3 )
     => ( ( member_a @ X @ A3 )
       => ( ( member_a @ Y @ A3 )
         => ( ( R3 @ X @ Y )
           => ~ ( R3 @ Y @ X ) ) ) ) ) ).

% asymp_onD
thf(fact_498_asymp__on__def,axiom,
    ( asymp_on_a
    = ( ^ [A4: set_a,R4: a > a > $o] :
        ! [X2: a] :
          ( ( member_a @ X2 @ A4 )
         => ! [Y3: a] :
              ( ( member_a @ Y3 @ A4 )
             => ( ( R4 @ X2 @ Y3 )
               => ~ ( R4 @ Y3 @ X2 ) ) ) ) ) ) ).

% asymp_on_def
thf(fact_499_irreflp__onD,axiom,
    ! [A3: set_Pr1696764298067387072list_a,R3: produc1392733710753813600list_a > produc1392733710753813600list_a > $o,A: produc1392733710753813600list_a] :
      ( ( irrefl2851044447983556410list_a @ A3 @ R3 )
     => ( ( member8740110733080369545list_a @ A @ A3 )
       => ~ ( R3 @ A @ A ) ) ) ).

% irreflp_onD
thf(fact_500_irreflp__onD,axiom,
    ! [A3: set_Pr7222558492679949814list_a,R3: produc2547433716817190550list_a > produc2547433716817190550list_a > $o,A: produc2547433716817190550list_a] :
      ( ( irrefl6236602579355916656list_a @ A3 @ R3 )
     => ( ( member852939521359900607list_a @ A @ A3 )
       => ~ ( R3 @ A @ A ) ) ) ).

% irreflp_onD
thf(fact_501_irreflp__onD,axiom,
    ! [A3: set_Product_prod_a_b,R3: product_prod_a_b > product_prod_a_b > $o,A: product_prod_a_b] :
      ( ( irrefl6748496301328320898od_a_b @ A3 @ R3 )
     => ( ( member1426531481828664017od_a_b @ A @ A3 )
       => ~ ( R3 @ A @ A ) ) ) ).

% irreflp_onD
thf(fact_502_irreflp__onD,axiom,
    ! [A3: set_list_a,R3: list_a > list_a > $o,A: list_a] :
      ( ( irreflp_on_list_a @ A3 @ R3 )
     => ( ( member_list_a @ A @ A3 )
       => ~ ( R3 @ A @ A ) ) ) ).

% irreflp_onD
thf(fact_503_irreflp__onD,axiom,
    ! [A3: set_a_a_o,R3: ( a > a > $o ) > ( a > a > $o ) > $o,A: a > a > $o] :
      ( ( irreflp_on_a_a_o @ A3 @ R3 )
     => ( ( member_a_a_o @ A @ A3 )
       => ~ ( R3 @ A @ A ) ) ) ).

% irreflp_onD
thf(fact_504_irreflp__onD,axiom,
    ! [A3: set_a,R3: a > a > $o,A: a] :
      ( ( irreflp_on_a @ A3 @ R3 )
     => ( ( member_a @ A @ A3 )
       => ~ ( R3 @ A @ A ) ) ) ).

% irreflp_onD
thf(fact_505_irreflp__onI,axiom,
    ! [A3: set_Pr1696764298067387072list_a,R3: produc1392733710753813600list_a > produc1392733710753813600list_a > $o] :
      ( ! [A2: produc1392733710753813600list_a] :
          ( ( member8740110733080369545list_a @ A2 @ A3 )
         => ~ ( R3 @ A2 @ A2 ) )
     => ( irrefl2851044447983556410list_a @ A3 @ R3 ) ) ).

% irreflp_onI
thf(fact_506_irreflp__onI,axiom,
    ! [A3: set_Pr7222558492679949814list_a,R3: produc2547433716817190550list_a > produc2547433716817190550list_a > $o] :
      ( ! [A2: produc2547433716817190550list_a] :
          ( ( member852939521359900607list_a @ A2 @ A3 )
         => ~ ( R3 @ A2 @ A2 ) )
     => ( irrefl6236602579355916656list_a @ A3 @ R3 ) ) ).

% irreflp_onI
thf(fact_507_irreflp__onI,axiom,
    ! [A3: set_Product_prod_a_b,R3: product_prod_a_b > product_prod_a_b > $o] :
      ( ! [A2: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ A2 @ A3 )
         => ~ ( R3 @ A2 @ A2 ) )
     => ( irrefl6748496301328320898od_a_b @ A3 @ R3 ) ) ).

% irreflp_onI
thf(fact_508_irreflp__onI,axiom,
    ! [A3: set_list_a,R3: list_a > list_a > $o] :
      ( ! [A2: list_a] :
          ( ( member_list_a @ A2 @ A3 )
         => ~ ( R3 @ A2 @ A2 ) )
     => ( irreflp_on_list_a @ A3 @ R3 ) ) ).

% irreflp_onI
thf(fact_509_irreflp__onI,axiom,
    ! [A3: set_a_a_o,R3: ( a > a > $o ) > ( a > a > $o ) > $o] :
      ( ! [A2: a > a > $o] :
          ( ( member_a_a_o @ A2 @ A3 )
         => ~ ( R3 @ A2 @ A2 ) )
     => ( irreflp_on_a_a_o @ A3 @ R3 ) ) ).

% irreflp_onI
thf(fact_510_irreflp__onI,axiom,
    ! [A3: set_a,R3: a > a > $o] :
      ( ! [A2: a] :
          ( ( member_a @ A2 @ A3 )
         => ~ ( R3 @ A2 @ A2 ) )
     => ( irreflp_on_a @ A3 @ R3 ) ) ).

% irreflp_onI
thf(fact_511_irreflp__on__def,axiom,
    ( irreflp_on_a
    = ( ^ [A4: set_a,R4: a > a > $o] :
        ! [X2: a] :
          ( ( member_a @ X2 @ A4 )
         => ~ ( R4 @ X2 @ X2 ) ) ) ) ).

% irreflp_on_def
thf(fact_512_map__upd__Some__unfold,axiom,
    ! [M: a > option_b,A: a,B: b,X: a,Y: b] :
      ( ( ( fun_upd_a_option_b @ M @ A @ ( some_b @ B ) @ X )
        = ( some_b @ Y ) )
      = ( ( ( X = A )
          & ( B = Y ) )
        | ( ( X != A )
          & ( ( M @ X )
            = ( some_b @ Y ) ) ) ) ) ).

% map_upd_Some_unfold
thf(fact_513_map__upd__Some__unfold,axiom,
    ! [M: product_prod_a_b > option_b,A: product_prod_a_b,B: b,X: product_prod_a_b,Y: b] :
      ( ( ( fun_up3276301394330262810tion_b @ M @ A @ ( some_b @ B ) @ X )
        = ( some_b @ Y ) )
      = ( ( ( X = A )
          & ( B = Y ) )
        | ( ( X != A )
          & ( ( M @ X )
            = ( some_b @ Y ) ) ) ) ) ).

% map_upd_Some_unfold
thf(fact_514_map__upd__triv,axiom,
    ! [T: a > option_b,K: a,X: b] :
      ( ( ( T @ K )
        = ( some_b @ X ) )
     => ( ( fun_upd_a_option_b @ T @ K @ ( some_b @ X ) )
        = T ) ) ).

% map_upd_triv
thf(fact_515_map__upd__triv,axiom,
    ! [T: product_prod_a_b > option_b,K: product_prod_a_b,X: b] :
      ( ( ( T @ K )
        = ( some_b @ X ) )
     => ( ( fun_up3276301394330262810tion_b @ T @ K @ ( some_b @ X ) )
        = T ) ) ).

% map_upd_triv
thf(fact_516_map__upd__eqD1,axiom,
    ! [M: a > option_b,A: a,X: b,N: a > option_b,Y: b] :
      ( ( ( fun_upd_a_option_b @ M @ A @ ( some_b @ X ) )
        = ( fun_upd_a_option_b @ N @ A @ ( some_b @ Y ) ) )
     => ( X = Y ) ) ).

% map_upd_eqD1
thf(fact_517_map__upd__eqD1,axiom,
    ! [M: product_prod_a_b > option_b,A: product_prod_a_b,X: b,N: product_prod_a_b > option_b,Y: b] :
      ( ( ( fun_up3276301394330262810tion_b @ M @ A @ ( some_b @ X ) )
        = ( fun_up3276301394330262810tion_b @ N @ A @ ( some_b @ Y ) ) )
     => ( X = Y ) ) ).

% map_upd_eqD1
thf(fact_518_irreflp__on__if__asymp__on,axiom,
    ! [A3: set_a,R: a > a > $o] :
      ( ( asymp_on_a @ A3 @ R )
     => ( irreflp_on_a @ A3 @ R ) ) ).

% irreflp_on_if_asymp_on
thf(fact_519_rbt__split__min__inv,axiom,
    ! [T: rBT_rb484615002580549772list_a,A: a,B: produc271001111946445293list_a,T3: rBT_rb484615002580549772list_a] :
      ( ( ( rBT_rb7388485119689856307list_a @ T )
        = ( produc9186377531069891279list_a @ A @ ( produc5162862577585215778list_a @ B @ T3 ) ) )
     => ( ( rBT_in516323401141568425list_a @ T )
       => ( ( T != rBT_Em8565251403037144576list_a )
         => ( rBT_in516323401141568425list_a @ T3 ) ) ) ) ).

% rbt_split_min_inv
thf(fact_520_rbt__split__min__inv,axiom,
    ! [T: rBT_rb9014005217275227734list_a,A: a > a > $o,B: produc9164743771328383783list_a,T3: rBT_rb9014005217275227734list_a] :
      ( ( ( rBT_rb7773194748379524093list_a @ T )
        = ( produc3214049537859106371list_a @ A @ ( produc7931496199330935334list_a @ B @ T3 ) ) )
     => ( ( rBT_in1422491598177194611list_a @ T )
       => ( ( T != rBT_Em6530526691317698762list_a )
         => ( rBT_in1422491598177194611list_a @ T3 ) ) ) ) ).

% rbt_split_min_inv
thf(fact_521_rbt__split__min__inv,axiom,
    ! [T: rBT_rb146355364677272861list_a,A: list_a,B: list_a,T3: rBT_rb146355364677272861list_a] :
      ( ( ( rBT_rb6817416403757366084list_a @ T )
        = ( produc3896882375441822717list_a @ A @ ( produc8592078595194200650list_a @ B @ T3 ) ) )
     => ( ( rBT_in6265342139143602618list_a @ T )
       => ( ( T != rBT_Em7612442972313006097list_a )
         => ( rBT_in6265342139143602618list_a @ T3 ) ) ) ) ).

% rbt_split_min_inv
thf(fact_522_rbt__split__min__inv,axiom,
    ! [T: rBT_rb6721489163287533103list_a,A: list_a,B: produc2547433716817190550list_a,T3: rBT_rb6721489163287533103list_a] :
      ( ( ( rBT_rb551397257639590486list_a @ T )
        = ( produc6988289231029734561list_a @ A @ ( produc5414369813378224622list_a @ B @ T3 ) ) )
     => ( ( rBT_in6455254765751373772list_a @ T )
       => ( ( T != rBT_Em8319540245955853219list_a )
         => ( rBT_in6455254765751373772list_a @ T3 ) ) ) ) ).

% rbt_split_min_inv
thf(fact_523_rbt__split__min__inv,axiom,
    ! [T: rBT_rbt_a_b,A: a,B: b,T3: rBT_rbt_a_b] :
      ( ( ( rBT_rb5461770324497761349in_a_b @ T )
        = ( produc3015636708179276019bt_a_b @ A @ ( produc8464718726981904454bt_a_b @ B @ T3 ) ) )
     => ( ( rBT_inv_12_a_b @ T )
       => ( ( T != rBT_Empty_a_b )
         => ( rBT_inv_12_a_b @ T3 ) ) ) ) ).

% rbt_split_min_inv
thf(fact_524_local_Orbt__minus__rec,axiom,
    ! [T1: rBT_rbt_a_b,T2: rBT_rbt_a_b] :
      ( ( rBT_inv_12_a_b @ T1 )
     => ( ( rBT_inv_12_a_b @ T2 )
       => ( rBT_inv_12_a_b @ ( rBT_rb8950846305964686124ec_a_b @ less @ T1 @ T2 ) ) ) ) ).

% local.rbt_minus_rec
thf(fact_525_local_Orbt__union__rec,axiom,
    ! [T1: rBT_rbt_a_b,T2: rBT_rbt_a_b,F: a > b > b > b] :
      ( ( rBT_inv_12_a_b @ T1 )
     => ( ( rBT_inv_12_a_b @ T2 )
       => ( rBT_inv_12_a_b @ ( rBT_rb1827089648477395117ec_a_b @ less @ F @ T1 @ T2 ) ) ) ) ).

% local.rbt_union_rec
thf(fact_526_ord_Orbt__lookup_Osimps_I1_J,axiom,
    ! [Less: ( a > a > $o ) > ( a > a > $o ) > $o,K: a > a > $o] :
      ( ( rBT_rb6483784951565284958list_a @ Less @ rBT_Em6530526691317698762list_a @ K )
      = none_P3970268120610169480list_a ) ).

% ord.rbt_lookup.simps(1)
thf(fact_527_ord_Orbt__lookup_Osimps_I1_J,axiom,
    ! [Less: list_a > list_a > $o,K: list_a] :
      ( ( rBT_rb5231746220636214693list_a @ Less @ rBT_Em7612442972313006097list_a @ K )
      = none_list_a ) ).

% ord.rbt_lookup.simps(1)
thf(fact_528_ord_Orbt__lookup_Osimps_I1_J,axiom,
    ! [Less: list_a > list_a > $o,K: list_a] :
      ( ( rBT_rb3107071546442335031list_a @ Less @ rBT_Em8319540245955853219list_a @ K )
      = none_P1759628928084742647list_a ) ).

% ord.rbt_lookup.simps(1)
thf(fact_529_ord_Orbt__lookup_Osimps_I1_J,axiom,
    ! [Less: a > a > $o,K: a] :
      ( ( rBT_rb4780624521278262164list_a @ Less @ rBT_Em8565251403037144576list_a @ K )
      = none_P4075134247637483086list_a ) ).

% ord.rbt_lookup.simps(1)
thf(fact_530_ord_Orbt__lookup_Osimps_I1_J,axiom,
    ! [Less: a > a > $o,K: a] :
      ( ( rBT_rbt_lookup_a_b @ Less @ rBT_Empty_a_b @ K )
      = none_b ) ).

% ord.rbt_lookup.simps(1)
thf(fact_531_ord_Orbt__lookup__Empty,axiom,
    ! [Less: ( a > a > $o ) > ( a > a > $o ) > $o] :
      ( ( rBT_rb6483784951565284958list_a @ Less @ rBT_Em6530526691317698762list_a )
      = ( ^ [X2: a > a > $o] : none_P3970268120610169480list_a ) ) ).

% ord.rbt_lookup_Empty
thf(fact_532_ord_Orbt__lookup__Empty,axiom,
    ! [Less: list_a > list_a > $o] :
      ( ( rBT_rb5231746220636214693list_a @ Less @ rBT_Em7612442972313006097list_a )
      = ( ^ [X2: list_a] : none_list_a ) ) ).

% ord.rbt_lookup_Empty
thf(fact_533_ord_Orbt__lookup__Empty,axiom,
    ! [Less: list_a > list_a > $o] :
      ( ( rBT_rb3107071546442335031list_a @ Less @ rBT_Em8319540245955853219list_a )
      = ( ^ [X2: list_a] : none_P1759628928084742647list_a ) ) ).

% ord.rbt_lookup_Empty
thf(fact_534_ord_Orbt__lookup__Empty,axiom,
    ! [Less: a > a > $o] :
      ( ( rBT_rb4780624521278262164list_a @ Less @ rBT_Em8565251403037144576list_a )
      = ( ^ [X2: a] : none_P4075134247637483086list_a ) ) ).

% ord.rbt_lookup_Empty
thf(fact_535_ord_Orbt__lookup__Empty,axiom,
    ! [Less: a > a > $o] :
      ( ( rBT_rbt_lookup_a_b @ Less @ rBT_Empty_a_b )
      = ( ^ [X2: a] : none_b ) ) ).

% ord.rbt_lookup_Empty
thf(fact_536_ord_Orbt__lookup_Osimps_I2_J,axiom,
    ! [Less: ( a > a > $o ) > ( a > a > $o ) > $o,K: a > a > $o,X: a > a > $o,Uu: rBT_color,L: rBT_rb9014005217275227734list_a,Y: produc9164743771328383783list_a,R: rBT_rb9014005217275227734list_a] :
      ( ( ( Less @ K @ X )
       => ( ( rBT_rb6483784951565284958list_a @ Less @ ( rBT_Br6721835470276586743list_a @ Uu @ L @ X @ Y @ R ) @ K )
          = ( rBT_rb6483784951565284958list_a @ Less @ L @ K ) ) )
      & ( ~ ( Less @ K @ X )
       => ( ( ( Less @ X @ K )
           => ( ( rBT_rb6483784951565284958list_a @ Less @ ( rBT_Br6721835470276586743list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( rBT_rb6483784951565284958list_a @ Less @ R @ K ) ) )
          & ( ~ ( Less @ X @ K )
           => ( ( rBT_rb6483784951565284958list_a @ Less @ ( rBT_Br6721835470276586743list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( some_P5335617695965928972list_a @ Y ) ) ) ) ) ) ).

% ord.rbt_lookup.simps(2)
thf(fact_537_ord_Orbt__lookup_Osimps_I2_J,axiom,
    ! [Less: list_a > list_a > $o,K: list_a,X: list_a,Uu: rBT_color,L: rBT_rb146355364677272861list_a,Y: list_a,R: rBT_rb146355364677272861list_a] :
      ( ( ( Less @ K @ X )
       => ( ( rBT_rb5231746220636214693list_a @ Less @ ( rBT_Br8372335562727726142list_a @ Uu @ L @ X @ Y @ R ) @ K )
          = ( rBT_rb5231746220636214693list_a @ Less @ L @ K ) ) )
      & ( ~ ( Less @ K @ X )
       => ( ( ( Less @ X @ K )
           => ( ( rBT_rb5231746220636214693list_a @ Less @ ( rBT_Br8372335562727726142list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( rBT_rb5231746220636214693list_a @ Less @ R @ K ) ) )
          & ( ~ ( Less @ X @ K )
           => ( ( rBT_rb5231746220636214693list_a @ Less @ ( rBT_Br8372335562727726142list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( some_list_a @ Y ) ) ) ) ) ) ).

% ord.rbt_lookup.simps(2)
thf(fact_538_ord_Orbt__lookup_Osimps_I2_J,axiom,
    ! [Less: list_a > list_a > $o,K: list_a,X: list_a,Uu: rBT_color,L: rBT_rb6721489163287533103list_a,Y: produc2547433716817190550list_a,R: rBT_rb6721489163287533103list_a] :
      ( ( ( Less @ K @ X )
       => ( ( rBT_rb3107071546442335031list_a @ Less @ ( rBT_Br1204270329063416912list_a @ Uu @ L @ X @ Y @ R ) @ K )
          = ( rBT_rb3107071546442335031list_a @ Less @ L @ K ) ) )
      & ( ~ ( Less @ K @ X )
       => ( ( ( Less @ X @ K )
           => ( ( rBT_rb3107071546442335031list_a @ Less @ ( rBT_Br1204270329063416912list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( rBT_rb3107071546442335031list_a @ Less @ R @ K ) ) )
          & ( ~ ( Less @ X @ K )
           => ( ( rBT_rb3107071546442335031list_a @ Less @ ( rBT_Br1204270329063416912list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( some_P3259385407967395195list_a @ Y ) ) ) ) ) ) ).

% ord.rbt_lookup.simps(2)
thf(fact_539_ord_Orbt__lookup_Osimps_I2_J,axiom,
    ! [Less: a > a > $o,K: a,X: a,Uu: rBT_color,L: rBT_rb484615002580549772list_a,Y: produc271001111946445293list_a,R: rBT_rb484615002580549772list_a] :
      ( ( ( Less @ K @ X )
       => ( ( rBT_rb4780624521278262164list_a @ Less @ ( rBT_Br706236987277315629list_a @ Uu @ L @ X @ Y @ R ) @ K )
          = ( rBT_rb4780624521278262164list_a @ Less @ L @ K ) ) )
      & ( ~ ( Less @ K @ X )
       => ( ( ( Less @ X @ K )
           => ( ( rBT_rb4780624521278262164list_a @ Less @ ( rBT_Br706236987277315629list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( rBT_rb4780624521278262164list_a @ Less @ R @ K ) ) )
          & ( ~ ( Less @ X @ K )
           => ( ( rBT_rb4780624521278262164list_a @ Less @ ( rBT_Br706236987277315629list_a @ Uu @ L @ X @ Y @ R ) @ K )
              = ( some_P1402039549733047762list_a @ Y ) ) ) ) ) ) ).

% ord.rbt_lookup.simps(2)
thf(fact_540_ord_Orbt__lookup_Osimps_I2_J,axiom,
    ! [Less: a > a > $o,K: a,X: a,Uu: rBT_color,L: rBT_rbt_a_b,Y: b,R: rBT_rbt_a_b] :
      ( ( ( Less @ K @ X )
       => ( ( rBT_rbt_lookup_a_b @ Less @ ( rBT_Branch_a_b @ Uu @ L @ X @ Y @ R ) @ K )
          = ( rBT_rbt_lookup_a_b @ Less @ L @ K ) ) )
      & ( ~ ( Less @ K @ X )
       => ( ( ( Less @ X @ K )
           => ( ( rBT_rbt_lookup_a_b @ Less @ ( rBT_Branch_a_b @ Uu @ L @ X @ Y @ R ) @ K )
              = ( rBT_rbt_lookup_a_b @ Less @ R @ K ) ) )
          & ( ~ ( Less @ X @ K )
           => ( ( rBT_rbt_lookup_a_b @ Less @ ( rBT_Branch_a_b @ Uu @ L @ X @ Y @ R ) @ K )
              = ( some_b @ Y ) ) ) ) ) ) ).

% ord.rbt_lookup.simps(2)
thf(fact_541_local_Orbt__lookup__rbt__map__entry,axiom,
    ! [K: a,F: produc271001111946445293list_a > produc271001111946445293list_a,T: rBT_rb484615002580549772list_a] :
      ( ( rBT_rb4780624521278262164list_a @ less @ ( rBT_rb9060310988641997584list_a @ less @ K @ F @ T ) )
      = ( fun_up8065330075003041074list_a @ ( rBT_rb4780624521278262164list_a @ less @ T ) @ K @ ( map_op5497283568914401227list_a @ F @ ( rBT_rb4780624521278262164list_a @ less @ T @ K ) ) ) ) ).

% local.rbt_lookup_rbt_map_entry
thf(fact_542_local_Orbt__lookup__rbt__map__entry,axiom,
    ! [K: a,F: b > b,T: rBT_rbt_a_b] :
      ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rb4700473395652492066ry_a_b @ less @ K @ F @ T ) )
      = ( fun_upd_a_option_b @ ( rBT_rbt_lookup_a_b @ less @ T ) @ K @ ( map_option_b_b @ F @ ( rBT_rbt_lookup_a_b @ less @ T @ K ) ) ) ) ).

% local.rbt_lookup_rbt_map_entry
thf(fact_543_rbt__split__comp_Ocases,axiom,
    ! [X: produc727794225953898581_a_b_a] :
      ( ! [K2: a] :
          ( X
         != ( produc2410990786219400071_a_b_a @ rBT_Empty_a_b @ K2 ) )
     => ~ ! [Uu2: rBT_color,L2: rBT_rbt_a_b,A2: a,B2: b,R2: rBT_rbt_a_b,X3: a] :
            ( X
           != ( produc2410990786219400071_a_b_a @ ( rBT_Branch_a_b @ Uu2 @ L2 @ A2 @ B2 @ R2 ) @ X3 ) ) ) ).

% rbt_split_comp.cases
thf(fact_544_rbt__split__comp_Ocases,axiom,
    ! [X: produc4172418568614467815st_a_a] :
      ( ! [K2: a] :
          ( X
         != ( produc5537483632639485721st_a_a @ rBT_Em8565251403037144576list_a @ K2 ) )
     => ~ ! [Uu2: rBT_color,L2: rBT_rb484615002580549772list_a,A2: a,B2: produc271001111946445293list_a,R2: rBT_rb484615002580549772list_a,X3: a] :
            ( X
           != ( produc5537483632639485721st_a_a @ ( rBT_Br706236987277315629list_a @ Uu2 @ L2 @ A2 @ B2 @ R2 ) @ X3 ) ) ) ).

% rbt_split_comp.cases
thf(fact_545_rbt__split__comp_Ocases,axiom,
    ! [X: produc7068643790369957645_a_a_o] :
      ( ! [K2: a > a > $o] :
          ( X
         != ( produc3805221970424485439_a_a_o @ rBT_Em6530526691317698762list_a @ K2 ) )
     => ~ ! [Uu2: rBT_color,L2: rBT_rb9014005217275227734list_a,A2: a > a > $o,B2: produc9164743771328383783list_a,R2: rBT_rb9014005217275227734list_a,X3: a > a > $o] :
            ( X
           != ( produc3805221970424485439_a_a_o @ ( rBT_Br6721835470276586743list_a @ Uu2 @ L2 @ A2 @ B2 @ R2 ) @ X3 ) ) ) ).

% rbt_split_comp.cases
thf(fact_546_rbt__split__comp_Ocases,axiom,
    ! [X: produc111643535050012060list_a] :
      ( ! [K2: list_a] :
          ( X
         != ( produc4881793773736933454list_a @ rBT_Em7612442972313006097list_a @ K2 ) )
     => ~ ! [Uu2: rBT_color,L2: rBT_rb146355364677272861list_a,A2: list_a,B2: list_a,R2: rBT_rb146355364677272861list_a,X3: list_a] :
            ( X
           != ( produc4881793773736933454list_a @ ( rBT_Br8372335562727726142list_a @ Uu2 @ L2 @ A2 @ B2 @ R2 ) @ X3 ) ) ) ).

% rbt_split_comp.cases
thf(fact_547_rbt__split__comp_Ocases,axiom,
    ! [X: produc5999062819672298378list_a] :
      ( ! [K2: list_a] :
          ( X
         != ( produc5316640063668597948list_a @ rBT_Em8319540245955853219list_a @ K2 ) )
     => ~ ! [Uu2: rBT_color,L2: rBT_rb6721489163287533103list_a,A2: list_a,B2: produc2547433716817190550list_a,R2: rBT_rb6721489163287533103list_a,X3: list_a] :
            ( X
           != ( produc5316640063668597948list_a @ ( rBT_Br1204270329063416912list_a @ Uu2 @ L2 @ A2 @ B2 @ R2 ) @ X3 ) ) ) ).

% rbt_split_comp.cases
thf(fact_548_ord__class_Orbt__split_Ocases,axiom,
    ! [X: produc7068643790369957645_a_a_o] :
      ( ! [K2: a > a > $o] :
          ( X
         != ( produc3805221970424485439_a_a_o @ rBT_Em6530526691317698762list_a @ K2 ) )
     => ~ ! [Uu2: rBT_color,L2: rBT_rb9014005217275227734list_a,A2: a > a > $o,B2: produc9164743771328383783list_a,R2: rBT_rb9014005217275227734list_a,X3: a > a > $o] :
            ( X
           != ( produc3805221970424485439_a_a_o @ ( rBT_Br6721835470276586743list_a @ Uu2 @ L2 @ A2 @ B2 @ R2 ) @ X3 ) ) ) ).

% ord_class.rbt_split.cases
thf(fact_549_rbt_Oinject,axiom,
    ! [X21: rBT_color,X222: rBT_rbt_a_b,X232: a,X24: b,X25: rBT_rbt_a_b,Y21: rBT_color,Y222: rBT_rbt_a_b,Y23: a,Y24: b,Y25: rBT_rbt_a_b] :
      ( ( ( rBT_Branch_a_b @ X21 @ X222 @ X232 @ X24 @ X25 )
        = ( rBT_Branch_a_b @ Y21 @ Y222 @ Y23 @ Y24 @ Y25 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 )
        & ( X232 = Y23 )
        & ( X24 = Y24 )
        & ( X25 = Y25 ) ) ) ).

% rbt.inject
thf(fact_550_rbt_Oinject,axiom,
    ! [X21: rBT_color,X222: rBT_rb484615002580549772list_a,X232: a,X24: produc271001111946445293list_a,X25: rBT_rb484615002580549772list_a,Y21: rBT_color,Y222: rBT_rb484615002580549772list_a,Y23: a,Y24: produc271001111946445293list_a,Y25: rBT_rb484615002580549772list_a] :
      ( ( ( rBT_Br706236987277315629list_a @ X21 @ X222 @ X232 @ X24 @ X25 )
        = ( rBT_Br706236987277315629list_a @ Y21 @ Y222 @ Y23 @ Y24 @ Y25 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 )
        & ( X232 = Y23 )
        & ( X24 = Y24 )
        & ( X25 = Y25 ) ) ) ).

% rbt.inject
thf(fact_551_rbt_Oinject,axiom,
    ! [X21: rBT_color,X222: rBT_rb9014005217275227734list_a,X232: a > a > $o,X24: produc9164743771328383783list_a,X25: rBT_rb9014005217275227734list_a,Y21: rBT_color,Y222: rBT_rb9014005217275227734list_a,Y23: a > a > $o,Y24: produc9164743771328383783list_a,Y25: rBT_rb9014005217275227734list_a] :
      ( ( ( rBT_Br6721835470276586743list_a @ X21 @ X222 @ X232 @ X24 @ X25 )
        = ( rBT_Br6721835470276586743list_a @ Y21 @ Y222 @ Y23 @ Y24 @ Y25 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 )
        & ( X232 = Y23 )
        & ( X24 = Y24 )
        & ( X25 = Y25 ) ) ) ).

% rbt.inject
thf(fact_552_rbt_Oinject,axiom,
    ! [X21: rBT_color,X222: rBT_rb146355364677272861list_a,X232: list_a,X24: list_a,X25: rBT_rb146355364677272861list_a,Y21: rBT_color,Y222: rBT_rb146355364677272861list_a,Y23: list_a,Y24: list_a,Y25: rBT_rb146355364677272861list_a] :
      ( ( ( rBT_Br8372335562727726142list_a @ X21 @ X222 @ X232 @ X24 @ X25 )
        = ( rBT_Br8372335562727726142list_a @ Y21 @ Y222 @ Y23 @ Y24 @ Y25 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 )
        & ( X232 = Y23 )
        & ( X24 = Y24 )
        & ( X25 = Y25 ) ) ) ).

% rbt.inject
thf(fact_553_rbt_Oinject,axiom,
    ! [X21: rBT_color,X222: rBT_rb6721489163287533103list_a,X232: list_a,X24: produc2547433716817190550list_a,X25: rBT_rb6721489163287533103list_a,Y21: rBT_color,Y222: rBT_rb6721489163287533103list_a,Y23: list_a,Y24: produc2547433716817190550list_a,Y25: rBT_rb6721489163287533103list_a] :
      ( ( ( rBT_Br1204270329063416912list_a @ X21 @ X222 @ X232 @ X24 @ X25 )
        = ( rBT_Br1204270329063416912list_a @ Y21 @ Y222 @ Y23 @ Y24 @ Y25 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 )
        & ( X232 = Y23 )
        & ( X24 = Y24 )
        & ( X25 = Y25 ) ) ) ).

% rbt.inject
thf(fact_554_local_Orbt__lookup__map,axiom,
    ! [F: a > produc271001111946445293list_a > b,T: rBT_rb484615002580549772list_a,X: a] :
      ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_ma2596417598595193579st_a_b @ F @ T ) @ X )
      = ( map_op4144660371050442461st_a_b @ ( F @ X ) @ ( rBT_rb4780624521278262164list_a @ less @ T @ X ) ) ) ).

% local.rbt_lookup_map
thf(fact_555_local_Orbt__lookup__map,axiom,
    ! [F: a > b > produc271001111946445293list_a,T: rBT_rbt_a_b,X: a] :
      ( ( rBT_rb4780624521278262164list_a @ less @ ( rBT_ma8342713975248396615list_a @ F @ T ) @ X )
      = ( map_op667584710848869689list_a @ ( F @ X ) @ ( rBT_rbt_lookup_a_b @ less @ T @ X ) ) ) ).

% local.rbt_lookup_map
thf(fact_556_local_Orbt__lookup__map,axiom,
    ! [F: a > produc271001111946445293list_a > produc271001111946445293list_a,T: rBT_rb484615002580549772list_a,X: a] :
      ( ( rBT_rb4780624521278262164list_a @ less @ ( rBT_ma6168333958713114073list_a @ F @ T ) @ X )
      = ( map_op5497283568914401227list_a @ ( F @ X ) @ ( rBT_rb4780624521278262164list_a @ less @ T @ X ) ) ) ).

% local.rbt_lookup_map
thf(fact_557_local_Orbt__lookup__map,axiom,
    ! [F: a > b > b,T: rBT_rbt_a_b,X: a] :
      ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_map_a_b_b @ F @ T ) @ X )
      = ( map_option_b_b @ ( F @ X ) @ ( rBT_rbt_lookup_a_b @ less @ T @ X ) ) ) ).

% local.rbt_lookup_map
thf(fact_558_map__option__eq__Some,axiom,
    ! [F: b > b,Xo: option_b,Y: b] :
      ( ( ( map_option_b_b @ F @ Xo )
        = ( some_b @ Y ) )
      = ( ? [Z2: b] :
            ( ( Xo
              = ( some_b @ Z2 ) )
            & ( ( F @ Z2 )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_559_map__option__eq__Some,axiom,
    ! [F: list_a > b,Xo: option_list_a,Y: b] :
      ( ( ( map_option_list_a_b @ F @ Xo )
        = ( some_b @ Y ) )
      = ( ? [Z2: list_a] :
            ( ( Xo
              = ( some_list_a @ Z2 ) )
            & ( ( F @ Z2 )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_560_map__option__eq__Some,axiom,
    ! [F: b > list_a,Xo: option_b,Y: list_a] :
      ( ( ( map_option_b_list_a @ F @ Xo )
        = ( some_list_a @ Y ) )
      = ( ? [Z2: b] :
            ( ( Xo
              = ( some_b @ Z2 ) )
            & ( ( F @ Z2 )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_561_map__option__eq__Some,axiom,
    ! [F: list_a > list_a,Xo: option_list_a,Y: list_a] :
      ( ( ( map_op1857795647477319691list_a @ F @ Xo )
        = ( some_list_a @ Y ) )
      = ( ? [Z2: list_a] :
            ( ( Xo
              = ( some_list_a @ Z2 ) )
            & ( ( F @ Z2 )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_562_map__option__eq__Some,axiom,
    ! [F: produc9164743771328383783list_a > b,Xo: option4277935495599621869list_a,Y: b] :
      ( ( ( map_op1273925508862086179st_a_b @ F @ Xo )
        = ( some_b @ Y ) )
      = ( ? [Z2: produc9164743771328383783list_a] :
            ( ( Xo
              = ( some_P5335617695965928972list_a @ Z2 ) )
            & ( ( F @ Z2 )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_563_map__option__eq__Some,axiom,
    ! [F: b > produc9164743771328383783list_a,Xo: option_b,Y: produc9164743771328383783list_a] :
      ( ( ( map_op8115135547206484595list_a @ F @ Xo )
        = ( some_P5335617695965928972list_a @ Y ) )
      = ( ? [Z2: b] :
            ( ( Xo
              = ( some_b @ Z2 ) )
            & ( ( F @ Z2 )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_564_map__option__eq__Some,axiom,
    ! [F: list_a > produc9164743771328383783list_a,Xo: option_list_a,Y: produc9164743771328383783list_a] :
      ( ( ( map_op6660605347205137710list_a @ F @ Xo )
        = ( some_P5335617695965928972list_a @ Y ) )
      = ( ? [Z2: list_a] :
            ( ( Xo
              = ( some_list_a @ Z2 ) )
            & ( ( F @ Z2 )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_565_map__option__eq__Some,axiom,
    ! [F: produc9164743771328383783list_a > list_a,Xo: option4277935495599621869list_a,Y: list_a] :
      ( ( ( map_op6316156699137132200list_a @ F @ Xo )
        = ( some_list_a @ Y ) )
      = ( ? [Z2: produc9164743771328383783list_a] :
            ( ( Xo
              = ( some_P5335617695965928972list_a @ Z2 ) )
            & ( ( F @ Z2 )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_566_map__option__eq__Some,axiom,
    ! [F: produc3854044251064639184list_a > b,Xo: option788170273582809878list_a,Y: b] :
      ( ( ( map_op8566905496903362746st_a_b @ F @ Xo )
        = ( some_b @ Y ) )
      = ( ? [Z2: produc3854044251064639184list_a] :
            ( ( Xo
              = ( some_P5354654743593010357list_a @ Z2 ) )
            & ( ( F @ Z2 )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_567_map__option__eq__Some,axiom,
    ! [F: b > produc3854044251064639184list_a,Xo: option_b,Y: produc3854044251064639184list_a] :
      ( ( ( map_op5185276952246331932list_a @ F @ Xo )
        = ( some_P5354654743593010357list_a @ Y ) )
      = ( ? [Z2: b] :
            ( ( Xo
              = ( some_b @ Z2 ) )
            & ( ( F @ Z2 )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_568_None__eq__map__option__iff,axiom,
    ! [F: b > b,X: option_b] :
      ( ( none_b
        = ( map_option_b_b @ F @ X ) )
      = ( X = none_b ) ) ).

% None_eq_map_option_iff
thf(fact_569_None__eq__map__option__iff,axiom,
    ! [F: produc3854044251064639184list_a > b,X: option788170273582809878list_a] :
      ( ( none_b
        = ( map_op8566905496903362746st_a_b @ F @ X ) )
      = ( X = none_P5893993846586699057list_a ) ) ).

% None_eq_map_option_iff
thf(fact_570_None__eq__map__option__iff,axiom,
    ! [F: b > produc3854044251064639184list_a,X: option_b] :
      ( ( none_P5893993846586699057list_a
        = ( map_op5185276952246331932list_a @ F @ X ) )
      = ( X = none_b ) ) ).

% None_eq_map_option_iff
thf(fact_571_None__eq__map__option__iff,axiom,
    ! [F: produc271001111946445293list_a > b,X: option110588449099563443list_a] :
      ( ( none_b
        = ( map_op4144660371050442461st_a_b @ F @ X ) )
      = ( X = none_P4075134247637483086list_a ) ) ).

% None_eq_map_option_iff
thf(fact_572_None__eq__map__option__iff,axiom,
    ! [F: b > produc271001111946445293list_a,X: option_b] :
      ( ( none_P4075134247637483086list_a
        = ( map_op667584710848869689list_a @ F @ X ) )
      = ( X = none_b ) ) ).

% None_eq_map_option_iff
thf(fact_573_None__eq__map__option__iff,axiom,
    ! [F: produc1700058555794149158od_a_b > b,X: option2417831327529529196od_a_b] :
      ( ( none_b
        = ( map_op6276588259745388004_a_b_b @ F @ X ) )
      = ( X = none_P912253767061701255od_a_b ) ) ).

% None_eq_map_option_iff
thf(fact_574_None__eq__map__option__iff,axiom,
    ! [F: produc3854044251064639184list_a > produc3854044251064639184list_a,X: option788170273582809878list_a] :
      ( ( none_P5893993846586699057list_a
        = ( map_op5315205157914692491list_a @ F @ X ) )
      = ( X = none_P5893993846586699057list_a ) ) ).

% None_eq_map_option_iff
thf(fact_575_None__eq__map__option__iff,axiom,
    ! [F: b > produc1700058555794149158od_a_b,X: option_b] :
      ( ( none_P912253767061701255od_a_b
        = ( map_op6268938394405918322od_a_b @ F @ X ) )
      = ( X = none_b ) ) ).

% None_eq_map_option_iff
thf(fact_576_None__eq__map__option__iff,axiom,
    ! [F: produc3854044251064639184list_a > produc271001111946445293list_a,X: option788170273582809878list_a] :
      ( ( none_P4075134247637483086list_a
        = ( map_op7937041826318256296list_a @ F @ X ) )
      = ( X = none_P5893993846586699057list_a ) ) ).

% None_eq_map_option_iff
thf(fact_577_None__eq__map__option__iff,axiom,
    ! [F: produc271001111946445293list_a > produc3854044251064639184list_a,X: option110588449099563443list_a] :
      ( ( none_P5893993846586699057list_a
        = ( map_op4537204130133803182list_a @ F @ X ) )
      = ( X = none_P4075134247637483086list_a ) ) ).

% None_eq_map_option_iff
thf(fact_578_map__option__is__None,axiom,
    ! [F: b > b,Opt: option_b] :
      ( ( ( map_option_b_b @ F @ Opt )
        = none_b )
      = ( Opt = none_b ) ) ).

% map_option_is_None
thf(fact_579_map__option__is__None,axiom,
    ! [F: produc3854044251064639184list_a > b,Opt: option788170273582809878list_a] :
      ( ( ( map_op8566905496903362746st_a_b @ F @ Opt )
        = none_b )
      = ( Opt = none_P5893993846586699057list_a ) ) ).

% map_option_is_None
thf(fact_580_map__option__is__None,axiom,
    ! [F: b > produc3854044251064639184list_a,Opt: option_b] :
      ( ( ( map_op5185276952246331932list_a @ F @ Opt )
        = none_P5893993846586699057list_a )
      = ( Opt = none_b ) ) ).

% map_option_is_None
thf(fact_581_map__option__is__None,axiom,
    ! [F: produc271001111946445293list_a > b,Opt: option110588449099563443list_a] :
      ( ( ( map_op4144660371050442461st_a_b @ F @ Opt )
        = none_b )
      = ( Opt = none_P4075134247637483086list_a ) ) ).

% map_option_is_None
thf(fact_582_map__option__is__None,axiom,
    ! [F: b > produc271001111946445293list_a,Opt: option_b] :
      ( ( ( map_op667584710848869689list_a @ F @ Opt )
        = none_P4075134247637483086list_a )
      = ( Opt = none_b ) ) ).

% map_option_is_None
thf(fact_583_map__option__is__None,axiom,
    ! [F: produc1700058555794149158od_a_b > b,Opt: option2417831327529529196od_a_b] :
      ( ( ( map_op6276588259745388004_a_b_b @ F @ Opt )
        = none_b )
      = ( Opt = none_P912253767061701255od_a_b ) ) ).

% map_option_is_None
thf(fact_584_map__option__is__None,axiom,
    ! [F: produc3854044251064639184list_a > produc3854044251064639184list_a,Opt: option788170273582809878list_a] :
      ( ( ( map_op5315205157914692491list_a @ F @ Opt )
        = none_P5893993846586699057list_a )
      = ( Opt = none_P5893993846586699057list_a ) ) ).

% map_option_is_None
thf(fact_585_map__option__is__None,axiom,
    ! [F: b > produc1700058555794149158od_a_b,Opt: option_b] :
      ( ( ( map_op6268938394405918322od_a_b @ F @ Opt )
        = none_P912253767061701255od_a_b )
      = ( Opt = none_b ) ) ).

% map_option_is_None
thf(fact_586_map__option__is__None,axiom,
    ! [F: produc3854044251064639184list_a > produc271001111946445293list_a,Opt: option788170273582809878list_a] :
      ( ( ( map_op7937041826318256296list_a @ F @ Opt )
        = none_P4075134247637483086list_a )
      = ( Opt = none_P5893993846586699057list_a ) ) ).

% map_option_is_None
thf(fact_587_map__option__is__None,axiom,
    ! [F: produc271001111946445293list_a > produc3854044251064639184list_a,Opt: option110588449099563443list_a] :
      ( ( ( map_op4537204130133803182list_a @ F @ Opt )
        = none_P5893993846586699057list_a )
      = ( Opt = none_P4075134247637483086list_a ) ) ).

% map_option_is_None
thf(fact_588_option_Omap__disc__iff,axiom,
    ! [F: b > b,A: option_b] :
      ( ( ( map_option_b_b @ F @ A )
        = none_b )
      = ( A = none_b ) ) ).

% option.map_disc_iff
thf(fact_589_option_Omap__disc__iff,axiom,
    ! [F: produc3854044251064639184list_a > b,A: option788170273582809878list_a] :
      ( ( ( map_op8566905496903362746st_a_b @ F @ A )
        = none_b )
      = ( A = none_P5893993846586699057list_a ) ) ).

% option.map_disc_iff
thf(fact_590_option_Omap__disc__iff,axiom,
    ! [F: b > produc3854044251064639184list_a,A: option_b] :
      ( ( ( map_op5185276952246331932list_a @ F @ A )
        = none_P5893993846586699057list_a )
      = ( A = none_b ) ) ).

% option.map_disc_iff
thf(fact_591_option_Omap__disc__iff,axiom,
    ! [F: produc271001111946445293list_a > b,A: option110588449099563443list_a] :
      ( ( ( map_op4144660371050442461st_a_b @ F @ A )
        = none_b )
      = ( A = none_P4075134247637483086list_a ) ) ).

% option.map_disc_iff
thf(fact_592_option_Omap__disc__iff,axiom,
    ! [F: b > produc271001111946445293list_a,A: option_b] :
      ( ( ( map_op667584710848869689list_a @ F @ A )
        = none_P4075134247637483086list_a )
      = ( A = none_b ) ) ).

% option.map_disc_iff
thf(fact_593_option_Omap__disc__iff,axiom,
    ! [F: produc1700058555794149158od_a_b > b,A: option2417831327529529196od_a_b] :
      ( ( ( map_op6276588259745388004_a_b_b @ F @ A )
        = none_b )
      = ( A = none_P912253767061701255od_a_b ) ) ).

% option.map_disc_iff
thf(fact_594_option_Omap__disc__iff,axiom,
    ! [F: produc3854044251064639184list_a > produc3854044251064639184list_a,A: option788170273582809878list_a] :
      ( ( ( map_op5315205157914692491list_a @ F @ A )
        = none_P5893993846586699057list_a )
      = ( A = none_P5893993846586699057list_a ) ) ).

% option.map_disc_iff
thf(fact_595_option_Omap__disc__iff,axiom,
    ! [F: b > produc1700058555794149158od_a_b,A: option_b] :
      ( ( ( map_op6268938394405918322od_a_b @ F @ A )
        = none_P912253767061701255od_a_b )
      = ( A = none_b ) ) ).

% option.map_disc_iff
thf(fact_596_option_Omap__disc__iff,axiom,
    ! [F: produc3854044251064639184list_a > produc271001111946445293list_a,A: option788170273582809878list_a] :
      ( ( ( map_op7937041826318256296list_a @ F @ A )
        = none_P4075134247637483086list_a )
      = ( A = none_P5893993846586699057list_a ) ) ).

% option.map_disc_iff
thf(fact_597_option_Omap__disc__iff,axiom,
    ! [F: produc271001111946445293list_a > produc3854044251064639184list_a,A: option110588449099563443list_a] :
      ( ( ( map_op4537204130133803182list_a @ F @ A )
        = none_P5893993846586699057list_a )
      = ( A = none_P4075134247637483086list_a ) ) ).

% option.map_disc_iff
thf(fact_598_local_Orbt__bulkload__is__rbt,axiom,
    ! [Xs: list_P5792129111969709340list_a] : ( rBT_is3736607230663913552list_a @ less @ ( rBT_rb6861261436685619186list_a @ less @ Xs ) ) ).

% local.rbt_bulkload_is_rbt
thf(fact_599_local_Orbt__bulkload__is__rbt,axiom,
    ! [Xs: list_P1467974523212632366od_a_b] : ( rBT_is_rbt_a_b @ less @ ( rBT_rbt_bulkload_a_b @ less @ Xs ) ) ).

% local.rbt_bulkload_is_rbt
thf(fact_600_ord_Orbt__bulkload_Ocong,axiom,
    rBT_rbt_bulkload_a_b = rBT_rbt_bulkload_a_b ).

% ord.rbt_bulkload.cong
thf(fact_601_ord_Orbt__union__rec,axiom,
    ! [T1: rBT_rbt_a_b,T2: rBT_rbt_a_b,Less: a > a > $o,F: a > b > b > b] :
      ( ( rBT_inv_12_a_b @ T1 )
     => ( ( rBT_inv_12_a_b @ T2 )
       => ( rBT_inv_12_a_b @ ( rBT_rb1827089648477395117ec_a_b @ Less @ F @ T1 @ T2 ) ) ) ) ).

% ord.rbt_union_rec
thf(fact_602_option_Osimps_I9_J,axiom,
    ! [F: b > b,X22: b] :
      ( ( map_option_b_b @ F @ ( some_b @ X22 ) )
      = ( some_b @ ( F @ X22 ) ) ) ).

% option.simps(9)
thf(fact_603_option_Osimps_I9_J,axiom,
    ! [F: b > list_a,X22: b] :
      ( ( map_option_b_list_a @ F @ ( some_b @ X22 ) )
      = ( some_list_a @ ( F @ X22 ) ) ) ).

% option.simps(9)
thf(fact_604_option_Osimps_I9_J,axiom,
    ! [F: list_a > b,X22: list_a] :
      ( ( map_option_list_a_b @ F @ ( some_list_a @ X22 ) )
      = ( some_b @ ( F @ X22 ) ) ) ).

% option.simps(9)
thf(fact_605_option_Osimps_I9_J,axiom,
    ! [F: list_a > list_a,X22: list_a] :
      ( ( map_op1857795647477319691list_a @ F @ ( some_list_a @ X22 ) )
      = ( some_list_a @ ( F @ X22 ) ) ) ).

% option.simps(9)
thf(fact_606_option_Osimps_I9_J,axiom,
    ! [F: b > produc9164743771328383783list_a,X22: b] :
      ( ( map_op8115135547206484595list_a @ F @ ( some_b @ X22 ) )
      = ( some_P5335617695965928972list_a @ ( F @ X22 ) ) ) ).

% option.simps(9)
thf(fact_607_option_Osimps_I9_J,axiom,
    ! [F: produc9164743771328383783list_a > b,X22: produc9164743771328383783list_a] :
      ( ( map_op1273925508862086179st_a_b @ F @ ( some_P5335617695965928972list_a @ X22 ) )
      = ( some_b @ ( F @ X22 ) ) ) ).

% option.simps(9)
thf(fact_608_option_Osimps_I9_J,axiom,
    ! [F: produc9164743771328383783list_a > list_a,X22: produc9164743771328383783list_a] :
      ( ( map_op6316156699137132200list_a @ F @ ( some_P5335617695965928972list_a @ X22 ) )
      = ( some_list_a @ ( F @ X22 ) ) ) ).

% option.simps(9)
thf(fact_609_option_Osimps_I9_J,axiom,
    ! [F: list_a > produc9164743771328383783list_a,X22: list_a] :
      ( ( map_op6660605347205137710list_a @ F @ ( some_list_a @ X22 ) )
      = ( some_P5335617695965928972list_a @ ( F @ X22 ) ) ) ).

% option.simps(9)
thf(fact_610_option_Osimps_I9_J,axiom,
    ! [F: b > produc3854044251064639184list_a,X22: b] :
      ( ( map_op5185276952246331932list_a @ F @ ( some_b @ X22 ) )
      = ( some_P5354654743593010357list_a @ ( F @ X22 ) ) ) ).

% option.simps(9)
thf(fact_611_option_Osimps_I9_J,axiom,
    ! [F: produc3854044251064639184list_a > b,X22: produc3854044251064639184list_a] :
      ( ( map_op8566905496903362746st_a_b @ F @ ( some_P5354654743593010357list_a @ X22 ) )
      = ( some_b @ ( F @ X22 ) ) ) ).

% option.simps(9)
thf(fact_612_map__option__cong,axiom,
    ! [X: option_b,Y: option_b,F: b > b,G: b > b] :
      ( ( X = Y )
     => ( ! [A2: b] :
            ( ( Y
              = ( some_b @ A2 ) )
           => ( ( F @ A2 )
              = ( G @ A2 ) ) )
       => ( ( map_option_b_b @ F @ X )
          = ( map_option_b_b @ G @ Y ) ) ) ) ).

% map_option_cong
thf(fact_613_option_Osimps_I8_J,axiom,
    ! [F: b > b] :
      ( ( map_option_b_b @ F @ none_b )
      = none_b ) ).

% option.simps(8)
thf(fact_614_option_Osimps_I8_J,axiom,
    ! [F: b > produc3854044251064639184list_a] :
      ( ( map_op5185276952246331932list_a @ F @ none_b )
      = none_P5893993846586699057list_a ) ).

% option.simps(8)
thf(fact_615_option_Osimps_I8_J,axiom,
    ! [F: produc3854044251064639184list_a > b] :
      ( ( map_op8566905496903362746st_a_b @ F @ none_P5893993846586699057list_a )
      = none_b ) ).

% option.simps(8)
thf(fact_616_option_Osimps_I8_J,axiom,
    ! [F: b > produc271001111946445293list_a] :
      ( ( map_op667584710848869689list_a @ F @ none_b )
      = none_P4075134247637483086list_a ) ).

% option.simps(8)
thf(fact_617_option_Osimps_I8_J,axiom,
    ! [F: produc271001111946445293list_a > b] :
      ( ( map_op4144660371050442461st_a_b @ F @ none_P4075134247637483086list_a )
      = none_b ) ).

% option.simps(8)
thf(fact_618_option_Osimps_I8_J,axiom,
    ! [F: b > produc1700058555794149158od_a_b] :
      ( ( map_op6268938394405918322od_a_b @ F @ none_b )
      = none_P912253767061701255od_a_b ) ).

% option.simps(8)
thf(fact_619_option_Osimps_I8_J,axiom,
    ! [F: produc3854044251064639184list_a > produc3854044251064639184list_a] :
      ( ( map_op5315205157914692491list_a @ F @ none_P5893993846586699057list_a )
      = none_P5893993846586699057list_a ) ).

% option.simps(8)
thf(fact_620_option_Osimps_I8_J,axiom,
    ! [F: produc1700058555794149158od_a_b > b] :
      ( ( map_op6276588259745388004_a_b_b @ F @ none_P912253767061701255od_a_b )
      = none_b ) ).

% option.simps(8)
thf(fact_621_option_Osimps_I8_J,axiom,
    ! [F: produc271001111946445293list_a > produc3854044251064639184list_a] :
      ( ( map_op4537204130133803182list_a @ F @ none_P4075134247637483086list_a )
      = none_P5893993846586699057list_a ) ).

% option.simps(8)
thf(fact_622_option_Osimps_I8_J,axiom,
    ! [F: produc3854044251064639184list_a > produc271001111946445293list_a] :
      ( ( map_op7937041826318256296list_a @ F @ none_P5893993846586699057list_a )
      = none_P4075134247637483086list_a ) ).

% option.simps(8)
thf(fact_623_ord_Ois__rbt_Ocong,axiom,
    rBT_is3736607230663913552list_a = rBT_is3736607230663913552list_a ).

% ord.is_rbt.cong
thf(fact_624_ord_Ois__rbt_Ocong,axiom,
    rBT_is_rbt_a_b = rBT_is_rbt_a_b ).

% ord.is_rbt.cong
thf(fact_625_ord_Orbt__sorted_Ocong,axiom,
    rBT_rbt_sorted_a_b = rBT_rbt_sorted_a_b ).

% ord.rbt_sorted.cong
thf(fact_626_ord_Orbt__sorted_Ocong,axiom,
    rBT_rb6864293667938249111list_a = rBT_rb6864293667938249111list_a ).

% ord.rbt_sorted.cong
thf(fact_627_ord_Orbt__lookup_Ocong,axiom,
    rBT_rb4780624521278262164list_a = rBT_rb4780624521278262164list_a ).

% ord.rbt_lookup.cong
thf(fact_628_ord_Orbt__lookup_Ocong,axiom,
    rBT_rbt_lookup_a_b = rBT_rbt_lookup_a_b ).

% ord.rbt_lookup.cong
thf(fact_629_ord_Orbt__map__entry_Ocong,axiom,
    rBT_rb4700473395652492066ry_a_b = rBT_rb4700473395652492066ry_a_b ).

% ord.rbt_map_entry.cong
thf(fact_630_ord_Orbt__insert_Ocong,axiom,
    rBT_rb4834844912256151443list_a = rBT_rb4834844912256151443list_a ).

% ord.rbt_insert.cong
thf(fact_631_ord_Orbt__insert_Ocong,axiom,
    rBT_rbt_insert_a_b = rBT_rbt_insert_a_b ).

% ord.rbt_insert.cong
thf(fact_632_ord_Orbt__delete_Ocong,axiom,
    rBT_rbt_delete_a_b = rBT_rbt_delete_a_b ).

% ord.rbt_delete.cong
thf(fact_633_ord_Orbt__delete_Ocong,axiom,
    rBT_rb9212334439554585221list_a = rBT_rb9212334439554585221list_a ).

% ord.rbt_delete.cong
thf(fact_634_ord_Orbt__union__with__key_Ocong,axiom,
    rBT_rb1234474489191313366ey_a_b = rBT_rb1234474489191313366ey_a_b ).

% ord.rbt_union_with_key.cong
thf(fact_635_ord_Orbt__inter__with__key_Ocong,axiom,
    rBT_rb935285606175209993ey_a_b = rBT_rb935285606175209993ey_a_b ).

% ord.rbt_inter_with_key.cong
thf(fact_636_ord_Orbt__minus_Ocong,axiom,
    rBT_rbt_minus_a_b = rBT_rbt_minus_a_b ).

% ord.rbt_minus.cong
thf(fact_637_ord_Orbt__insert__with_Ocong,axiom,
    rBT_rb5519417620787648267th_a_b = rBT_rb5519417620787648267th_a_b ).

% ord.rbt_insert_with.cong
thf(fact_638_ord_Orbt__union_Ocong,axiom,
    rBT_rbt_union_a_b = rBT_rbt_union_a_b ).

% ord.rbt_union.cong
thf(fact_639_ord_Orbt__inter_Ocong,axiom,
    rBT_rbt_inter_a_b = rBT_rbt_inter_a_b ).

% ord.rbt_inter.cong
thf(fact_640_ord_Orbt__union__with_Ocong,axiom,
    rBT_rb7608173054702221355th_a_b = rBT_rb7608173054702221355th_a_b ).

% ord.rbt_union_with.cong
thf(fact_641_ord_Orbt__inter__with_Ocong,axiom,
    rBT_rb6197565798871309880th_a_b = rBT_rb6197565798871309880th_a_b ).

% ord.rbt_inter_with.cong
thf(fact_642_rbt__split__min_Ocases,axiom,
    ! [X: rBT_rbt_a_b] :
      ( ( X != rBT_Empty_a_b )
     => ~ ! [Uu2: rBT_color,L2: rBT_rbt_a_b,A2: a,B2: b,R2: rBT_rbt_a_b] :
            ( X
           != ( rBT_Branch_a_b @ Uu2 @ L2 @ A2 @ B2 @ R2 ) ) ) ).

% rbt_split_min.cases
thf(fact_643_rbt__split__min_Ocases,axiom,
    ! [X: rBT_rb484615002580549772list_a] :
      ( ( X != rBT_Em8565251403037144576list_a )
     => ~ ! [Uu2: rBT_color,L2: rBT_rb484615002580549772list_a,A2: a,B2: produc271001111946445293list_a,R2: rBT_rb484615002580549772list_a] :
            ( X
           != ( rBT_Br706236987277315629list_a @ Uu2 @ L2 @ A2 @ B2 @ R2 ) ) ) ).

% rbt_split_min.cases
thf(fact_644_rbt__split__min_Ocases,axiom,
    ! [X: rBT_rb9014005217275227734list_a] :
      ( ( X != rBT_Em6530526691317698762list_a )
     => ~ ! [Uu2: rBT_color,L2: rBT_rb9014005217275227734list_a,A2: a > a > $o,B2: produc9164743771328383783list_a,R2: rBT_rb9014005217275227734list_a] :
            ( X
           != ( rBT_Br6721835470276586743list_a @ Uu2 @ L2 @ A2 @ B2 @ R2 ) ) ) ).

% rbt_split_min.cases
thf(fact_645_rbt__split__min_Ocases,axiom,
    ! [X: rBT_rb146355364677272861list_a] :
      ( ( X != rBT_Em7612442972313006097list_a )
     => ~ ! [Uu2: rBT_color,L2: rBT_rb146355364677272861list_a,A2: list_a,B2: list_a,R2: rBT_rb146355364677272861list_a] :
            ( X
           != ( rBT_Br8372335562727726142list_a @ Uu2 @ L2 @ A2 @ B2 @ R2 ) ) ) ).

% rbt_split_min.cases
thf(fact_646_rbt__split__min_Ocases,axiom,
    ! [X: rBT_rb6721489163287533103list_a] :
      ( ( X != rBT_Em8319540245955853219list_a )
     => ~ ! [Uu2: rBT_color,L2: rBT_rb6721489163287533103list_a,A2: list_a,B2: produc2547433716817190550list_a,R2: rBT_rb6721489163287533103list_a] :
            ( X
           != ( rBT_Br1204270329063416912list_a @ Uu2 @ L2 @ A2 @ B2 @ R2 ) ) ) ).

% rbt_split_min.cases
thf(fact_647_rbt_Oexhaust,axiom,
    ! [Y: rBT_rbt_a_b] :
      ( ( Y != rBT_Empty_a_b )
     => ~ ! [X212: rBT_color,X223: rBT_rbt_a_b,X233: a,X242: b,X252: rBT_rbt_a_b] :
            ( Y
           != ( rBT_Branch_a_b @ X212 @ X223 @ X233 @ X242 @ X252 ) ) ) ).

% rbt.exhaust
thf(fact_648_rbt_Oexhaust,axiom,
    ! [Y: rBT_rb484615002580549772list_a] :
      ( ( Y != rBT_Em8565251403037144576list_a )
     => ~ ! [X212: rBT_color,X223: rBT_rb484615002580549772list_a,X233: a,X242: produc271001111946445293list_a,X252: rBT_rb484615002580549772list_a] :
            ( Y
           != ( rBT_Br706236987277315629list_a @ X212 @ X223 @ X233 @ X242 @ X252 ) ) ) ).

% rbt.exhaust
thf(fact_649_rbt_Oexhaust,axiom,
    ! [Y: rBT_rb9014005217275227734list_a] :
      ( ( Y != rBT_Em6530526691317698762list_a )
     => ~ ! [X212: rBT_color,X223: rBT_rb9014005217275227734list_a,X233: a > a > $o,X242: produc9164743771328383783list_a,X252: rBT_rb9014005217275227734list_a] :
            ( Y
           != ( rBT_Br6721835470276586743list_a @ X212 @ X223 @ X233 @ X242 @ X252 ) ) ) ).

% rbt.exhaust
thf(fact_650_rbt_Oexhaust,axiom,
    ! [Y: rBT_rb146355364677272861list_a] :
      ( ( Y != rBT_Em7612442972313006097list_a )
     => ~ ! [X212: rBT_color,X223: rBT_rb146355364677272861list_a,X233: list_a,X242: list_a,X252: rBT_rb146355364677272861list_a] :
            ( Y
           != ( rBT_Br8372335562727726142list_a @ X212 @ X223 @ X233 @ X242 @ X252 ) ) ) ).

% rbt.exhaust
thf(fact_651_rbt_Oexhaust,axiom,
    ! [Y: rBT_rb6721489163287533103list_a] :
      ( ( Y != rBT_Em8319540245955853219list_a )
     => ~ ! [X212: rBT_color,X223: rBT_rb6721489163287533103list_a,X233: list_a,X242: produc2547433716817190550list_a,X252: rBT_rb6721489163287533103list_a] :
            ( Y
           != ( rBT_Br1204270329063416912list_a @ X212 @ X223 @ X233 @ X242 @ X252 ) ) ) ).

% rbt.exhaust
thf(fact_652_rbt_Odistinct_I1_J,axiom,
    ! [X21: rBT_color,X222: rBT_rbt_a_b,X232: a,X24: b,X25: rBT_rbt_a_b] :
      ( rBT_Empty_a_b
     != ( rBT_Branch_a_b @ X21 @ X222 @ X232 @ X24 @ X25 ) ) ).

% rbt.distinct(1)
thf(fact_653_rbt_Odistinct_I1_J,axiom,
    ! [X21: rBT_color,X222: rBT_rb484615002580549772list_a,X232: a,X24: produc271001111946445293list_a,X25: rBT_rb484615002580549772list_a] :
      ( rBT_Em8565251403037144576list_a
     != ( rBT_Br706236987277315629list_a @ X21 @ X222 @ X232 @ X24 @ X25 ) ) ).

% rbt.distinct(1)
thf(fact_654_rbt_Odistinct_I1_J,axiom,
    ! [X21: rBT_color,X222: rBT_rb9014005217275227734list_a,X232: a > a > $o,X24: produc9164743771328383783list_a,X25: rBT_rb9014005217275227734list_a] :
      ( rBT_Em6530526691317698762list_a
     != ( rBT_Br6721835470276586743list_a @ X21 @ X222 @ X232 @ X24 @ X25 ) ) ).

% rbt.distinct(1)
thf(fact_655_rbt_Odistinct_I1_J,axiom,
    ! [X21: rBT_color,X222: rBT_rb146355364677272861list_a,X232: list_a,X24: list_a,X25: rBT_rb146355364677272861list_a] :
      ( rBT_Em7612442972313006097list_a
     != ( rBT_Br8372335562727726142list_a @ X21 @ X222 @ X232 @ X24 @ X25 ) ) ).

% rbt.distinct(1)
thf(fact_656_rbt_Odistinct_I1_J,axiom,
    ! [X21: rBT_color,X222: rBT_rb6721489163287533103list_a,X232: list_a,X24: produc2547433716817190550list_a,X25: rBT_rb6721489163287533103list_a] :
      ( rBT_Em8319540245955853219list_a
     != ( rBT_Br1204270329063416912list_a @ X21 @ X222 @ X232 @ X24 @ X25 ) ) ).

% rbt.distinct(1)
thf(fact_657_rbt__joinR_Ocases,axiom,
    ! [X: produc3513110283169999992bt_a_b] :
      ~ ! [L2: rBT_rbt_a_b,A2: a,B2: b,R2: rBT_rbt_a_b] :
          ( X
         != ( produc4801245906263185456bt_a_b @ L2 @ ( produc3015636708179276019bt_a_b @ A2 @ ( produc8464718726981904454bt_a_b @ B2 @ R2 ) ) ) ) ).

% rbt_joinR.cases
thf(fact_658_ord_OEmpty__is__rbt,axiom,
    ! [Less: list_a > list_a > $o] : ( rBT_is2145532136008089075list_a @ Less @ rBT_Em8319540245955853219list_a ) ).

% ord.Empty_is_rbt
thf(fact_659_ord_OEmpty__is__rbt,axiom,
    ! [Less: a > a > $o] : ( rBT_is3736607230663913552list_a @ Less @ rBT_Em8565251403037144576list_a ) ).

% ord.Empty_is_rbt
thf(fact_660_ord_OEmpty__is__rbt,axiom,
    ! [Less: a > a > $o] : ( rBT_is_rbt_a_b @ Less @ rBT_Empty_a_b ) ).

% ord.Empty_is_rbt
thf(fact_661_ord_Ois__rbt__rbt__sorted,axiom,
    ! [Less: a > a > $o,T: rBT_rbt_a_b] :
      ( ( rBT_is_rbt_a_b @ Less @ T )
     => ( rBT_rbt_sorted_a_b @ Less @ T ) ) ).

% ord.is_rbt_rbt_sorted
thf(fact_662_ord_Omap__is__rbt,axiom,
    ! [Less: a > a > $o,F: a > b > b,T: rBT_rbt_a_b] :
      ( ( rBT_is_rbt_a_b @ Less @ ( rBT_map_a_b_b @ F @ T ) )
      = ( rBT_is_rbt_a_b @ Less @ T ) ) ).

% ord.map_is_rbt
thf(fact_663_ord_Orbt__map__entry__is__rbt,axiom,
    ! [Less: a > a > $o,K: a,F: b > b,T: rBT_rbt_a_b] :
      ( ( rBT_is_rbt_a_b @ Less @ ( rBT_rb4700473395652492066ry_a_b @ Less @ K @ F @ T ) )
      = ( rBT_is_rbt_a_b @ Less @ T ) ) ).

% ord.rbt_map_entry_is_rbt
thf(fact_664_local_Orbt__insertwk__is__rbt,axiom,
    ! [T: rBT_rbt_a_b,F: a > b > b > b,K: a,X: b] :
      ( ( rBT_is_rbt_a_b @ less @ T )
     => ( rBT_is_rbt_a_b @ less @ ( rBT_rb2461657779293087990ey_a_b @ less @ F @ K @ X @ T ) ) ) ).

% local.rbt_insertwk_is_rbt
thf(fact_665_local_Ototalp__on__less,axiom,
    ! [A3: set_a] : ( totalp_on_a @ A3 @ less ) ).

% local.totalp_on_less
thf(fact_666_totalp__on__def,axiom,
    ( totalp_on_a
    = ( ^ [A4: set_a,R4: a > a > $o] :
        ! [X2: a] :
          ( ( member_a @ X2 @ A4 )
         => ! [Y3: a] :
              ( ( member_a @ Y3 @ A4 )
             => ( ( X2 != Y3 )
               => ( ( R4 @ X2 @ Y3 )
                  | ( R4 @ Y3 @ X2 ) ) ) ) ) ) ) ).

% totalp_on_def
thf(fact_667_totalp__onI,axiom,
    ! [A3: set_a,R3: a > a > $o] :
      ( ! [X3: a,Y2: a] :
          ( ( member_a @ X3 @ A3 )
         => ( ( member_a @ Y2 @ A3 )
           => ( ( X3 != Y2 )
             => ( ( R3 @ X3 @ Y2 )
                | ( R3 @ Y2 @ X3 ) ) ) ) )
     => ( totalp_on_a @ A3 @ R3 ) ) ).

% totalp_onI
thf(fact_668_totalp__onD,axiom,
    ! [A3: set_a,R3: a > a > $o,X: a,Y: a] :
      ( ( totalp_on_a @ A3 @ R3 )
     => ( ( member_a @ X @ A3 )
       => ( ( member_a @ Y @ A3 )
         => ( ( X != Y )
           => ( ( R3 @ X @ Y )
              | ( R3 @ Y @ X ) ) ) ) ) ) ).

% totalp_onD
thf(fact_669_totalp__on__empty,axiom,
    ! [R3: product_prod_a_b > product_prod_a_b > $o] : ( totalp2285046867861603700od_a_b @ bot_bo3428410327501040244od_a_b @ R3 ) ).

% totalp_on_empty
thf(fact_670_totalp__on__empty,axiom,
    ! [R3: a > a > $o] : ( totalp_on_a @ bot_bot_set_a @ R3 ) ).

% totalp_on_empty
thf(fact_671_totalp__on__singleton,axiom,
    ! [X: product_prod_a_b,R3: product_prod_a_b > product_prod_a_b > $o] : ( totalp2285046867861603700od_a_b @ ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) @ R3 ) ).

% totalp_on_singleton
thf(fact_672_totalp__on__singleton,axiom,
    ! [X: a,R3: a > a > $o] : ( totalp_on_a @ ( insert_a @ X @ bot_bot_set_a ) @ R3 ) ).

% totalp_on_singleton
thf(fact_673_prod__induct6,axiom,
    ! [P: produc1828243372476419385list_a > $o,X: produc1828243372476419385list_a] :
      ( ! [A2: list_a,B2: a,C3: list_a,D2: list_a,E: list_a,F2: list_a] : ( P @ ( produc8961369442823515561list_a @ A2 @ ( produc4562151247771834630list_a @ B2 @ ( produc2289259375147936733list_a @ C3 @ ( produc1910438886824740410list_a @ D2 @ ( produc6837034575241423639list_a @ E @ F2 ) ) ) ) ) )
     => ( P @ X ) ) ).

% prod_induct6
thf(fact_674_prod__induct5,axiom,
    ! [P: produc1828243372476419385list_a > $o,X: produc1828243372476419385list_a] :
      ( ! [A2: list_a,B2: a,C3: list_a,D2: list_a,E: produc9164743771328383783list_a] : ( P @ ( produc8961369442823515561list_a @ A2 @ ( produc4562151247771834630list_a @ B2 @ ( produc2289259375147936733list_a @ C3 @ ( produc1910438886824740410list_a @ D2 @ E ) ) ) ) )
     => ( P @ X ) ) ).

% prod_induct5
thf(fact_675_prod__induct5,axiom,
    ! [P: produc2547433716817190550list_a > $o,X: produc2547433716817190550list_a] :
      ( ! [A2: a,B2: list_a,C3: list_a,D2: list_a,E: list_a] : ( P @ ( produc4562151247771834630list_a @ A2 @ ( produc2289259375147936733list_a @ B2 @ ( produc1910438886824740410list_a @ C3 @ ( produc6837034575241423639list_a @ D2 @ E ) ) ) ) )
     => ( P @ X ) ) ).

% prod_induct5
thf(fact_676_prod__induct4,axiom,
    ! [P: produc1828243372476419385list_a > $o,X: produc1828243372476419385list_a] :
      ( ! [A2: list_a,B2: a,C3: list_a,D2: produc1553995403754578250list_a] : ( P @ ( produc8961369442823515561list_a @ A2 @ ( produc4562151247771834630list_a @ B2 @ ( produc2289259375147936733list_a @ C3 @ D2 ) ) ) )
     => ( P @ X ) ) ).

% prod_induct4
thf(fact_677_prod__induct4,axiom,
    ! [P: produc2547433716817190550list_a > $o,X: produc2547433716817190550list_a] :
      ( ! [A2: a,B2: list_a,C3: list_a,D2: produc9164743771328383783list_a] : ( P @ ( produc4562151247771834630list_a @ A2 @ ( produc2289259375147936733list_a @ B2 @ ( produc1910438886824740410list_a @ C3 @ D2 ) ) ) )
     => ( P @ X ) ) ).

% prod_induct4
thf(fact_678_prod__cases6,axiom,
    ! [Y: produc1828243372476419385list_a] :
      ~ ! [A2: list_a,B2: a,C3: list_a,D2: list_a,E: list_a,F2: list_a] :
          ( Y
         != ( produc8961369442823515561list_a @ A2 @ ( produc4562151247771834630list_a @ B2 @ ( produc2289259375147936733list_a @ C3 @ ( produc1910438886824740410list_a @ D2 @ ( produc6837034575241423639list_a @ E @ F2 ) ) ) ) ) ) ).

% prod_cases6
thf(fact_679_prod__cases5,axiom,
    ! [Y: produc1828243372476419385list_a] :
      ~ ! [A2: list_a,B2: a,C3: list_a,D2: list_a,E: produc9164743771328383783list_a] :
          ( Y
         != ( produc8961369442823515561list_a @ A2 @ ( produc4562151247771834630list_a @ B2 @ ( produc2289259375147936733list_a @ C3 @ ( produc1910438886824740410list_a @ D2 @ E ) ) ) ) ) ).

% prod_cases5
thf(fact_680_prod__cases5,axiom,
    ! [Y: produc2547433716817190550list_a] :
      ~ ! [A2: a,B2: list_a,C3: list_a,D2: list_a,E: list_a] :
          ( Y
         != ( produc4562151247771834630list_a @ A2 @ ( produc2289259375147936733list_a @ B2 @ ( produc1910438886824740410list_a @ C3 @ ( produc6837034575241423639list_a @ D2 @ E ) ) ) ) ) ).

% prod_cases5
thf(fact_681_prod_Oinject,axiom,
    ! [X1: list_a,X22: produc2547433716817190550list_a,Y1: list_a,Y22: produc2547433716817190550list_a] :
      ( ( ( produc8961369442823515561list_a @ X1 @ X22 )
        = ( produc8961369442823515561list_a @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_682_prod_Oinject,axiom,
    ! [X1: list_a,X22: list_a,Y1: list_a,Y22: list_a] :
      ( ( ( produc6837034575241423639list_a @ X1 @ X22 )
        = ( produc6837034575241423639list_a @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_683_prod_Oinject,axiom,
    ! [X1: a > a > $o,X22: produc9164743771328383783list_a,Y1: a > a > $o,Y22: produc9164743771328383783list_a] :
      ( ( ( produc8684083420967988176list_a @ X1 @ X22 )
        = ( produc8684083420967988176list_a @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_684_prod_Oinject,axiom,
    ! [X1: a,X22: produc271001111946445293list_a,Y1: a,Y22: produc271001111946445293list_a] :
      ( ( ( produc4562151247771834630list_a @ X1 @ X22 )
        = ( produc4562151247771834630list_a @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_685_prod_Oinject,axiom,
    ! [X1: a,X22: b,Y1: a,Y22: b] :
      ( ( ( product_Pair_a_b @ X1 @ X22 )
        = ( product_Pair_a_b @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_686_old_Oprod_Oinject,axiom,
    ! [A: list_a,B: produc2547433716817190550list_a,A5: list_a,B5: produc2547433716817190550list_a] :
      ( ( ( produc8961369442823515561list_a @ A @ B )
        = ( produc8961369442823515561list_a @ A5 @ B5 ) )
      = ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_687_old_Oprod_Oinject,axiom,
    ! [A: list_a,B: list_a,A5: list_a,B5: list_a] :
      ( ( ( produc6837034575241423639list_a @ A @ B )
        = ( produc6837034575241423639list_a @ A5 @ B5 ) )
      = ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_688_old_Oprod_Oinject,axiom,
    ! [A: a > a > $o,B: produc9164743771328383783list_a,A5: a > a > $o,B5: produc9164743771328383783list_a] :
      ( ( ( produc8684083420967988176list_a @ A @ B )
        = ( produc8684083420967988176list_a @ A5 @ B5 ) )
      = ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_689_old_Oprod_Oinject,axiom,
    ! [A: a,B: produc271001111946445293list_a,A5: a,B5: produc271001111946445293list_a] :
      ( ( ( produc4562151247771834630list_a @ A @ B )
        = ( produc4562151247771834630list_a @ A5 @ B5 ) )
      = ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_690_old_Oprod_Oinject,axiom,
    ! [A: a,B: b,A5: a,B5: b] :
      ( ( ( product_Pair_a_b @ A @ B )
        = ( product_Pair_a_b @ A5 @ B5 ) )
      = ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_691_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1828243372476419385list_a] :
      ~ ! [A2: list_a,B2: produc2547433716817190550list_a] :
          ( Y
         != ( produc8961369442823515561list_a @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_692_old_Oprod_Oexhaust,axiom,
    ! [Y: produc9164743771328383783list_a] :
      ~ ! [A2: list_a,B2: list_a] :
          ( Y
         != ( produc6837034575241423639list_a @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_693_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1392733710753813600list_a] :
      ~ ! [A2: a > a > $o,B2: produc9164743771328383783list_a] :
          ( Y
         != ( produc8684083420967988176list_a @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_694_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2547433716817190550list_a] :
      ~ ! [A2: a,B2: produc271001111946445293list_a] :
          ( Y
         != ( produc4562151247771834630list_a @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_695_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_a_b] :
      ~ ! [A2: a,B2: b] :
          ( Y
         != ( product_Pair_a_b @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_696_surj__pair,axiom,
    ! [P4: produc1828243372476419385list_a] :
    ? [X3: list_a,Y2: produc2547433716817190550list_a] :
      ( P4
      = ( produc8961369442823515561list_a @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_697_surj__pair,axiom,
    ! [P4: produc9164743771328383783list_a] :
    ? [X3: list_a,Y2: list_a] :
      ( P4
      = ( produc6837034575241423639list_a @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_698_surj__pair,axiom,
    ! [P4: produc1392733710753813600list_a] :
    ? [X3: a > a > $o,Y2: produc9164743771328383783list_a] :
      ( P4
      = ( produc8684083420967988176list_a @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_699_surj__pair,axiom,
    ! [P4: produc2547433716817190550list_a] :
    ? [X3: a,Y2: produc271001111946445293list_a] :
      ( P4
      = ( produc4562151247771834630list_a @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_700_surj__pair,axiom,
    ! [P4: product_prod_a_b] :
    ? [X3: a,Y2: b] :
      ( P4
      = ( product_Pair_a_b @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_701_prod__cases,axiom,
    ! [P: produc1828243372476419385list_a > $o,P4: produc1828243372476419385list_a] :
      ( ! [A2: list_a,B2: produc2547433716817190550list_a] : ( P @ ( produc8961369442823515561list_a @ A2 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_702_prod__cases,axiom,
    ! [P: produc9164743771328383783list_a > $o,P4: produc9164743771328383783list_a] :
      ( ! [A2: list_a,B2: list_a] : ( P @ ( produc6837034575241423639list_a @ A2 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_703_prod__cases,axiom,
    ! [P: produc1392733710753813600list_a > $o,P4: produc1392733710753813600list_a] :
      ( ! [A2: a > a > $o,B2: produc9164743771328383783list_a] : ( P @ ( produc8684083420967988176list_a @ A2 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_704_prod__cases,axiom,
    ! [P: produc2547433716817190550list_a > $o,P4: produc2547433716817190550list_a] :
      ( ! [A2: a,B2: produc271001111946445293list_a] : ( P @ ( produc4562151247771834630list_a @ A2 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_705_prod__cases,axiom,
    ! [P: product_prod_a_b > $o,P4: product_prod_a_b] :
      ( ! [A2: a,B2: b] : ( P @ ( product_Pair_a_b @ A2 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_706_Pair__inject,axiom,
    ! [A: list_a,B: produc2547433716817190550list_a,A5: list_a,B5: produc2547433716817190550list_a] :
      ( ( ( produc8961369442823515561list_a @ A @ B )
        = ( produc8961369442823515561list_a @ A5 @ B5 ) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_707_Pair__inject,axiom,
    ! [A: list_a,B: list_a,A5: list_a,B5: list_a] :
      ( ( ( produc6837034575241423639list_a @ A @ B )
        = ( produc6837034575241423639list_a @ A5 @ B5 ) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_708_Pair__inject,axiom,
    ! [A: a > a > $o,B: produc9164743771328383783list_a,A5: a > a > $o,B5: produc9164743771328383783list_a] :
      ( ( ( produc8684083420967988176list_a @ A @ B )
        = ( produc8684083420967988176list_a @ A5 @ B5 ) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_709_Pair__inject,axiom,
    ! [A: a,B: produc271001111946445293list_a,A5: a,B5: produc271001111946445293list_a] :
      ( ( ( produc4562151247771834630list_a @ A @ B )
        = ( produc4562151247771834630list_a @ A5 @ B5 ) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_710_Pair__inject,axiom,
    ! [A: a,B: b,A5: a,B5: b] :
      ( ( ( product_Pair_a_b @ A @ B )
        = ( product_Pair_a_b @ A5 @ B5 ) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_711_prod__cases3,axiom,
    ! [Y: produc1828243372476419385list_a] :
      ~ ! [A2: list_a,B2: a,C3: produc271001111946445293list_a] :
          ( Y
         != ( produc8961369442823515561list_a @ A2 @ ( produc4562151247771834630list_a @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_712_prod__cases3,axiom,
    ! [Y: produc1392733710753813600list_a] :
      ~ ! [A2: a > a > $o,B2: list_a,C3: list_a] :
          ( Y
         != ( produc8684083420967988176list_a @ A2 @ ( produc6837034575241423639list_a @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_713_prod__cases3,axiom,
    ! [Y: produc2547433716817190550list_a] :
      ~ ! [A2: a,B2: list_a,C3: produc1553995403754578250list_a] :
          ( Y
         != ( produc4562151247771834630list_a @ A2 @ ( produc2289259375147936733list_a @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_714_prod__induct3,axiom,
    ! [P: produc1828243372476419385list_a > $o,X: produc1828243372476419385list_a] :
      ( ! [A2: list_a,B2: a,C3: produc271001111946445293list_a] : ( P @ ( produc8961369442823515561list_a @ A2 @ ( produc4562151247771834630list_a @ B2 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_715_prod__induct3,axiom,
    ! [P: produc1392733710753813600list_a > $o,X: produc1392733710753813600list_a] :
      ( ! [A2: a > a > $o,B2: list_a,C3: list_a] : ( P @ ( produc8684083420967988176list_a @ A2 @ ( produc6837034575241423639list_a @ B2 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_716_prod__induct3,axiom,
    ! [P: produc2547433716817190550list_a > $o,X: produc2547433716817190550list_a] :
      ( ! [A2: a,B2: list_a,C3: produc1553995403754578250list_a] : ( P @ ( produc4562151247771834630list_a @ A2 @ ( produc2289259375147936733list_a @ B2 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_717_prod__cases4,axiom,
    ! [Y: produc1828243372476419385list_a] :
      ~ ! [A2: list_a,B2: a,C3: list_a,D2: produc1553995403754578250list_a] :
          ( Y
         != ( produc8961369442823515561list_a @ A2 @ ( produc4562151247771834630list_a @ B2 @ ( produc2289259375147936733list_a @ C3 @ D2 ) ) ) ) ).

% prod_cases4
thf(fact_718_prod__cases4,axiom,
    ! [Y: produc2547433716817190550list_a] :
      ~ ! [A2: a,B2: list_a,C3: list_a,D2: produc9164743771328383783list_a] :
          ( Y
         != ( produc4562151247771834630list_a @ A2 @ ( produc2289259375147936733list_a @ B2 @ ( produc1910438886824740410list_a @ C3 @ D2 ) ) ) ) ).

% prod_cases4
thf(fact_719_local_Ois__rbt__fold__rbt__delete,axiom,
    ! [T: rBT_rbt_a_b,Xs: list_a] :
      ( ( rBT_is_rbt_a_b @ less @ T )
     => ( rBT_is_rbt_a_b @ less @ ( fold_a_RBT_rbt_a_b @ ( rBT_rbt_delete_a_b @ less ) @ Xs @ T ) ) ) ).

% local.is_rbt_fold_rbt_delete
thf(fact_720_local_Ois__rbt__fold__rbt__insert__impl,axiom,
    ! [T: rBT_rbt_a_b,T3: rBT_rbt_a_b] :
      ( ( rBT_is_rbt_a_b @ less @ T )
     => ( rBT_is_rbt_a_b @ less @ ( rBT_fo372896895925944313bt_a_b @ ( rBT_rbt_insert_a_b @ less ) @ T3 @ T ) ) ) ).

% local.is_rbt_fold_rbt_insert_impl
thf(fact_721_local_Ois__rbt__fold__rbt__insertwk,axiom,
    ! [T1: rBT_rbt_a_b,F: a > b > b > b,T2: rBT_rbt_a_b] :
      ( ( rBT_is_rbt_a_b @ less @ T1 )
     => ( rBT_is_rbt_a_b @ less @ ( rBT_fo372896895925944313bt_a_b @ ( rBT_rb2461657779293087990ey_a_b @ less @ F ) @ T2 @ T1 ) ) ) ).

% local.is_rbt_fold_rbt_insertwk
thf(fact_722_local_Orbt__sorted__fold__insert,axiom,
    ! [T: rBT_rbt_a_b,T3: rBT_rbt_a_b] :
      ( ( rBT_rbt_sorted_a_b @ less @ T )
     => ( rBT_rbt_sorted_a_b @ less @ ( rBT_fo372896895925944313bt_a_b @ ( rBT_rbt_insert_a_b @ less ) @ T3 @ T ) ) ) ).

% local.rbt_sorted_fold_insert
thf(fact_723_bot__empty__eq,axiom,
    ( bot_bo1372834404344936809_a_b_o
    = ( ^ [X2: product_prod_a_b] : ( member1426531481828664017od_a_b @ X2 @ bot_bo3428410327501040244od_a_b ) ) ) ).

% bot_empty_eq
thf(fact_724_bot__empty__eq,axiom,
    ( bot_bot_a_o
    = ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).

% bot_empty_eq
thf(fact_725_bot__set__def,axiom,
    ( bot_bo3428410327501040244od_a_b
    = ( collec3336397801687681299od_a_b @ bot_bo1372834404344936809_a_b_o ) ) ).

% bot_set_def
thf(fact_726_bot__set__def,axiom,
    ( bot_bot_set_a
    = ( collect_a @ bot_bot_a_o ) ) ).

% bot_set_def
thf(fact_727_local_Orbt__lookup__fold__rbt__insert__impl,axiom,
    ! [T2: rBT_rbt_a_b,T1: rBT_rbt_a_b] :
      ( ( rBT_rbt_sorted_a_b @ less @ T2 )
     => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_fo372896895925944313bt_a_b @ ( rBT_rbt_insert_a_b @ less ) @ T1 @ T2 ) )
        = ( map_add_a_b @ ( rBT_rbt_lookup_a_b @ less @ T2 ) @ ( map_of_a_b @ ( rev_Product_prod_a_b @ ( rBT_entries_a_b @ T1 ) ) ) ) ) ) ).

% local.rbt_lookup_fold_rbt_insert_impl
thf(fact_728_local_Oantisymp__on__less,axiom,
    ! [A3: set_a] : ( antisymp_on_a @ A3 @ less ) ).

% local.antisymp_on_less
thf(fact_729_is__singletonI,axiom,
    ! [X: product_prod_a_b] : ( is_sin3171834910201899932od_a_b @ ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) ) ).

% is_singletonI
thf(fact_730_is__singletonI,axiom,
    ! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).

% is_singletonI
thf(fact_731_set__rev,axiom,
    ! [Xs: list_P1467974523212632366od_a_b] :
      ( ( set_Product_prod_a_b2 @ ( rev_Product_prod_a_b @ Xs ) )
      = ( set_Product_prod_a_b2 @ Xs ) ) ).

% set_rev
thf(fact_732_set__rev,axiom,
    ! [Xs: list_a] :
      ( ( set_a2 @ ( rev_a @ Xs ) )
      = ( set_a2 @ Xs ) ) ).

% set_rev
thf(fact_733_antisymp__onD,axiom,
    ! [A3: set_a,R3: a > a > $o,X: a,Y: a] :
      ( ( antisymp_on_a @ A3 @ R3 )
     => ( ( member_a @ X @ A3 )
       => ( ( member_a @ Y @ A3 )
         => ( ( R3 @ X @ Y )
           => ( ( R3 @ Y @ X )
             => ( X = Y ) ) ) ) ) ) ).

% antisymp_onD
thf(fact_734_antisymp__onI,axiom,
    ! [A3: set_a,R3: a > a > $o] :
      ( ! [X3: a,Y2: a] :
          ( ( member_a @ X3 @ A3 )
         => ( ( member_a @ Y2 @ A3 )
           => ( ( R3 @ X3 @ Y2 )
             => ( ( R3 @ Y2 @ X3 )
               => ( X3 = Y2 ) ) ) ) )
     => ( antisymp_on_a @ A3 @ R3 ) ) ).

% antisymp_onI
thf(fact_735_antisymp__on__def,axiom,
    ( antisymp_on_a
    = ( ^ [A4: set_a,R4: a > a > $o] :
        ! [X2: a] :
          ( ( member_a @ X2 @ A4 )
         => ! [Y3: a] :
              ( ( member_a @ Y3 @ A4 )
             => ( ( R4 @ X2 @ Y3 )
               => ( ( R4 @ Y3 @ X2 )
                 => ( X2 = Y3 ) ) ) ) ) ) ) ).

% antisymp_on_def
thf(fact_736_antisymp__on__if__asymp__on,axiom,
    ! [A3: set_a,R3: a > a > $o] :
      ( ( asymp_on_a @ A3 @ R3 )
     => ( antisymp_on_a @ A3 @ R3 ) ) ).

% antisymp_on_if_asymp_on
thf(fact_737_is__singletonI_H,axiom,
    ! [A3: set_Product_prod_a_b] :
      ( ( A3 != bot_bo3428410327501040244od_a_b )
     => ( ! [X3: product_prod_a_b,Y2: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X3 @ A3 )
           => ( ( member1426531481828664017od_a_b @ Y2 @ A3 )
             => ( X3 = Y2 ) ) )
       => ( is_sin3171834910201899932od_a_b @ A3 ) ) ) ).

% is_singletonI'
thf(fact_738_is__singletonI_H,axiom,
    ! [A3: set_a] :
      ( ( A3 != bot_bot_set_a )
     => ( ! [X3: a,Y2: a] :
            ( ( member_a @ X3 @ A3 )
           => ( ( member_a @ Y2 @ A3 )
             => ( X3 = Y2 ) ) )
       => ( is_singleton_a @ A3 ) ) ) ).

% is_singletonI'
thf(fact_739_is__singletonE,axiom,
    ! [A3: set_Product_prod_a_b] :
      ( ( is_sin3171834910201899932od_a_b @ A3 )
     => ~ ! [X3: product_prod_a_b] :
            ( A3
           != ( insert4534936386344385144od_a_b @ X3 @ bot_bo3428410327501040244od_a_b ) ) ) ).

% is_singletonE
thf(fact_740_is__singletonE,axiom,
    ! [A3: set_a] :
      ( ( is_singleton_a @ A3 )
     => ~ ! [X3: a] :
            ( A3
           != ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).

% is_singletonE
thf(fact_741_is__singleton__def,axiom,
    ( is_sin3171834910201899932od_a_b
    = ( ^ [A4: set_Product_prod_a_b] :
        ? [X2: product_prod_a_b] :
          ( A4
          = ( insert4534936386344385144od_a_b @ X2 @ bot_bo3428410327501040244od_a_b ) ) ) ) ).

% is_singleton_def
thf(fact_742_is__singleton__def,axiom,
    ( is_singleton_a
    = ( ^ [A4: set_a] :
        ? [X2: a] :
          ( A4
          = ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).

% is_singleton_def
thf(fact_743_local_Orbt__lookup__iff__keys_I2_J,axiom,
    ! [T: rBT_rbt_a_b,K: a] :
      ( ( rBT_rbt_sorted_a_b @ less @ T )
     => ( ( ( rBT_rbt_lookup_a_b @ less @ T @ K )
          = none_b )
        = ( ~ ( member_a @ K @ ( set_a2 @ ( rBT_keys_a_b @ T ) ) ) ) ) ) ).

% local.rbt_lookup_iff_keys(2)
thf(fact_744_local_Orbt__lookup__iff__keys_I3_J,axiom,
    ! [T: rBT_rbt_a_b,K: a] :
      ( ( rBT_rbt_sorted_a_b @ less @ T )
     => ( ( ? [V4: b] :
              ( ( rBT_rbt_lookup_a_b @ less @ T @ K )
              = ( some_b @ V4 ) ) )
        = ( member_a @ K @ ( set_a2 @ ( rBT_keys_a_b @ T ) ) ) ) ) ).

% local.rbt_lookup_iff_keys(3)
thf(fact_745_Collect__empty__eq__bot,axiom,
    ! [P: product_prod_a_b > $o] :
      ( ( ( collec3336397801687681299od_a_b @ P )
        = bot_bo3428410327501040244od_a_b )
      = ( P = bot_bo1372834404344936809_a_b_o ) ) ).

% Collect_empty_eq_bot
thf(fact_746_Collect__empty__eq__bot,axiom,
    ! [P: a > $o] :
      ( ( ( collect_a @ P )
        = bot_bot_set_a )
      = ( P = bot_bot_a_o ) ) ).

% Collect_empty_eq_bot
thf(fact_747_local_Orbt__lookup__rbt__greater,axiom,
    ! [K: a,T: rBT_rbt_a_b] :
      ( ( rBT_rbt_greater_a_b @ less @ K @ T )
     => ( ( rBT_rbt_lookup_a_b @ less @ T @ K )
        = none_b ) ) ).

% local.rbt_lookup_rbt_greater
thf(fact_748_ord_Orbt__lookup__rbt__greater,axiom,
    ! [Less: a > a > $o,K: a,T: rBT_rbt_a_b] :
      ( ( rBT_rbt_greater_a_b @ Less @ K @ T )
     => ( ( rBT_rbt_lookup_a_b @ Less @ T @ K )
        = none_b ) ) ).

% ord.rbt_lookup_rbt_greater
thf(fact_749_entry__in__tree__keys,axiom,
    ! [K: list_a,V: produc2547433716817190550list_a,T: rBT_rb6721489163287533103list_a] :
      ( ( member5573719349037835490list_a @ ( produc8961369442823515561list_a @ K @ V ) @ ( set_Pr7132271252953855310list_a @ ( rBT_en3328903274962959370list_a @ T ) ) )
     => ( member_list_a @ K @ ( set_list_a2 @ ( rBT_ke8879216618931534142list_a @ T ) ) ) ) ).

% entry_in_tree_keys
thf(fact_750_entry__in__tree__keys,axiom,
    ! [K: list_a,V: list_a,T: rBT_rb146355364677272861list_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ K @ V ) @ ( set_Pr2906193453920572092list_a @ ( rBT_en6256609917331000056list_a @ T ) ) )
     => ( member_list_a @ K @ ( set_list_a2 @ ( rBT_ke4638300239729038892list_a @ T ) ) ) ) ).

% entry_in_tree_keys
thf(fact_751_entry__in__tree__keys,axiom,
    ! [K: a > a > $o,V: produc9164743771328383783list_a,T: rBT_rb9014005217275227734list_a] :
      ( ( member8740110733080369545list_a @ ( produc8684083420967988176list_a @ K @ V ) @ ( set_Pr6051718963112299509list_a @ ( rBT_en1829592632444517297list_a @ T ) ) )
     => ( member_a_a_o @ K @ ( set_a_a_o2 @ ( rBT_ke5174640127303026405list_a @ T ) ) ) ) ).

% entry_in_tree_keys
thf(fact_752_entry__in__tree__keys,axiom,
    ! [K: a,V: produc271001111946445293list_a,T: rBT_rb484615002580549772list_a] :
      ( ( member852939521359900607list_a @ ( produc4562151247771834630list_a @ K @ V ) @ ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ T ) ) )
     => ( member_a @ K @ ( set_a2 @ ( rBT_ke6113686706786570267list_a @ T ) ) ) ) ).

% entry_in_tree_keys
thf(fact_753_entry__in__tree__keys,axiom,
    ! [K: a,V: b,T: rBT_rbt_a_b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K @ V ) @ ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ T ) ) )
     => ( member_a @ K @ ( set_a2 @ ( rBT_keys_a_b @ T ) ) ) ) ).

% entry_in_tree_keys
thf(fact_754_keys__entries,axiom,
    ! [K: list_a,T: rBT_rb6721489163287533103list_a] :
      ( ( member_list_a @ K @ ( set_list_a2 @ ( rBT_ke8879216618931534142list_a @ T ) ) )
      = ( ? [V4: produc2547433716817190550list_a] : ( member5573719349037835490list_a @ ( produc8961369442823515561list_a @ K @ V4 ) @ ( set_Pr7132271252953855310list_a @ ( rBT_en3328903274962959370list_a @ T ) ) ) ) ) ).

% keys_entries
thf(fact_755_keys__entries,axiom,
    ! [K: list_a,T: rBT_rb146355364677272861list_a] :
      ( ( member_list_a @ K @ ( set_list_a2 @ ( rBT_ke4638300239729038892list_a @ T ) ) )
      = ( ? [V4: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ K @ V4 ) @ ( set_Pr2906193453920572092list_a @ ( rBT_en6256609917331000056list_a @ T ) ) ) ) ) ).

% keys_entries
thf(fact_756_keys__entries,axiom,
    ! [K: a > a > $o,T: rBT_rb9014005217275227734list_a] :
      ( ( member_a_a_o @ K @ ( set_a_a_o2 @ ( rBT_ke5174640127303026405list_a @ T ) ) )
      = ( ? [V4: produc9164743771328383783list_a] : ( member8740110733080369545list_a @ ( produc8684083420967988176list_a @ K @ V4 ) @ ( set_Pr6051718963112299509list_a @ ( rBT_en1829592632444517297list_a @ T ) ) ) ) ) ).

% keys_entries
thf(fact_757_keys__entries,axiom,
    ! [K: a,T: rBT_rb484615002580549772list_a] :
      ( ( member_a @ K @ ( set_a2 @ ( rBT_ke6113686706786570267list_a @ T ) ) )
      = ( ? [V4: produc271001111946445293list_a] : ( member852939521359900607list_a @ ( produc4562151247771834630list_a @ K @ V4 ) @ ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ T ) ) ) ) ) ).

% keys_entries
thf(fact_758_keys__entries,axiom,
    ! [K: a,T: rBT_rbt_a_b] :
      ( ( member_a @ K @ ( set_a2 @ ( rBT_keys_a_b @ T ) ) )
      = ( ? [V4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K @ V4 ) @ ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ T ) ) ) ) ) ).

% keys_entries
thf(fact_759_local_Orbt__lookup__rbt__join,axiom,
    ! [L: rBT_rbt_a_b,R: rBT_rbt_a_b,A: a,K: a,B: b] :
      ( ( rBT_rbt_sorted_a_b @ less @ L )
     => ( ( rBT_rbt_sorted_a_b @ less @ R )
       => ( ( rBT_rbt_less_a_b @ less @ A @ L )
         => ( ( rBT_rbt_greater_a_b @ less @ A @ R )
           => ( ( ( less @ K @ A )
               => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_join_a_b @ L @ A @ B @ R ) @ K )
                  = ( rBT_rbt_lookup_a_b @ less @ L @ K ) ) )
              & ( ~ ( less @ K @ A )
               => ( ( ( K = A )
                   => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_join_a_b @ L @ A @ B @ R ) @ K )
                      = ( some_b @ B ) ) )
                  & ( ( K != A )
                   => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_join_a_b @ L @ A @ B @ R ) @ K )
                      = ( rBT_rbt_lookup_a_b @ less @ R @ K ) ) ) ) ) ) ) ) ) ) ).

% local.rbt_lookup_rbt_join
thf(fact_760_neq__Black,axiom,
    ! [C: rBT_color] :
      ( ( C != rBT_B )
      = ( C = rBT_R ) ) ).

% neq_Black
thf(fact_761_local_Orbt__lookup__rbt__less,axiom,
    ! [K: a,T: rBT_rbt_a_b] :
      ( ( rBT_rbt_less_a_b @ less @ K @ T )
     => ( ( rBT_rbt_lookup_a_b @ less @ T @ K )
        = none_b ) ) ).

% local.rbt_lookup_rbt_less
thf(fact_762_color_Odistinct_I1_J,axiom,
    rBT_R != rBT_B ).

% color.distinct(1)
thf(fact_763_color_Oexhaust,axiom,
    ! [Y: rBT_color] :
      ( ( Y != rBT_R )
     => ( Y = rBT_B ) ) ).

% color.exhaust
thf(fact_764_ord_Orbt__lookup__rbt__less,axiom,
    ! [Less: a > a > $o,K: a,T: rBT_rbt_a_b] :
      ( ( rBT_rbt_less_a_b @ Less @ K @ T )
     => ( ( rBT_rbt_lookup_a_b @ Less @ T @ K )
        = none_b ) ) ).

% ord.rbt_lookup_rbt_less
thf(fact_765_local_Orbt__lookup__rbt__joinL,axiom,
    ! [L: rBT_rbt_a_b,R: rBT_rbt_a_b,A: a,K: a,B: b] :
      ( ( rBT_rbt_sorted_a_b @ less @ L )
     => ( ( rBT_rbt_sorted_a_b @ less @ R )
       => ( ( rBT_rbt_less_a_b @ less @ A @ L )
         => ( ( rBT_rbt_greater_a_b @ less @ A @ R )
           => ( ( ( less @ K @ A )
               => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_joinL_a_b @ L @ A @ B @ R ) @ K )
                  = ( rBT_rbt_lookup_a_b @ less @ L @ K ) ) )
              & ( ~ ( less @ K @ A )
               => ( ( ( K = A )
                   => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_joinL_a_b @ L @ A @ B @ R ) @ K )
                      = ( some_b @ B ) ) )
                  & ( ( K != A )
                   => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_joinL_a_b @ L @ A @ B @ R ) @ K )
                      = ( rBT_rbt_lookup_a_b @ less @ R @ K ) ) ) ) ) ) ) ) ) ) ).

% local.rbt_lookup_rbt_joinL
thf(fact_766_local_Orbt__lookup__rbt__joinR,axiom,
    ! [L: rBT_rbt_a_b,R: rBT_rbt_a_b,A: a,K: a,B: b] :
      ( ( rBT_rbt_sorted_a_b @ less @ L )
     => ( ( rBT_rbt_sorted_a_b @ less @ R )
       => ( ( rBT_rbt_less_a_b @ less @ A @ L )
         => ( ( rBT_rbt_greater_a_b @ less @ A @ R )
           => ( ( ( less @ K @ A )
               => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_joinR_a_b @ L @ A @ B @ R ) @ K )
                  = ( rBT_rbt_lookup_a_b @ less @ L @ K ) ) )
              & ( ~ ( less @ K @ A )
               => ( ( ( K = A )
                   => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_joinR_a_b @ L @ A @ B @ R ) @ K )
                      = ( some_b @ B ) ) )
                  & ( ( K != A )
                   => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_joinR_a_b @ L @ A @ B @ R ) @ K )
                      = ( rBT_rbt_lookup_a_b @ less @ R @ K ) ) ) ) ) ) ) ) ) ) ).

% local.rbt_lookup_rbt_joinR
thf(fact_767_local_Orbt__lookup__rbt__baliL,axiom,
    ! [L: rBT_rbt_a_b,R: rBT_rbt_a_b,A: a,K: a,B: b] :
      ( ( rBT_rbt_sorted_a_b @ less @ L )
     => ( ( rBT_rbt_sorted_a_b @ less @ R )
       => ( ( rBT_rbt_less_a_b @ less @ A @ L )
         => ( ( rBT_rbt_greater_a_b @ less @ A @ R )
           => ( ( ( less @ K @ A )
               => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_baliL_a_b @ L @ A @ B @ R ) @ K )
                  = ( rBT_rbt_lookup_a_b @ less @ L @ K ) ) )
              & ( ~ ( less @ K @ A )
               => ( ( ( K = A )
                   => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_baliL_a_b @ L @ A @ B @ R ) @ K )
                      = ( some_b @ B ) ) )
                  & ( ( K != A )
                   => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_baliL_a_b @ L @ A @ B @ R ) @ K )
                      = ( rBT_rbt_lookup_a_b @ less @ R @ K ) ) ) ) ) ) ) ) ) ) ).

% local.rbt_lookup_rbt_baliL
thf(fact_768_local_Orbt__lookup__rbt__baliR,axiom,
    ! [L: rBT_rbt_a_b,R: rBT_rbt_a_b,A: a,K: a,B: b] :
      ( ( rBT_rbt_sorted_a_b @ less @ L )
     => ( ( rBT_rbt_sorted_a_b @ less @ R )
       => ( ( rBT_rbt_less_a_b @ less @ A @ L )
         => ( ( rBT_rbt_greater_a_b @ less @ A @ R )
           => ( ( ( less @ K @ A )
               => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_baliR_a_b @ L @ A @ B @ R ) @ K )
                  = ( rBT_rbt_lookup_a_b @ less @ L @ K ) ) )
              & ( ~ ( less @ K @ A )
               => ( ( ( K = A )
                   => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_baliR_a_b @ L @ A @ B @ R ) @ K )
                      = ( some_b @ B ) ) )
                  & ( ( K != A )
                   => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_baliR_a_b @ L @ A @ B @ R ) @ K )
                      = ( rBT_rbt_lookup_a_b @ less @ R @ K ) ) ) ) ) ) ) ) ) ) ).

% local.rbt_lookup_rbt_baliR
thf(fact_769_Un__iff,axiom,
    ! [C: a,A3: set_a,B3: set_a] :
      ( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) )
      = ( ( member_a @ C @ A3 )
        | ( member_a @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_770_UnCI,axiom,
    ! [C: a,B3: set_a,A3: set_a] :
      ( ( ~ ( member_a @ C @ B3 )
       => ( member_a @ C @ A3 ) )
     => ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_771_sup__bot__left,axiom,
    ! [X: set_Product_prod_a_b] :
      ( ( sup_su3119292821646003676od_a_b @ bot_bo3428410327501040244od_a_b @ X )
      = X ) ).

% sup_bot_left
thf(fact_772_sup__bot__left,axiom,
    ! [X: set_a] :
      ( ( sup_sup_set_a @ bot_bot_set_a @ X )
      = X ) ).

% sup_bot_left
thf(fact_773_sup__bot__right,axiom,
    ! [X: set_Product_prod_a_b] :
      ( ( sup_su3119292821646003676od_a_b @ X @ bot_bo3428410327501040244od_a_b )
      = X ) ).

% sup_bot_right
thf(fact_774_sup__bot__right,axiom,
    ! [X: set_a] :
      ( ( sup_sup_set_a @ X @ bot_bot_set_a )
      = X ) ).

% sup_bot_right
thf(fact_775_bot__eq__sup__iff,axiom,
    ! [X: set_Product_prod_a_b,Y: set_Product_prod_a_b] :
      ( ( bot_bo3428410327501040244od_a_b
        = ( sup_su3119292821646003676od_a_b @ X @ Y ) )
      = ( ( X = bot_bo3428410327501040244od_a_b )
        & ( Y = bot_bo3428410327501040244od_a_b ) ) ) ).

% bot_eq_sup_iff
thf(fact_776_bot__eq__sup__iff,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( bot_bot_set_a
        = ( sup_sup_set_a @ X @ Y ) )
      = ( ( X = bot_bot_set_a )
        & ( Y = bot_bot_set_a ) ) ) ).

% bot_eq_sup_iff
thf(fact_777_sup__eq__bot__iff,axiom,
    ! [X: set_Product_prod_a_b,Y: set_Product_prod_a_b] :
      ( ( ( sup_su3119292821646003676od_a_b @ X @ Y )
        = bot_bo3428410327501040244od_a_b )
      = ( ( X = bot_bo3428410327501040244od_a_b )
        & ( Y = bot_bo3428410327501040244od_a_b ) ) ) ).

% sup_eq_bot_iff
thf(fact_778_sup__eq__bot__iff,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( ( sup_sup_set_a @ X @ Y )
        = bot_bot_set_a )
      = ( ( X = bot_bot_set_a )
        & ( Y = bot_bot_set_a ) ) ) ).

% sup_eq_bot_iff
thf(fact_779_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Product_prod_a_b,B: set_Product_prod_a_b] :
      ( ( ( sup_su3119292821646003676od_a_b @ A @ B )
        = bot_bo3428410327501040244od_a_b )
      = ( ( A = bot_bo3428410327501040244od_a_b )
        & ( B = bot_bo3428410327501040244od_a_b ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_780_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ( sup_sup_set_a @ A @ B )
        = bot_bot_set_a )
      = ( ( A = bot_bot_set_a )
        & ( B = bot_bot_set_a ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_781_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Product_prod_a_b] :
      ( ( sup_su3119292821646003676od_a_b @ bot_bo3428410327501040244od_a_b @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_782_sup__bot_Oleft__neutral,axiom,
    ! [A: set_a] :
      ( ( sup_sup_set_a @ bot_bot_set_a @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_783_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Product_prod_a_b,B: set_Product_prod_a_b] :
      ( ( bot_bo3428410327501040244od_a_b
        = ( sup_su3119292821646003676od_a_b @ A @ B ) )
      = ( ( A = bot_bo3428410327501040244od_a_b )
        & ( B = bot_bo3428410327501040244od_a_b ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_784_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_a,B: set_a] :
      ( ( bot_bot_set_a
        = ( sup_sup_set_a @ A @ B ) )
      = ( ( A = bot_bot_set_a )
        & ( B = bot_bot_set_a ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_785_sup__bot_Oright__neutral,axiom,
    ! [A: set_Product_prod_a_b] :
      ( ( sup_su3119292821646003676od_a_b @ A @ bot_bo3428410327501040244od_a_b )
      = A ) ).

% sup_bot.right_neutral
thf(fact_786_sup__bot_Oright__neutral,axiom,
    ! [A: set_a] :
      ( ( sup_sup_set_a @ A @ bot_bot_set_a )
      = A ) ).

% sup_bot.right_neutral
thf(fact_787_Un__empty,axiom,
    ! [A3: set_Product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ( ( sup_su3119292821646003676od_a_b @ A3 @ B3 )
        = bot_bo3428410327501040244od_a_b )
      = ( ( A3 = bot_bo3428410327501040244od_a_b )
        & ( B3 = bot_bo3428410327501040244od_a_b ) ) ) ).

% Un_empty
thf(fact_788_Un__empty,axiom,
    ! [A3: set_a,B3: set_a] :
      ( ( ( sup_sup_set_a @ A3 @ B3 )
        = bot_bot_set_a )
      = ( ( A3 = bot_bot_set_a )
        & ( B3 = bot_bot_set_a ) ) ) ).

% Un_empty
thf(fact_789_Un__insert__right,axiom,
    ! [A3: set_Product_prod_a_b,A: product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ( sup_su3119292821646003676od_a_b @ A3 @ ( insert4534936386344385144od_a_b @ A @ B3 ) )
      = ( insert4534936386344385144od_a_b @ A @ ( sup_su3119292821646003676od_a_b @ A3 @ B3 ) ) ) ).

% Un_insert_right
thf(fact_790_Un__insert__right,axiom,
    ! [A3: set_a,A: a,B3: set_a] :
      ( ( sup_sup_set_a @ A3 @ ( insert_a @ A @ B3 ) )
      = ( insert_a @ A @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).

% Un_insert_right
thf(fact_791_Un__insert__left,axiom,
    ! [A: product_prod_a_b,B3: set_Product_prod_a_b,C4: set_Product_prod_a_b] :
      ( ( sup_su3119292821646003676od_a_b @ ( insert4534936386344385144od_a_b @ A @ B3 ) @ C4 )
      = ( insert4534936386344385144od_a_b @ A @ ( sup_su3119292821646003676od_a_b @ B3 @ C4 ) ) ) ).

% Un_insert_left
thf(fact_792_Un__insert__left,axiom,
    ! [A: a,B3: set_a,C4: set_a] :
      ( ( sup_sup_set_a @ ( insert_a @ A @ B3 ) @ C4 )
      = ( insert_a @ A @ ( sup_sup_set_a @ B3 @ C4 ) ) ) ).

% Un_insert_left
thf(fact_793_Un__Diff__cancel2,axiom,
    ! [B3: set_Product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ( sup_su3119292821646003676od_a_b @ ( minus_6888070959853530735od_a_b @ B3 @ A3 ) @ A3 )
      = ( sup_su3119292821646003676od_a_b @ B3 @ A3 ) ) ).

% Un_Diff_cancel2
thf(fact_794_Un__Diff__cancel2,axiom,
    ! [B3: set_a,A3: set_a] :
      ( ( sup_sup_set_a @ ( minus_minus_set_a @ B3 @ A3 ) @ A3 )
      = ( sup_sup_set_a @ B3 @ A3 ) ) ).

% Un_Diff_cancel2
thf(fact_795_Un__Diff__cancel,axiom,
    ! [A3: set_Product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ( sup_su3119292821646003676od_a_b @ A3 @ ( minus_6888070959853530735od_a_b @ B3 @ A3 ) )
      = ( sup_su3119292821646003676od_a_b @ A3 @ B3 ) ) ).

% Un_Diff_cancel
thf(fact_796_Un__Diff__cancel,axiom,
    ! [A3: set_a,B3: set_a] :
      ( ( sup_sup_set_a @ A3 @ ( minus_minus_set_a @ B3 @ A3 ) )
      = ( sup_sup_set_a @ A3 @ B3 ) ) ).

% Un_Diff_cancel
thf(fact_797_local_Opaint__rbt__lookup,axiom,
    ! [C: rBT_color,T: rBT_rbt_a_b] :
      ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_paint_a_b @ C @ T ) )
      = ( rBT_rbt_lookup_a_b @ less @ T ) ) ).

% local.paint_rbt_lookup
thf(fact_798_paint__in__tree,axiom,
    ! [K: list_a,X: produc2547433716817190550list_a,C: rBT_color,T: rBT_rb6721489163287533103list_a] :
      ( ( member5573719349037835490list_a @ ( produc8961369442823515561list_a @ K @ X ) @ ( set_Pr7132271252953855310list_a @ ( rBT_en3328903274962959370list_a @ ( rBT_pa6697467494130406808list_a @ C @ T ) ) ) )
      = ( member5573719349037835490list_a @ ( produc8961369442823515561list_a @ K @ X ) @ ( set_Pr7132271252953855310list_a @ ( rBT_en3328903274962959370list_a @ T ) ) ) ) ).

% paint_in_tree
thf(fact_799_paint__in__tree,axiom,
    ! [K: list_a,X: list_a,C: rBT_color,T: rBT_rb146355364677272861list_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ K @ X ) @ ( set_Pr2906193453920572092list_a @ ( rBT_en6256609917331000056list_a @ ( rBT_pa6702398345666083206list_a @ C @ T ) ) ) )
      = ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ K @ X ) @ ( set_Pr2906193453920572092list_a @ ( rBT_en6256609917331000056list_a @ T ) ) ) ) ).

% paint_in_tree
thf(fact_800_paint__in__tree,axiom,
    ! [K: a > a > $o,X: produc9164743771328383783list_a,C: rBT_color,T: rBT_rb9014005217275227734list_a] :
      ( ( member8740110733080369545list_a @ ( produc8684083420967988176list_a @ K @ X ) @ ( set_Pr6051718963112299509list_a @ ( rBT_en1829592632444517297list_a @ ( rBT_pa804577256866744895list_a @ C @ T ) ) ) )
      = ( member8740110733080369545list_a @ ( produc8684083420967988176list_a @ K @ X ) @ ( set_Pr6051718963112299509list_a @ ( rBT_en1829592632444517297list_a @ T ) ) ) ) ).

% paint_in_tree
thf(fact_801_paint__in__tree,axiom,
    ! [K: a,X: produc271001111946445293list_a,C: rBT_color,T: rBT_rb484615002580549772list_a] :
      ( ( member852939521359900607list_a @ ( produc4562151247771834630list_a @ K @ X ) @ ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ ( rBT_pa7206040881279535989list_a @ C @ T ) ) ) )
      = ( member852939521359900607list_a @ ( produc4562151247771834630list_a @ K @ X ) @ ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ T ) ) ) ) ).

% paint_in_tree
thf(fact_802_paint__in__tree,axiom,
    ! [K: a,X: b,C: rBT_color,T: rBT_rbt_a_b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K @ X ) @ ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ ( rBT_paint_a_b @ C @ T ) ) ) )
      = ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K @ X ) @ ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ T ) ) ) ) ).

% paint_in_tree
thf(fact_803_local_Orbt__lookup__balance,axiom,
    ! [L: rBT_rbt_a_b,R: rBT_rbt_a_b,K: a,V: b,X: a] :
      ( ( rBT_rbt_sorted_a_b @ less @ L )
     => ( ( rBT_rbt_sorted_a_b @ less @ R )
       => ( ( rBT_rbt_less_a_b @ less @ K @ L )
         => ( ( rBT_rbt_greater_a_b @ less @ K @ R )
           => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_balance_a_b @ L @ K @ V @ R ) @ X )
              = ( rBT_rbt_lookup_a_b @ less @ ( rBT_Branch_a_b @ rBT_B @ L @ K @ V @ R ) @ X ) ) ) ) ) ) ).

% local.rbt_lookup_balance
thf(fact_804_Un__empty__left,axiom,
    ! [B3: set_Product_prod_a_b] :
      ( ( sup_su3119292821646003676od_a_b @ bot_bo3428410327501040244od_a_b @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_805_Un__empty__left,axiom,
    ! [B3: set_a] :
      ( ( sup_sup_set_a @ bot_bot_set_a @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_806_Un__empty__right,axiom,
    ! [A3: set_Product_prod_a_b] :
      ( ( sup_su3119292821646003676od_a_b @ A3 @ bot_bo3428410327501040244od_a_b )
      = A3 ) ).

% Un_empty_right
thf(fact_807_Un__empty__right,axiom,
    ! [A3: set_a] :
      ( ( sup_sup_set_a @ A3 @ bot_bot_set_a )
      = A3 ) ).

% Un_empty_right
thf(fact_808_Un__Diff,axiom,
    ! [A3: set_Product_prod_a_b,B3: set_Product_prod_a_b,C4: set_Product_prod_a_b] :
      ( ( minus_6888070959853530735od_a_b @ ( sup_su3119292821646003676od_a_b @ A3 @ B3 ) @ C4 )
      = ( sup_su3119292821646003676od_a_b @ ( minus_6888070959853530735od_a_b @ A3 @ C4 ) @ ( minus_6888070959853530735od_a_b @ B3 @ C4 ) ) ) ).

% Un_Diff
thf(fact_809_Un__Diff,axiom,
    ! [A3: set_a,B3: set_a,C4: set_a] :
      ( ( minus_minus_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C4 )
      = ( sup_sup_set_a @ ( minus_minus_set_a @ A3 @ C4 ) @ ( minus_minus_set_a @ B3 @ C4 ) ) ) ).

% Un_Diff
thf(fact_810_Un__left__commute,axiom,
    ! [A3: set_a,B3: set_a,C4: set_a] :
      ( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C4 ) )
      = ( sup_sup_set_a @ B3 @ ( sup_sup_set_a @ A3 @ C4 ) ) ) ).

% Un_left_commute
thf(fact_811_Un__left__absorb,axiom,
    ! [A3: set_a,B3: set_a] :
      ( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B3 ) )
      = ( sup_sup_set_a @ A3 @ B3 ) ) ).

% Un_left_absorb
thf(fact_812_Un__commute,axiom,
    ( sup_sup_set_a
    = ( ^ [A4: set_a,B6: set_a] : ( sup_sup_set_a @ B6 @ A4 ) ) ) ).

% Un_commute
thf(fact_813_Un__absorb,axiom,
    ! [A3: set_a] :
      ( ( sup_sup_set_a @ A3 @ A3 )
      = A3 ) ).

% Un_absorb
thf(fact_814_Un__assoc,axiom,
    ! [A3: set_a,B3: set_a,C4: set_a] :
      ( ( sup_sup_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C4 )
      = ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C4 ) ) ) ).

% Un_assoc
thf(fact_815_ball__Un,axiom,
    ! [A3: set_a,B3: set_a,P: a > $o] :
      ( ( ! [X2: a] :
            ( ( member_a @ X2 @ ( sup_sup_set_a @ A3 @ B3 ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: a] :
            ( ( member_a @ X2 @ A3 )
           => ( P @ X2 ) )
        & ! [X2: a] :
            ( ( member_a @ X2 @ B3 )
           => ( P @ X2 ) ) ) ) ).

% ball_Un
thf(fact_816_bex__Un,axiom,
    ! [A3: set_a,B3: set_a,P: a > $o] :
      ( ( ? [X2: a] :
            ( ( member_a @ X2 @ ( sup_sup_set_a @ A3 @ B3 ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: a] :
            ( ( member_a @ X2 @ A3 )
            & ( P @ X2 ) )
        | ? [X2: a] :
            ( ( member_a @ X2 @ B3 )
            & ( P @ X2 ) ) ) ) ).

% bex_Un
thf(fact_817_UnI2,axiom,
    ! [C: a,B3: set_a,A3: set_a] :
      ( ( member_a @ C @ B3 )
     => ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_818_UnI1,axiom,
    ! [C: a,A3: set_a,B3: set_a] :
      ( ( member_a @ C @ A3 )
     => ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_819_UnE,axiom,
    ! [C: a,A3: set_a,B3: set_a] :
      ( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) )
     => ( ~ ( member_a @ C @ A3 )
       => ( member_a @ C @ B3 ) ) ) ).

% UnE
thf(fact_820_ord_Opaint__rbt__lookup,axiom,
    ! [Less: a > a > $o,C: rBT_color,T: rBT_rbt_a_b] :
      ( ( rBT_rbt_lookup_a_b @ Less @ ( rBT_paint_a_b @ C @ T ) )
      = ( rBT_rbt_lookup_a_b @ Less @ T ) ) ).

% ord.paint_rbt_lookup
thf(fact_821_singleton__Un__iff,axiom,
    ! [X: product_prod_a_b,A3: set_Product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ( ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b )
        = ( sup_su3119292821646003676od_a_b @ A3 @ B3 ) )
      = ( ( ( A3 = bot_bo3428410327501040244od_a_b )
          & ( B3
            = ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) ) )
        | ( ( A3
            = ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) )
          & ( B3 = bot_bo3428410327501040244od_a_b ) )
        | ( ( A3
            = ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) )
          & ( B3
            = ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_822_singleton__Un__iff,axiom,
    ! [X: a,A3: set_a,B3: set_a] :
      ( ( ( insert_a @ X @ bot_bot_set_a )
        = ( sup_sup_set_a @ A3 @ B3 ) )
      = ( ( ( A3 = bot_bot_set_a )
          & ( B3
            = ( insert_a @ X @ bot_bot_set_a ) ) )
        | ( ( A3
            = ( insert_a @ X @ bot_bot_set_a ) )
          & ( B3 = bot_bot_set_a ) )
        | ( ( A3
            = ( insert_a @ X @ bot_bot_set_a ) )
          & ( B3
            = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_823_Un__singleton__iff,axiom,
    ! [A3: set_Product_prod_a_b,B3: set_Product_prod_a_b,X: product_prod_a_b] :
      ( ( ( sup_su3119292821646003676od_a_b @ A3 @ B3 )
        = ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) )
      = ( ( ( A3 = bot_bo3428410327501040244od_a_b )
          & ( B3
            = ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) ) )
        | ( ( A3
            = ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) )
          & ( B3 = bot_bo3428410327501040244od_a_b ) )
        | ( ( A3
            = ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) )
          & ( B3
            = ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_824_Un__singleton__iff,axiom,
    ! [A3: set_a,B3: set_a,X: a] :
      ( ( ( sup_sup_set_a @ A3 @ B3 )
        = ( insert_a @ X @ bot_bot_set_a ) )
      = ( ( ( A3 = bot_bot_set_a )
          & ( B3
            = ( insert_a @ X @ bot_bot_set_a ) ) )
        | ( ( A3
            = ( insert_a @ X @ bot_bot_set_a ) )
          & ( B3 = bot_bot_set_a ) )
        | ( ( A3
            = ( insert_a @ X @ bot_bot_set_a ) )
          & ( B3
            = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_825_insert__is__Un,axiom,
    ( insert4534936386344385144od_a_b
    = ( ^ [A6: product_prod_a_b] : ( sup_su3119292821646003676od_a_b @ ( insert4534936386344385144od_a_b @ A6 @ bot_bo3428410327501040244od_a_b ) ) ) ) ).

% insert_is_Un
thf(fact_826_insert__is__Un,axiom,
    ( insert_a
    = ( ^ [A6: a] : ( sup_sup_set_a @ ( insert_a @ A6 @ bot_bot_set_a ) ) ) ) ).

% insert_is_Un
thf(fact_827_union__set__fold,axiom,
    ! [Xs: list_P1467974523212632366od_a_b,A3: set_Product_prod_a_b] :
      ( ( sup_su3119292821646003676od_a_b @ ( set_Product_prod_a_b2 @ Xs ) @ A3 )
      = ( fold_P1237120252314792299od_a_b @ insert4534936386344385144od_a_b @ Xs @ A3 ) ) ).

% union_set_fold
thf(fact_828_union__set__fold,axiom,
    ! [Xs: list_a,A3: set_a] :
      ( ( sup_sup_set_a @ ( set_a2 @ Xs ) @ A3 )
      = ( fold_a_set_a @ insert_a @ Xs @ A3 ) ) ).

% union_set_fold
thf(fact_829_balance__in__tree,axiom,
    ! [K: list_a,X: produc2547433716817190550list_a,L: rBT_rb6721489163287533103list_a,V: list_a,Y: produc2547433716817190550list_a,R: rBT_rb6721489163287533103list_a] :
      ( ( member5573719349037835490list_a @ ( produc8961369442823515561list_a @ K @ X ) @ ( set_Pr7132271252953855310list_a @ ( rBT_en3328903274962959370list_a @ ( rBT_ba3573774157313868950list_a @ L @ V @ Y @ R ) ) ) )
      = ( ( member5573719349037835490list_a @ ( produc8961369442823515561list_a @ K @ X ) @ ( set_Pr7132271252953855310list_a @ ( rBT_en3328903274962959370list_a @ L ) ) )
        | ( ( K = V )
          & ( X = Y ) )
        | ( member5573719349037835490list_a @ ( produc8961369442823515561list_a @ K @ X ) @ ( set_Pr7132271252953855310list_a @ ( rBT_en3328903274962959370list_a @ R ) ) ) ) ) ).

% balance_in_tree
thf(fact_830_balance__in__tree,axiom,
    ! [K: list_a,X: list_a,L: rBT_rb146355364677272861list_a,V: list_a,Y: list_a,R: rBT_rb146355364677272861list_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ K @ X ) @ ( set_Pr2906193453920572092list_a @ ( rBT_en6256609917331000056list_a @ ( rBT_ba132907544946337156list_a @ L @ V @ Y @ R ) ) ) )
      = ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ K @ X ) @ ( set_Pr2906193453920572092list_a @ ( rBT_en6256609917331000056list_a @ L ) ) )
        | ( ( K = V )
          & ( X = Y ) )
        | ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ K @ X ) @ ( set_Pr2906193453920572092list_a @ ( rBT_en6256609917331000056list_a @ R ) ) ) ) ) ).

% balance_in_tree
thf(fact_831_balance__in__tree,axiom,
    ! [K: a > a > $o,X: produc9164743771328383783list_a,L: rBT_rb9014005217275227734list_a,V: a > a > $o,Y: produc9164743771328383783list_a,R: rBT_rb9014005217275227734list_a] :
      ( ( member8740110733080369545list_a @ ( produc8684083420967988176list_a @ K @ X ) @ ( set_Pr6051718963112299509list_a @ ( rBT_en1829592632444517297list_a @ ( rBT_ba3932203811964311101list_a @ L @ V @ Y @ R ) ) ) )
      = ( ( member8740110733080369545list_a @ ( produc8684083420967988176list_a @ K @ X ) @ ( set_Pr6051718963112299509list_a @ ( rBT_en1829592632444517297list_a @ L ) ) )
        | ( ( K = V )
          & ( X = Y ) )
        | ( member8740110733080369545list_a @ ( produc8684083420967988176list_a @ K @ X ) @ ( set_Pr6051718963112299509list_a @ ( rBT_en1829592632444517297list_a @ R ) ) ) ) ) ).

% balance_in_tree
thf(fact_832_balance__in__tree,axiom,
    ! [K: a,X: produc271001111946445293list_a,L: rBT_rb484615002580549772list_a,V: a,Y: produc271001111946445293list_a,R: rBT_rb484615002580549772list_a] :
      ( ( member852939521359900607list_a @ ( produc4562151247771834630list_a @ K @ X ) @ ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ ( rBT_ba225581376578813811list_a @ L @ V @ Y @ R ) ) ) )
      = ( ( member852939521359900607list_a @ ( produc4562151247771834630list_a @ K @ X ) @ ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ L ) ) )
        | ( ( K = V )
          & ( X = Y ) )
        | ( member852939521359900607list_a @ ( produc4562151247771834630list_a @ K @ X ) @ ( set_Pr4782379691873284651list_a @ ( rBT_en3875316870356833511list_a @ R ) ) ) ) ) ).

% balance_in_tree
thf(fact_833_balance__in__tree,axiom,
    ! [K: a,X: b,L: rBT_rbt_a_b,V: a,Y: b,R: rBT_rbt_a_b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K @ X ) @ ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ ( rBT_balance_a_b @ L @ V @ Y @ R ) ) ) )
      = ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K @ X ) @ ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ L ) ) )
        | ( ( K = V )
          & ( X = Y ) )
        | ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K @ X ) @ ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ R ) ) ) ) ) ).

% balance_in_tree
thf(fact_834_set__union,axiom,
    ! [Xs: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( set_Product_prod_a_b2 @ ( union_7798659292840801954od_a_b @ Xs @ Ys ) )
      = ( sup_su3119292821646003676od_a_b @ ( set_Product_prod_a_b2 @ Xs ) @ ( set_Product_prod_a_b2 @ Ys ) ) ) ).

% set_union
thf(fact_835_set__union,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( set_a2 @ ( union_a @ Xs @ Ys ) )
      = ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).

% set_union
thf(fact_836_local_Odom__rbt__lookup__Branch,axiom,
    ! [C: rBT_color,T1: rBT_rbt_a_b,K: a,V: b,T2: rBT_rbt_a_b] :
      ( ( rBT_rbt_sorted_a_b @ less @ ( rBT_Branch_a_b @ C @ T1 @ K @ V @ T2 ) )
     => ( ( dom_a_b @ ( rBT_rbt_lookup_a_b @ less @ ( rBT_Branch_a_b @ C @ T1 @ K @ V @ T2 ) ) )
        = ( insert_a @ K @ ( sup_sup_set_a @ ( dom_a_b @ ( rBT_rbt_lookup_a_b @ less @ T1 ) ) @ ( dom_a_b @ ( rBT_rbt_lookup_a_b @ less @ T2 ) ) ) ) ) ) ).

% local.dom_rbt_lookup_Branch
thf(fact_837_local_Orbt__lookup__keys,axiom,
    ! [T: rBT_rbt_a_b] :
      ( ( rBT_rbt_sorted_a_b @ less @ T )
     => ( ( dom_a_b @ ( rBT_rbt_lookup_a_b @ less @ T ) )
        = ( set_a2 @ ( rBT_keys_a_b @ T ) ) ) ) ).

% local.rbt_lookup_keys
thf(fact_838_domD,axiom,
    ! [A: a,M: a > option_b] :
      ( ( member_a @ A @ ( dom_a_b @ M ) )
     => ? [B2: b] :
          ( ( M @ A )
          = ( some_b @ B2 ) ) ) ).

% domD
thf(fact_839_domI,axiom,
    ! [M: a > option_b,A: a,B: b] :
      ( ( ( M @ A )
        = ( some_b @ B ) )
     => ( member_a @ A @ ( dom_a_b @ M ) ) ) ).

% domI
thf(fact_840_insert__dom,axiom,
    ! [F: product_prod_a_b > option_b,X: product_prod_a_b,Y: b] :
      ( ( ( F @ X )
        = ( some_b @ Y ) )
     => ( ( insert4534936386344385144od_a_b @ X @ ( dom_Pr5742538987700077097_a_b_b @ F ) )
        = ( dom_Pr5742538987700077097_a_b_b @ F ) ) ) ).

% insert_dom
thf(fact_841_insert__dom,axiom,
    ! [F: a > option_b,X: a,Y: b] :
      ( ( ( F @ X )
        = ( some_b @ Y ) )
     => ( ( insert_a @ X @ ( dom_a_b @ F ) )
        = ( dom_a_b @ F ) ) ) ).

% insert_dom
thf(fact_842_map__add__upd__left,axiom,
    ! [M: a,E2: a > option_b,E1: a > option_b,U1: b] :
      ( ~ ( member_a @ M @ ( dom_a_b @ E2 ) )
     => ( ( map_add_a_b @ ( fun_upd_a_option_b @ E1 @ M @ ( some_b @ U1 ) ) @ E2 )
        = ( fun_upd_a_option_b @ ( map_add_a_b @ E1 @ E2 ) @ M @ ( some_b @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_843_local_Orbt__lookup__rbt__insertw,axiom,
    ! [T: rBT_rbt_a_b,F: b > b > b,K: a,V: b] :
      ( ( rBT_is_rbt_a_b @ less @ T )
     => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rb5519417620787648267th_a_b @ less @ F @ K @ V @ T ) )
        = ( fun_upd_a_option_b @ ( rBT_rbt_lookup_a_b @ less @ T ) @ K @ ( some_b @ ( if_b @ ( member_a @ K @ ( dom_a_b @ ( rBT_rbt_lookup_a_b @ less @ T ) ) ) @ ( F @ ( the_b @ ( rBT_rbt_lookup_a_b @ less @ T @ K ) ) @ V ) @ V ) ) ) ) ) ).

% local.rbt_lookup_rbt_insertw
thf(fact_844_local_Orbt__lookup__rbt__inter,axiom,
    ! [T1: rBT_rbt_a_b,T2: rBT_rbt_a_b] :
      ( ( rBT_rbt_sorted_a_b @ less @ T1 )
     => ( ( rBT_rbt_sorted_a_b @ less @ T2 )
       => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_inter_a_b @ less @ T1 @ T2 ) )
          = ( restrict_map_a_b @ ( rBT_rbt_lookup_a_b @ less @ T2 ) @ ( dom_a_b @ ( rBT_rbt_lookup_a_b @ less @ T1 ) ) ) ) ) ) ).

% local.rbt_lookup_rbt_inter
thf(fact_845_option_Ocollapse,axiom,
    ! [Option: option_b] :
      ( ( Option != none_b )
     => ( ( some_b @ ( the_b @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_846_option_Osel,axiom,
    ! [X22: b] :
      ( ( the_b @ ( some_b @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_847_option_Oexhaust__sel,axiom,
    ! [Option: option_b] :
      ( ( Option != none_b )
     => ( Option
        = ( some_b @ ( the_b @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_848_local_Orbt__lookup__rbt__delete,axiom,
    ! [T: rBT_rbt_a_b,K: a] :
      ( ( rBT_is_rbt_a_b @ less @ T )
     => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_delete_a_b @ less @ K @ T ) )
        = ( restrict_map_a_b @ ( rBT_rbt_lookup_a_b @ less @ T ) @ ( uminus_uminus_set_a @ ( insert_a @ K @ bot_bot_set_a ) ) ) ) ) ).

% local.rbt_lookup_rbt_delete
thf(fact_849_local_Orbt__lookup__rbt__minus,axiom,
    ! [T1: rBT_rbt_a_b,T2: rBT_rbt_a_b] :
      ( ( rBT_is_rbt_a_b @ less @ T1 )
     => ( ( rBT_is_rbt_a_b @ less @ T2 )
       => ( ( rBT_rbt_lookup_a_b @ less @ ( rBT_rbt_minus_a_b @ less @ T1 @ T2 ) )
          = ( restrict_map_a_b @ ( rBT_rbt_lookup_a_b @ less @ T1 ) @ ( uminus_uminus_set_a @ ( dom_a_b @ ( rBT_rbt_lookup_a_b @ less @ T2 ) ) ) ) ) ) ) ).

% local.rbt_lookup_rbt_minus
thf(fact_850_local_Orbt__lookup__fold__rbt__delete,axiom,
    ! [T: rBT_rbt_a_b,Xs: list_a] :
      ( ( rBT_is_rbt_a_b @ less @ T )
     => ( ( rBT_rbt_lookup_a_b @ less @ ( fold_a_RBT_rbt_a_b @ ( rBT_rbt_delete_a_b @ less ) @ Xs @ T ) )
        = ( restrict_map_a_b @ ( rBT_rbt_lookup_a_b @ less @ T ) @ ( uminus_uminus_set_a @ ( set_a2 @ Xs ) ) ) ) ) ).

% local.rbt_lookup_fold_rbt_delete
thf(fact_851_ComplI,axiom,
    ! [C: a,A3: set_a] :
      ( ~ ( member_a @ C @ A3 )
     => ( member_a @ C @ ( uminus_uminus_set_a @ A3 ) ) ) ).

% ComplI
thf(fact_852_Compl__iff,axiom,
    ! [C: a,A3: set_a] :
      ( ( member_a @ C @ ( uminus_uminus_set_a @ A3 ) )
      = ( ~ ( member_a @ C @ A3 ) ) ) ).

% Compl_iff
thf(fact_853_Compl__eq__Compl__iff,axiom,
    ! [A3: set_a,B3: set_a] :
      ( ( ( uminus_uminus_set_a @ A3 )
        = ( uminus_uminus_set_a @ B3 ) )
      = ( A3 = B3 ) ) ).

% Compl_eq_Compl_iff
thf(fact_854_Compl__Diff__eq,axiom,
    ! [A3: set_Product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ( uminus5601964437033819903od_a_b @ ( minus_6888070959853530735od_a_b @ A3 @ B3 ) )
      = ( sup_su3119292821646003676od_a_b @ ( uminus5601964437033819903od_a_b @ A3 ) @ B3 ) ) ).

% Compl_Diff_eq
thf(fact_855_Compl__Diff__eq,axiom,
    ! [A3: set_a,B3: set_a] :
      ( ( uminus_uminus_set_a @ ( minus_minus_set_a @ A3 @ B3 ) )
      = ( sup_sup_set_a @ ( uminus_uminus_set_a @ A3 ) @ B3 ) ) ).

% Compl_Diff_eq
thf(fact_856_restrict__upd__same,axiom,
    ! [M: product_prod_a_b > option_b,X: product_prod_a_b,Y: b] :
      ( ( restri3210584575095474951_a_b_b @ ( fun_up3276301394330262810tion_b @ M @ X @ ( some_b @ Y ) ) @ ( uminus5601964437033819903od_a_b @ ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) ) )
      = ( restri3210584575095474951_a_b_b @ M @ ( uminus5601964437033819903od_a_b @ ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) ) ) ) ).

% restrict_upd_same
thf(fact_857_restrict__upd__same,axiom,
    ! [M: a > option_b,X: a,Y: b] :
      ( ( restrict_map_a_b @ ( fun_upd_a_option_b @ M @ X @ ( some_b @ Y ) ) @ ( uminus_uminus_set_a @ ( insert_a @ X @ bot_bot_set_a ) ) )
      = ( restrict_map_a_b @ M @ ( uminus_uminus_set_a @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).

% restrict_upd_same
thf(fact_858_ComplD,axiom,
    ! [C: a,A3: set_a] :
      ( ( member_a @ C @ ( uminus_uminus_set_a @ A3 ) )
     => ~ ( member_a @ C @ A3 ) ) ).

% ComplD
thf(fact_859_double__complement,axiom,
    ! [A3: set_a] :
      ( ( uminus_uminus_set_a @ ( uminus_uminus_set_a @ A3 ) )
      = A3 ) ).

% double_complement
thf(fact_860_Compl__insert,axiom,
    ! [X: product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ( uminus5601964437033819903od_a_b @ ( insert4534936386344385144od_a_b @ X @ A3 ) )
      = ( minus_6888070959853530735od_a_b @ ( uminus5601964437033819903od_a_b @ A3 ) @ ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) ) ) ).

% Compl_insert
thf(fact_861_Compl__insert,axiom,
    ! [X: a,A3: set_a] :
      ( ( uminus_uminus_set_a @ ( insert_a @ X @ A3 ) )
      = ( minus_minus_set_a @ ( uminus_uminus_set_a @ A3 ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ).

% Compl_insert
thf(fact_862_local_Ofinite__linorder__max__induct,axiom,
    ! [A3: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ A3 )
     => ( ( P @ bot_bot_set_a )
       => ( ! [B2: a,A7: set_a] :
              ( ( finite_finite_a @ A7 )
             => ( ! [X5: a] :
                    ( ( member_a @ X5 @ A7 )
                   => ( less @ X5 @ B2 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_a @ B2 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% local.finite_linorder_max_induct
thf(fact_863_local_Oinfinite__growing,axiom,
    ! [X6: set_a] :
      ( ( X6 != bot_bot_set_a )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ X6 )
           => ? [Xa: a] :
                ( ( member_a @ Xa @ X6 )
                & ( less @ X3 @ Xa ) ) )
       => ~ ( finite_finite_a @ X6 ) ) ) ).

% local.infinite_growing
thf(fact_864_local_Ofinite__linorder__min__induct,axiom,
    ! [A3: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ A3 )
     => ( ( P @ bot_bot_set_a )
       => ( ! [B2: a,A7: set_a] :
              ( ( finite_finite_a @ A7 )
             => ( ! [X5: a] :
                    ( ( member_a @ X5 @ A7 )
                   => ( less @ B2 @ X5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_a @ B2 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% local.finite_linorder_min_induct
thf(fact_865_List_Ofinite__set,axiom,
    ! [Xs: list_P1467974523212632366od_a_b] : ( finite6544458599311216081od_a_b @ ( set_Product_prod_a_b2 @ Xs ) ) ).

% List.finite_set
thf(fact_866_List_Ofinite__set,axiom,
    ! [Xs: list_a] : ( finite_finite_a @ ( set_a2 @ Xs ) ) ).

% List.finite_set
thf(fact_867_local_Ofinite__dom__rbt__lookup,axiom,
    ! [T: rBT_rbt_a_b] : ( finite_finite_a @ ( dom_a_b @ ( rBT_rbt_lookup_a_b @ less @ T ) ) ) ).

% local.finite_dom_rbt_lookup
thf(fact_868_finite__list,axiom,
    ! [A3: set_Product_prod_a_b] :
      ( ( finite6544458599311216081od_a_b @ A3 )
     => ? [Xs2: list_P1467974523212632366od_a_b] :
          ( ( set_Product_prod_a_b2 @ Xs2 )
          = A3 ) ) ).

% finite_list
thf(fact_869_finite__list,axiom,
    ! [A3: set_a] :
      ( ( finite_finite_a @ A3 )
     => ? [Xs2: list_a] :
          ( ( set_a2 @ Xs2 )
          = A3 ) ) ).

% finite_list
thf(fact_870_finite__Diff__insert,axiom,
    ! [A3: set_Product_prod_a_b,A: product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ( finite6544458599311216081od_a_b @ ( minus_6888070959853530735od_a_b @ A3 @ ( insert4534936386344385144od_a_b @ A @ B3 ) ) )
      = ( finite6544458599311216081od_a_b @ ( minus_6888070959853530735od_a_b @ A3 @ B3 ) ) ) ).

% finite_Diff_insert
thf(fact_871_finite__Diff__insert,axiom,
    ! [A3: set_a,A: a,B3: set_a] :
      ( ( finite_finite_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ B3 ) ) )
      = ( finite_finite_a @ ( minus_minus_set_a @ A3 @ B3 ) ) ) ).

% finite_Diff_insert
thf(fact_872_finite__Diff,axiom,
    ! [A3: set_Product_prod_a_b,B3: set_Product_prod_a_b] :
      ( ( finite6544458599311216081od_a_b @ A3 )
     => ( finite6544458599311216081od_a_b @ ( minus_6888070959853530735od_a_b @ A3 @ B3 ) ) ) ).

% finite_Diff
thf(fact_873_finite__Diff,axiom,
    ! [A3: set_a,B3: set_a] :
      ( ( finite_finite_a @ A3 )
     => ( finite_finite_a @ ( minus_minus_set_a @ A3 @ B3 ) ) ) ).

% finite_Diff
thf(fact_874_finite__Diff2,axiom,
    ! [B3: set_Product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ( finite6544458599311216081od_a_b @ B3 )
     => ( ( finite6544458599311216081od_a_b @ ( minus_6888070959853530735od_a_b @ A3 @ B3 ) )
        = ( finite6544458599311216081od_a_b @ A3 ) ) ) ).

% finite_Diff2
thf(fact_875_finite__Diff2,axiom,
    ! [B3: set_a,A3: set_a] :
      ( ( finite_finite_a @ B3 )
     => ( ( finite_finite_a @ ( minus_minus_set_a @ A3 @ B3 ) )
        = ( finite_finite_a @ A3 ) ) ) ).

% finite_Diff2
thf(fact_876_finite__insert,axiom,
    ! [A: product_prod_a_b,A3: set_Product_prod_a_b] :
      ( ( finite6544458599311216081od_a_b @ ( insert4534936386344385144od_a_b @ A @ A3 ) )
      = ( finite6544458599311216081od_a_b @ A3 ) ) ).

% finite_insert
thf(fact_877_finite__insert,axiom,
    ! [A: a,A3: set_a] :
      ( ( finite_finite_a @ ( insert_a @ A @ A3 ) )
      = ( finite_finite_a @ A3 ) ) ).

% finite_insert
thf(fact_878_finite_OemptyI,axiom,
    finite6544458599311216081od_a_b @ bot_bo3428410327501040244od_a_b ).

% finite.emptyI
thf(fact_879_finite_OemptyI,axiom,
    finite_finite_a @ bot_bot_set_a ).

% finite.emptyI
thf(fact_880_infinite__imp__nonempty,axiom,
    ! [S3: set_Product_prod_a_b] :
      ( ~ ( finite6544458599311216081od_a_b @ S3 )
     => ( S3 != bot_bo3428410327501040244od_a_b ) ) ).

% infinite_imp_nonempty
thf(fact_881_infinite__imp__nonempty,axiom,
    ! [S3: set_a] :
      ( ~ ( finite_finite_a @ S3 )
     => ( S3 != bot_bot_set_a ) ) ).

% infinite_imp_nonempty
thf(fact_882_finite_OinsertI,axiom,
    ! [A3: set_Product_prod_a_b,A: product_prod_a_b] :
      ( ( finite6544458599311216081od_a_b @ A3 )
     => ( finite6544458599311216081od_a_b @ ( insert4534936386344385144od_a_b @ A @ A3 ) ) ) ).

% finite.insertI
thf(fact_883_finite_OinsertI,axiom,
    ! [A3: set_a,A: a] :
      ( ( finite_finite_a @ A3 )
     => ( finite_finite_a @ ( insert_a @ A @ A3 ) ) ) ).

% finite.insertI
thf(fact_884_Diff__infinite__finite,axiom,
    ! [T4: set_Product_prod_a_b,S3: set_Product_prod_a_b] :
      ( ( finite6544458599311216081od_a_b @ T4 )
     => ( ~ ( finite6544458599311216081od_a_b @ S3 )
       => ~ ( finite6544458599311216081od_a_b @ ( minus_6888070959853530735od_a_b @ S3 @ T4 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_885_Diff__infinite__finite,axiom,
    ! [T4: set_a,S3: set_a] :
      ( ( finite_finite_a @ T4 )
     => ( ~ ( finite_finite_a @ S3 )
       => ~ ( finite_finite_a @ ( minus_minus_set_a @ S3 @ T4 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_886_infinite__finite__induct,axiom,
    ! [P: set_Product_prod_a_b > $o,A3: set_Product_prod_a_b] :
      ( ! [A7: set_Product_prod_a_b] :
          ( ~ ( finite6544458599311216081od_a_b @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bo3428410327501040244od_a_b )
       => ( ! [X3: product_prod_a_b,F3: set_Product_prod_a_b] :
              ( ( finite6544458599311216081od_a_b @ F3 )
             => ( ~ ( member1426531481828664017od_a_b @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert4534936386344385144od_a_b @ X3 @ F3 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_887_infinite__finite__induct,axiom,
    ! [P: set_a > $o,A3: set_a] :
      ( ! [A7: set_a] :
          ( ~ ( finite_finite_a @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_a )
       => ( ! [X3: a,F3: set_a] :
              ( ( finite_finite_a @ F3 )
             => ( ~ ( member_a @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_888_finite__ne__induct,axiom,
    ! [F4: set_Product_prod_a_b,P: set_Product_prod_a_b > $o] :
      ( ( finite6544458599311216081od_a_b @ F4 )
     => ( ( F4 != bot_bo3428410327501040244od_a_b )
       => ( ! [X3: product_prod_a_b] : ( P @ ( insert4534936386344385144od_a_b @ X3 @ bot_bo3428410327501040244od_a_b ) )
         => ( ! [X3: product_prod_a_b,F3: set_Product_prod_a_b] :
                ( ( finite6544458599311216081od_a_b @ F3 )
               => ( ( F3 != bot_bo3428410327501040244od_a_b )
                 => ( ~ ( member1426531481828664017od_a_b @ X3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert4534936386344385144od_a_b @ X3 @ F3 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_889_finite__ne__induct,axiom,
    ! [F4: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F4 )
     => ( ( F4 != bot_bot_set_a )
       => ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
         => ( ! [X3: a,F3: set_a] :
                ( ( finite_finite_a @ F3 )
               => ( ( F3 != bot_bot_set_a )
                 => ( ~ ( member_a @ X3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_a @ X3 @ F3 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_890_finite__induct,axiom,
    ! [F4: set_Product_prod_a_b,P: set_Product_prod_a_b > $o] :
      ( ( finite6544458599311216081od_a_b @ F4 )
     => ( ( P @ bot_bo3428410327501040244od_a_b )
       => ( ! [X3: product_prod_a_b,F3: set_Product_prod_a_b] :
              ( ( finite6544458599311216081od_a_b @ F3 )
             => ( ~ ( member1426531481828664017od_a_b @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert4534936386344385144od_a_b @ X3 @ F3 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_891_finite__induct,axiom,
    ! [F4: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F4 )
     => ( ( P @ bot_bot_set_a )
       => ( ! [X3: a,F3: set_a] :
              ( ( finite_finite_a @ F3 )
             => ( ~ ( member_a @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_892_finite_Osimps,axiom,
    ( finite6544458599311216081od_a_b
    = ( ^ [A6: set_Product_prod_a_b] :
          ( ( A6 = bot_bo3428410327501040244od_a_b )
          | ? [A4: set_Product_prod_a_b,B7: product_prod_a_b] :
              ( ( A6
                = ( insert4534936386344385144od_a_b @ B7 @ A4 ) )
              & ( finite6544458599311216081od_a_b @ A4 ) ) ) ) ) ).

% finite.simps
thf(fact_893_finite_Osimps,axiom,
    ( finite_finite_a
    = ( ^ [A6: set_a] :
          ( ( A6 = bot_bot_set_a )
          | ? [A4: set_a,B7: a] :
              ( ( A6
                = ( insert_a @ B7 @ A4 ) )
              & ( finite_finite_a @ A4 ) ) ) ) ) ).

% finite.simps
thf(fact_894_finite_Ocases,axiom,
    ! [A: set_Product_prod_a_b] :
      ( ( finite6544458599311216081od_a_b @ A )
     => ( ( A != bot_bo3428410327501040244od_a_b )
       => ~ ! [A7: set_Product_prod_a_b] :
              ( ? [A2: product_prod_a_b] :
                  ( A
                  = ( insert4534936386344385144od_a_b @ A2 @ A7 ) )
             => ~ ( finite6544458599311216081od_a_b @ A7 ) ) ) ) ).

% finite.cases
thf(fact_895_finite_Ocases,axiom,
    ! [A: set_a] :
      ( ( finite_finite_a @ A )
     => ( ( A != bot_bot_set_a )
       => ~ ! [A7: set_a] :
              ( ? [A2: a] :
                  ( A
                  = ( insert_a @ A2 @ A7 ) )
             => ~ ( finite_finite_a @ A7 ) ) ) ) ).

% finite.cases
thf(fact_896_infinite__remove,axiom,
    ! [S3: set_Product_prod_a_b,A: product_prod_a_b] :
      ( ~ ( finite6544458599311216081od_a_b @ S3 )
     => ~ ( finite6544458599311216081od_a_b @ ( minus_6888070959853530735od_a_b @ S3 @ ( insert4534936386344385144od_a_b @ A @ bot_bo3428410327501040244od_a_b ) ) ) ) ).

% infinite_remove
thf(fact_897_infinite__remove,axiom,
    ! [S3: set_a,A: a] :
      ( ~ ( finite_finite_a @ S3 )
     => ~ ( finite_finite_a @ ( minus_minus_set_a @ S3 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).

% infinite_remove
thf(fact_898_infinite__coinduct,axiom,
    ! [X6: set_Product_prod_a_b > $o,A3: set_Product_prod_a_b] :
      ( ( X6 @ A3 )
     => ( ! [A7: set_Product_prod_a_b] :
            ( ( X6 @ A7 )
           => ? [X5: product_prod_a_b] :
                ( ( member1426531481828664017od_a_b @ X5 @ A7 )
                & ( ( X6 @ ( minus_6888070959853530735od_a_b @ A7 @ ( insert4534936386344385144od_a_b @ X5 @ bot_bo3428410327501040244od_a_b ) ) )
                  | ~ ( finite6544458599311216081od_a_b @ ( minus_6888070959853530735od_a_b @ A7 @ ( insert4534936386344385144od_a_b @ X5 @ bot_bo3428410327501040244od_a_b ) ) ) ) ) )
       => ~ ( finite6544458599311216081od_a_b @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_899_infinite__coinduct,axiom,
    ! [X6: set_a > $o,A3: set_a] :
      ( ( X6 @ A3 )
     => ( ! [A7: set_a] :
            ( ( X6 @ A7 )
           => ? [X5: a] :
                ( ( member_a @ X5 @ A7 )
                & ( ( X6 @ ( minus_minus_set_a @ A7 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
                  | ~ ( finite_finite_a @ ( minus_minus_set_a @ A7 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
       => ~ ( finite_finite_a @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_900_finite__empty__induct,axiom,
    ! [A3: set_Product_prod_a_b,P: set_Product_prod_a_b > $o] :
      ( ( finite6544458599311216081od_a_b @ A3 )
     => ( ( P @ A3 )
       => ( ! [A2: product_prod_a_b,A7: set_Product_prod_a_b] :
              ( ( finite6544458599311216081od_a_b @ A7 )
             => ( ( member1426531481828664017od_a_b @ A2 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_6888070959853530735od_a_b @ A7 @ ( insert4534936386344385144od_a_b @ A2 @ bot_bo3428410327501040244od_a_b ) ) ) ) ) )
         => ( P @ bot_bo3428410327501040244od_a_b ) ) ) ) ).

% finite_empty_induct
thf(fact_901_finite__empty__induct,axiom,
    ! [A3: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ A3 )
     => ( ( P @ A3 )
       => ( ! [A2: a,A7: set_a] :
              ( ( finite_finite_a @ A7 )
             => ( ( member_a @ A2 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ) )
         => ( P @ bot_bot_set_a ) ) ) ) ).

% finite_empty_induct
thf(fact_902_local_Osorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [Xs: list_a] :
      ( ( distinct_a @ Xs )
     => ( distinct_a @ Xs ) ) ).

% local.sorted_list_of_set.distinct_if_distinct_map
thf(fact_903_finite__distinct__list,axiom,
    ! [A3: set_Product_prod_a_b] :
      ( ( finite6544458599311216081od_a_b @ A3 )
     => ? [Xs2: list_P1467974523212632366od_a_b] :
          ( ( ( set_Product_prod_a_b2 @ Xs2 )
            = A3 )
          & ( distin132333874345289761od_a_b @ Xs2 ) ) ) ).

% finite_distinct_list
thf(fact_904_finite__distinct__list,axiom,
    ! [A3: set_a] :
      ( ( finite_finite_a @ A3 )
     => ? [Xs2: list_a] :
          ( ( ( set_a2 @ Xs2 )
            = A3 )
          & ( distinct_a @ Xs2 ) ) ) ).

% finite_distinct_list
thf(fact_905_local_Oset__quicksort__part,axiom,
    ! [Ac: list_a,X: a,Lts: list_a,Eqs: list_a,Gts: list_a,Zs: list_a] :
      ( ( set_a2 @ ( set_quicksort_part_a @ less @ Ac @ X @ Lts @ Eqs @ Gts @ Zs ) )
      = ( sup_sup_set_a @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( set_a2 @ Ac ) @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( set_a2 @ Lts ) ) @ ( set_a2 @ Eqs ) ) @ ( set_a2 @ Gts ) ) @ ( set_a2 @ Zs ) ) ) ).

% local.set_quicksort_part
thf(fact_906_ord_Oset__quicksort__part,axiom,
    ! [Less: product_prod_a_b > product_prod_a_b > $o,Ac: list_P1467974523212632366od_a_b,X: product_prod_a_b,Lts: list_P1467974523212632366od_a_b,Eqs: list_P1467974523212632366od_a_b,Gts: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b] :
      ( ( set_Product_prod_a_b2 @ ( set_qu4258237464756766404od_a_b @ Less @ Ac @ X @ Lts @ Eqs @ Gts @ Zs ) )
      = ( sup_su3119292821646003676od_a_b @ ( sup_su3119292821646003676od_a_b @ ( sup_su3119292821646003676od_a_b @ ( sup_su3119292821646003676od_a_b @ ( sup_su3119292821646003676od_a_b @ ( set_Product_prod_a_b2 @ Ac ) @ ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) ) @ ( set_Product_prod_a_b2 @ Lts ) ) @ ( set_Product_prod_a_b2 @ Eqs ) ) @ ( set_Product_prod_a_b2 @ Gts ) ) @ ( set_Product_prod_a_b2 @ Zs ) ) ) ).

% ord.set_quicksort_part
thf(fact_907_ord_Oset__quicksort__part,axiom,
    ! [Less: a > a > $o,Ac: list_a,X: a,Lts: list_a,Eqs: list_a,Gts: list_a,Zs: list_a] :
      ( ( set_a2 @ ( set_quicksort_part_a @ Less @ Ac @ X @ Lts @ Eqs @ Gts @ Zs ) )
      = ( sup_sup_set_a @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( set_a2 @ Ac ) @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( set_a2 @ Lts ) ) @ ( set_a2 @ Eqs ) ) @ ( set_a2 @ Gts ) ) @ ( set_a2 @ Zs ) ) ) ).

% ord.set_quicksort_part
thf(fact_908_local_Oquicksort__part_Osimps_I2_J,axiom,
    ! [X: a,Z: a,Ac: list_a,Lts: list_a,Eqs: list_a,Gts: list_a,Zs: list_a] :
      ( ( ( less @ X @ Z )
       => ( ( set_quicksort_part_a @ less @ Ac @ X @ Lts @ Eqs @ Gts @ ( cons_a @ Z @ Zs ) )
          = ( set_quicksort_part_a @ less @ Ac @ X @ Lts @ Eqs @ ( cons_a @ Z @ Gts ) @ Zs ) ) )
      & ( ~ ( less @ X @ Z )
       => ( ( ( less @ Z @ X )
           => ( ( set_quicksort_part_a @ less @ Ac @ X @ Lts @ Eqs @ Gts @ ( cons_a @ Z @ Zs ) )
              = ( set_quicksort_part_a @ less @ Ac @ X @ ( cons_a @ Z @ Lts ) @ Eqs @ Gts @ Zs ) ) )
          & ( ~ ( less @ Z @ X )
           => ( ( set_quicksort_part_a @ less @ Ac @ X @ Lts @ Eqs @ Gts @ ( cons_a @ Z @ Zs ) )
              = ( set_quicksort_part_a @ less @ Ac @ X @ Lts @ ( cons_a @ Z @ Eqs ) @ Gts @ Zs ) ) ) ) ) ) ).

% local.quicksort_part.simps(2)
thf(fact_909_set__remove1__eq,axiom,
    ! [Xs: list_P1467974523212632366od_a_b,X: product_prod_a_b] :
      ( ( distin132333874345289761od_a_b @ Xs )
     => ( ( set_Product_prod_a_b2 @ ( remove3530114610544164932od_a_b @ X @ Xs ) )
        = ( minus_6888070959853530735od_a_b @ ( set_Product_prod_a_b2 @ Xs ) @ ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) ) ) ) ).

% set_remove1_eq
thf(fact_910_set__remove1__eq,axiom,
    ! [Xs: list_a,X: a] :
      ( ( distinct_a @ Xs )
     => ( ( set_a2 @ ( remove1_a @ X @ Xs ) )
        = ( minus_minus_set_a @ ( set_a2 @ Xs ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).

% set_remove1_eq
thf(fact_911_in__set__remove1,axiom,
    ! [A: product_prod_a_b,B: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
      ( ( A != B )
     => ( ( member1426531481828664017od_a_b @ A @ ( set_Product_prod_a_b2 @ ( remove3530114610544164932od_a_b @ B @ Xs ) ) )
        = ( member1426531481828664017od_a_b @ A @ ( set_Product_prod_a_b2 @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_912_in__set__remove1,axiom,
    ! [A: a,B: a,Xs: list_a] :
      ( ( A != B )
     => ( ( member_a @ A @ ( set_a2 @ ( remove1_a @ B @ Xs ) ) )
        = ( member_a @ A @ ( set_a2 @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_913_list_Osimps_I15_J,axiom,
    ! [X21: product_prod_a_b,X222: list_P1467974523212632366od_a_b] :
      ( ( set_Product_prod_a_b2 @ ( cons_P7316939131009794654od_a_b @ X21 @ X222 ) )
      = ( insert4534936386344385144od_a_b @ X21 @ ( set_Product_prod_a_b2 @ X222 ) ) ) ).

% list.simps(15)
thf(fact_914_list_Osimps_I15_J,axiom,
    ! [X21: a,X222: list_a] :
      ( ( set_a2 @ ( cons_a @ X21 @ X222 ) )
      = ( insert_a @ X21 @ ( set_a2 @ X222 ) ) ) ).

% list.simps(15)
thf(fact_915_list_Oset__intros_I2_J,axiom,
    ! [Y: product_prod_a_b,X222: list_P1467974523212632366od_a_b,X21: product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ Y @ ( set_Product_prod_a_b2 @ X222 ) )
     => ( member1426531481828664017od_a_b @ Y @ ( set_Product_prod_a_b2 @ ( cons_P7316939131009794654od_a_b @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_916_list_Oset__intros_I2_J,axiom,
    ! [Y: a,X222: list_a,X21: a] :
      ( ( member_a @ Y @ ( set_a2 @ X222 ) )
     => ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_917_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_a_b,X222: list_P1467974523212632366od_a_b] : ( member1426531481828664017od_a_b @ X21 @ ( set_Product_prod_a_b2 @ ( cons_P7316939131009794654od_a_b @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_918_list_Oset__intros_I1_J,axiom,
    ! [X21: a,X222: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_919_list_Oset__cases,axiom,
    ! [E3: product_prod_a_b,A: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ E3 @ ( set_Product_prod_a_b2 @ A ) )
     => ( ! [Z22: list_P1467974523212632366od_a_b] :
            ( A
           != ( cons_P7316939131009794654od_a_b @ E3 @ Z22 ) )
       => ~ ! [Z1: product_prod_a_b,Z22: list_P1467974523212632366od_a_b] :
              ( ( A
                = ( cons_P7316939131009794654od_a_b @ Z1 @ Z22 ) )
             => ~ ( member1426531481828664017od_a_b @ E3 @ ( set_Product_prod_a_b2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_920_list_Oset__cases,axiom,
    ! [E3: a,A: list_a] :
      ( ( member_a @ E3 @ ( set_a2 @ A ) )
     => ( ! [Z22: list_a] :
            ( A
           != ( cons_a @ E3 @ Z22 ) )
       => ~ ! [Z1: a,Z22: list_a] :
              ( ( A
                = ( cons_a @ Z1 @ Z22 ) )
             => ~ ( member_a @ E3 @ ( set_a2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_921_set__ConsD,axiom,
    ! [Y: product_prod_a_b,X: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ Y @ ( set_Product_prod_a_b2 @ ( cons_P7316939131009794654od_a_b @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member1426531481828664017od_a_b @ Y @ ( set_Product_prod_a_b2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_922_set__ConsD,axiom,
    ! [Y: a,X: a,Xs: list_a] :
      ( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_923_notin__set__remove1,axiom,
    ! [X: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Y: product_prod_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs ) )
     => ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ ( remove3530114610544164932od_a_b @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_924_notin__set__remove1,axiom,
    ! [X: a,Xs: list_a,Y: a] :
      ( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
     => ~ ( member_a @ X @ ( set_a2 @ ( remove1_a @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_925_remove1__idem,axiom,
    ! [X: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs ) )
     => ( ( remove3530114610544164932od_a_b @ X @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_926_remove1__idem,axiom,
    ! [X: a,Xs: list_a] :
      ( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
     => ( ( remove1_a @ X @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_927_distinct_Osimps_I2_J,axiom,
    ! [X: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
      ( ( distin132333874345289761od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs ) )
      = ( ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs ) )
        & ( distin132333874345289761od_a_b @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_928_distinct_Osimps_I2_J,axiom,
    ! [X: a,Xs: list_a] :
      ( ( distinct_a @ ( cons_a @ X @ Xs ) )
      = ( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
        & ( distinct_a @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_929_local_Oremdups__sorted_Osimps_I3_J,axiom,
    ! [X: a,Y: a,Xs: list_a] :
      ( ( ( less @ X @ Y )
       => ( ( set_remdups_sorted_a @ less @ ( cons_a @ X @ ( cons_a @ Y @ Xs ) ) )
          = ( cons_a @ X @ ( set_remdups_sorted_a @ less @ ( cons_a @ Y @ Xs ) ) ) ) )
      & ( ~ ( less @ X @ Y )
       => ( ( set_remdups_sorted_a @ less @ ( cons_a @ X @ ( cons_a @ Y @ Xs ) ) )
          = ( set_remdups_sorted_a @ less @ ( cons_a @ Y @ Xs ) ) ) ) ) ).

% local.remdups_sorted.simps(3)
thf(fact_930_local_Oremdups__sorted_Ocases,axiom,
    ! [X: list_a] :
      ( ( X != nil_a )
     => ( ! [X3: a] :
            ( X
           != ( cons_a @ X3 @ nil_a ) )
       => ~ ! [X3: a,Y2: a,Xs2: list_a] :
              ( X
             != ( cons_a @ X3 @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ).

% local.remdups_sorted.cases
thf(fact_931_local_Osorted__list__subset_Ocases,axiom,
    ! [X: produc1392733710753813600list_a] :
      ( ! [Eq: a > a > $o,Ys2: list_a] :
          ( X
         != ( produc8684083420967988176list_a @ Eq @ ( produc6837034575241423639list_a @ nil_a @ Ys2 ) ) )
     => ( ! [Eq: a > a > $o,X3: a,Xs2: list_a] :
            ( X
           != ( produc8684083420967988176list_a @ Eq @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ nil_a ) ) )
       => ~ ! [Eq: a > a > $o,X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
              ( X
             != ( produc8684083420967988176list_a @ Eq @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ) ).

% local.sorted_list_subset.cases
thf(fact_932_local_Oquicksort__part_Ocases,axiom,
    ! [X: produc1828243372476419385list_a] :
      ( ! [Ac2: list_a,X3: a,Lts2: list_a,Eqs2: list_a,Gts2: list_a] :
          ( X
         != ( produc8961369442823515561list_a @ Ac2 @ ( produc4562151247771834630list_a @ X3 @ ( produc2289259375147936733list_a @ Lts2 @ ( produc1910438886824740410list_a @ Eqs2 @ ( produc6837034575241423639list_a @ Gts2 @ nil_a ) ) ) ) ) )
     => ~ ! [Ac2: list_a,X3: a,Lts2: list_a,Eqs2: list_a,Gts2: list_a,Z3: a,Zs2: list_a] :
            ( X
           != ( produc8961369442823515561list_a @ Ac2 @ ( produc4562151247771834630list_a @ X3 @ ( produc2289259375147936733list_a @ Lts2 @ ( produc1910438886824740410list_a @ Eqs2 @ ( produc6837034575241423639list_a @ Gts2 @ ( cons_a @ Z3 @ Zs2 ) ) ) ) ) ) ) ) ).

% local.quicksort_part.cases
thf(fact_933_local_Oquicksort__acc_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ( ! [Ac2: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ Ac2 @ nil_a ) )
     => ( ! [Ac2: list_a,X3: a] :
            ( X
           != ( produc6837034575241423639list_a @ Ac2 @ ( cons_a @ X3 @ nil_a ) ) )
       => ~ ! [Ac2: list_a,X3: a,V2: a,Va: list_a] :
              ( X
             != ( produc6837034575241423639list_a @ Ac2 @ ( cons_a @ X3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ).

% local.quicksort_acc.cases
thf(fact_934_local_Oremdups__sorted_Osimps_I1_J,axiom,
    ( ( set_remdups_sorted_a @ less @ nil_a )
    = nil_a ) ).

% local.remdups_sorted.simps(1)
thf(fact_935_local_Osingle__list__def,axiom,
    ( set_single_list_a
    = ( ^ [A6: a] : ( cons_a @ A6 @ nil_a ) ) ) ).

% local.single_list_def
thf(fact_936_local_Oremdups__sorted_Oelims,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( ( set_remdups_sorted_a @ less @ X )
        = Y )
     => ( ( ( X = nil_a )
         => ( Y != nil_a ) )
       => ( ! [X3: a] :
              ( ( X
                = ( cons_a @ X3 @ nil_a ) )
             => ( Y
               != ( cons_a @ X3 @ nil_a ) ) )
         => ~ ! [X3: a,Y2: a,Xs2: list_a] :
                ( ( X
                  = ( cons_a @ X3 @ ( cons_a @ Y2 @ Xs2 ) ) )
               => ~ ( ( ( less @ X3 @ Y2 )
                     => ( Y
                        = ( cons_a @ X3 @ ( set_remdups_sorted_a @ less @ ( cons_a @ Y2 @ Xs2 ) ) ) ) )
                    & ( ~ ( less @ X3 @ Y2 )
                     => ( Y
                        = ( set_remdups_sorted_a @ less @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ).

% local.remdups_sorted.elims
thf(fact_937_local_Oremdups__sorted_Osimps_I2_J,axiom,
    ! [X: a] :
      ( ( set_remdups_sorted_a @ less @ ( cons_a @ X @ nil_a ) )
      = ( cons_a @ X @ nil_a ) ) ).

% local.remdups_sorted.simps(2)
thf(fact_938_List_Oset__empty,axiom,
    ! [Xs: list_P1467974523212632366od_a_b] :
      ( ( ( set_Product_prod_a_b2 @ Xs )
        = bot_bo3428410327501040244od_a_b )
      = ( Xs = nil_Product_prod_a_b ) ) ).

% List.set_empty
thf(fact_939_List_Oset__empty,axiom,
    ! [Xs: list_a] :
      ( ( ( set_a2 @ Xs )
        = bot_bot_set_a )
      = ( Xs = nil_a ) ) ).

% List.set_empty
thf(fact_940_set__empty2,axiom,
    ! [Xs: list_P1467974523212632366od_a_b] :
      ( ( bot_bo3428410327501040244od_a_b
        = ( set_Product_prod_a_b2 @ Xs ) )
      = ( Xs = nil_Product_prod_a_b ) ) ).

% set_empty2
thf(fact_941_set__empty2,axiom,
    ! [Xs: list_a] :
      ( ( bot_bot_set_a
        = ( set_a2 @ Xs ) )
      = ( Xs = nil_a ) ) ).

% set_empty2
thf(fact_942_ord_Oquicksort__acc_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ( ! [Ac2: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ Ac2 @ nil_a ) )
     => ( ! [Ac2: list_a,X3: a] :
            ( X
           != ( produc6837034575241423639list_a @ Ac2 @ ( cons_a @ X3 @ nil_a ) ) )
       => ~ ! [Ac2: list_a,X3: a,V2: a,Va: list_a] :
              ( X
             != ( produc6837034575241423639list_a @ Ac2 @ ( cons_a @ X3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ).

% ord.quicksort_acc.cases
thf(fact_943_ord_Oquicksort__part_Ocases,axiom,
    ! [X: produc1828243372476419385list_a] :
      ( ! [Ac2: list_a,X3: a,Lts2: list_a,Eqs2: list_a,Gts2: list_a] :
          ( X
         != ( produc8961369442823515561list_a @ Ac2 @ ( produc4562151247771834630list_a @ X3 @ ( produc2289259375147936733list_a @ Lts2 @ ( produc1910438886824740410list_a @ Eqs2 @ ( produc6837034575241423639list_a @ Gts2 @ nil_a ) ) ) ) ) )
     => ~ ! [Ac2: list_a,X3: a,Lts2: list_a,Eqs2: list_a,Gts2: list_a,Z3: a,Zs2: list_a] :
            ( X
           != ( produc8961369442823515561list_a @ Ac2 @ ( produc4562151247771834630list_a @ X3 @ ( produc2289259375147936733list_a @ Lts2 @ ( produc1910438886824740410list_a @ Eqs2 @ ( produc6837034575241423639list_a @ Gts2 @ ( cons_a @ Z3 @ Zs2 ) ) ) ) ) ) ) ) ).

% ord.quicksort_part.cases
thf(fact_944_ord_Osorted__list__subset_Ocases,axiom,
    ! [X: produc1392733710753813600list_a] :
      ( ! [Eq: a > a > $o,Ys2: list_a] :
          ( X
         != ( produc8684083420967988176list_a @ Eq @ ( produc6837034575241423639list_a @ nil_a @ Ys2 ) ) )
     => ( ! [Eq: a > a > $o,X3: a,Xs2: list_a] :
            ( X
           != ( produc8684083420967988176list_a @ Eq @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ nil_a ) ) )
       => ~ ! [Eq: a > a > $o,X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
              ( X
             != ( produc8684083420967988176list_a @ Eq @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ) ).

% ord.sorted_list_subset.cases
thf(fact_945_map__tailrec__rev_Ocases,axiom,
    ! [X: produc1473018763691903991list_a] :
      ( ! [F2: a > a,Bs: list_a] :
          ( X
         != ( produc8643929849434629545list_a @ F2 @ ( produc6837034575241423639list_a @ nil_a @ Bs ) ) )
     => ~ ! [F2: a > a,A2: a,As: list_a,Bs: list_a] :
            ( X
           != ( produc8643929849434629545list_a @ F2 @ ( produc6837034575241423639list_a @ ( cons_a @ A2 @ As ) @ Bs ) ) ) ) ).

% map_tailrec_rev.cases
thf(fact_946_successively_Ocases,axiom,
    ! [X: produc5032551385658279741list_a] :
      ( ! [P5: a > a > $o] :
          ( X
         != ( produc8111569692950616493list_a @ P5 @ nil_a ) )
     => ( ! [P5: a > a > $o,X3: a] :
            ( X
           != ( produc8111569692950616493list_a @ P5 @ ( cons_a @ X3 @ nil_a ) ) )
       => ~ ! [P5: a > a > $o,X3: a,Y2: a,Xs2: list_a] :
              ( X
             != ( produc8111569692950616493list_a @ P5 @ ( cons_a @ X3 @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_947_sorted__wrt_Ocases,axiom,
    ! [X: produc5032551385658279741list_a] :
      ( ! [P5: a > a > $o] :
          ( X
         != ( produc8111569692950616493list_a @ P5 @ nil_a ) )
     => ~ ! [P5: a > a > $o,X3: a,Ys2: list_a] :
            ( X
           != ( produc8111569692950616493list_a @ P5 @ ( cons_a @ X3 @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_948_shuffles_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ( ! [Ys2: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
     => ( ! [Xs2: list_a] :
            ( X
           != ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
       => ~ ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
              ( X
             != ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_949_splice_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ( ! [Ys2: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
     => ~ ! [X3: a,Xs2: list_a,Ys2: list_a] :
            ( X
           != ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_950_empty__set,axiom,
    ( bot_bo3428410327501040244od_a_b
    = ( set_Product_prod_a_b2 @ nil_Product_prod_a_b ) ) ).

% empty_set
thf(fact_951_empty__set,axiom,
    ( bot_bot_set_a
    = ( set_a2 @ nil_a ) ) ).

% empty_set
thf(fact_952_map__of__Cons__code_I2_J,axiom,
    ! [L: list_a,K: list_a,V: produc2547433716817190550list_a,Ps: list_P1945613012453903807list_a] :
      ( ( ( L = K )
       => ( ( map_of4460908170151070709list_a @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ L @ V ) @ Ps ) @ K )
          = ( some_P3259385407967395195list_a @ V ) ) )
      & ( ( L != K )
       => ( ( map_of4460908170151070709list_a @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ L @ V ) @ Ps ) @ K )
          = ( map_of4460908170151070709list_a @ Ps @ K ) ) ) ) ).

% map_of_Cons_code(2)
thf(fact_953_map__of__Cons__code_I2_J,axiom,
    ! [L: list_a,K: list_a,V: list_a,Ps: list_P321204300973800749list_a] :
      ( ( ( L = K )
       => ( ( map_of_list_a_list_a @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ L @ V ) @ Ps ) @ K )
          = ( some_list_a @ V ) ) )
      & ( ( L != K )
       => ( ( map_of_list_a_list_a @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ L @ V ) @ Ps ) @ K )
          = ( map_of_list_a_list_a @ Ps @ K ) ) ) ) ).

% map_of_Cons_code(2)
thf(fact_954_map__of__Cons__code_I2_J,axiom,
    ! [L: a > a > $o,K: a > a > $o,V: produc9164743771328383783list_a,Ps: list_P1472896444292205030list_a] :
      ( ( ( L = K )
       => ( ( map_of1656090001276894236list_a @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ L @ V ) @ Ps ) @ K )
          = ( some_P5335617695965928972list_a @ V ) ) )
      & ( ( L != K )
       => ( ( map_of1656090001276894236list_a @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ L @ V ) @ Ps ) @ K )
          = ( map_of1656090001276894236list_a @ Ps @ K ) ) ) ) ).

% map_of_Cons_code(2)
thf(fact_955_map__of__Cons__code_I2_J,axiom,
    ! [L: a,K: a,V: produc271001111946445293list_a,Ps: list_P5792129111969709340list_a] :
      ( ( ( L = K )
       => ( ( map_of4901303744667269458list_a @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ L @ V ) @ Ps ) @ K )
          = ( some_P1402039549733047762list_a @ V ) ) )
      & ( ( L != K )
       => ( ( map_of4901303744667269458list_a @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ L @ V ) @ Ps ) @ K )
          = ( map_of4901303744667269458list_a @ Ps @ K ) ) ) ) ).

% map_of_Cons_code(2)
thf(fact_956_map__of__Cons__code_I2_J,axiom,
    ! [L: a,K: a,V: b,Ps: list_P1467974523212632366od_a_b] :
      ( ( ( L = K )
       => ( ( map_of_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ L @ V ) @ Ps ) @ K )
          = ( some_b @ V ) ) )
      & ( ( L != K )
       => ( ( map_of_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ L @ V ) @ Ps ) @ K )
          = ( map_of_a_b @ Ps @ K ) ) ) ) ).

% map_of_Cons_code(2)
thf(fact_957_local_Odistinct__quicksort__part,axiom,
    ! [Ac: list_a,X: a,Lts: list_a,Eqs: list_a,Gts: list_a,Zs: list_a] :
      ( ( distinct_a @ ( set_quicksort_part_a @ less @ Ac @ X @ Lts @ Eqs @ Gts @ Zs ) )
      = ( distinct_a @ ( append_a @ Ac @ ( append_a @ ( cons_a @ X @ nil_a ) @ ( append_a @ Lts @ ( append_a @ Eqs @ ( append_a @ Gts @ Zs ) ) ) ) ) ) ) ).

% local.distinct_quicksort_part
thf(fact_958_local_Oquicksort__acc_Osimps_I3_J,axiom,
    ! [Ac: list_a,X: a,V: a,Va2: list_a] :
      ( ( set_quicksort_acc_a @ less @ Ac @ ( cons_a @ X @ ( cons_a @ V @ Va2 ) ) )
      = ( set_quicksort_part_a @ less @ Ac @ X @ nil_a @ nil_a @ nil_a @ ( cons_a @ V @ Va2 ) ) ) ).

% local.quicksort_acc.simps(3)
thf(fact_959_local_Oquicksort__acc_Oelims,axiom,
    ! [X: list_a,Xa2: list_a,Y: list_a] :
      ( ( ( set_quicksort_acc_a @ less @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_a )
         => ( Y != X ) )
       => ( ! [X3: a] :
              ( ( Xa2
                = ( cons_a @ X3 @ nil_a ) )
             => ( Y
               != ( cons_a @ X3 @ X ) ) )
         => ~ ! [X3: a,V2: a,Va: list_a] :
                ( ( Xa2
                  = ( cons_a @ X3 @ ( cons_a @ V2 @ Va ) ) )
               => ( Y
                 != ( set_quicksort_part_a @ less @ X @ X3 @ nil_a @ nil_a @ nil_a @ ( cons_a @ V2 @ Va ) ) ) ) ) ) ) ).

% local.quicksort_acc.elims
thf(fact_960_local_Osunion__with_Ocases,axiom,
    ! [X: produc7336345174136042802list_a] :
      ( ! [F2: a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a,K2: a,V2: produc271001111946445293list_a,As: list_P5792129111969709340list_a,K3: a,V5: produc271001111946445293list_a,Bs: list_P5792129111969709340list_a] :
          ( X
         != ( produc9035104894279545956list_a @ F2 @ ( produc9181221412137986839list_a @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K2 @ V2 ) @ As ) @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K3 @ V5 ) @ Bs ) ) ) )
     => ( ! [F2: a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a,Bs: list_P5792129111969709340list_a] :
            ( X
           != ( produc9035104894279545956list_a @ F2 @ ( produc9181221412137986839list_a @ nil_Pr347010806919629468list_a @ Bs ) ) )
       => ~ ! [F2: a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a,As: list_P5792129111969709340list_a] :
              ( X
             != ( produc9035104894279545956list_a @ F2 @ ( produc9181221412137986839list_a @ As @ nil_Pr347010806919629468list_a ) ) ) ) ) ).

% local.sunion_with.cases
thf(fact_961_local_Osunion__with_Ocases,axiom,
    ! [X: produc6420972542830929824od_a_b] :
      ( ! [F2: a > b > b > b,K2: a,V2: b,As: list_P1467974523212632366od_a_b,K3: a,V5: b,Bs: list_P1467974523212632366od_a_b] :
          ( X
         != ( produc7079246663768801234od_a_b @ F2 @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ As ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K3 @ V5 ) @ Bs ) ) ) )
     => ( ! [F2: a > b > b > b,Bs: list_P1467974523212632366od_a_b] :
            ( X
           != ( produc7079246663768801234od_a_b @ F2 @ ( produc5288560138351607191od_a_b @ nil_Product_prod_a_b @ Bs ) ) )
       => ~ ! [F2: a > b > b > b,As: list_P1467974523212632366od_a_b] :
              ( X
             != ( produc7079246663768801234od_a_b @ F2 @ ( produc5288560138351607191od_a_b @ As @ nil_Product_prod_a_b ) ) ) ) ) ).

% local.sunion_with.cases
thf(fact_962_local_Oquicksort__acc_Osimps_I1_J,axiom,
    ! [Ac: list_a] :
      ( ( set_quicksort_acc_a @ less @ Ac @ nil_a )
      = Ac ) ).

% local.quicksort_acc.simps(1)
thf(fact_963_local_Oto__single__list,axiom,
    ( cons_a
    = ( ^ [X2: a] : ( append_a @ ( set_single_list_a @ X2 ) ) ) ) ).

% local.to_single_list
thf(fact_964_local_Oquicksort__acc_Osimps_I2_J,axiom,
    ! [Ac: list_a,X: a] :
      ( ( set_quicksort_acc_a @ less @ Ac @ ( cons_a @ X @ nil_a ) )
      = ( cons_a @ X @ Ac ) ) ).

% local.quicksort_acc.simps(2)
thf(fact_965_local_Odistinct__quicksort__acc,axiom,
    ! [Ac: list_a,Xs: list_a] :
      ( ( distinct_a @ ( set_quicksort_acc_a @ less @ Ac @ Xs ) )
      = ( distinct_a @ ( append_a @ Ac @ Xs ) ) ) ).

% local.distinct_quicksort_acc
thf(fact_966_local_Oquicksort__part_Osimps_I1_J,axiom,
    ! [Ac: list_a,X: a,Lts: list_a,Eqs: list_a,Gts: list_a] :
      ( ( set_quicksort_part_a @ less @ Ac @ X @ Lts @ Eqs @ Gts @ nil_a )
      = ( set_quicksort_acc_a @ less @ ( append_a @ Eqs @ ( cons_a @ X @ ( set_quicksort_acc_a @ less @ Ac @ Gts ) ) ) @ Lts ) ) ).

% local.quicksort_part.simps(1)
thf(fact_967_local_Oquicksort__part_Oelims,axiom,
    ! [X: list_a,Xa2: a,Xb: list_a,Xc: list_a,Xd: list_a,Xe: list_a,Y: list_a] :
      ( ( ( set_quicksort_part_a @ less @ X @ Xa2 @ Xb @ Xc @ Xd @ Xe )
        = Y )
     => ( ( ( Xe = nil_a )
         => ( Y
           != ( set_quicksort_acc_a @ less @ ( append_a @ Xc @ ( cons_a @ Xa2 @ ( set_quicksort_acc_a @ less @ X @ Xd ) ) ) @ Xb ) ) )
       => ~ ! [Z3: a,Zs2: list_a] :
              ( ( Xe
                = ( cons_a @ Z3 @ Zs2 ) )
             => ~ ( ( ( less @ Xa2 @ Z3 )
                   => ( Y
                      = ( set_quicksort_part_a @ less @ X @ Xa2 @ Xb @ Xc @ ( cons_a @ Z3 @ Xd ) @ Zs2 ) ) )
                  & ( ~ ( less @ Xa2 @ Z3 )
                   => ( ( ( less @ Z3 @ Xa2 )
                       => ( Y
                          = ( set_quicksort_part_a @ less @ X @ Xa2 @ ( cons_a @ Z3 @ Xb ) @ Xc @ Xd @ Zs2 ) ) )
                      & ( ~ ( less @ Z3 @ Xa2 )
                       => ( Y
                          = ( set_quicksort_part_a @ less @ X @ Xa2 @ Xb @ ( cons_a @ Z3 @ Xc ) @ Xd @ Zs2 ) ) ) ) ) ) ) ) ) ).

% local.quicksort_part.elims
thf(fact_968_set__append,axiom,
    ! [Xs: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( set_Product_prod_a_b2 @ ( append5335208823350062147od_a_b @ Xs @ Ys ) )
      = ( sup_su3119292821646003676od_a_b @ ( set_Product_prod_a_b2 @ Xs ) @ ( set_Product_prod_a_b2 @ Ys ) ) ) ).

% set_append
thf(fact_969_set__append,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( set_a2 @ ( append_a @ Xs @ Ys ) )
      = ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).

% set_append
thf(fact_970_local_Oset__quicksort__acc,axiom,
    ! [Ac: list_a,Xs: list_a] :
      ( ( set_a2 @ ( set_quicksort_acc_a @ less @ Ac @ Xs ) )
      = ( sup_sup_set_a @ ( set_a2 @ Ac ) @ ( set_a2 @ Xs ) ) ) ).

% local.set_quicksort_acc
thf(fact_971_split__list,axiom,
    ! [X: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs ) )
     => ? [Ys2: list_P1467974523212632366od_a_b,Zs2: list_P1467974523212632366od_a_b] :
          ( Xs
          = ( append5335208823350062147od_a_b @ Ys2 @ ( cons_P7316939131009794654od_a_b @ X @ Zs2 ) ) ) ) ).

% split_list
thf(fact_972_split__list,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
     => ? [Ys2: list_a,Zs2: list_a] :
          ( Xs
          = ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) ) ) ).

% split_list
thf(fact_973_split__list__last,axiom,
    ! [X: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs ) )
     => ? [Ys2: list_P1467974523212632366od_a_b,Zs2: list_P1467974523212632366od_a_b] :
          ( ( Xs
            = ( append5335208823350062147od_a_b @ Ys2 @ ( cons_P7316939131009794654od_a_b @ X @ Zs2 ) ) )
          & ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_974_split__list__last,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
     => ? [Ys2: list_a,Zs2: list_a] :
          ( ( Xs
            = ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
          & ~ ( member_a @ X @ ( set_a2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_975_split__list__prop,axiom,
    ! [Xs: list_P1467974523212632366od_a_b,P: product_prod_a_b > $o] :
      ( ? [X5: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X5 @ ( set_Product_prod_a_b2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_P1467974523212632366od_a_b,X3: product_prod_a_b] :
          ( ? [Zs2: list_P1467974523212632366od_a_b] :
              ( Xs
              = ( append5335208823350062147od_a_b @ Ys2 @ ( cons_P7316939131009794654od_a_b @ X3 @ Zs2 ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_976_split__list__prop,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_a,X3: a] :
          ( ? [Zs2: list_a] :
              ( Xs
              = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_977_split__list__first,axiom,
    ! [X: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs ) )
     => ? [Ys2: list_P1467974523212632366od_a_b,Zs2: list_P1467974523212632366od_a_b] :
          ( ( Xs
            = ( append5335208823350062147od_a_b @ Ys2 @ ( cons_P7316939131009794654od_a_b @ X @ Zs2 ) ) )
          & ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_978_split__list__first,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
     => ? [Ys2: list_a,Zs2: list_a] :
          ( ( Xs
            = ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
          & ~ ( member_a @ X @ ( set_a2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_979_split__list__propE,axiom,
    ! [Xs: list_P1467974523212632366od_a_b,P: product_prod_a_b > $o] :
      ( ? [X5: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X5 @ ( set_Product_prod_a_b2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_P1467974523212632366od_a_b,X3: product_prod_a_b] :
            ( ? [Zs2: list_P1467974523212632366od_a_b] :
                ( Xs
                = ( append5335208823350062147od_a_b @ Ys2 @ ( cons_P7316939131009794654od_a_b @ X3 @ Zs2 ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_980_split__list__propE,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_a,X3: a] :
            ( ? [Zs2: list_a] :
                ( Xs
                = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_981_append__Cons__eq__iff,axiom,
    ! [X: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Xs3: list_P1467974523212632366od_a_b,Ys3: list_P1467974523212632366od_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs ) )
     => ( ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Ys ) )
       => ( ( ( append5335208823350062147od_a_b @ Xs @ ( cons_P7316939131009794654od_a_b @ X @ Ys ) )
            = ( append5335208823350062147od_a_b @ Xs3 @ ( cons_P7316939131009794654od_a_b @ X @ Ys3 ) ) )
          = ( ( Xs = Xs3 )
            & ( Ys = Ys3 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_982_append__Cons__eq__iff,axiom,
    ! [X: a,Xs: list_a,Ys: list_a,Xs3: list_a,Ys3: list_a] :
      ( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
     => ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
       => ( ( ( append_a @ Xs @ ( cons_a @ X @ Ys ) )
            = ( append_a @ Xs3 @ ( cons_a @ X @ Ys3 ) ) )
          = ( ( Xs = Xs3 )
            & ( Ys = Ys3 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_983_in__set__conv__decomp,axiom,
    ! [X: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs ) )
      = ( ? [Ys4: list_P1467974523212632366od_a_b,Zs3: list_P1467974523212632366od_a_b] :
            ( Xs
            = ( append5335208823350062147od_a_b @ Ys4 @ ( cons_P7316939131009794654od_a_b @ X @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_984_in__set__conv__decomp,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
      = ( ? [Ys4: list_a,Zs3: list_a] :
            ( Xs
            = ( append_a @ Ys4 @ ( cons_a @ X @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_985_split__list__last__prop,axiom,
    ! [Xs: list_P1467974523212632366od_a_b,P: product_prod_a_b > $o] :
      ( ? [X5: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X5 @ ( set_Product_prod_a_b2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_P1467974523212632366od_a_b,X3: product_prod_a_b,Zs2: list_P1467974523212632366od_a_b] :
          ( ( Xs
            = ( append5335208823350062147od_a_b @ Ys2 @ ( cons_P7316939131009794654od_a_b @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa: product_prod_a_b] :
              ( ( member1426531481828664017od_a_b @ Xa @ ( set_Product_prod_a_b2 @ Zs2 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_986_split__list__last__prop,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_a,X3: a,Zs2: list_a] :
          ( ( Xs
            = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa: a] :
              ( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_987_split__list__first__prop,axiom,
    ! [Xs: list_P1467974523212632366od_a_b,P: product_prod_a_b > $o] :
      ( ? [X5: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X5 @ ( set_Product_prod_a_b2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_P1467974523212632366od_a_b,X3: product_prod_a_b] :
          ( ? [Zs2: list_P1467974523212632366od_a_b] :
              ( Xs
              = ( append5335208823350062147od_a_b @ Ys2 @ ( cons_P7316939131009794654od_a_b @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa: product_prod_a_b] :
              ( ( member1426531481828664017od_a_b @ Xa @ ( set_Product_prod_a_b2 @ Ys2 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_988_split__list__first__prop,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_a,X3: a] :
          ( ? [Zs2: list_a] :
              ( Xs
              = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa: a] :
              ( ( member_a @ Xa @ ( set_a2 @ Ys2 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_989_split__list__last__propE,axiom,
    ! [Xs: list_P1467974523212632366od_a_b,P: product_prod_a_b > $o] :
      ( ? [X5: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X5 @ ( set_Product_prod_a_b2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_P1467974523212632366od_a_b,X3: product_prod_a_b,Zs2: list_P1467974523212632366od_a_b] :
            ( ( Xs
              = ( append5335208823350062147od_a_b @ Ys2 @ ( cons_P7316939131009794654od_a_b @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa: product_prod_a_b] :
                    ( ( member1426531481828664017od_a_b @ Xa @ ( set_Product_prod_a_b2 @ Zs2 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_990_split__list__last__propE,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_a,X3: a,Zs2: list_a] :
            ( ( Xs
              = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa: a] :
                    ( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_991_split__list__first__propE,axiom,
    ! [Xs: list_P1467974523212632366od_a_b,P: product_prod_a_b > $o] :
      ( ? [X5: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X5 @ ( set_Product_prod_a_b2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_P1467974523212632366od_a_b,X3: product_prod_a_b] :
            ( ? [Zs2: list_P1467974523212632366od_a_b] :
                ( Xs
                = ( append5335208823350062147od_a_b @ Ys2 @ ( cons_P7316939131009794654od_a_b @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa: product_prod_a_b] :
                    ( ( member1426531481828664017od_a_b @ Xa @ ( set_Product_prod_a_b2 @ Ys2 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_992_split__list__first__propE,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_a,X3: a] :
            ( ? [Zs2: list_a] :
                ( Xs
                = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa: a] :
                    ( ( member_a @ Xa @ ( set_a2 @ Ys2 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_993_in__set__conv__decomp__last,axiom,
    ! [X: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs ) )
      = ( ? [Ys4: list_P1467974523212632366od_a_b,Zs3: list_P1467974523212632366od_a_b] :
            ( ( Xs
              = ( append5335208823350062147od_a_b @ Ys4 @ ( cons_P7316939131009794654od_a_b @ X @ Zs3 ) ) )
            & ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_994_in__set__conv__decomp__last,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
      = ( ? [Ys4: list_a,Zs3: list_a] :
            ( ( Xs
              = ( append_a @ Ys4 @ ( cons_a @ X @ Zs3 ) ) )
            & ~ ( member_a @ X @ ( set_a2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_995_in__set__conv__decomp__first,axiom,
    ! [X: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs ) )
      = ( ? [Ys4: list_P1467974523212632366od_a_b,Zs3: list_P1467974523212632366od_a_b] :
            ( ( Xs
              = ( append5335208823350062147od_a_b @ Ys4 @ ( cons_P7316939131009794654od_a_b @ X @ Zs3 ) ) )
            & ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Ys4 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_996_in__set__conv__decomp__first,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
      = ( ? [Ys4: list_a,Zs3: list_a] :
            ( ( Xs
              = ( append_a @ Ys4 @ ( cons_a @ X @ Zs3 ) ) )
            & ~ ( member_a @ X @ ( set_a2 @ Ys4 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_997_split__list__last__prop__iff,axiom,
    ! [Xs: list_P1467974523212632366od_a_b,P: product_prod_a_b > $o] :
      ( ( ? [X2: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X2 @ ( set_Product_prod_a_b2 @ Xs ) )
            & ( P @ X2 ) ) )
      = ( ? [Ys4: list_P1467974523212632366od_a_b,X2: product_prod_a_b,Zs3: list_P1467974523212632366od_a_b] :
            ( ( Xs
              = ( append5335208823350062147od_a_b @ Ys4 @ ( cons_P7316939131009794654od_a_b @ X2 @ Zs3 ) ) )
            & ( P @ X2 )
            & ! [Y3: product_prod_a_b] :
                ( ( member1426531481828664017od_a_b @ Y3 @ ( set_Product_prod_a_b2 @ Zs3 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_998_split__list__last__prop__iff,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ( ? [X2: a] :
            ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
            & ( P @ X2 ) ) )
      = ( ? [Ys4: list_a,X2: a,Zs3: list_a] :
            ( ( Xs
              = ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) )
            & ( P @ X2 )
            & ! [Y3: a] :
                ( ( member_a @ Y3 @ ( set_a2 @ Zs3 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_999_split__list__first__prop__iff,axiom,
    ! [Xs: list_P1467974523212632366od_a_b,P: product_prod_a_b > $o] :
      ( ( ? [X2: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X2 @ ( set_Product_prod_a_b2 @ Xs ) )
            & ( P @ X2 ) ) )
      = ( ? [Ys4: list_P1467974523212632366od_a_b,X2: product_prod_a_b] :
            ( ? [Zs3: list_P1467974523212632366od_a_b] :
                ( Xs
                = ( append5335208823350062147od_a_b @ Ys4 @ ( cons_P7316939131009794654od_a_b @ X2 @ Zs3 ) ) )
            & ( P @ X2 )
            & ! [Y3: product_prod_a_b] :
                ( ( member1426531481828664017od_a_b @ Y3 @ ( set_Product_prod_a_b2 @ Ys4 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1000_split__list__first__prop__iff,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ( ? [X2: a] :
            ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
            & ( P @ X2 ) ) )
      = ( ? [Ys4: list_a,X2: a] :
            ( ? [Zs3: list_a] :
                ( Xs
                = ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) )
            & ( P @ X2 )
            & ! [Y3: a] :
                ( ( member_a @ Y3 @ ( set_a2 @ Ys4 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1001_remove1__append,axiom,
    ! [X: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs ) )
       => ( ( remove3530114610544164932od_a_b @ X @ ( append5335208823350062147od_a_b @ Xs @ Ys ) )
          = ( append5335208823350062147od_a_b @ ( remove3530114610544164932od_a_b @ X @ Xs ) @ Ys ) ) )
      & ( ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs ) )
       => ( ( remove3530114610544164932od_a_b @ X @ ( append5335208823350062147od_a_b @ Xs @ Ys ) )
          = ( append5335208823350062147od_a_b @ Xs @ ( remove3530114610544164932od_a_b @ X @ Ys ) ) ) ) ) ).

% remove1_append
thf(fact_1002_remove1__append,axiom,
    ! [X: a,Xs: list_a,Ys: list_a] :
      ( ( ( member_a @ X @ ( set_a2 @ Xs ) )
       => ( ( remove1_a @ X @ ( append_a @ Xs @ Ys ) )
          = ( append_a @ ( remove1_a @ X @ Xs ) @ Ys ) ) )
      & ( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
       => ( ( remove1_a @ X @ ( append_a @ Xs @ Ys ) )
          = ( append_a @ Xs @ ( remove1_a @ X @ Ys ) ) ) ) ) ).

% remove1_append
thf(fact_1003_comp__sunion__with_Ocases,axiom,
    ! [X: produc6790509112560320207list_a] :
      ( ! [F2: list_a > produc2547433716817190550list_a > produc2547433716817190550list_a > produc2547433716817190550list_a,K2: list_a,V2: produc2547433716817190550list_a,As: list_P1945613012453903807list_a,K3: list_a,V5: produc2547433716817190550list_a,Bs: list_P1945613012453903807list_a] :
          ( X
         != ( produc846896688571326337list_a @ F2 @ ( produc6682309408470888791list_a @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K2 @ V2 ) @ As ) @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K3 @ V5 ) @ Bs ) ) ) )
     => ( ! [F2: list_a > produc2547433716817190550list_a > produc2547433716817190550list_a > produc2547433716817190550list_a,Bs: list_P1945613012453903807list_a] :
            ( X
           != ( produc846896688571326337list_a @ F2 @ ( produc6682309408470888791list_a @ nil_Pr3732201366458353727list_a @ Bs ) ) )
       => ~ ! [F2: list_a > produc2547433716817190550list_a > produc2547433716817190550list_a > produc2547433716817190550list_a,As: list_P1945613012453903807list_a] :
              ( X
             != ( produc846896688571326337list_a @ F2 @ ( produc6682309408470888791list_a @ As @ nil_Pr3732201366458353727list_a ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_1004_comp__sunion__with_Ocases,axiom,
    ! [X: produc1586653734184941793list_a] :
      ( ! [F2: list_a > list_a > list_a > list_a,K2: list_a,V2: list_a,As: list_P321204300973800749list_a,K3: list_a,V5: list_a,Bs: list_P321204300973800749list_a] :
          ( X
         != ( produc4938186435958756627list_a @ F2 @ ( produc6622907533990562775list_a @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K2 @ V2 ) @ As ) @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K3 @ V5 ) @ Bs ) ) ) )
     => ( ! [F2: list_a > list_a > list_a > list_a,Bs: list_P321204300973800749list_a] :
            ( X
           != ( produc4938186435958756627list_a @ F2 @ ( produc6622907533990562775list_a @ nil_Pr3188421586756112173list_a @ Bs ) ) )
       => ~ ! [F2: list_a > list_a > list_a > list_a,As: list_P321204300973800749list_a] :
              ( X
             != ( produc4938186435958756627list_a @ F2 @ ( produc6622907533990562775list_a @ As @ nil_Pr3188421586756112173list_a ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_1005_comp__sunion__with_Ocases,axiom,
    ! [X: produc6421213437698858984list_a] :
      ( ! [F2: ( a > a > $o ) > produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,K2: a > a > $o,V2: produc9164743771328383783list_a,As: list_P1472896444292205030list_a,K3: a > a > $o,V5: produc9164743771328383783list_a,Bs: list_P1472896444292205030list_a] :
          ( X
         != ( produc5942803683472797978list_a @ F2 @ ( produc8604296732560830103list_a @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K2 @ V2 ) @ As ) @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K3 @ V5 ) @ Bs ) ) ) )
     => ( ! [F2: ( a > a > $o ) > produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,Bs: list_P1472896444292205030list_a] :
            ( X
           != ( produc5942803683472797978list_a @ F2 @ ( produc8604296732560830103list_a @ nil_Pr3115398151409596518list_a @ Bs ) ) )
       => ~ ! [F2: ( a > a > $o ) > produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,As: list_P1472896444292205030list_a] :
              ( X
             != ( produc5942803683472797978list_a @ F2 @ ( produc8604296732560830103list_a @ As @ nil_Pr3115398151409596518list_a ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_1006_comp__sunion__with_Ocases,axiom,
    ! [X: produc7336345174136042802list_a] :
      ( ! [F2: a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a,K2: a,V2: produc271001111946445293list_a,As: list_P5792129111969709340list_a,K3: a,V5: produc271001111946445293list_a,Bs: list_P5792129111969709340list_a] :
          ( X
         != ( produc9035104894279545956list_a @ F2 @ ( produc9181221412137986839list_a @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K2 @ V2 ) @ As ) @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K3 @ V5 ) @ Bs ) ) ) )
     => ( ! [F2: a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a,Bs: list_P5792129111969709340list_a] :
            ( X
           != ( produc9035104894279545956list_a @ F2 @ ( produc9181221412137986839list_a @ nil_Pr347010806919629468list_a @ Bs ) ) )
       => ~ ! [F2: a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a,As: list_P5792129111969709340list_a] :
              ( X
             != ( produc9035104894279545956list_a @ F2 @ ( produc9181221412137986839list_a @ As @ nil_Pr347010806919629468list_a ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_1007_comp__sunion__with_Ocases,axiom,
    ! [X: produc6420972542830929824od_a_b] :
      ( ! [F2: a > b > b > b,K2: a,V2: b,As: list_P1467974523212632366od_a_b,K3: a,V5: b,Bs: list_P1467974523212632366od_a_b] :
          ( X
         != ( produc7079246663768801234od_a_b @ F2 @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ As ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K3 @ V5 ) @ Bs ) ) ) )
     => ( ! [F2: a > b > b > b,Bs: list_P1467974523212632366od_a_b] :
            ( X
           != ( produc7079246663768801234od_a_b @ F2 @ ( produc5288560138351607191od_a_b @ nil_Product_prod_a_b @ Bs ) ) )
       => ~ ! [F2: a > b > b > b,As: list_P1467974523212632366od_a_b] :
              ( X
             != ( produc7079246663768801234od_a_b @ F2 @ ( produc5288560138351607191od_a_b @ As @ nil_Product_prod_a_b ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_1008_ord__class_Osunion__with_Ocases,axiom,
    ! [X: produc6421213437698858984list_a] :
      ( ! [F2: ( a > a > $o ) > produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,K2: a > a > $o,V2: produc9164743771328383783list_a,As: list_P1472896444292205030list_a,K3: a > a > $o,V5: produc9164743771328383783list_a,Bs: list_P1472896444292205030list_a] :
          ( X
         != ( produc5942803683472797978list_a @ F2 @ ( produc8604296732560830103list_a @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K2 @ V2 ) @ As ) @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K3 @ V5 ) @ Bs ) ) ) )
     => ( ! [F2: ( a > a > $o ) > produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,Bs: list_P1472896444292205030list_a] :
            ( X
           != ( produc5942803683472797978list_a @ F2 @ ( produc8604296732560830103list_a @ nil_Pr3115398151409596518list_a @ Bs ) ) )
       => ~ ! [F2: ( a > a > $o ) > produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,As: list_P1472896444292205030list_a] :
              ( X
             != ( produc5942803683472797978list_a @ F2 @ ( produc8604296732560830103list_a @ As @ nil_Pr3115398151409596518list_a ) ) ) ) ) ).

% ord_class.sunion_with.cases
thf(fact_1009_entries_Osimps_I1_J,axiom,
    ( ( rBT_entries_a_b @ rBT_Empty_a_b )
    = nil_Product_prod_a_b ) ).

% entries.simps(1)
thf(fact_1010_ord_Oset__quicksort__acc,axiom,
    ! [Less: product_prod_a_b > product_prod_a_b > $o,Ac: list_P1467974523212632366od_a_b,Xs: list_P1467974523212632366od_a_b] :
      ( ( set_Product_prod_a_b2 @ ( set_qu2587572545967408182od_a_b @ Less @ Ac @ Xs ) )
      = ( sup_su3119292821646003676od_a_b @ ( set_Product_prod_a_b2 @ Ac ) @ ( set_Product_prod_a_b2 @ Xs ) ) ) ).

% ord.set_quicksort_acc
thf(fact_1011_ord_Oset__quicksort__acc,axiom,
    ! [Less: a > a > $o,Ac: list_a,Xs: list_a] :
      ( ( set_a2 @ ( set_quicksort_acc_a @ Less @ Ac @ Xs ) )
      = ( sup_sup_set_a @ ( set_a2 @ Ac ) @ ( set_a2 @ Xs ) ) ) ).

% ord.set_quicksort_acc
thf(fact_1012_not__distinct__conv__prefix,axiom,
    ! [As2: list_P1467974523212632366od_a_b] :
      ( ( ~ ( distin132333874345289761od_a_b @ As2 ) )
      = ( ? [Xs4: list_P1467974523212632366od_a_b,Y3: product_prod_a_b,Ys4: list_P1467974523212632366od_a_b] :
            ( ( member1426531481828664017od_a_b @ Y3 @ ( set_Product_prod_a_b2 @ Xs4 ) )
            & ( distin132333874345289761od_a_b @ Xs4 )
            & ( As2
              = ( append5335208823350062147od_a_b @ Xs4 @ ( cons_P7316939131009794654od_a_b @ Y3 @ Ys4 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1013_not__distinct__conv__prefix,axiom,
    ! [As2: list_a] :
      ( ( ~ ( distinct_a @ As2 ) )
      = ( ? [Xs4: list_a,Y3: a,Ys4: list_a] :
            ( ( member_a @ Y3 @ ( set_a2 @ Xs4 ) )
            & ( distinct_a @ Xs4 )
            & ( As2
              = ( append_a @ Xs4 @ ( cons_a @ Y3 @ Ys4 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1014_remove1__split,axiom,
    ! [A: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ A @ ( set_Product_prod_a_b2 @ Xs ) )
     => ( ( ( remove3530114610544164932od_a_b @ A @ Xs )
          = Ys )
        = ( ? [Ls: list_P1467974523212632366od_a_b,Rs: list_P1467974523212632366od_a_b] :
              ( ( Xs
                = ( append5335208823350062147od_a_b @ Ls @ ( cons_P7316939131009794654od_a_b @ A @ Rs ) ) )
              & ~ ( member1426531481828664017od_a_b @ A @ ( set_Product_prod_a_b2 @ Ls ) )
              & ( Ys
                = ( append5335208823350062147od_a_b @ Ls @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1015_remove1__split,axiom,
    ! [A: a,Xs: list_a,Ys: list_a] :
      ( ( member_a @ A @ ( set_a2 @ Xs ) )
     => ( ( ( remove1_a @ A @ Xs )
          = Ys )
        = ( ? [Ls: list_a,Rs: list_a] :
              ( ( Xs
                = ( append_a @ Ls @ ( cons_a @ A @ Rs ) ) )
              & ~ ( member_a @ A @ ( set_a2 @ Ls ) )
              & ( Ys
                = ( append_a @ Ls @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1016_local_Oquicksort__def,axiom,
    ( ( set_quicksort_a @ less )
    = ( set_quicksort_acc_a @ less @ nil_a ) ) ).

% local.quicksort_def
thf(fact_1017_entries__balance,axiom,
    ! [L: rBT_rb6721489163287533103list_a,K: list_a,V: produc2547433716817190550list_a,R: rBT_rb6721489163287533103list_a] :
      ( ( rBT_en3328903274962959370list_a @ ( rBT_ba3573774157313868950list_a @ L @ K @ V @ R ) )
      = ( append4972384724332675412list_a @ ( rBT_en3328903274962959370list_a @ L ) @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K @ V ) @ ( rBT_en3328903274962959370list_a @ R ) ) ) ) ).

% entries_balance
thf(fact_1018_entries__balance,axiom,
    ! [L: rBT_rb146355364677272861list_a,K: list_a,V: list_a,R: rBT_rb146355364677272861list_a] :
      ( ( rBT_en6256609917331000056list_a @ ( rBT_ba132907544946337156list_a @ L @ K @ V @ R ) )
      = ( append622457529216362434list_a @ ( rBT_en6256609917331000056list_a @ L ) @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K @ V ) @ ( rBT_en6256609917331000056list_a @ R ) ) ) ) ).

% entries_balance
thf(fact_1019_entries__balance,axiom,
    ! [L: rBT_rb9014005217275227734list_a,K: a > a > $o,V: produc9164743771328383783list_a,R: rBT_rb9014005217275227734list_a] :
      ( ( rBT_en1829592632444517297list_a @ ( rBT_ba3932203811964311101list_a @ L @ K @ V @ R ) )
      = ( append61665529288884475list_a @ ( rBT_en1829592632444517297list_a @ L ) @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K @ V ) @ ( rBT_en1829592632444517297list_a @ R ) ) ) ) ).

% entries_balance
thf(fact_1020_entries__balance,axiom,
    ! [L: rBT_rb484615002580549772list_a,K: a,V: produc271001111946445293list_a,R: rBT_rb484615002580549772list_a] :
      ( ( rBT_en3875316870356833511list_a @ ( rBT_ba225581376578813811list_a @ L @ K @ V @ R ) )
      = ( append1302148889043496753list_a @ ( rBT_en3875316870356833511list_a @ L ) @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ ( rBT_en3875316870356833511list_a @ R ) ) ) ) ).

% entries_balance
thf(fact_1021_entries__balance,axiom,
    ! [L: rBT_rbt_a_b,K: a,V: b,R: rBT_rbt_a_b] :
      ( ( rBT_entries_a_b @ ( rBT_balance_a_b @ L @ K @ V @ R ) )
      = ( append5335208823350062147od_a_b @ ( rBT_entries_a_b @ L ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ ( rBT_entries_a_b @ R ) ) ) ) ).

% entries_balance
thf(fact_1022_local_Oset__quicksort,axiom,
    ! [Xs: list_a] :
      ( ( set_a2 @ ( set_quicksort_a @ less @ Xs ) )
      = ( set_a2 @ Xs ) ) ).

% local.set_quicksort
thf(fact_1023_local_Odistinct__quicksort,axiom,
    ! [Xs: list_a] :
      ( ( distinct_a @ ( set_quicksort_a @ less @ Xs ) )
      = ( distinct_a @ Xs ) ) ).

% local.distinct_quicksort
thf(fact_1024_ord_Oset__quicksort,axiom,
    ! [Less: product_prod_a_b > product_prod_a_b > $o,Xs: list_P1467974523212632366od_a_b] :
      ( ( set_Product_prod_a_b2 @ ( set_qu3445024020691032247od_a_b @ Less @ Xs ) )
      = ( set_Product_prod_a_b2 @ Xs ) ) ).

% ord.set_quicksort
thf(fact_1025_ord_Oset__quicksort,axiom,
    ! [Less: a > a > $o,Xs: list_a] :
      ( ( set_a2 @ ( set_quicksort_a @ Less @ Xs ) )
      = ( set_a2 @ Xs ) ) ).

% ord.set_quicksort
thf(fact_1026_entries_Osimps_I2_J,axiom,
    ! [Uu: rBT_color,L: rBT_rb6721489163287533103list_a,K: list_a,V: produc2547433716817190550list_a,R: rBT_rb6721489163287533103list_a] :
      ( ( rBT_en3328903274962959370list_a @ ( rBT_Br1204270329063416912list_a @ Uu @ L @ K @ V @ R ) )
      = ( append4972384724332675412list_a @ ( rBT_en3328903274962959370list_a @ L ) @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K @ V ) @ ( rBT_en3328903274962959370list_a @ R ) ) ) ) ).

% entries.simps(2)
thf(fact_1027_entries_Osimps_I2_J,axiom,
    ! [Uu: rBT_color,L: rBT_rb146355364677272861list_a,K: list_a,V: list_a,R: rBT_rb146355364677272861list_a] :
      ( ( rBT_en6256609917331000056list_a @ ( rBT_Br8372335562727726142list_a @ Uu @ L @ K @ V @ R ) )
      = ( append622457529216362434list_a @ ( rBT_en6256609917331000056list_a @ L ) @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K @ V ) @ ( rBT_en6256609917331000056list_a @ R ) ) ) ) ).

% entries.simps(2)
thf(fact_1028_entries_Osimps_I2_J,axiom,
    ! [Uu: rBT_color,L: rBT_rb9014005217275227734list_a,K: a > a > $o,V: produc9164743771328383783list_a,R: rBT_rb9014005217275227734list_a] :
      ( ( rBT_en1829592632444517297list_a @ ( rBT_Br6721835470276586743list_a @ Uu @ L @ K @ V @ R ) )
      = ( append61665529288884475list_a @ ( rBT_en1829592632444517297list_a @ L ) @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K @ V ) @ ( rBT_en1829592632444517297list_a @ R ) ) ) ) ).

% entries.simps(2)
thf(fact_1029_entries_Osimps_I2_J,axiom,
    ! [Uu: rBT_color,L: rBT_rb484615002580549772list_a,K: a,V: produc271001111946445293list_a,R: rBT_rb484615002580549772list_a] :
      ( ( rBT_en3875316870356833511list_a @ ( rBT_Br706236987277315629list_a @ Uu @ L @ K @ V @ R ) )
      = ( append1302148889043496753list_a @ ( rBT_en3875316870356833511list_a @ L ) @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ ( rBT_en3875316870356833511list_a @ R ) ) ) ) ).

% entries.simps(2)
thf(fact_1030_entries_Osimps_I2_J,axiom,
    ! [Uu: rBT_color,L: rBT_rbt_a_b,K: a,V: b,R: rBT_rbt_a_b] :
      ( ( rBT_entries_a_b @ ( rBT_Branch_a_b @ Uu @ L @ K @ V @ R ) )
      = ( append5335208823350062147od_a_b @ ( rBT_entries_a_b @ L ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ ( rBT_entries_a_b @ R ) ) ) ) ).

% entries.simps(2)
thf(fact_1031_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc3989028204896995298list_a] :
      ( ! [K2: list_a] :
          ( X
         != ( produc2273375156964460242list_a @ K2 @ nil_Pr3732201366458353727list_a ) )
     => ~ ! [K2: list_a,K3: list_a,V2: produc2547433716817190550list_a,Xs2: list_P1945613012453903807list_a] :
            ( X
           != ( produc2273375156964460242list_a @ K2 @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K3 @ V2 ) @ Xs2 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_1032_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc4823114226738795600list_a] :
      ( ! [K2: list_a] :
          ( X
         != ( produc3757311814099817152list_a @ K2 @ nil_Pr3188421586756112173list_a ) )
     => ~ ! [K2: list_a,K3: list_a,V2: list_a,Xs2: list_P321204300973800749list_a] :
            ( X
           != ( produc3757311814099817152list_a @ K2 @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K3 @ V2 ) @ Xs2 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_1033_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc7336575416149648543list_a] :
      ( ! [K2: a > a > $o] :
          ( X
         != ( produc5309338222365962767list_a @ K2 @ nil_Pr3115398151409596518list_a ) )
     => ~ ! [K2: a > a > $o,K3: a > a > $o,V2: produc9164743771328383783list_a,Xs2: list_P1472896444292205030list_a] :
            ( X
           != ( produc5309338222365962767list_a @ K2 @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K3 @ V2 ) @ Xs2 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_1034_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc7879897333239164229list_a] :
      ( ! [K2: a] :
          ( X
         != ( produc4159846287338058421list_a @ K2 @ nil_Pr347010806919629468list_a ) )
     => ~ ! [K2: a,K3: a,V2: produc271001111946445293list_a,Xs2: list_P5792129111969709340list_a] :
            ( X
           != ( produc4159846287338058421list_a @ K2 @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K3 @ V2 ) @ Xs2 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_1035_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc2639161875728825943od_a_b] :
      ( ! [K2: a] :
          ( X
         != ( produc6643264353547226055od_a_b @ K2 @ nil_Product_prod_a_b ) )
     => ~ ! [K2: a,K3: a,V2: b,Xs2: list_P1467974523212632366od_a_b] :
            ( X
           != ( produc6643264353547226055od_a_b @ K2 @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K3 @ V2 ) @ Xs2 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_1036_local_Osunion__with_Oelims,axiom,
    ! [X: a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a,Xa2: list_P5792129111969709340list_a,Xb: list_P5792129111969709340list_a,Y: list_P5792129111969709340list_a] :
      ( ( ( rBT_su2724219885554377002list_a @ less @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [K2: a,V2: produc271001111946445293list_a,As: list_P5792129111969709340list_a] :
            ( ( Xa2
              = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K2 @ V2 ) @ As ) )
           => ! [K3: a,V5: produc271001111946445293list_a,Bs: list_P5792129111969709340list_a] :
                ( ( Xb
                  = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K3 @ V5 ) @ Bs ) )
               => ~ ( ( ( less @ K3 @ K2 )
                     => ( Y
                        = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K3 @ V5 ) @ ( rBT_su2724219885554377002list_a @ less @ X @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K2 @ V2 ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( less @ K3 @ K2 )
                     => ( ( ( less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K2 @ V2 ) @ ( rBT_su2724219885554377002list_a @ less @ X @ As @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K3 @ V5 ) @ Bs ) ) ) ) )
                        & ( ~ ( less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K2 @ ( X @ K2 @ V2 @ V5 ) ) @ ( rBT_su2724219885554377002list_a @ less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr347010806919629468list_a )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr347010806919629468list_a )
             => ( Y != Xa2 ) ) ) ) ) ).

% local.sunion_with.elims
thf(fact_1037_local_Osunion__with_Oelims,axiom,
    ! [X: a > b > b > b,Xa2: list_P1467974523212632366od_a_b,Xb: list_P1467974523212632366od_a_b,Y: list_P1467974523212632366od_a_b] :
      ( ( ( rBT_sunion_with_a_b @ less @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [K2: a,V2: b,As: list_P1467974523212632366od_a_b] :
            ( ( Xa2
              = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ As ) )
           => ! [K3: a,V5: b,Bs: list_P1467974523212632366od_a_b] :
                ( ( Xb
                  = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K3 @ V5 ) @ Bs ) )
               => ~ ( ( ( less @ K3 @ K2 )
                     => ( Y
                        = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K3 @ V5 ) @ ( rBT_sunion_with_a_b @ less @ X @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( less @ K3 @ K2 )
                     => ( ( ( less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ ( rBT_sunion_with_a_b @ less @ X @ As @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K3 @ V5 ) @ Bs ) ) ) ) )
                        & ( ~ ( less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ ( X @ K2 @ V2 @ V5 ) ) @ ( rBT_sunion_with_a_b @ less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Product_prod_a_b )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Product_prod_a_b )
             => ( Y != Xa2 ) ) ) ) ) ).

% local.sunion_with.elims
thf(fact_1038_local_Osunion__with_Osimps_I1_J,axiom,
    ! [K4: a,K: a,F: a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a,V: produc271001111946445293list_a,As2: list_P5792129111969709340list_a,V3: produc271001111946445293list_a,Bs2: list_P5792129111969709340list_a] :
      ( ( ( less @ K4 @ K )
       => ( ( rBT_su2724219885554377002list_a @ less @ F @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ As2 ) @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ Bs2 ) )
          = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ ( rBT_su2724219885554377002list_a @ less @ F @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( less @ K4 @ K )
       => ( ( ( less @ K @ K4 )
           => ( ( rBT_su2724219885554377002list_a @ less @ F @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ As2 ) @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ ( rBT_su2724219885554377002list_a @ less @ F @ As2 @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ Bs2 ) ) ) ) )
          & ( ~ ( less @ K @ K4 )
           => ( ( rBT_su2724219885554377002list_a @ less @ F @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ As2 ) @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ ( F @ K @ V @ V3 ) ) @ ( rBT_su2724219885554377002list_a @ less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% local.sunion_with.simps(1)
thf(fact_1039_local_Osunion__with_Osimps_I1_J,axiom,
    ! [K4: a,K: a,F: a > b > b > b,V: b,As2: list_P1467974523212632366od_a_b,V3: b,Bs2: list_P1467974523212632366od_a_b] :
      ( ( ( less @ K4 @ K )
       => ( ( rBT_sunion_with_a_b @ less @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ As2 ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ Bs2 ) )
          = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ ( rBT_sunion_with_a_b @ less @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( less @ K4 @ K )
       => ( ( ( less @ K @ K4 )
           => ( ( rBT_sunion_with_a_b @ less @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ As2 ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ ( rBT_sunion_with_a_b @ less @ F @ As2 @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ Bs2 ) ) ) ) )
          & ( ~ ( less @ K @ K4 )
           => ( ( rBT_sunion_with_a_b @ less @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ As2 ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ ( F @ K @ V @ V3 ) ) @ ( rBT_sunion_with_a_b @ less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% local.sunion_with.simps(1)
thf(fact_1040_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: list_a > list_a > $o,K4: list_a,K: list_a,F: list_a > produc2547433716817190550list_a > produc2547433716817190550list_a > produc2547433716817190550list_a,V: produc2547433716817190550list_a,As2: list_P1945613012453903807list_a,V3: produc2547433716817190550list_a,Bs2: list_P1945613012453903807list_a] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_su605796190718627789list_a @ Less @ F @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K @ V ) @ As2 ) @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K4 @ V3 ) @ Bs2 ) )
          = ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K4 @ V3 ) @ ( rBT_su605796190718627789list_a @ Less @ F @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K @ V ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_su605796190718627789list_a @ Less @ F @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K @ V ) @ As2 ) @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K @ V ) @ ( rBT_su605796190718627789list_a @ Less @ F @ As2 @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K4 @ V3 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_su605796190718627789list_a @ Less @ F @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K @ V ) @ As2 ) @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K @ ( F @ K @ V @ V3 ) ) @ ( rBT_su605796190718627789list_a @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_1041_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: list_a > list_a > $o,K4: list_a,K: list_a,F: list_a > list_a > list_a > list_a,V: list_a,As2: list_P321204300973800749list_a,V3: list_a,Bs2: list_P321204300973800749list_a] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_su5562828613431444283list_a @ Less @ F @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K @ V ) @ As2 ) @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K4 @ V3 ) @ Bs2 ) )
          = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K4 @ V3 ) @ ( rBT_su5562828613431444283list_a @ Less @ F @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K @ V ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_su5562828613431444283list_a @ Less @ F @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K @ V ) @ As2 ) @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K @ V ) @ ( rBT_su5562828613431444283list_a @ Less @ F @ As2 @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K4 @ V3 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_su5562828613431444283list_a @ Less @ F @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K @ V ) @ As2 ) @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K @ ( F @ K @ V @ V3 ) ) @ ( rBT_su5562828613431444283list_a @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_1042_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: ( a > a > $o ) > ( a > a > $o ) > $o,K4: a > a > $o,K: a > a > $o,F: ( a > a > $o ) > produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,V: produc9164743771328383783list_a,As2: list_P1472896444292205030list_a,V3: produc9164743771328383783list_a,Bs2: list_P1472896444292205030list_a] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_su4068828071963083764list_a @ Less @ F @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K @ V ) @ As2 ) @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K4 @ V3 ) @ Bs2 ) )
          = ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K4 @ V3 ) @ ( rBT_su4068828071963083764list_a @ Less @ F @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K @ V ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_su4068828071963083764list_a @ Less @ F @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K @ V ) @ As2 ) @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K @ V ) @ ( rBT_su4068828071963083764list_a @ Less @ F @ As2 @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K4 @ V3 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_su4068828071963083764list_a @ Less @ F @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K @ V ) @ As2 ) @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K @ ( F @ K @ V @ V3 ) ) @ ( rBT_su4068828071963083764list_a @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_1043_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: a > a > $o,K4: a,K: a,F: a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a,V: produc271001111946445293list_a,As2: list_P5792129111969709340list_a,V3: produc271001111946445293list_a,Bs2: list_P5792129111969709340list_a] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_su2724219885554377002list_a @ Less @ F @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ As2 ) @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ Bs2 ) )
          = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ ( rBT_su2724219885554377002list_a @ Less @ F @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_su2724219885554377002list_a @ Less @ F @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ As2 ) @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ ( rBT_su2724219885554377002list_a @ Less @ F @ As2 @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_su2724219885554377002list_a @ Less @ F @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ As2 ) @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ ( F @ K @ V @ V3 ) ) @ ( rBT_su2724219885554377002list_a @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_1044_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: a > a > $o,K4: a,K: a,F: a > b > b > b,V: b,As2: list_P1467974523212632366od_a_b,V3: b,Bs2: list_P1467974523212632366od_a_b] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_sunion_with_a_b @ Less @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ As2 ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ Bs2 ) )
          = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ ( rBT_sunion_with_a_b @ Less @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_sunion_with_a_b @ Less @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ As2 ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ ( rBT_sunion_with_a_b @ Less @ F @ As2 @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_sunion_with_a_b @ Less @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ As2 ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ ( F @ K @ V @ V3 ) ) @ ( rBT_sunion_with_a_b @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_1045_ord_Osunion__with_Oelims,axiom,
    ! [Less: list_a > list_a > $o,X: list_a > produc2547433716817190550list_a > produc2547433716817190550list_a > produc2547433716817190550list_a,Xa2: list_P1945613012453903807list_a,Xb: list_P1945613012453903807list_a,Y: list_P1945613012453903807list_a] :
      ( ( ( rBT_su605796190718627789list_a @ Less @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [K2: list_a,V2: produc2547433716817190550list_a,As: list_P1945613012453903807list_a] :
            ( ( Xa2
              = ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K2 @ V2 ) @ As ) )
           => ! [K3: list_a,V5: produc2547433716817190550list_a,Bs: list_P1945613012453903807list_a] :
                ( ( Xb
                  = ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K3 @ V5 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K3 @ V5 ) @ ( rBT_su605796190718627789list_a @ Less @ X @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K2 @ V2 ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K2 @ V2 ) @ ( rBT_su605796190718627789list_a @ Less @ X @ As @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K3 @ V5 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K2 @ ( X @ K2 @ V2 @ V5 ) ) @ ( rBT_su605796190718627789list_a @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr3732201366458353727list_a )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr3732201366458353727list_a )
             => ( Y != Xa2 ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_1046_ord_Osunion__with_Oelims,axiom,
    ! [Less: list_a > list_a > $o,X: list_a > list_a > list_a > list_a,Xa2: list_P321204300973800749list_a,Xb: list_P321204300973800749list_a,Y: list_P321204300973800749list_a] :
      ( ( ( rBT_su5562828613431444283list_a @ Less @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [K2: list_a,V2: list_a,As: list_P321204300973800749list_a] :
            ( ( Xa2
              = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K2 @ V2 ) @ As ) )
           => ! [K3: list_a,V5: list_a,Bs: list_P321204300973800749list_a] :
                ( ( Xb
                  = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K3 @ V5 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K3 @ V5 ) @ ( rBT_su5562828613431444283list_a @ Less @ X @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K2 @ V2 ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K2 @ V2 ) @ ( rBT_su5562828613431444283list_a @ Less @ X @ As @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K3 @ V5 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K2 @ ( X @ K2 @ V2 @ V5 ) ) @ ( rBT_su5562828613431444283list_a @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr3188421586756112173list_a )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr3188421586756112173list_a )
             => ( Y != Xa2 ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_1047_ord_Osunion__with_Oelims,axiom,
    ! [Less: ( a > a > $o ) > ( a > a > $o ) > $o,X: ( a > a > $o ) > produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,Xa2: list_P1472896444292205030list_a,Xb: list_P1472896444292205030list_a,Y: list_P1472896444292205030list_a] :
      ( ( ( rBT_su4068828071963083764list_a @ Less @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [K2: a > a > $o,V2: produc9164743771328383783list_a,As: list_P1472896444292205030list_a] :
            ( ( Xa2
              = ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K2 @ V2 ) @ As ) )
           => ! [K3: a > a > $o,V5: produc9164743771328383783list_a,Bs: list_P1472896444292205030list_a] :
                ( ( Xb
                  = ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K3 @ V5 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K3 @ V5 ) @ ( rBT_su4068828071963083764list_a @ Less @ X @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K2 @ V2 ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K2 @ V2 ) @ ( rBT_su4068828071963083764list_a @ Less @ X @ As @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K3 @ V5 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K2 @ ( X @ K2 @ V2 @ V5 ) ) @ ( rBT_su4068828071963083764list_a @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr3115398151409596518list_a )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr3115398151409596518list_a )
             => ( Y != Xa2 ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_1048_ord_Osunion__with_Oelims,axiom,
    ! [Less: a > a > $o,X: a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a,Xa2: list_P5792129111969709340list_a,Xb: list_P5792129111969709340list_a,Y: list_P5792129111969709340list_a] :
      ( ( ( rBT_su2724219885554377002list_a @ Less @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [K2: a,V2: produc271001111946445293list_a,As: list_P5792129111969709340list_a] :
            ( ( Xa2
              = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K2 @ V2 ) @ As ) )
           => ! [K3: a,V5: produc271001111946445293list_a,Bs: list_P5792129111969709340list_a] :
                ( ( Xb
                  = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K3 @ V5 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K3 @ V5 ) @ ( rBT_su2724219885554377002list_a @ Less @ X @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K2 @ V2 ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K2 @ V2 ) @ ( rBT_su2724219885554377002list_a @ Less @ X @ As @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K3 @ V5 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K2 @ ( X @ K2 @ V2 @ V5 ) ) @ ( rBT_su2724219885554377002list_a @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr347010806919629468list_a )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr347010806919629468list_a )
             => ( Y != Xa2 ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_1049_ord_Osunion__with_Oelims,axiom,
    ! [Less: a > a > $o,X: a > b > b > b,Xa2: list_P1467974523212632366od_a_b,Xb: list_P1467974523212632366od_a_b,Y: list_P1467974523212632366od_a_b] :
      ( ( ( rBT_sunion_with_a_b @ Less @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [K2: a,V2: b,As: list_P1467974523212632366od_a_b] :
            ( ( Xa2
              = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ As ) )
           => ! [K3: a,V5: b,Bs: list_P1467974523212632366od_a_b] :
                ( ( Xb
                  = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K3 @ V5 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K3 @ V5 ) @ ( rBT_sunion_with_a_b @ Less @ X @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ ( rBT_sunion_with_a_b @ Less @ X @ As @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K3 @ V5 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ ( X @ K2 @ V2 @ V5 ) ) @ ( rBT_sunion_with_a_b @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Product_prod_a_b )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Product_prod_a_b )
             => ( Y != Xa2 ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_1050_gen__entries__simps_I3_J,axiom,
    ! [Kvts: list_P6367721468888276775list_a,C: rBT_color,L: rBT_rb6721489163287533103list_a,K: list_a,V: produc2547433716817190550list_a,R: rBT_rb6721489163287533103list_a] :
      ( ( rBT_ge5911799955945291786list_a @ Kvts @ ( rBT_Br1204270329063416912list_a @ C @ L @ K @ V @ R ) )
      = ( rBT_ge5911799955945291786list_a @ ( cons_P3144650002224114017list_a @ ( produc5396506868210886417list_a @ ( produc8961369442823515561list_a @ K @ V ) @ R ) @ Kvts ) @ L ) ) ).

% gen_entries_simps(3)
thf(fact_1051_gen__entries__simps_I3_J,axiom,
    ! [Kvts: list_P868163099566772739list_a,C: rBT_color,L: rBT_rb146355364677272861list_a,K: list_a,V: list_a,R: rBT_rb146355364677272861list_a] :
      ( ( rBT_ge6589554342944444152list_a @ Kvts @ ( rBT_Br8372335562727726142list_a @ C @ L @ K @ V @ R ) )
      = ( rBT_ge6589554342944444152list_a @ ( cons_P695356969528146237list_a @ ( produc5288693782868576493list_a @ ( produc6837034575241423639list_a @ K @ V ) @ R ) @ Kvts ) @ L ) ) ).

% gen_entries_simps(3)
thf(fact_1052_gen__entries__simps_I3_J,axiom,
    ! [Kvts: list_P7334024422598439925list_a,C: rBT_color,L: rBT_rb9014005217275227734list_a,K: a > a > $o,V: produc9164743771328383783list_a,R: rBT_rb9014005217275227734list_a] :
      ( ( rBT_ge5942196056055463857list_a @ Kvts @ ( rBT_Br6721835470276586743list_a @ C @ L @ K @ V @ R ) )
      = ( rBT_ge5942196056055463857list_a @ ( cons_P4323844787983635503list_a @ ( produc7258404890345721951list_a @ ( produc8684083420967988176list_a @ K @ V ) @ R ) @ Kvts ) @ L ) ) ).

% gen_entries_simps(3)
thf(fact_1053_gen__entries__simps_I3_J,axiom,
    ! [Kvts: list_P7781133056824549729list_a,C: rBT_color,L: rBT_rb484615002580549772list_a,K: a,V: produc271001111946445293list_a,R: rBT_rb484615002580549772list_a] :
      ( ( rBT_ge6786174521314728167list_a @ Kvts @ ( rBT_Br706236987277315629list_a @ C @ L @ K @ V @ R ) )
      = ( rBT_ge6786174521314728167list_a @ ( cons_P4107865082250473371list_a @ ( produc2500210828402294475list_a @ ( produc4562151247771834630list_a @ K @ V ) @ R ) @ Kvts ) @ L ) ) ).

% gen_entries_simps(3)
thf(fact_1054_gen__entries__simps_I3_J,axiom,
    ! [Kvts: list_P2302744184456665989bt_a_b,C: rBT_color,L: rBT_rbt_a_b,K: a,V: b,R: rBT_rbt_a_b] :
      ( ( rBT_gen_entries_a_b @ Kvts @ ( rBT_Branch_a_b @ C @ L @ K @ V @ R ) )
      = ( rBT_gen_entries_a_b @ ( cons_P4763212896715341247bt_a_b @ ( produc2347283123186229231bt_a_b @ ( product_Pair_a_b @ K @ V ) @ R ) @ Kvts ) @ L ) ) ).

% gen_entries_simps(3)
thf(fact_1055_entries__code,axiom,
    ( rBT_entries_a_b
    = ( rBT_gen_entries_a_b @ nil_Pr7081231462618039663bt_a_b ) ) ).

% entries_code
thf(fact_1056_local_Osinter__with_Oelims,axiom,
    ! [X: a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a,Xa2: list_P5792129111969709340list_a,Xb: list_P5792129111969709340list_a,Y: list_P5792129111969709340list_a] :
      ( ( ( rBT_si6934442379753512503list_a @ less @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [K2: a,V2: produc271001111946445293list_a,As: list_P5792129111969709340list_a] :
            ( ( Xa2
              = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K2 @ V2 ) @ As ) )
           => ! [K3: a,V5: produc271001111946445293list_a,Bs: list_P5792129111969709340list_a] :
                ( ( Xb
                  = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K3 @ V5 ) @ Bs ) )
               => ~ ( ( ( less @ K3 @ K2 )
                     => ( Y
                        = ( rBT_si6934442379753512503list_a @ less @ X @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K2 @ V2 ) @ As ) @ Bs ) ) )
                    & ( ~ ( less @ K3 @ K2 )
                     => ( ( ( less @ K2 @ K3 )
                         => ( Y
                            = ( rBT_si6934442379753512503list_a @ less @ X @ As @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K3 @ V5 ) @ Bs ) ) ) )
                        & ( ~ ( less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K2 @ ( X @ K2 @ V2 @ V5 ) ) @ ( rBT_si6934442379753512503list_a @ less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr347010806919629468list_a )
           => ( Y != nil_Pr347010806919629468list_a ) )
         => ~ ( ( Xb = nil_Pr347010806919629468list_a )
             => ( Y != nil_Pr347010806919629468list_a ) ) ) ) ) ).

% local.sinter_with.elims
thf(fact_1057_local_Osinter__with_Oelims,axiom,
    ! [X: a > b > b > b,Xa2: list_P1467974523212632366od_a_b,Xb: list_P1467974523212632366od_a_b,Y: list_P1467974523212632366od_a_b] :
      ( ( ( rBT_sinter_with_a_b @ less @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [K2: a,V2: b,As: list_P1467974523212632366od_a_b] :
            ( ( Xa2
              = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ As ) )
           => ! [K3: a,V5: b,Bs: list_P1467974523212632366od_a_b] :
                ( ( Xb
                  = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K3 @ V5 ) @ Bs ) )
               => ~ ( ( ( less @ K3 @ K2 )
                     => ( Y
                        = ( rBT_sinter_with_a_b @ less @ X @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ As ) @ Bs ) ) )
                    & ( ~ ( less @ K3 @ K2 )
                     => ( ( ( less @ K2 @ K3 )
                         => ( Y
                            = ( rBT_sinter_with_a_b @ less @ X @ As @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K3 @ V5 ) @ Bs ) ) ) )
                        & ( ~ ( less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ ( X @ K2 @ V2 @ V5 ) ) @ ( rBT_sinter_with_a_b @ less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Product_prod_a_b )
           => ( Y != nil_Product_prod_a_b ) )
         => ~ ( ( Xb = nil_Product_prod_a_b )
             => ( Y != nil_Product_prod_a_b ) ) ) ) ) ).

% local.sinter_with.elims
thf(fact_1058_local_Osinter__with_Osimps_I1_J,axiom,
    ! [K4: a,K: a,F: a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a,V: produc271001111946445293list_a,As2: list_P5792129111969709340list_a,V3: produc271001111946445293list_a,Bs2: list_P5792129111969709340list_a] :
      ( ( ( less @ K4 @ K )
       => ( ( rBT_si6934442379753512503list_a @ less @ F @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ As2 ) @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ Bs2 ) )
          = ( rBT_si6934442379753512503list_a @ less @ F @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( less @ K4 @ K )
       => ( ( ( less @ K @ K4 )
           => ( ( rBT_si6934442379753512503list_a @ less @ F @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ As2 ) @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( rBT_si6934442379753512503list_a @ less @ F @ As2 @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ Bs2 ) ) ) )
          & ( ~ ( less @ K @ K4 )
           => ( ( rBT_si6934442379753512503list_a @ less @ F @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ As2 ) @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ ( F @ K @ V @ V3 ) ) @ ( rBT_si6934442379753512503list_a @ less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% local.sinter_with.simps(1)
thf(fact_1059_local_Osinter__with_Osimps_I1_J,axiom,
    ! [K4: a,K: a,F: a > b > b > b,V: b,As2: list_P1467974523212632366od_a_b,V3: b,Bs2: list_P1467974523212632366od_a_b] :
      ( ( ( less @ K4 @ K )
       => ( ( rBT_sinter_with_a_b @ less @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ As2 ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ Bs2 ) )
          = ( rBT_sinter_with_a_b @ less @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( less @ K4 @ K )
       => ( ( ( less @ K @ K4 )
           => ( ( rBT_sinter_with_a_b @ less @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ As2 ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ Bs2 ) )
              = ( rBT_sinter_with_a_b @ less @ F @ As2 @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ Bs2 ) ) ) )
          & ( ~ ( less @ K @ K4 )
           => ( ( rBT_sinter_with_a_b @ less @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ As2 ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ ( F @ K @ V @ V3 ) ) @ ( rBT_sinter_with_a_b @ less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% local.sinter_with.simps(1)
thf(fact_1060_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: list_a > list_a > $o,K4: list_a,K: list_a,F: list_a > produc2547433716817190550list_a > produc2547433716817190550list_a > produc2547433716817190550list_a,V: produc2547433716817190550list_a,As2: list_P1945613012453903807list_a,V3: produc2547433716817190550list_a,Bs2: list_P1945613012453903807list_a] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_si5295904238279282010list_a @ Less @ F @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K @ V ) @ As2 ) @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K4 @ V3 ) @ Bs2 ) )
          = ( rBT_si5295904238279282010list_a @ Less @ F @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K @ V ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_si5295904238279282010list_a @ Less @ F @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K @ V ) @ As2 ) @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( rBT_si5295904238279282010list_a @ Less @ F @ As2 @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K4 @ V3 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_si5295904238279282010list_a @ Less @ F @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K @ V ) @ As2 ) @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K @ ( F @ K @ V @ V3 ) ) @ ( rBT_si5295904238279282010list_a @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_1061_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: list_a > list_a > $o,K4: list_a,K: list_a,F: list_a > list_a > list_a > list_a,V: list_a,As2: list_P321204300973800749list_a,V3: list_a,Bs2: list_P321204300973800749list_a] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_si8725346231877325896list_a @ Less @ F @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K @ V ) @ As2 ) @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K4 @ V3 ) @ Bs2 ) )
          = ( rBT_si8725346231877325896list_a @ Less @ F @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K @ V ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_si8725346231877325896list_a @ Less @ F @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K @ V ) @ As2 ) @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( rBT_si8725346231877325896list_a @ Less @ F @ As2 @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K4 @ V3 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_si8725346231877325896list_a @ Less @ F @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K @ V ) @ As2 ) @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K @ ( F @ K @ V @ V3 ) ) @ ( rBT_si8725346231877325896list_a @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_1062_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: ( a > a > $o ) > ( a > a > $o ) > $o,K4: a > a > $o,K: a > a > $o,F: ( a > a > $o ) > produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,V: produc9164743771328383783list_a,As2: list_P1472896444292205030list_a,V3: produc9164743771328383783list_a,Bs2: list_P1472896444292205030list_a] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_si4727287429423564033list_a @ Less @ F @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K @ V ) @ As2 ) @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K4 @ V3 ) @ Bs2 ) )
          = ( rBT_si4727287429423564033list_a @ Less @ F @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K @ V ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_si4727287429423564033list_a @ Less @ F @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K @ V ) @ As2 ) @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( rBT_si4727287429423564033list_a @ Less @ F @ As2 @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K4 @ V3 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_si4727287429423564033list_a @ Less @ F @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K @ V ) @ As2 ) @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K @ ( F @ K @ V @ V3 ) ) @ ( rBT_si4727287429423564033list_a @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_1063_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: a > a > $o,K4: a,K: a,F: a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a,V: produc271001111946445293list_a,As2: list_P5792129111969709340list_a,V3: produc271001111946445293list_a,Bs2: list_P5792129111969709340list_a] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_si6934442379753512503list_a @ Less @ F @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ As2 ) @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ Bs2 ) )
          = ( rBT_si6934442379753512503list_a @ Less @ F @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_si6934442379753512503list_a @ Less @ F @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ As2 ) @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( rBT_si6934442379753512503list_a @ Less @ F @ As2 @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_si6934442379753512503list_a @ Less @ F @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ V ) @ As2 ) @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K @ ( F @ K @ V @ V3 ) ) @ ( rBT_si6934442379753512503list_a @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_1064_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: a > a > $o,K4: a,K: a,F: a > b > b > b,V: b,As2: list_P1467974523212632366od_a_b,V3: b,Bs2: list_P1467974523212632366od_a_b] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_sinter_with_a_b @ Less @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ As2 ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ Bs2 ) )
          = ( rBT_sinter_with_a_b @ Less @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_sinter_with_a_b @ Less @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ As2 ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ Bs2 ) )
              = ( rBT_sinter_with_a_b @ Less @ F @ As2 @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_sinter_with_a_b @ Less @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ V ) @ As2 ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K4 @ V3 ) @ Bs2 ) )
              = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K @ ( F @ K @ V @ V3 ) ) @ ( rBT_sinter_with_a_b @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_1065_ord_Osinter__with_Oelims,axiom,
    ! [Less: list_a > list_a > $o,X: list_a > produc2547433716817190550list_a > produc2547433716817190550list_a > produc2547433716817190550list_a,Xa2: list_P1945613012453903807list_a,Xb: list_P1945613012453903807list_a,Y: list_P1945613012453903807list_a] :
      ( ( ( rBT_si5295904238279282010list_a @ Less @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [K2: list_a,V2: produc2547433716817190550list_a,As: list_P1945613012453903807list_a] :
            ( ( Xa2
              = ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K2 @ V2 ) @ As ) )
           => ! [K3: list_a,V5: produc2547433716817190550list_a,Bs: list_P1945613012453903807list_a] :
                ( ( Xb
                  = ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K3 @ V5 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( rBT_si5295904238279282010list_a @ Less @ X @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K2 @ V2 ) @ As ) @ Bs ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( rBT_si5295904238279282010list_a @ Less @ X @ As @ ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K3 @ V5 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P8246450671169330159list_a @ ( produc8961369442823515561list_a @ K2 @ ( X @ K2 @ V2 @ V5 ) ) @ ( rBT_si5295904238279282010list_a @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr3732201366458353727list_a )
           => ( Y != nil_Pr3732201366458353727list_a ) )
         => ~ ( ( Xb = nil_Pr3732201366458353727list_a )
             => ( Y != nil_Pr3732201366458353727list_a ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_1066_ord_Osinter__with_Oelims,axiom,
    ! [Less: list_a > list_a > $o,X: list_a > list_a > list_a > list_a,Xa2: list_P321204300973800749list_a,Xb: list_P321204300973800749list_a,Y: list_P321204300973800749list_a] :
      ( ( ( rBT_si8725346231877325896list_a @ Less @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [K2: list_a,V2: list_a,As: list_P321204300973800749list_a] :
            ( ( Xa2
              = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K2 @ V2 ) @ As ) )
           => ! [K3: list_a,V5: list_a,Bs: list_P321204300973800749list_a] :
                ( ( Xb
                  = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K3 @ V5 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( rBT_si8725346231877325896list_a @ Less @ X @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K2 @ V2 ) @ As ) @ Bs ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( rBT_si8725346231877325896list_a @ Less @ X @ As @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K3 @ V5 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ K2 @ ( X @ K2 @ V2 @ V5 ) ) @ ( rBT_si8725346231877325896list_a @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr3188421586756112173list_a )
           => ( Y != nil_Pr3188421586756112173list_a ) )
         => ~ ( ( Xb = nil_Pr3188421586756112173list_a )
             => ( Y != nil_Pr3188421586756112173list_a ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_1067_ord_Osinter__with_Oelims,axiom,
    ! [Less: ( a > a > $o ) > ( a > a > $o ) > $o,X: ( a > a > $o ) > produc9164743771328383783list_a > produc9164743771328383783list_a > produc9164743771328383783list_a,Xa2: list_P1472896444292205030list_a,Xb: list_P1472896444292205030list_a,Y: list_P1472896444292205030list_a] :
      ( ( ( rBT_si4727287429423564033list_a @ Less @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [K2: a > a > $o,V2: produc9164743771328383783list_a,As: list_P1472896444292205030list_a] :
            ( ( Xa2
              = ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K2 @ V2 ) @ As ) )
           => ! [K3: a > a > $o,V5: produc9164743771328383783list_a,Bs: list_P1472896444292205030list_a] :
                ( ( Xb
                  = ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K3 @ V5 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( rBT_si4727287429423564033list_a @ Less @ X @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K2 @ V2 ) @ As ) @ Bs ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( rBT_si4727287429423564033list_a @ Less @ X @ As @ ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K3 @ V5 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P42116974143214614list_a @ ( produc8684083420967988176list_a @ K2 @ ( X @ K2 @ V2 @ V5 ) ) @ ( rBT_si4727287429423564033list_a @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr3115398151409596518list_a )
           => ( Y != nil_Pr3115398151409596518list_a ) )
         => ~ ( ( Xb = nil_Pr3115398151409596518list_a )
             => ( Y != nil_Pr3115398151409596518list_a ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_1068_ord_Osinter__with_Oelims,axiom,
    ! [Less: a > a > $o,X: a > produc271001111946445293list_a > produc271001111946445293list_a > produc271001111946445293list_a,Xa2: list_P5792129111969709340list_a,Xb: list_P5792129111969709340list_a,Y: list_P5792129111969709340list_a] :
      ( ( ( rBT_si6934442379753512503list_a @ Less @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [K2: a,V2: produc271001111946445293list_a,As: list_P5792129111969709340list_a] :
            ( ( Xa2
              = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K2 @ V2 ) @ As ) )
           => ! [K3: a,V5: produc271001111946445293list_a,Bs: list_P5792129111969709340list_a] :
                ( ( Xb
                  = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K3 @ V5 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( rBT_si6934442379753512503list_a @ Less @ X @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K2 @ V2 ) @ As ) @ Bs ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( rBT_si6934442379753512503list_a @ Less @ X @ As @ ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K3 @ V5 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P5518497943414203980list_a @ ( produc4562151247771834630list_a @ K2 @ ( X @ K2 @ V2 @ V5 ) ) @ ( rBT_si6934442379753512503list_a @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr347010806919629468list_a )
           => ( Y != nil_Pr347010806919629468list_a ) )
         => ~ ( ( Xb = nil_Pr347010806919629468list_a )
             => ( Y != nil_Pr347010806919629468list_a ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_1069_ord_Osinter__with_Oelims,axiom,
    ! [Less: a > a > $o,X: a > b > b > b,Xa2: list_P1467974523212632366od_a_b,Xb: list_P1467974523212632366od_a_b,Y: list_P1467974523212632366od_a_b] :
      ( ( ( rBT_sinter_with_a_b @ Less @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [K2: a,V2: b,As: list_P1467974523212632366od_a_b] :
            ( ( Xa2
              = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ As ) )
           => ! [K3: a,V5: b,Bs: list_P1467974523212632366od_a_b] :
                ( ( Xb
                  = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K3 @ V5 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( rBT_sinter_with_a_b @ Less @ X @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ As ) @ Bs ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( rBT_sinter_with_a_b @ Less @ X @ As @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K3 @ V5 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ ( X @ K2 @ V2 @ V5 ) ) @ ( rBT_sinter_with_a_b @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Product_prod_a_b )
           => ( Y != nil_Product_prod_a_b ) )
         => ~ ( ( Xb = nil_Product_prod_a_b )
             => ( Y != nil_Product_prod_a_b ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_1070_proper__intrvl_Oexhaustive__above_Ocases,axiom,
    ! [X: produc8685980395799941037list_a] :
      ( ! [X3: a] :
          ( X
         != ( produc6670463072477821725list_a @ X3 @ nil_a ) )
     => ~ ! [X3: a,Y2: a,Ys2: list_a] :
            ( X
           != ( produc6670463072477821725list_a @ X3 @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ).

% proper_intrvl.exhaustive_above.cases
thf(fact_1071_proper__intrvl_OCompl__set__less__eq__aux_Ocases,axiom,
    ! [X: produc5345840553016159882list_a] :
      ( ! [Ao: option_a] :
          ( X
         != ( produc489741553574343802list_a @ Ao @ ( produc6837034575241423639list_a @ nil_a @ nil_a ) ) )
     => ( ! [Ao: option_a,Y2: a,Ys2: list_a] :
            ( X
           != ( produc489741553574343802list_a @ Ao @ ( produc6837034575241423639list_a @ nil_a @ ( cons_a @ Y2 @ Ys2 ) ) ) )
       => ( ! [Ao: option_a,X3: a,Xs2: list_a] :
              ( X
             != ( produc489741553574343802list_a @ Ao @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ nil_a ) ) )
         => ~ ! [Ao: option_a,X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
                ( X
               != ( produc489741553574343802list_a @ Ao @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ) ) ).

% proper_intrvl.Compl_set_less_eq_aux.cases
thf(fact_1072_proper__intrvl_Oset__less__eq__aux__Compl_Ocases,axiom,
    ! [X: produc5345840553016159882list_a] :
      ( ! [Ao: option_a,Ys2: list_a] :
          ( X
         != ( produc489741553574343802list_a @ Ao @ ( produc6837034575241423639list_a @ nil_a @ Ys2 ) ) )
     => ( ! [Ao: option_a,Xs2: list_a] :
            ( X
           != ( produc489741553574343802list_a @ Ao @ ( produc6837034575241423639list_a @ Xs2 @ nil_a ) ) )
       => ~ ! [Ao: option_a,X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
              ( X
             != ( produc489741553574343802list_a @ Ao @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ) ).

% proper_intrvl.set_less_eq_aux_Compl.cases
thf(fact_1073_proper__intrvl_Oproper__interval__set__aux_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ( ! [Xs2: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
     => ( ! [Y2: a,Ys2: list_a] :
            ( X
           != ( produc6837034575241423639list_a @ nil_a @ ( cons_a @ Y2 @ Ys2 ) ) )
       => ~ ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
              ( X
             != ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ).

% proper_intrvl.proper_interval_set_aux.cases
thf(fact_1074_proper__intrvl_Oproper__interval__Compl__set__aux_Ocases,axiom,
    ! [X: produc5345840553016159882list_a] :
      ( ! [Ao: option_a,X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
          ( X
         != ( produc489741553574343802list_a @ Ao @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
     => ( ! [Ao: option_a,Uv: list_a] :
            ( X
           != ( produc489741553574343802list_a @ Ao @ ( produc6837034575241423639list_a @ nil_a @ Uv ) ) )
       => ~ ! [Ao: option_a,Uu2: list_a] :
              ( X
             != ( produc489741553574343802list_a @ Ao @ ( produc6837034575241423639list_a @ Uu2 @ nil_a ) ) ) ) ) ).

% proper_intrvl.proper_interval_Compl_set_aux.cases
thf(fact_1075_proper__intrvl_Oproper__interval__set__Compl__aux_Ocases,axiom,
    ! [X: produc486642197419069375list_a] :
      ( ! [Ao: option_a,N2: nat] :
          ( X
         != ( produc702244377756350009list_a @ Ao @ ( produc7260490323852791550list_a @ N2 @ ( produc6837034575241423639list_a @ nil_a @ nil_a ) ) ) )
     => ( ! [Ao: option_a,N2: nat,Y2: a,Ys2: list_a] :
            ( X
           != ( produc702244377756350009list_a @ Ao @ ( produc7260490323852791550list_a @ N2 @ ( produc6837034575241423639list_a @ nil_a @ ( cons_a @ Y2 @ Ys2 ) ) ) ) )
       => ( ! [Ao: option_a,N2: nat,X3: a,Xs2: list_a] :
              ( X
             != ( produc702244377756350009list_a @ Ao @ ( produc7260490323852791550list_a @ N2 @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ nil_a ) ) ) )
         => ~ ! [Ao: option_a,N2: nat,X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
                ( X
               != ( produc702244377756350009list_a @ Ao @ ( produc7260490323852791550list_a @ N2 @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ) ) ) ).

% proper_intrvl.proper_interval_set_Compl_aux.cases
thf(fact_1076_in__set__simps_I2_J,axiom,
    ! [X: product_prod_a_b,Y: product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ ( cons_P7316939131009794654od_a_b @ Y @ nil_Product_prod_a_b ) ) )
      = ( X = Y ) ) ).

% in_set_simps(2)
thf(fact_1077_in__set__simps_I2_J,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( set_a2 @ ( cons_a @ Y @ nil_a ) ) )
      = ( X = Y ) ) ).

% in_set_simps(2)
thf(fact_1078_extract__SomeE,axiom,
    ! [P: product_prod_a_b > $o,Xs: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Y: product_prod_a_b,Zs: list_P1467974523212632366od_a_b] :
      ( ( ( extrac5577817238722109616od_a_b @ P @ Xs )
        = ( some_P2412010246944353803od_a_b @ ( produc1447209203954967446od_a_b @ Ys @ ( produc1872688513461243805od_a_b @ Y @ Zs ) ) ) )
     => ( ( Xs
          = ( append5335208823350062147od_a_b @ Ys @ ( cons_P7316939131009794654od_a_b @ Y @ Zs ) ) )
        & ( P @ Y )
        & ~ ? [X5: product_prod_a_b] :
              ( ( member1426531481828664017od_a_b @ X5 @ ( set_Product_prod_a_b2 @ Ys ) )
              & ( P @ X5 ) ) ) ) ).

% extract_SomeE
thf(fact_1079_extract__SomeE,axiom,
    ! [P: a > $o,Xs: list_a,Ys: list_a,Y: a,Zs: list_a] :
      ( ( ( extract_a @ P @ Xs )
        = ( some_P5354654743593010357list_a @ ( produc3204708664006668352list_a @ Ys @ ( produc6670463072477821725list_a @ Y @ Zs ) ) ) )
     => ( ( Xs
          = ( append_a @ Ys @ ( cons_a @ Y @ Zs ) ) )
        & ( P @ Y )
        & ~ ? [X5: a] :
              ( ( member_a @ X5 @ ( set_a2 @ Ys ) )
              & ( P @ X5 ) ) ) ) ).

% extract_SomeE
thf(fact_1080_extract__None__iff,axiom,
    ! [P: product_prod_a_b > $o,Xs: list_P1467974523212632366od_a_b] :
      ( ( ( extrac5577817238722109616od_a_b @ P @ Xs )
        = none_P912253767061701255od_a_b )
      = ( ~ ? [X2: product_prod_a_b] :
              ( ( member1426531481828664017od_a_b @ X2 @ ( set_Product_prod_a_b2 @ Xs ) )
              & ( P @ X2 ) ) ) ) ).

% extract_None_iff
thf(fact_1081_extract__None__iff,axiom,
    ! [P: a > $o,Xs: list_a] :
      ( ( ( extract_a @ P @ Xs )
        = none_P5893993846586699057list_a )
      = ( ~ ? [X2: a] :
              ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
              & ( P @ X2 ) ) ) ) ).

% extract_None_iff
thf(fact_1082_in__set__simps_I1_J,axiom,
    ! [X: product_prod_a_b,Y: product_prod_a_b,Z: product_prod_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ ( cons_P7316939131009794654od_a_b @ Y @ ( cons_P7316939131009794654od_a_b @ Z @ Ys ) ) ) )
      = ( ( X = Y )
        | ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ ( cons_P7316939131009794654od_a_b @ Z @ Ys ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_1083_in__set__simps_I1_J,axiom,
    ! [X: a,Y: a,Z: a,Ys: list_a] :
      ( ( member_a @ X @ ( set_a2 @ ( cons_a @ Y @ ( cons_a @ Z @ Ys ) ) ) )
      = ( ( X = Y )
        | ( member_a @ X @ ( set_a2 @ ( cons_a @ Z @ Ys ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_1084_in__set__simps_I3_J,axiom,
    ! [X: product_prod_a_b] :
      ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ nil_Product_prod_a_b ) ) ).

% in_set_simps(3)
thf(fact_1085_in__set__simps_I3_J,axiom,
    ! [X: a] :
      ~ ( member_a @ X @ ( set_a2 @ nil_a ) ) ).

% in_set_simps(3)
thf(fact_1086_extract__Some__iff,axiom,
    ! [P: product_prod_a_b > $o,Xs: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Y: product_prod_a_b,Zs: list_P1467974523212632366od_a_b] :
      ( ( ( extrac5577817238722109616od_a_b @ P @ Xs )
        = ( some_P2412010246944353803od_a_b @ ( produc1447209203954967446od_a_b @ Ys @ ( produc1872688513461243805od_a_b @ Y @ Zs ) ) ) )
      = ( ( Xs
          = ( append5335208823350062147od_a_b @ Ys @ ( cons_P7316939131009794654od_a_b @ Y @ Zs ) ) )
        & ( P @ Y )
        & ~ ? [X2: product_prod_a_b] :
              ( ( member1426531481828664017od_a_b @ X2 @ ( set_Product_prod_a_b2 @ Ys ) )
              & ( P @ X2 ) ) ) ) ).

% extract_Some_iff
thf(fact_1087_extract__Some__iff,axiom,
    ! [P: a > $o,Xs: list_a,Ys: list_a,Y: a,Zs: list_a] :
      ( ( ( extract_a @ P @ Xs )
        = ( some_P5354654743593010357list_a @ ( produc3204708664006668352list_a @ Ys @ ( produc6670463072477821725list_a @ Y @ Zs ) ) ) )
      = ( ( Xs
          = ( append_a @ Ys @ ( cons_a @ Y @ Zs ) ) )
        & ( P @ Y )
        & ~ ? [X2: a] :
              ( ( member_a @ X2 @ ( set_a2 @ Ys ) )
              & ( P @ X2 ) ) ) ) ).

% extract_Some_iff
thf(fact_1088_local_Osorted__list__subset_Osimps_I2_J,axiom,
    ! [Eq2: a > a > $o,X: a,Xs: list_a] :
      ~ ( set_so53198715357033701bset_a @ less @ Eq2 @ ( cons_a @ X @ Xs ) @ nil_a ) ).

% local.sorted_list_subset.simps(2)
thf(fact_1089_local_Osorted__list__subset_Oelims_I1_J,axiom,
    ! [X: a > a > $o,Xa2: list_a,Xb: list_a,Y: $o] :
      ( ( ( set_so53198715357033701bset_a @ less @ X @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = nil_a )
         => ~ Y )
       => ( ( ? [X3: a,Xs2: list_a] :
                ( Xa2
                = ( cons_a @ X3 @ Xs2 ) )
           => ( ( Xb = nil_a )
             => Y ) )
         => ~ ! [X3: a,Xs2: list_a] :
                ( ( Xa2
                  = ( cons_a @ X3 @ Xs2 ) )
               => ! [Y2: a,Ys2: list_a] :
                    ( ( Xb
                      = ( cons_a @ Y2 @ Ys2 ) )
                   => ( Y
                      = ( ~ ( ( ( X @ X3 @ Y2 )
                             => ( set_so53198715357033701bset_a @ less @ X @ Xs2 @ Ys2 ) )
                            & ( ~ ( X @ X3 @ Y2 )
                             => ( ( less @ Y2 @ X3 )
                                & ( set_so53198715357033701bset_a @ less @ X @ ( cons_a @ X3 @ Xs2 ) @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ).

% local.sorted_list_subset.elims(1)
thf(fact_1090_local_Osorted__list__subset_Osimps_I3_J,axiom,
    ! [Eq2: a > a > $o,X: a,Xs: list_a,Y: a,Ys: list_a] :
      ( ( set_so53198715357033701bset_a @ less @ Eq2 @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) )
      = ( ( ( Eq2 @ X @ Y )
         => ( set_so53198715357033701bset_a @ less @ Eq2 @ Xs @ Ys ) )
        & ( ~ ( Eq2 @ X @ Y )
         => ( ( less @ Y @ X )
            & ( set_so53198715357033701bset_a @ less @ Eq2 @ ( cons_a @ X @ Xs ) @ Ys ) ) ) ) ) ).

% local.sorted_list_subset.simps(3)
thf(fact_1091_local_Osorted__list__subset_Osimps_I1_J,axiom,
    ! [Eq2: a > a > $o,Ys: list_a] : ( set_so53198715357033701bset_a @ less @ Eq2 @ nil_a @ Ys ) ).

% local.sorted_list_subset.simps(1)
thf(fact_1092_local_Osorted__list__subset_Oelims_I3_J,axiom,
    ! [X: a > a > $o,Xa2: list_a,Xb: list_a] :
      ( ~ ( set_so53198715357033701bset_a @ less @ X @ Xa2 @ Xb )
     => ( ( ? [X3: a,Xs2: list_a] :
              ( Xa2
              = ( cons_a @ X3 @ Xs2 ) )
         => ( Xb != nil_a ) )
       => ~ ! [X3: a,Xs2: list_a] :
              ( ( Xa2
                = ( cons_a @ X3 @ Xs2 ) )
             => ! [Y2: a,Ys2: list_a] :
                  ( ( Xb
                    = ( cons_a @ Y2 @ Ys2 ) )
                 => ( ( ( X @ X3 @ Y2 )
                     => ( set_so53198715357033701bset_a @ less @ X @ Xs2 @ Ys2 ) )
                    & ( ~ ( X @ X3 @ Y2 )
                     => ( ( less @ Y2 @ X3 )
                        & ( set_so53198715357033701bset_a @ less @ X @ ( cons_a @ X3 @ Xs2 ) @ Ys2 ) ) ) ) ) ) ) ) ).

% local.sorted_list_subset.elims(3)
thf(fact_1093_local_Osorted__list__subset_Oelims_I2_J,axiom,
    ! [X: a > a > $o,Xa2: list_a,Xb: list_a] :
      ( ( set_so53198715357033701bset_a @ less @ X @ Xa2 @ Xb )
     => ( ( Xa2 != nil_a )
       => ~ ! [X3: a,Xs2: list_a] :
              ( ( Xa2
                = ( cons_a @ X3 @ Xs2 ) )
             => ! [Y2: a,Ys2: list_a] :
                  ( ( Xb
                    = ( cons_a @ Y2 @ Ys2 ) )
                 => ~ ( ( ( X @ X3 @ Y2 )
                       => ( set_so53198715357033701bset_a @ less @ X @ Xs2 @ Ys2 ) )
                      & ( ~ ( X @ X3 @ Y2 )
                       => ( ( less @ Y2 @ X3 )
                          & ( set_so53198715357033701bset_a @ less @ X @ ( cons_a @ X3 @ Xs2 ) @ Ys2 ) ) ) ) ) ) ) ) ).

% local.sorted_list_subset.elims(2)
thf(fact_1094_local_Olexordp__eq_Osimps,axiom,
    ! [A1: list_a,A22: list_a] :
      ( ( lexordp_eq_a @ less @ A1 @ A22 )
      = ( ? [Ys4: list_a] :
            ( ( A1 = nil_a )
            & ( A22 = Ys4 ) )
        | ? [X2: a,Y3: a,Xs4: list_a,Ys4: list_a] :
            ( ( A1
              = ( cons_a @ X2 @ Xs4 ) )
            & ( A22
              = ( cons_a @ Y3 @ Ys4 ) )
            & ( less @ X2 @ Y3 ) )
        | ? [X2: a,Y3: a,Xs4: list_a,Ys4: list_a] :
            ( ( A1
              = ( cons_a @ X2 @ Xs4 ) )
            & ( A22
              = ( cons_a @ Y3 @ Ys4 ) )
            & ~ ( less @ X2 @ Y3 )
            & ~ ( less @ Y3 @ X2 )
            & ( lexordp_eq_a @ less @ Xs4 @ Ys4 ) ) ) ) ).

% local.lexordp_eq.simps
thf(fact_1095_local_Olexordp__eq__antisym,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( lexordp_eq_a @ less @ Xs @ Ys )
     => ( ( lexordp_eq_a @ less @ Ys @ Xs )
       => ( Xs = Ys ) ) ) ).

% local.lexordp_eq_antisym
thf(fact_1096_local_Olexordp__eq__linear,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( lexordp_eq_a @ less @ Xs @ Ys )
      | ( lexordp_eq_a @ less @ Ys @ Xs ) ) ).

% local.lexordp_eq_linear
thf(fact_1097_local_Olexordp__eq__refl,axiom,
    ! [Xs: list_a] : ( lexordp_eq_a @ less @ Xs @ Xs ) ).

% local.lexordp_eq_refl
thf(fact_1098_local_Olexordp__eq__trans,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ( lexordp_eq_a @ less @ Xs @ Ys )
     => ( ( lexordp_eq_a @ less @ Ys @ Zs )
       => ( lexordp_eq_a @ less @ Xs @ Zs ) ) ) ).

% local.lexordp_eq_trans
thf(fact_1099_local_Olexordp__eq_OCons__eq,axiom,
    ! [X: a,Y: a,Xs: list_a,Ys: list_a] :
      ( ~ ( less @ X @ Y )
     => ( ~ ( less @ Y @ X )
       => ( ( lexordp_eq_a @ less @ Xs @ Ys )
         => ( lexordp_eq_a @ less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ) ) ).

% local.lexordp_eq.Cons_eq
thf(fact_1100_local_Olexordp__eq_OCons,axiom,
    ! [X: a,Y: a,Xs: list_a,Ys: list_a] :
      ( ( less @ X @ Y )
     => ( lexordp_eq_a @ less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ).

% local.lexordp_eq.Cons
thf(fact_1101_local_Olexordp__eq_ONil,axiom,
    ! [Ys: list_a] : ( lexordp_eq_a @ less @ nil_a @ Ys ) ).

% local.lexordp_eq.Nil
thf(fact_1102_local_Olexordp__eq__pref,axiom,
    ! [U: list_a,V: list_a] : ( lexordp_eq_a @ less @ U @ ( append_a @ U @ V ) ) ).

% local.lexordp_eq_pref
thf(fact_1103_local_Olexordp__eq_Ocases,axiom,
    ! [A1: list_a,A22: list_a] :
      ( ( lexordp_eq_a @ less @ A1 @ A22 )
     => ( ( A1 != nil_a )
       => ( ! [X3: a] :
              ( ? [Xs2: list_a] :
                  ( A1
                  = ( cons_a @ X3 @ Xs2 ) )
             => ! [Y2: a] :
                  ( ? [Ys2: list_a] :
                      ( A22
                      = ( cons_a @ Y2 @ Ys2 ) )
                 => ~ ( less @ X3 @ Y2 ) ) )
         => ~ ! [X3: a,Y2: a,Xs2: list_a] :
                ( ( A1
                  = ( cons_a @ X3 @ Xs2 ) )
               => ! [Ys2: list_a] :
                    ( ( A22
                      = ( cons_a @ Y2 @ Ys2 ) )
                   => ( ~ ( less @ X3 @ Y2 )
                     => ( ~ ( less @ Y2 @ X3 )
                       => ~ ( lexordp_eq_a @ less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).

% local.lexordp_eq.cases
thf(fact_1104_local_Olexordp__eq__simps_I4_J,axiom,
    ! [X: a,Xs: list_a,Y: a,Ys: list_a] :
      ( ( lexordp_eq_a @ less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) )
      = ( ( less @ X @ Y )
        | ( ~ ( less @ Y @ X )
          & ( lexordp_eq_a @ less @ Xs @ Ys ) ) ) ) ).

% local.lexordp_eq_simps(4)
thf(fact_1105_local_Olexordp__eq__simps_I1_J,axiom,
    ! [Ys: list_a] : ( lexordp_eq_a @ less @ nil_a @ Ys ) ).

% local.lexordp_eq_simps(1)
thf(fact_1106_local_Olexordp__eq__simps_I2_J,axiom,
    ! [Xs: list_a] :
      ( ( lexordp_eq_a @ less @ Xs @ nil_a )
      = ( Xs = nil_a ) ) ).

% local.lexordp_eq_simps(2)
thf(fact_1107_local_Olexordp__eq__simps_I3_J,axiom,
    ! [X: a,Xs: list_a] :
      ~ ( lexordp_eq_a @ less @ ( cons_a @ X @ Xs ) @ nil_a ) ).

% local.lexordp_eq_simps(3)
thf(fact_1108_rbt__entries__next_Osimps_I2_J,axiom,
    ! [Kts: list_P6367721468888276775list_a,C: rBT_color,L: rBT_rb6721489163287533103list_a,K: list_a,V: produc2547433716817190550list_a,R: rBT_rb6721489163287533103list_a] :
      ( ( rBT_rb183187240731308954list_a @ ( produc5503654560254416879list_a @ Kts @ ( rBT_Br1204270329063416912list_a @ C @ L @ K @ V @ R ) ) )
      = ( rBT_rb183187240731308954list_a @ ( produc5503654560254416879list_a @ ( cons_P3144650002224114017list_a @ ( produc5396506868210886417list_a @ ( produc8961369442823515561list_a @ K @ V ) @ R ) @ Kts ) @ L ) ) ) ).

% rbt_entries_next.simps(2)
thf(fact_1109_rbt__entries__next_Osimps_I2_J,axiom,
    ! [Kts: list_P868163099566772739list_a,C: rBT_color,L: rBT_rb146355364677272861list_a,K: list_a,V: list_a,R: rBT_rb146355364677272861list_a] :
      ( ( rBT_rb2202894292051942024list_a @ ( produc1400523206701116857list_a @ Kts @ ( rBT_Br8372335562727726142list_a @ C @ L @ K @ V @ R ) ) )
      = ( rBT_rb2202894292051942024list_a @ ( produc1400523206701116857list_a @ ( cons_P695356969528146237list_a @ ( produc5288693782868576493list_a @ ( produc6837034575241423639list_a @ K @ V ) @ R ) @ Kts ) @ L ) ) ) ).

% rbt_entries_next.simps(2)
thf(fact_1110_rbt__entries__next_Osimps_I2_J,axiom,
    ! [Kts: list_P7334024422598439925list_a,C: rBT_color,L: rBT_rb9014005217275227734list_a,K: a > a > $o,V: produc9164743771328383783list_a,R: rBT_rb9014005217275227734list_a] :
      ( ( rBT_rb6329126105604721473list_a @ ( produc8990773972949836516list_a @ Kts @ ( rBT_Br6721835470276586743list_a @ C @ L @ K @ V @ R ) ) )
      = ( rBT_rb6329126105604721473list_a @ ( produc8990773972949836516list_a @ ( cons_P4323844787983635503list_a @ ( produc7258404890345721951list_a @ ( produc8684083420967988176list_a @ K @ V ) @ R ) @ Kts ) @ L ) ) ) ).

% rbt_entries_next.simps(2)
thf(fact_1111_rbt__entries__next_Osimps_I2_J,axiom,
    ! [Kts: list_P7781133056824549729list_a,C: rBT_color,L: rBT_rb484615002580549772list_a,K: a,V: produc271001111946445293list_a,R: rBT_rb484615002580549772list_a] :
      ( ( rBT_rb8531841597279888503list_a @ ( produc3814750783646602886list_a @ Kts @ ( rBT_Br706236987277315629list_a @ C @ L @ K @ V @ R ) ) )
      = ( rBT_rb8531841597279888503list_a @ ( produc3814750783646602886list_a @ ( cons_P4107865082250473371list_a @ ( produc2500210828402294475list_a @ ( produc4562151247771834630list_a @ K @ V ) @ R ) @ Kts ) @ L ) ) ) ).

% rbt_entries_next.simps(2)
thf(fact_1112_rbt__entries__next_Osimps_I2_J,axiom,
    ! [Kts: list_P2302744184456665989bt_a_b,C: rBT_color,L: rBT_rbt_a_b,K: a,V: b,R: rBT_rbt_a_b] :
      ( ( rBT_rb1003684898779318153xt_a_b @ ( produc4569715891362637500bt_a_b @ Kts @ ( rBT_Branch_a_b @ C @ L @ K @ V @ R ) ) )
      = ( rBT_rb1003684898779318153xt_a_b @ ( produc4569715891362637500bt_a_b @ ( cons_P4763212896715341247bt_a_b @ ( produc2347283123186229231bt_a_b @ ( product_Pair_a_b @ K @ V ) @ R ) @ Kts ) @ L ) ) ) ).

% rbt_entries_next.simps(2)
thf(fact_1113_local_Olexordp__append__rightI,axiom,
    ! [Ys: list_a,Xs: list_a] :
      ( ( Ys != nil_a )
     => ( lexordp_a @ less @ Xs @ ( append_a @ Xs @ Ys ) ) ) ).

% local.lexordp_append_rightI
thf(fact_1114_local_Olexordp__trans,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ( lexordp_a @ less @ Xs @ Ys )
     => ( ( lexordp_a @ less @ Ys @ Zs )
       => ( lexordp_a @ less @ Xs @ Zs ) ) ) ).

% local.lexordp_trans
thf(fact_1115_local_Olexordp__linear,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( lexordp_a @ less @ Xs @ Ys )
      | ( Xs = Ys )
      | ( lexordp_a @ less @ Ys @ Xs ) ) ).

% local.lexordp_linear
thf(fact_1116_local_Olexordp__irreflexive_H,axiom,
    ! [Xs: list_a] :
      ~ ( lexordp_a @ less @ Xs @ Xs ) ).

% local.lexordp_irreflexive'
thf(fact_1117_local_Olexordp__irreflexive,axiom,
    ! [Xs: list_a] :
      ( ! [X3: a] :
          ~ ( less @ X3 @ X3 )
     => ~ ( lexordp_a @ less @ Xs @ Xs ) ) ).

% local.lexordp_irreflexive
thf(fact_1118_local_Olexordp__antisym,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( lexordp_a @ less @ Xs @ Ys )
     => ~ ( lexordp_a @ less @ Ys @ Xs ) ) ).

% local.lexordp_antisym
thf(fact_1119_local_Olexordp_OCons,axiom,
    ! [X: a,Y: a,Xs: list_a,Ys: list_a] :
      ( ( less @ X @ Y )
     => ( lexordp_a @ less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ).

% local.lexordp.Cons
thf(fact_1120_local_Olexordp_OCons__eq,axiom,
    ! [X: a,Y: a,Xs: list_a,Ys: list_a] :
      ( ~ ( less @ X @ Y )
     => ( ~ ( less @ Y @ X )
       => ( ( lexordp_a @ less @ Xs @ Ys )
         => ( lexordp_a @ less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ) ) ).

% local.lexordp.Cons_eq
thf(fact_1121_local_Olexordp__append__leftD,axiom,
    ! [Xs: list_a,Us: list_a,Vs: list_a] :
      ( ( lexordp_a @ less @ ( append_a @ Xs @ Us ) @ ( append_a @ Xs @ Vs ) )
     => ( ! [A2: a] :
            ~ ( less @ A2 @ A2 )
       => ( lexordp_a @ less @ Us @ Vs ) ) ) ).

% local.lexordp_append_leftD
thf(fact_1122_local_Olexordp__append__leftI,axiom,
    ! [Us: list_a,Vs: list_a,Xs: list_a] :
      ( ( lexordp_a @ less @ Us @ Vs )
     => ( lexordp_a @ less @ ( append_a @ Xs @ Us ) @ ( append_a @ Xs @ Vs ) ) ) ).

% local.lexordp_append_leftI
thf(fact_1123_local_Olexordp__conv__lexordp__eq,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( lexordp_a @ less @ Xs @ Ys )
      = ( ( lexordp_eq_a @ less @ Xs @ Ys )
        & ~ ( lexordp_eq_a @ less @ Ys @ Xs ) ) ) ).

% local.lexordp_conv_lexordp_eq
thf(fact_1124_local_Olexordp__eq__conv__lexord,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( lexordp_eq_a @ less @ Xs @ Ys )
      = ( ( Xs = Ys )
        | ( lexordp_a @ less @ Xs @ Ys ) ) ) ).

% local.lexordp_eq_conv_lexord
thf(fact_1125_local_Olexordp__into__lexordp__eq,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( lexordp_a @ less @ Xs @ Ys )
     => ( lexordp_eq_a @ less @ Xs @ Ys ) ) ).

% local.lexordp_into_lexordp_eq
thf(fact_1126_local_Olexordp__induct,axiom,
    ! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
      ( ( lexordp_a @ less @ Xs @ Ys )
     => ( ! [Y2: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys2 ) )
       => ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
              ( ( less @ X3 @ Y2 )
             => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
         => ( ! [X3: a,Xs2: list_a,Ys2: list_a] :
                ( ( lexordp_a @ less @ Xs2 @ Ys2 )
               => ( ( P @ Xs2 @ Ys2 )
                 => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ X3 @ Ys2 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% local.lexordp_induct
thf(fact_1127_local_Olexordp__cases,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( lexordp_a @ less @ Xs @ Ys )
     => ( ( ( Xs = nil_a )
         => ! [Y2: a,Ys5: list_a] :
              ( Ys
             != ( cons_a @ Y2 @ Ys5 ) ) )
       => ( ! [X3: a] :
              ( ? [Xs5: list_a] :
                  ( Xs
                  = ( cons_a @ X3 @ Xs5 ) )
             => ! [Y2: a] :
                  ( ? [Ys5: list_a] :
                      ( Ys
                      = ( cons_a @ Y2 @ Ys5 ) )
                 => ~ ( less @ X3 @ Y2 ) ) )
         => ~ ! [X3: a,Xs5: list_a] :
                ( ( Xs
                  = ( cons_a @ X3 @ Xs5 ) )
               => ! [Ys5: list_a] :
                    ( ( Ys
                      = ( cons_a @ X3 @ Ys5 ) )
                   => ~ ( lexordp_a @ less @ Xs5 @ Ys5 ) ) ) ) ) ) ).

% local.lexordp_cases
thf(fact_1128_local_Olexordp_Osimps,axiom,
    ! [A1: list_a,A22: list_a] :
      ( ( lexordp_a @ less @ A1 @ A22 )
      = ( ? [Y3: a,Ys4: list_a] :
            ( ( A1 = nil_a )
            & ( A22
              = ( cons_a @ Y3 @ Ys4 ) ) )
        | ? [X2: a,Y3: a,Xs4: list_a,Ys4: list_a] :
            ( ( A1
              = ( cons_a @ X2 @ Xs4 ) )
            & ( A22
              = ( cons_a @ Y3 @ Ys4 ) )
            & ( less @ X2 @ Y3 ) )
        | ? [X2: a,Y3: a,Xs4: list_a,Ys4: list_a] :
            ( ( A1
              = ( cons_a @ X2 @ Xs4 ) )
            & ( A22
              = ( cons_a @ Y3 @ Ys4 ) )
            & ~ ( less @ X2 @ Y3 )
            & ~ ( less @ Y3 @ X2 )
            & ( lexordp_a @ less @ Xs4 @ Ys4 ) ) ) ) ).

% local.lexordp.simps
thf(fact_1129_local_Olexordp_Ocases,axiom,
    ! [A1: list_a,A22: list_a] :
      ( ( lexordp_a @ less @ A1 @ A22 )
     => ( ( ( A1 = nil_a )
         => ! [Y2: a,Ys2: list_a] :
              ( A22
             != ( cons_a @ Y2 @ Ys2 ) ) )
       => ( ! [X3: a] :
              ( ? [Xs2: list_a] :
                  ( A1
                  = ( cons_a @ X3 @ Xs2 ) )
             => ! [Y2: a] :
                  ( ? [Ys2: list_a] :
                      ( A22
                      = ( cons_a @ Y2 @ Ys2 ) )
                 => ~ ( less @ X3 @ Y2 ) ) )
         => ~ ! [X3: a,Y2: a,Xs2: list_a] :
                ( ( A1
                  = ( cons_a @ X3 @ Xs2 ) )
               => ! [Ys2: list_a] :
                    ( ( A22
                      = ( cons_a @ Y2 @ Ys2 ) )
                   => ( ~ ( less @ X3 @ Y2 )
                     => ( ~ ( less @ Y2 @ X3 )
                       => ~ ( lexordp_a @ less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).

% local.lexordp.cases
thf(fact_1130_local_Olexordp_ONil,axiom,
    ! [Y: a,Ys: list_a] : ( lexordp_a @ less @ nil_a @ ( cons_a @ Y @ Ys ) ) ).

% local.lexordp.Nil
thf(fact_1131_local_Olexordp__append__left__rightI,axiom,
    ! [X: a,Y: a,Us: list_a,Xs: list_a,Ys: list_a] :
      ( ( less @ X @ Y )
     => ( lexordp_a @ less @ ( append_a @ Us @ ( cons_a @ X @ Xs ) ) @ ( append_a @ Us @ ( cons_a @ Y @ Ys ) ) ) ) ).

% local.lexordp_append_left_rightI
thf(fact_1132_local_Olexordp__iff,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( lexordp_a @ less @ Xs @ Ys )
      = ( ? [X2: a,Vs2: list_a] :
            ( Ys
            = ( append_a @ Xs @ ( cons_a @ X2 @ Vs2 ) ) )
        | ? [Us2: list_a,A6: a,B7: a,Vs2: list_a,Ws: list_a] :
            ( ( less @ A6 @ B7 )
            & ( Xs
              = ( append_a @ Us2 @ ( cons_a @ A6 @ Vs2 ) ) )
            & ( Ys
              = ( append_a @ Us2 @ ( cons_a @ B7 @ Ws ) ) ) ) ) ) ).

% local.lexordp_iff
thf(fact_1133_local_Olexordp__simps_I3_J,axiom,
    ! [X: a,Xs: list_a,Y: a,Ys: list_a] :
      ( ( lexordp_a @ less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) )
      = ( ( less @ X @ Y )
        | ( ~ ( less @ Y @ X )
          & ( lexordp_a @ less @ Xs @ Ys ) ) ) ) ).

% local.lexordp_simps(3)
thf(fact_1134_local_Olexordp__simps_I2_J,axiom,
    ! [Xs: list_a] :
      ~ ( lexordp_a @ less @ Xs @ nil_a ) ).

% local.lexordp_simps(2)
thf(fact_1135_local_Olexordp__simps_I1_J,axiom,
    ! [Ys: list_a] :
      ( ( lexordp_a @ less @ nil_a @ Ys )
      = ( Ys != nil_a ) ) ).

% local.lexordp_simps(1)
thf(fact_1136_rbt__entries__next_Oelims,axiom,
    ! [X: produc5302566010284363575list_a,Y: produc7323266412867044453list_a] :
      ( ( ( rBT_rb183187240731308954list_a @ X )
        = Y )
     => ( ! [Kv: produc1828243372476419385list_a,T5: rBT_rb6721489163287533103list_a,Kts2: list_P6367721468888276775list_a] :
            ( ( X
              = ( produc5503654560254416879list_a @ ( cons_P3144650002224114017list_a @ ( produc5396506868210886417list_a @ Kv @ T5 ) @ Kts2 ) @ rBT_Em8319540245955853219list_a ) )
           => ( Y
             != ( produc5747529839659839189list_a @ Kv @ ( produc5503654560254416879list_a @ Kts2 @ T5 ) ) ) )
       => ( ! [Kts2: list_P6367721468888276775list_a,C3: rBT_color,L2: rBT_rb6721489163287533103list_a,K2: list_a,V2: produc2547433716817190550list_a,R2: rBT_rb6721489163287533103list_a] :
              ( ( X
                = ( produc5503654560254416879list_a @ Kts2 @ ( rBT_Br1204270329063416912list_a @ C3 @ L2 @ K2 @ V2 @ R2 ) ) )
             => ( Y
               != ( rBT_rb183187240731308954list_a @ ( produc5503654560254416879list_a @ ( cons_P3144650002224114017list_a @ ( produc5396506868210886417list_a @ ( produc8961369442823515561list_a @ K2 @ V2 ) @ R2 ) @ Kts2 ) @ L2 ) ) ) )
         => ~ ( ( X
                = ( produc5503654560254416879list_a @ nil_Pr6553021886619091217list_a @ rBT_Em8319540245955853219list_a ) )
             => ( Y != undefi818404855949225677list_a ) ) ) ) ) ).

% rbt_entries_next.elims
thf(fact_1137_rbt__entries__next_Oelims,axiom,
    ! [X: produc8014987678242552961list_a,Y: produc7756919208082593089list_a] :
      ( ( ( rBT_rb2202894292051942024list_a @ X )
        = Y )
     => ( ! [Kv: produc9164743771328383783list_a,T5: rBT_rb146355364677272861list_a,Kts2: list_P868163099566772739list_a] :
            ( ( X
              = ( produc1400523206701116857list_a @ ( cons_P695356969528146237list_a @ ( produc5288693782868576493list_a @ Kv @ T5 ) @ Kts2 ) @ rBT_Em7612442972313006097list_a ) )
           => ( Y
             != ( produc2864690913915763889list_a @ Kv @ ( produc1400523206701116857list_a @ Kts2 @ T5 ) ) ) )
       => ( ! [Kts2: list_P868163099566772739list_a,C3: rBT_color,L2: rBT_rb146355364677272861list_a,K2: list_a,V2: list_a,R2: rBT_rb146355364677272861list_a] :
              ( ( X
                = ( produc1400523206701116857list_a @ Kts2 @ ( rBT_Br8372335562727726142list_a @ C3 @ L2 @ K2 @ V2 @ R2 ) ) )
             => ( Y
               != ( rBT_rb2202894292051942024list_a @ ( produc1400523206701116857list_a @ ( cons_P695356969528146237list_a @ ( produc5288693782868576493list_a @ ( produc6837034575241423639list_a @ K2 @ V2 ) @ R2 ) @ Kts2 ) @ L2 ) ) ) )
         => ~ ( ( X
                = ( produc1400523206701116857list_a @ nil_Pr7336071211429847277list_a @ rBT_Em7612442972313006097list_a ) )
             => ( Y != undefi8980584808961208233list_a ) ) ) ) ) ).

% rbt_entries_next.elims
thf(fact_1138_rbt__entries__next_Oelims,axiom,
    ! [X: produc7226406167899075884list_a,Y: produc2580682116604856691list_a] :
      ( ( ( rBT_rb6329126105604721473list_a @ X )
        = Y )
     => ( ! [Kv: produc1392733710753813600list_a,T5: rBT_rb9014005217275227734list_a,Kts2: list_P7334024422598439925list_a] :
            ( ( X
              = ( produc8990773972949836516list_a @ ( cons_P4323844787983635503list_a @ ( produc7258404890345721951list_a @ Kv @ T5 ) @ Kts2 ) @ rBT_Em6530526691317698762list_a ) )
           => ( Y
             != ( produc1295700230449419875list_a @ Kv @ ( produc8990773972949836516list_a @ Kts2 @ T5 ) ) ) )
       => ( ! [Kts2: list_P7334024422598439925list_a,C3: rBT_color,L2: rBT_rb9014005217275227734list_a,K2: a > a > $o,V2: produc9164743771328383783list_a,R2: rBT_rb9014005217275227734list_a] :
              ( ( X
                = ( produc8990773972949836516list_a @ Kts2 @ ( rBT_Br6721835470276586743list_a @ C3 @ L2 @ K2 @ V2 @ R2 ) ) )
             => ( Y
               != ( rBT_rb6329126105604721473list_a @ ( produc8990773972949836516list_a @ ( cons_P4323844787983635503list_a @ ( produc7258404890345721951list_a @ ( produc8684083420967988176list_a @ K2 @ V2 ) @ R2 ) @ Kts2 ) @ L2 ) ) ) )
         => ~ ( ( X
                = ( produc8990773972949836516list_a @ nil_Pr1415508891854699487list_a @ rBT_Em6530526691317698762list_a ) )
             => ( Y != undefi5487801273968337883list_a ) ) ) ) ) ).

% rbt_entries_next.elims
thf(fact_1139_rbt__entries__next_Oelims,axiom,
    ! [X: produc6729519863575003598list_a,Y: produc5384522425006519391list_a] :
      ( ( ( rBT_rb8531841597279888503list_a @ X )
        = Y )
     => ( ! [Kv: produc2547433716817190550list_a,T5: rBT_rb484615002580549772list_a,Kts2: list_P7781133056824549729list_a] :
            ( ( X
              = ( produc3814750783646602886list_a @ ( cons_P4107865082250473371list_a @ ( produc2500210828402294475list_a @ Kv @ T5 ) @ Kts2 ) @ rBT_Em8565251403037144576list_a ) )
           => ( Y
             != ( produc64810836902687311list_a @ Kv @ ( produc3814750783646602886list_a @ Kts2 @ T5 ) ) ) )
       => ( ! [Kts2: list_P7781133056824549729list_a,C3: rBT_color,L2: rBT_rb484615002580549772list_a,K2: a,V2: produc271001111946445293list_a,R2: rBT_rb484615002580549772list_a] :
              ( ( X
                = ( produc3814750783646602886list_a @ Kts2 @ ( rBT_Br706236987277315629list_a @ C3 @ L2 @ K2 @ V2 @ R2 ) ) )
             => ( Y
               != ( rBT_rb8531841597279888503list_a @ ( produc3814750783646602886list_a @ ( cons_P4107865082250473371list_a @ ( produc2500210828402294475list_a @ ( produc4562151247771834630list_a @ K2 @ V2 ) @ R2 ) @ Kts2 ) @ L2 ) ) ) )
         => ~ ( ( X
                = ( produc3814750783646602886list_a @ nil_Pr5096462914871187275list_a @ rBT_Em8565251403037144576list_a ) )
             => ( Y != undefi6661219117333548743list_a ) ) ) ) ) ).

% rbt_entries_next.elims
thf(fact_1140_rbt__entries__next_Oelims,axiom,
    ! [X: produc3543412085927525380bt_a_b,Y: produc2973455828936588035bt_a_b] :
      ( ( ( rBT_rb1003684898779318153xt_a_b @ X )
        = Y )
     => ( ! [Kv: product_prod_a_b,T5: rBT_rbt_a_b,Kts2: list_P2302744184456665989bt_a_b] :
            ( ( X
              = ( produc4569715891362637500bt_a_b @ ( cons_P4763212896715341247bt_a_b @ ( produc2347283123186229231bt_a_b @ Kv @ T5 ) @ Kts2 ) @ rBT_Empty_a_b ) )
           => ( Y
             != ( produc7977303101459006451bt_a_b @ Kv @ ( produc4569715891362637500bt_a_b @ Kts2 @ T5 ) ) ) )
       => ( ! [Kts2: list_P2302744184456665989bt_a_b,C3: rBT_color,L2: rBT_rbt_a_b,K2: a,V2: b,R2: rBT_rbt_a_b] :
              ( ( X
                = ( produc4569715891362637500bt_a_b @ Kts2 @ ( rBT_Branch_a_b @ C3 @ L2 @ K2 @ V2 @ R2 ) ) )
             => ( Y
               != ( rBT_rb1003684898779318153xt_a_b @ ( produc4569715891362637500bt_a_b @ ( cons_P4763212896715341247bt_a_b @ ( produc2347283123186229231bt_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ R2 ) @ Kts2 ) @ L2 ) ) ) )
         => ~ ( ( X
                = ( produc4569715891362637500bt_a_b @ nil_Pr7081231462618039663bt_a_b @ rBT_Empty_a_b ) )
             => ( Y != undefi2915438512828003691bt_a_b ) ) ) ) ) ).

% rbt_entries_next.elims
thf(fact_1141_rbt__entries__next_Opelims,axiom,
    ! [X: produc5302566010284363575list_a,Y: produc7323266412867044453list_a] :
      ( ( ( rBT_rb183187240731308954list_a @ X )
        = Y )
     => ( ( accp_P6422504809503731904list_a @ rBT_rb6375575558407650051list_a @ X )
       => ( ! [Kv: produc1828243372476419385list_a,T5: rBT_rb6721489163287533103list_a,Kts2: list_P6367721468888276775list_a] :
              ( ( X
                = ( produc5503654560254416879list_a @ ( cons_P3144650002224114017list_a @ ( produc5396506868210886417list_a @ Kv @ T5 ) @ Kts2 ) @ rBT_Em8319540245955853219list_a ) )
             => ( ( Y
                  = ( produc5747529839659839189list_a @ Kv @ ( produc5503654560254416879list_a @ Kts2 @ T5 ) ) )
               => ~ ( accp_P6422504809503731904list_a @ rBT_rb6375575558407650051list_a @ ( produc5503654560254416879list_a @ ( cons_P3144650002224114017list_a @ ( produc5396506868210886417list_a @ Kv @ T5 ) @ Kts2 ) @ rBT_Em8319540245955853219list_a ) ) ) )
         => ( ! [Kts2: list_P6367721468888276775list_a,C3: rBT_color,L2: rBT_rb6721489163287533103list_a,K2: list_a,V2: produc2547433716817190550list_a,R2: rBT_rb6721489163287533103list_a] :
                ( ( X
                  = ( produc5503654560254416879list_a @ Kts2 @ ( rBT_Br1204270329063416912list_a @ C3 @ L2 @ K2 @ V2 @ R2 ) ) )
               => ( ( Y
                    = ( rBT_rb183187240731308954list_a @ ( produc5503654560254416879list_a @ ( cons_P3144650002224114017list_a @ ( produc5396506868210886417list_a @ ( produc8961369442823515561list_a @ K2 @ V2 ) @ R2 ) @ Kts2 ) @ L2 ) ) )
                 => ~ ( accp_P6422504809503731904list_a @ rBT_rb6375575558407650051list_a @ ( produc5503654560254416879list_a @ Kts2 @ ( rBT_Br1204270329063416912list_a @ C3 @ L2 @ K2 @ V2 @ R2 ) ) ) ) )
           => ~ ( ( X
                  = ( produc5503654560254416879list_a @ nil_Pr6553021886619091217list_a @ rBT_Em8319540245955853219list_a ) )
               => ( ( Y = undefi818404855949225677list_a )
                 => ~ ( accp_P6422504809503731904list_a @ rBT_rb6375575558407650051list_a @ ( produc5503654560254416879list_a @ nil_Pr6553021886619091217list_a @ rBT_Em8319540245955853219list_a ) ) ) ) ) ) ) ) ).

% rbt_entries_next.pelims
thf(fact_1142_rbt__entries__next_Opelims,axiom,
    ! [X: produc8014987678242552961list_a,Y: produc7756919208082593089list_a] :
      ( ( ( rBT_rb2202894292051942024list_a @ X )
        = Y )
     => ( ( accp_P7092546708261160586list_a @ rBT_rb3736807956228538225list_a @ X )
       => ( ! [Kv: produc9164743771328383783list_a,T5: rBT_rb146355364677272861list_a,Kts2: list_P868163099566772739list_a] :
              ( ( X
                = ( produc1400523206701116857list_a @ ( cons_P695356969528146237list_a @ ( produc5288693782868576493list_a @ Kv @ T5 ) @ Kts2 ) @ rBT_Em7612442972313006097list_a ) )
             => ( ( Y
                  = ( produc2864690913915763889list_a @ Kv @ ( produc1400523206701116857list_a @ Kts2 @ T5 ) ) )
               => ~ ( accp_P7092546708261160586list_a @ rBT_rb3736807956228538225list_a @ ( produc1400523206701116857list_a @ ( cons_P695356969528146237list_a @ ( produc5288693782868576493list_a @ Kv @ T5 ) @ Kts2 ) @ rBT_Em7612442972313006097list_a ) ) ) )
         => ( ! [Kts2: list_P868163099566772739list_a,C3: rBT_color,L2: rBT_rb146355364677272861list_a,K2: list_a,V2: list_a,R2: rBT_rb146355364677272861list_a] :
                ( ( X
                  = ( produc1400523206701116857list_a @ Kts2 @ ( rBT_Br8372335562727726142list_a @ C3 @ L2 @ K2 @ V2 @ R2 ) ) )
               => ( ( Y
                    = ( rBT_rb2202894292051942024list_a @ ( produc1400523206701116857list_a @ ( cons_P695356969528146237list_a @ ( produc5288693782868576493list_a @ ( produc6837034575241423639list_a @ K2 @ V2 ) @ R2 ) @ Kts2 ) @ L2 ) ) )
                 => ~ ( accp_P7092546708261160586list_a @ rBT_rb3736807956228538225list_a @ ( produc1400523206701116857list_a @ Kts2 @ ( rBT_Br8372335562727726142list_a @ C3 @ L2 @ K2 @ V2 @ R2 ) ) ) ) )
           => ~ ( ( X
                  = ( produc1400523206701116857list_a @ nil_Pr7336071211429847277list_a @ rBT_Em7612442972313006097list_a ) )
               => ( ( Y = undefi8980584808961208233list_a )
                 => ~ ( accp_P7092546708261160586list_a @ rBT_rb3736807956228538225list_a @ ( produc1400523206701116857list_a @ nil_Pr7336071211429847277list_a @ rBT_Em7612442972313006097list_a ) ) ) ) ) ) ) ) ).

% rbt_entries_next.pelims
thf(fact_1143_rbt__entries__next_Opelims,axiom,
    ! [X: produc7226406167899075884list_a,Y: produc2580682116604856691list_a] :
      ( ( ( rBT_rb6329126105604721473list_a @ X )
        = Y )
     => ( ( accp_P2260255140058976053list_a @ rBT_rb3347100673370300458list_a @ X )
       => ( ! [Kv: produc1392733710753813600list_a,T5: rBT_rb9014005217275227734list_a,Kts2: list_P7334024422598439925list_a] :
              ( ( X
                = ( produc8990773972949836516list_a @ ( cons_P4323844787983635503list_a @ ( produc7258404890345721951list_a @ Kv @ T5 ) @ Kts2 ) @ rBT_Em6530526691317698762list_a ) )
             => ( ( Y
                  = ( produc1295700230449419875list_a @ Kv @ ( produc8990773972949836516list_a @ Kts2 @ T5 ) ) )
               => ~ ( accp_P2260255140058976053list_a @ rBT_rb3347100673370300458list_a @ ( produc8990773972949836516list_a @ ( cons_P4323844787983635503list_a @ ( produc7258404890345721951list_a @ Kv @ T5 ) @ Kts2 ) @ rBT_Em6530526691317698762list_a ) ) ) )
         => ( ! [Kts2: list_P7334024422598439925list_a,C3: rBT_color,L2: rBT_rb9014005217275227734list_a,K2: a > a > $o,V2: produc9164743771328383783list_a,R2: rBT_rb9014005217275227734list_a] :
                ( ( X
                  = ( produc8990773972949836516list_a @ Kts2 @ ( rBT_Br6721835470276586743list_a @ C3 @ L2 @ K2 @ V2 @ R2 ) ) )
               => ( ( Y
                    = ( rBT_rb6329126105604721473list_a @ ( produc8990773972949836516list_a @ ( cons_P4323844787983635503list_a @ ( produc7258404890345721951list_a @ ( produc8684083420967988176list_a @ K2 @ V2 ) @ R2 ) @ Kts2 ) @ L2 ) ) )
                 => ~ ( accp_P2260255140058976053list_a @ rBT_rb3347100673370300458list_a @ ( produc8990773972949836516list_a @ Kts2 @ ( rBT_Br6721835470276586743list_a @ C3 @ L2 @ K2 @ V2 @ R2 ) ) ) ) )
           => ~ ( ( X
                  = ( produc8990773972949836516list_a @ nil_Pr1415508891854699487list_a @ rBT_Em6530526691317698762list_a ) )
               => ( ( Y = undefi5487801273968337883list_a )
                 => ~ ( accp_P2260255140058976053list_a @ rBT_rb3347100673370300458list_a @ ( produc8990773972949836516list_a @ nil_Pr1415508891854699487list_a @ rBT_Em6530526691317698762list_a ) ) ) ) ) ) ) ) ).

% rbt_entries_next.pelims
thf(fact_1144_rbt__entries__next_Opelims,axiom,
    ! [X: produc6729519863575003598list_a,Y: produc5384522425006519391list_a] :
      ( ( ( rBT_rb8531841597279888503list_a @ X )
        = Y )
     => ( ( accp_P6495670621881728983list_a @ rBT_rb1916594010287293792list_a @ X )
       => ( ! [Kv: produc2547433716817190550list_a,T5: rBT_rb484615002580549772list_a,Kts2: list_P7781133056824549729list_a] :
              ( ( X
                = ( produc3814750783646602886list_a @ ( cons_P4107865082250473371list_a @ ( produc2500210828402294475list_a @ Kv @ T5 ) @ Kts2 ) @ rBT_Em8565251403037144576list_a ) )
             => ( ( Y
                  = ( produc64810836902687311list_a @ Kv @ ( produc3814750783646602886list_a @ Kts2 @ T5 ) ) )
               => ~ ( accp_P6495670621881728983list_a @ rBT_rb1916594010287293792list_a @ ( produc3814750783646602886list_a @ ( cons_P4107865082250473371list_a @ ( produc2500210828402294475list_a @ Kv @ T5 ) @ Kts2 ) @ rBT_Em8565251403037144576list_a ) ) ) )
         => ( ! [Kts2: list_P7781133056824549729list_a,C3: rBT_color,L2: rBT_rb484615002580549772list_a,K2: a,V2: produc271001111946445293list_a,R2: rBT_rb484615002580549772list_a] :
                ( ( X
                  = ( produc3814750783646602886list_a @ Kts2 @ ( rBT_Br706236987277315629list_a @ C3 @ L2 @ K2 @ V2 @ R2 ) ) )
               => ( ( Y
                    = ( rBT_rb8531841597279888503list_a @ ( produc3814750783646602886list_a @ ( cons_P4107865082250473371list_a @ ( produc2500210828402294475list_a @ ( produc4562151247771834630list_a @ K2 @ V2 ) @ R2 ) @ Kts2 ) @ L2 ) ) )
                 => ~ ( accp_P6495670621881728983list_a @ rBT_rb1916594010287293792list_a @ ( produc3814750783646602886list_a @ Kts2 @ ( rBT_Br706236987277315629list_a @ C3 @ L2 @ K2 @ V2 @ R2 ) ) ) ) )
           => ~ ( ( X
                  = ( produc3814750783646602886list_a @ nil_Pr5096462914871187275list_a @ rBT_Em8565251403037144576list_a ) )
               => ( ( Y = undefi6661219117333548743list_a )
                 => ~ ( accp_P6495670621881728983list_a @ rBT_rb1916594010287293792list_a @ ( produc3814750783646602886list_a @ nil_Pr5096462914871187275list_a @ rBT_Em8565251403037144576list_a ) ) ) ) ) ) ) ) ).

% rbt_entries_next.pelims
thf(fact_1145_rbt__entries__next_Opelims,axiom,
    ! [X: produc3543412085927525380bt_a_b,Y: produc2973455828936588035bt_a_b] :
      ( ( ( rBT_rb1003684898779318153xt_a_b @ X )
        = Y )
     => ( ( accp_P2315051374951340301bt_a_b @ rBT_rb6568690042909900146el_a_b @ X )
       => ( ! [Kv: product_prod_a_b,T5: rBT_rbt_a_b,Kts2: list_P2302744184456665989bt_a_b] :
              ( ( X
                = ( produc4569715891362637500bt_a_b @ ( cons_P4763212896715341247bt_a_b @ ( produc2347283123186229231bt_a_b @ Kv @ T5 ) @ Kts2 ) @ rBT_Empty_a_b ) )
             => ( ( Y
                  = ( produc7977303101459006451bt_a_b @ Kv @ ( produc4569715891362637500bt_a_b @ Kts2 @ T5 ) ) )
               => ~ ( accp_P2315051374951340301bt_a_b @ rBT_rb6568690042909900146el_a_b @ ( produc4569715891362637500bt_a_b @ ( cons_P4763212896715341247bt_a_b @ ( produc2347283123186229231bt_a_b @ Kv @ T5 ) @ Kts2 ) @ rBT_Empty_a_b ) ) ) )
         => ( ! [Kts2: list_P2302744184456665989bt_a_b,C3: rBT_color,L2: rBT_rbt_a_b,K2: a,V2: b,R2: rBT_rbt_a_b] :
                ( ( X
                  = ( produc4569715891362637500bt_a_b @ Kts2 @ ( rBT_Branch_a_b @ C3 @ L2 @ K2 @ V2 @ R2 ) ) )
               => ( ( Y
                    = ( rBT_rb1003684898779318153xt_a_b @ ( produc4569715891362637500bt_a_b @ ( cons_P4763212896715341247bt_a_b @ ( produc2347283123186229231bt_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ R2 ) @ Kts2 ) @ L2 ) ) )
                 => ~ ( accp_P2315051374951340301bt_a_b @ rBT_rb6568690042909900146el_a_b @ ( produc4569715891362637500bt_a_b @ Kts2 @ ( rBT_Branch_a_b @ C3 @ L2 @ K2 @ V2 @ R2 ) ) ) ) )
           => ~ ( ( X
                  = ( produc4569715891362637500bt_a_b @ nil_Pr7081231462618039663bt_a_b @ rBT_Empty_a_b ) )
               => ( ( Y = undefi2915438512828003691bt_a_b )
                 => ~ ( accp_P2315051374951340301bt_a_b @ rBT_rb6568690042909900146el_a_b @ ( produc4569715891362637500bt_a_b @ nil_Pr7081231462618039663bt_a_b @ rBT_Empty_a_b ) ) ) ) ) ) ) ) ).

% rbt_entries_next.pelims
thf(fact_1146_local_Ostrict__sorted__simps_I2_J,axiom,
    ! [X: a,Ys: list_a] :
      ( ( sorted_wrt_a @ less @ ( cons_a @ X @ Ys ) )
      = ( ! [X2: a] :
            ( ( member_a @ X2 @ ( set_a2 @ Ys ) )
           => ( less @ X @ X2 ) )
        & ( sorted_wrt_a @ less @ Ys ) ) ) ).

% local.strict_sorted_simps(2)
thf(fact_1147_local_Orbt__split__lookup,axiom,
    ! [T: rBT_rbt_a_b,X: a,L: rBT_rbt_a_b,Beta: option_b,R: rBT_rbt_a_b,K: a] :
      ( ( ( rBT_rbt_split_a_b @ less @ T @ X )
        = ( produc2350881659922173257bt_a_b @ L @ ( produc6524864345732101644bt_a_b @ Beta @ R ) ) )
     => ( ( rBT_rbt_sorted_a_b @ less @ T )
       => ( ( ( less @ K @ X )
           => ( ( rBT_rbt_lookup_a_b @ less @ T @ K )
              = ( rBT_rbt_lookup_a_b @ less @ L @ K ) ) )
          & ( ~ ( less @ K @ X )
           => ( ( ( K = X )
               => ( ( rBT_rbt_lookup_a_b @ less @ T @ K )
                  = Beta ) )
              & ( ( K != X )
               => ( ( rBT_rbt_lookup_a_b @ less @ T @ K )
                  = ( rBT_rbt_lookup_a_b @ less @ R @ K ) ) ) ) ) ) ) ) ).

% local.rbt_split_lookup
thf(fact_1148_local_Ostrict__sorted__simps_I1_J,axiom,
    sorted_wrt_a @ less @ nil_a ).

% local.strict_sorted_simps(1)
thf(fact_1149_local_Ostrict__sorted__equal,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( sorted_wrt_a @ less @ Xs )
     => ( ( sorted_wrt_a @ less @ Ys )
       => ( ( ( set_a2 @ Ys )
            = ( set_a2 @ Xs ) )
         => ( Ys = Xs ) ) ) ) ).

% local.strict_sorted_equal
thf(fact_1150_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_P1467974523212632366od_a_b,P: product_prod_a_b > product_prod_a_b > $o,Q: product_prod_a_b > product_prod_a_b > $o] :
      ( ! [X3: product_prod_a_b,Y2: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X3 @ ( set_Product_prod_a_b2 @ Xs ) )
         => ( ( member1426531481828664017od_a_b @ Y2 @ ( set_Product_prod_a_b2 @ Xs ) )
           => ( ( P @ X3 @ Y2 )
             => ( Q @ X3 @ Y2 ) ) ) )
     => ( ( sorted7021363373903539861od_a_b @ P @ Xs )
       => ( sorted7021363373903539861od_a_b @ Q @ Xs ) ) ) ).

% sorted_wrt_mono_rel
thf(fact_1151_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_a,P: a > a > $o,Q: a > a > $o] :
      ( ! [X3: a,Y2: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ( ( member_a @ Y2 @ ( set_a2 @ Xs ) )
           => ( ( P @ X3 @ Y2 )
             => ( Q @ X3 @ Y2 ) ) ) )
     => ( ( sorted_wrt_a @ P @ Xs )
       => ( sorted_wrt_a @ Q @ Xs ) ) ) ).

% sorted_wrt_mono_rel
thf(fact_1152_sorted__wrt__append,axiom,
    ! [P: product_prod_a_b > product_prod_a_b > $o,Xs: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( sorted7021363373903539861od_a_b @ P @ ( append5335208823350062147od_a_b @ Xs @ Ys ) )
      = ( ( sorted7021363373903539861od_a_b @ P @ Xs )
        & ( sorted7021363373903539861od_a_b @ P @ Ys )
        & ! [X2: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X2 @ ( set_Product_prod_a_b2 @ Xs ) )
           => ! [Y3: product_prod_a_b] :
                ( ( member1426531481828664017od_a_b @ Y3 @ ( set_Product_prod_a_b2 @ Ys ) )
               => ( P @ X2 @ Y3 ) ) ) ) ) ).

% sorted_wrt_append
thf(fact_1153_sorted__wrt__append,axiom,
    ! [P: a > a > $o,Xs: list_a,Ys: list_a] :
      ( ( sorted_wrt_a @ P @ ( append_a @ Xs @ Ys ) )
      = ( ( sorted_wrt_a @ P @ Xs )
        & ( sorted_wrt_a @ P @ Ys )
        & ! [X2: a] :
            ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
           => ! [Y3: a] :
                ( ( member_a @ Y3 @ ( set_a2 @ Ys ) )
               => ( P @ X2 @ Y3 ) ) ) ) ) ).

% sorted_wrt_append
thf(fact_1154_local_Osorted__quicksort__part,axiom,
    ! [Ac: list_a,Lts: list_a,X: a,Eqs: list_a,Gts: list_a,Zs: list_a] :
      ( ( sorted_wrt_a @ less_eq @ Ac )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( set_a2 @ Lts ) @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( set_a2 @ Eqs ) ) @ ( set_a2 @ Gts ) ) @ ( set_a2 @ Zs ) ) )
           => ! [Xa3: a] :
                ( ( member_a @ Xa3 @ ( set_a2 @ Ac ) )
               => ( less @ X3 @ Xa3 ) ) )
       => ( ! [X3: a] :
              ( ( member_a @ X3 @ ( set_a2 @ Lts ) )
             => ( less @ X3 @ X ) )
         => ( ! [X3: a] :
                ( ( member_a @ X3 @ ( set_a2 @ Eqs ) )
               => ( X3 = X ) )
           => ( ! [X3: a] :
                  ( ( member_a @ X3 @ ( set_a2 @ Gts ) )
                 => ( less @ X @ X3 ) )
             => ( sorted_wrt_a @ less_eq @ ( set_quicksort_part_a @ less @ Ac @ X @ Lts @ Eqs @ Gts @ Zs ) ) ) ) ) ) ) ).

% local.sorted_quicksort_part
thf(fact_1155_local_Owlog__le,axiom,
    ! [P: a > a > $o,B: a,A: a] :
      ( ! [A2: a,B2: a] :
          ( ( P @ A2 @ B2 )
         => ( P @ B2 @ A2 ) )
     => ( ! [A2: a,B2: a] :
            ( ( less_eq @ A2 @ B2 )
           => ( P @ A2 @ B2 ) )
       => ( P @ B @ A ) ) ) ).

% local.wlog_le
thf(fact_1156_local_Oorder__trans,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( less_eq @ X @ Y )
     => ( ( less_eq @ Y @ Z )
       => ( less_eq @ X @ Z ) ) ) ).

% local.order_trans
thf(fact_1157_local_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: a,Z4: a] : ( Y4 = Z4 ) )
    = ( ^ [X2: a,Y3: a] :
          ( ( less_eq @ X2 @ Y3 )
          & ( less_eq @ Y3 @ X2 ) ) ) ) ).

% local.order_eq_iff
thf(fact_1158_local_Oorder__antisym,axiom,
    ! [X: a,Y: a] :
      ( ( less_eq @ X @ Y )
     => ( ( less_eq @ Y @ X )
       => ( X = Y ) ) ) ).

% local.order_antisym
thf(fact_1159_local_Oord__le__eq__trans,axiom,
    ! [A: a,B: a,C: a] :
      ( ( less_eq @ A @ B )
     => ( ( B = C )
       => ( less_eq @ A @ C ) ) ) ).

% local.ord_le_eq_trans
thf(fact_1160_local_Oord__eq__le__trans,axiom,
    ! [A: a,B: a,C: a] :
      ( ( A = B )
     => ( ( less_eq @ B @ C )
       => ( less_eq @ A @ C ) ) ) ).

% local.ord_eq_le_trans
thf(fact_1161_local_Onle__le,axiom,
    ! [A: a,B: a] :
      ( ( ~ ( less_eq @ A @ B ) )
      = ( ( less_eq @ B @ A )
        & ( B != A ) ) ) ).

% local.nle_le
thf(fact_1162_local_Olinorder__wlog,axiom,
    ! [P: a > a > $o,A: a,B: a] :
      ( ! [A2: a,B2: a] :
          ( ( less_eq @ A2 @ B2 )
         => ( P @ A2 @ B2 ) )
     => ( ! [A2: a,B2: a] :
            ( ( P @ B2 @ A2 )
           => ( P @ A2 @ B2 ) )
       => ( P @ A @ B ) ) ) ).

% local.linorder_wlog
thf(fact_1163_local_Olinear,axiom,
    ! [X: a,Y: a] :
      ( ( less_eq @ X @ Y )
      | ( less_eq @ Y @ X ) ) ).

% local.linear
thf(fact_1164_local_Ole__cases3,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( ( less_eq @ X @ Y )
       => ~ ( less_eq @ Y @ Z ) )
     => ( ( ( less_eq @ Y @ X )
         => ~ ( less_eq @ X @ Z ) )
       => ( ( ( less_eq @ X @ Z )
           => ~ ( less_eq @ Z @ Y ) )
         => ( ( ( less_eq @ Z @ Y )
             => ~ ( less_eq @ Y @ X ) )
           => ( ( ( less_eq @ Y @ Z )
               => ~ ( less_eq @ Z @ X ) )
             => ~ ( ( less_eq @ Z @ X )
                 => ~ ( less_eq @ X @ Y ) ) ) ) ) ) ) ).

% local.le_cases3
thf(fact_1165_local_Ole__cases,axiom,
    ! [X: a,Y: a] :
      ( ~ ( less_eq @ X @ Y )
     => ( less_eq @ Y @ X ) ) ).

% local.le_cases
thf(fact_1166_local_Oeq__refl,axiom,
    ! [X: a,Y: a] :
      ( ( X = Y )
     => ( less_eq @ X @ Y ) ) ).

% local.eq_refl
thf(fact_1167_local_Oantisym__conv,axiom,
    ! [Y: a,X: a] :
      ( ( less_eq @ Y @ X )
     => ( ( less_eq @ X @ Y )
        = ( X = Y ) ) ) ).

% local.antisym_conv
thf(fact_1168_local_Oorder_Otrans,axiom,
    ! [A: a,B: a,C: a] :
      ( ( less_eq @ A @ B )
     => ( ( less_eq @ B @ C )
       => ( less_eq @ A @ C ) ) ) ).

% local.order.trans
thf(fact_1169_local_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y4: a,Z4: a] : ( Y4 = Z4 ) )
    = ( ^ [A6: a,B7: a] :
          ( ( less_eq @ A6 @ B7 )
          & ( less_eq @ B7 @ A6 ) ) ) ) ).

% local.order.eq_iff
thf(fact_1170_local_Oorder_Oantisym,axiom,
    ! [A: a,B: a] :
      ( ( less_eq @ A @ B )
     => ( ( less_eq @ B @ A )
       => ( A = B ) ) ) ).

% local.order.antisym
thf(fact_1171_local_Odual__order_Otrans,axiom,
    ! [B: a,A: a,C: a] :
      ( ( less_eq @ B @ A )
     => ( ( less_eq @ C @ B )
       => ( less_eq @ C @ A ) ) ) ).

% local.dual_order.trans
thf(fact_1172_local_Odual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: a,Z4: a] : ( Y4 = Z4 ) )
    = ( ^ [A6: a,B7: a] :
          ( ( less_eq @ B7 @ A6 )
          & ( less_eq @ A6 @ B7 ) ) ) ) ).

% local.dual_order.eq_iff
thf(fact_1173_local_Odual__order_Oantisym,axiom,
    ! [B: a,A: a] :
      ( ( less_eq @ B @ A )
     => ( ( less_eq @ A @ B )
       => ( A = B ) ) ) ).

% local.dual_order.antisym
thf(fact_1174_local_Osorted__remove1,axiom,
    ! [Xs: list_a,A: a] :
      ( ( sorted_wrt_a @ less_eq @ Xs )
     => ( sorted_wrt_a @ less_eq @ ( remove1_a @ A @ Xs ) ) ) ).

% local.sorted_remove1
thf(fact_1175_local_Ofinite__has__minimal2,axiom,
    ! [A3: set_a,A: a] :
      ( ( finite_finite_a @ A3 )
     => ( ( member_a @ A @ A3 )
       => ? [X3: a] :
            ( ( member_a @ X3 @ A3 )
            & ( less_eq @ X3 @ A )
            & ! [Xa: a] :
                ( ( member_a @ Xa @ A3 )
               => ( ( less_eq @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% local.finite_has_minimal2
thf(fact_1176_local_Ofinite__has__maximal2,axiom,
    ! [A3: set_a,A: a] :
      ( ( finite_finite_a @ A3 )
     => ( ( member_a @ A @ A3 )
       => ? [X3: a] :
            ( ( member_a @ X3 @ A3 )
            & ( less_eq @ A @ X3 )
            & ! [Xa: a] :
                ( ( member_a @ Xa @ A3 )
               => ( ( less_eq @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% local.finite_has_maximal2
thf(fact_1177_local_Onot__less,axiom,
    ! [X: a,Y: a] :
      ( ( ~ ( less @ X @ Y ) )
      = ( less_eq @ Y @ X ) ) ).

% local.not_less
thf(fact_1178_local_Onot__le__imp__less,axiom,
    ! [Y: a,X: a] :
      ( ~ ( less_eq @ Y @ X )
     => ( less @ X @ Y ) ) ).

% local.not_le_imp_less
thf(fact_1179_local_Onot__le,axiom,
    ! [X: a,Y: a] :
      ( ( ~ ( less_eq @ X @ Y ) )
      = ( less @ Y @ X ) ) ).

% local.not_le
thf(fact_1180_local_Onless__le,axiom,
    ! [A: a,B: a] :
      ( ( ~ ( less @ A @ B ) )
      = ( ~ ( less_eq @ A @ B )
        | ( A = B ) ) ) ).

% local.nless_le
thf(fact_1181_local_Oneq__le__trans,axiom,
    ! [A: a,B: a] :
      ( ( A != B )
     => ( ( less_eq @ A @ B )
       => ( less @ A @ B ) ) ) ).

% local.neq_le_trans
thf(fact_1182_local_Oless__le__trans,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( less @ X @ Y )
     => ( ( less_eq @ Y @ Z )
       => ( less @ X @ Z ) ) ) ).

% local.less_le_trans
thf(fact_1183_local_Oless__le__not__le,axiom,
    ! [X: a,Y: a] :
      ( ( less @ X @ Y )
      = ( ( less_eq @ X @ Y )
        & ~ ( less_eq @ Y @ X ) ) ) ).

% local.less_le_not_le
thf(fact_1184_local_Oless__le,axiom,
    ! [X: a,Y: a] :
      ( ( less @ X @ Y )
      = ( ( less_eq @ X @ Y )
        & ( X != Y ) ) ) ).

% local.less_le
thf(fact_1185_local_Oless__imp__le,axiom,
    ! [X: a,Y: a] :
      ( ( less @ X @ Y )
     => ( less_eq @ X @ Y ) ) ).

% local.less_imp_le
thf(fact_1186_local_Ole__neq__trans,axiom,
    ! [A: a,B: a] :
      ( ( less_eq @ A @ B )
     => ( ( A != B )
       => ( less @ A @ B ) ) ) ).

% local.le_neq_trans
thf(fact_1187_local_Ole__less__trans,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( less_eq @ X @ Y )
     => ( ( less @ Y @ Z )
       => ( less @ X @ Z ) ) ) ).

% local.le_less_trans
thf(fact_1188_local_Ole__less__linear,axiom,
    ! [X: a,Y: a] :
      ( ( less_eq @ X @ Y )
      | ( less @ Y @ X ) ) ).

% local.le_less_linear
thf(fact_1189_local_Ole__less,axiom,
    ! [X: a,Y: a] :
      ( ( less_eq @ X @ Y )
      = ( ( less @ X @ Y )
        | ( X = Y ) ) ) ).

% local.le_less
thf(fact_1190_local_Ole__imp__less__or__eq,axiom,
    ! [X: a,Y: a] :
      ( ( less_eq @ X @ Y )
     => ( ( less @ X @ Y )
        | ( X = Y ) ) ) ).

% local.le_imp_less_or_eq
thf(fact_1191_local_OleI,axiom,
    ! [X: a,Y: a] :
      ( ~ ( less @ X @ Y )
     => ( less_eq @ Y @ X ) ) ).

% local.leI
thf(fact_1192_local_OleD,axiom,
    ! [Y: a,X: a] :
      ( ( less_eq @ Y @ X )
     => ~ ( less @ X @ Y ) ) ).

% local.leD
thf(fact_1193_local_Oantisym__conv2,axiom,
    ! [X: a,Y: a] :
      ( ( less_eq @ X @ Y )
     => ( ( ~ ( less @ X @ Y ) )
        = ( X = Y ) ) ) ).

% local.antisym_conv2
thf(fact_1194_local_Oantisym__conv1,axiom,
    ! [X: a,Y: a] :
      ( ~ ( less @ X @ Y )
     => ( ( less_eq @ X @ Y )
        = ( X = Y ) ) ) ).

% local.antisym_conv1
thf(fact_1195_local_Oorder_Ostrict__trans2,axiom,
    ! [A: a,B: a,C: a] :
      ( ( less @ A @ B )
     => ( ( less_eq @ B @ C )
       => ( less @ A @ C ) ) ) ).

% local.order.strict_trans2
thf(fact_1196_local_Oorder_Ostrict__trans1,axiom,
    ! [A: a,B: a,C: a] :
      ( ( less_eq @ A @ B )
     => ( ( less @ B @ C )
       => ( less @ A @ C ) ) ) ).

% local.order.strict_trans1
thf(fact_1197_local_Oorder_Ostrict__implies__order,axiom,
    ! [A: a,B: a] :
      ( ( less @ A @ B )
     => ( less_eq @ A @ B ) ) ).

% local.order.strict_implies_order
thf(fact_1198_local_Oorder_Ostrict__iff__order,axiom,
    ! [A: a,B: a] :
      ( ( less @ A @ B )
      = ( ( less_eq @ A @ B )
        & ( A != B ) ) ) ).

% local.order.strict_iff_order
thf(fact_1199_local_Oorder_Ostrict__iff__not,axiom,
    ! [A: a,B: a] :
      ( ( less @ A @ B )
      = ( ( less_eq @ A @ B )
        & ~ ( less_eq @ B @ A ) ) ) ).

% local.order.strict_iff_not
thf(fact_1200_local_Oorder_Oorder__iff__strict,axiom,
    ! [A: a,B: a] :
      ( ( less_eq @ A @ B )
      = ( ( less @ A @ B )
        | ( A = B ) ) ) ).

% local.order.order_iff_strict
thf(fact_1201_local_Odual__order_Ostrict__trans2,axiom,
    ! [B: a,A: a,C: a] :
      ( ( less @ B @ A )
     => ( ( less_eq @ C @ B )
       => ( less @ C @ A ) ) ) ).

% local.dual_order.strict_trans2
thf(fact_1202_local_Odual__order_Ostrict__trans1,axiom,
    ! [B: a,A: a,C: a] :
      ( ( less_eq @ B @ A )
     => ( ( less @ C @ B )
       => ( less @ C @ A ) ) ) ).

% local.dual_order.strict_trans1
thf(fact_1203_local_Odual__order_Ostrict__implies__order,axiom,
    ! [B: a,A: a] :
      ( ( less @ B @ A )
     => ( less_eq @ B @ A ) ) ).

% local.dual_order.strict_implies_order
thf(fact_1204_local_Odual__order_Ostrict__iff__order,axiom,
    ! [B: a,A: a] :
      ( ( less @ B @ A )
      = ( ( less_eq @ B @ A )
        & ( A != B ) ) ) ).

% local.dual_order.strict_iff_order
thf(fact_1205_local_Odual__order_Ostrict__iff__not,axiom,
    ! [B: a,A: a] :
      ( ( less @ B @ A )
      = ( ( less_eq @ B @ A )
        & ~ ( less_eq @ A @ B ) ) ) ).

% local.dual_order.strict_iff_not
thf(fact_1206_local_Odual__order_Oorder__iff__strict,axiom,
    ! [B: a,A: a] :
      ( ( less_eq @ B @ A )
      = ( ( less @ B @ A )
        | ( A = B ) ) ) ).

% local.dual_order.order_iff_strict
thf(fact_1207_local_Odual__order_Onot__eq__order__implies__strict,axiom,
    ! [A: a,B: a] :
      ( ( A != B )
     => ( ( less_eq @ B @ A )
       => ( less @ B @ A ) ) ) ).

% local.dual_order.not_eq_order_implies_strict
thf(fact_1208_local_Ototalp__on__le,axiom,
    ! [A3: set_a] : ( totalp_on_a @ A3 @ less_eq ) ).

% local.totalp_on_le
thf(fact_1209_local_Oantisymp__on__le,axiom,
    ! [A3: set_a] : ( antisymp_on_a @ A3 @ less_eq ) ).

% local.antisymp_on_le
thf(fact_1210_local_Osorted2,axiom,
    ! [X: a,Y: a,Zs: list_a] :
      ( ( sorted_wrt_a @ less_eq @ ( cons_a @ X @ ( cons_a @ Y @ Zs ) ) )
      = ( ( less_eq @ X @ Y )
        & ( sorted_wrt_a @ less_eq @ ( cons_a @ Y @ Zs ) ) ) ) ).

% local.sorted2
thf(fact_1211_local_Osorted0,axiom,
    sorted_wrt_a @ less_eq @ nil_a ).

% local.sorted0
thf(fact_1212_local_Ofinite__has__maximal,axiom,
    ! [A3: set_a] :
      ( ( finite_finite_a @ A3 )
     => ( ( A3 != bot_bot_set_a )
       => ? [X3: a] :
            ( ( member_a @ X3 @ A3 )
            & ! [Xa: a] :
                ( ( member_a @ Xa @ A3 )
               => ( ( less_eq @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% local.finite_has_maximal
thf(fact_1213_local_Ofinite__has__minimal,axiom,
    ! [A3: set_a] :
      ( ( finite_finite_a @ A3 )
     => ( ( A3 != bot_bot_set_a )
       => ? [X3: a] :
            ( ( member_a @ X3 @ A3 )
            & ! [Xa: a] :
                ( ( member_a @ Xa @ A3 )
               => ( ( less_eq @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% local.finite_has_minimal
thf(fact_1214_local_Ostrict__sorted__imp__sorted,axiom,
    ! [Xs: list_a] :
      ( ( sorted_wrt_a @ less @ Xs )
     => ( sorted_wrt_a @ less_eq @ Xs ) ) ).

% local.strict_sorted_imp_sorted
thf(fact_1215_local_Ofinite__ranking__induct,axiom,
    ! [S3: set_Product_prod_a_b,P: set_Product_prod_a_b > $o,F: product_prod_a_b > a] :
      ( ( finite6544458599311216081od_a_b @ S3 )
     => ( ( P @ bot_bo3428410327501040244od_a_b )
       => ( ! [X3: product_prod_a_b,S4: set_Product_prod_a_b] :
              ( ( finite6544458599311216081od_a_b @ S4 )
             => ( ! [Y5: product_prod_a_b] :
                    ( ( member1426531481828664017od_a_b @ Y5 @ S4 )
                   => ( less_eq @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert4534936386344385144od_a_b @ X3 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% local.finite_ranking_induct
thf(fact_1216_local_Ofinite__ranking__induct,axiom,
    ! [S3: set_a,P: set_a > $o,F: a > a] :
      ( ( finite_finite_a @ S3 )
     => ( ( P @ bot_bot_set_a )
       => ( ! [X3: a,S4: set_a] :
              ( ( finite_finite_a @ S4 )
             => ( ! [Y5: a] :
                    ( ( member_a @ Y5 @ S4 )
                   => ( less_eq @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_a @ X3 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% local.finite_ranking_induct
thf(fact_1217_local_Osorted1,axiom,
    ! [X: a] : ( sorted_wrt_a @ less_eq @ ( cons_a @ X @ nil_a ) ) ).

% local.sorted1
thf(fact_1218_local_Osorted__simps_I2_J,axiom,
    ! [X: a,Ys: list_a] :
      ( ( sorted_wrt_a @ less_eq @ ( cons_a @ X @ Ys ) )
      = ( ! [X2: a] :
            ( ( member_a @ X2 @ ( set_a2 @ Ys ) )
           => ( less_eq @ X @ X2 ) )
        & ( sorted_wrt_a @ less_eq @ Ys ) ) ) ).

% local.sorted_simps(2)
thf(fact_1219_local_Osorted__append,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( sorted_wrt_a @ less_eq @ ( append_a @ Xs @ Ys ) )
      = ( ( sorted_wrt_a @ less_eq @ Xs )
        & ( sorted_wrt_a @ less_eq @ Ys )
        & ! [X2: a] :
            ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
           => ! [Y3: a] :
                ( ( member_a @ Y3 @ ( set_a2 @ Ys ) )
               => ( less_eq @ X2 @ Y3 ) ) ) ) ) ).

% local.sorted_append
thf(fact_1220_local_Osorted__distinct__set__unique,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( sorted_wrt_a @ less_eq @ Xs )
     => ( ( distinct_a @ Xs )
       => ( ( sorted_wrt_a @ less_eq @ Ys )
         => ( ( distinct_a @ Ys )
           => ( ( ( set_a2 @ Xs )
                = ( set_a2 @ Ys ) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% local.sorted_distinct_set_unique
thf(fact_1221_local_Ostrict__sorted__iff,axiom,
    ! [L: list_a] :
      ( ( sorted_wrt_a @ less @ L )
      = ( ( sorted_wrt_a @ less_eq @ L )
        & ( distinct_a @ L ) ) ) ).

% local.strict_sorted_iff
thf(fact_1222_local_Osorted__remdups__sorted,axiom,
    ! [Xs: list_a] :
      ( ( sorted_wrt_a @ less_eq @ Xs )
     => ( sorted_wrt_a @ less_eq @ ( set_remdups_sorted_a @ less @ Xs ) ) ) ).

% local.sorted_remdups_sorted
thf(fact_1223_local_Osorted__quicksort,axiom,
    ! [Xs: list_a] : ( sorted_wrt_a @ less_eq @ ( set_quicksort_a @ less @ Xs ) ) ).

% local.sorted_quicksort
thf(fact_1224_local_Ofinite__sorted__distinct__unique,axiom,
    ! [A3: set_a] :
      ( ( finite_finite_a @ A3 )
     => ? [X3: list_a] :
          ( ( ( set_a2 @ X3 )
            = A3 )
          & ( sorted_wrt_a @ less_eq @ X3 )
          & ( distinct_a @ X3 )
          & ! [Y5: list_a] :
              ( ( ( ( set_a2 @ Y5 )
                  = A3 )
                & ( sorted_wrt_a @ less_eq @ Y5 )
                & ( distinct_a @ Y5 ) )
             => ( Y5 = X3 ) ) ) ) ).

% local.finite_sorted_distinct_unique
thf(fact_1225_local_Osorted__quicksort__acc,axiom,
    ! [Ac: list_a,Xs: list_a] :
      ( ( sorted_wrt_a @ less_eq @ Ac )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
           => ! [Xa3: a] :
                ( ( member_a @ Xa3 @ ( set_a2 @ Ac ) )
               => ( less @ X3 @ Xa3 ) ) )
       => ( sorted_wrt_a @ less_eq @ ( set_quicksort_acc_a @ less @ Ac @ Xs ) ) ) ) ).

% local.sorted_quicksort_acc
thf(fact_1226_local_Oorder__refl,axiom,
    ! [X: a] : ( less_eq @ X @ X ) ).

% local.order_refl
thf(fact_1227_local_Oorder_Orefl,axiom,
    ! [A: a] : ( less_eq @ A @ A ) ).

% local.order.refl
thf(fact_1228_local_Odistinct__remdups__sorted,axiom,
    ! [Xs: list_a] :
      ( ( sorted_wrt_a @ less_eq @ Xs )
     => ( distinct_a @ ( set_remdups_sorted_a @ less @ Xs ) ) ) ).

% local.distinct_remdups_sorted
thf(fact_1229_local_OGreatest__equality,axiom,
    ! [P: a > $o,X: a] :
      ( ( P @ X )
     => ( ! [Y2: a] :
            ( ( P @ Y2 )
           => ( less_eq @ Y2 @ X ) )
       => ( ( greatest_a @ less_eq @ P )
          = X ) ) ) ).

% local.Greatest_equality
thf(fact_1230_local_OGreatestI2__order,axiom,
    ! [P: a > $o,X: a,Q: a > $o] :
      ( ( P @ X )
     => ( ! [Y2: a] :
            ( ( P @ Y2 )
           => ( less_eq @ Y2 @ X ) )
       => ( ! [X3: a] :
              ( ( P @ X3 )
             => ( ! [Y5: a] :
                    ( ( P @ Y5 )
                   => ( less_eq @ Y5 @ X3 ) )
               => ( Q @ X3 ) ) )
         => ( Q @ ( greatest_a @ less_eq @ P ) ) ) ) ) ).

% local.GreatestI2_order
thf(fact_1231_local_Odistinct__insort__insert,axiom,
    ! [Xs: list_a,F: a > a,X: a] :
      ( ( distinct_a @ Xs )
     => ( distinct_a @ ( insort238848570091376780ey_a_a @ less_eq @ F @ X @ Xs ) ) ) ).

% local.distinct_insort_insert
thf(fact_1232_local_Oset__remdups__sorted,axiom,
    ! [Xs: list_a] :
      ( ( sorted_wrt_a @ less_eq @ Xs )
     => ( ( set_a2 @ ( set_remdups_sorted_a @ less @ Xs ) )
        = ( set_a2 @ Xs ) ) ) ).

% local.set_remdups_sorted
thf(fact_1233_local_OLeast__equality,axiom,
    ! [P: a > $o,X: a] :
      ( ( P @ X )
     => ( ! [Y2: a] :
            ( ( P @ Y2 )
           => ( less_eq @ X @ Y2 ) )
       => ( ( least_a @ less_eq @ P )
          = X ) ) ) ).

% local.Least_equality
thf(fact_1234_local_OLeastI2__order,axiom,
    ! [P: a > $o,X: a,Q: a > $o] :
      ( ( P @ X )
     => ( ! [Y2: a] :
            ( ( P @ Y2 )
           => ( less_eq @ X @ Y2 ) )
       => ( ! [X3: a] :
              ( ( P @ X3 )
             => ( ! [Y5: a] :
                    ( ( P @ Y5 )
                   => ( less_eq @ X3 @ Y5 ) )
               => ( Q @ X3 ) ) )
         => ( Q @ ( least_a @ less_eq @ P ) ) ) ) ) ).

% local.LeastI2_order
thf(fact_1235_local_OLeast1__le,axiom,
    ! [P: a > $o,Z: a] :
      ( ? [X5: a] :
          ( ( P @ X5 )
          & ! [Y2: a] :
              ( ( P @ Y2 )
             => ( less_eq @ X5 @ Y2 ) )
          & ! [Y2: a] :
              ( ( ( P @ Y2 )
                & ! [Ya: a] :
                    ( ( P @ Ya )
                   => ( less_eq @ Y2 @ Ya ) ) )
             => ( Y2 = X5 ) ) )
     => ( ( P @ Z )
       => ( less_eq @ ( least_a @ less_eq @ P ) @ Z ) ) ) ).

% local.Least1_le
thf(fact_1236_local_OLeast1I,axiom,
    ! [P: a > $o] :
      ( ? [X5: a] :
          ( ( P @ X5 )
          & ! [Y2: a] :
              ( ( P @ Y2 )
             => ( less_eq @ X5 @ Y2 ) )
          & ! [Y2: a] :
              ( ( ( P @ Y2 )
                & ! [Ya: a] :
                    ( ( P @ Ya )
                   => ( less_eq @ Y2 @ Ya ) ) )
             => ( Y2 = X5 ) ) )
     => ( P @ ( least_a @ less_eq @ P ) ) ) ).

% local.Least1I
thf(fact_1237_local_Osort__key__append,axiom,
    ! [Xs: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,F: product_prod_a_b > a] :
      ( ! [X3: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X3 @ ( set_Product_prod_a_b2 @ Xs ) )
         => ! [Xa3: product_prod_a_b] :
              ( ( member1426531481828664017od_a_b @ Xa3 @ ( set_Product_prod_a_b2 @ Ys ) )
             => ( less @ ( F @ X3 ) @ ( F @ Xa3 ) ) ) )
     => ( ( sort_k678281911440829620od_a_b @ less_eq @ F @ ( append5335208823350062147od_a_b @ Xs @ Ys ) )
        = ( append5335208823350062147od_a_b @ ( sort_k678281911440829620od_a_b @ less_eq @ F @ Xs ) @ ( sort_k678281911440829620od_a_b @ less_eq @ F @ Ys ) ) ) ) ).

% local.sort_key_append
thf(fact_1238_local_Osort__key__append,axiom,
    ! [Xs: list_a,Ys: list_a,F: a > a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ! [Xa3: a] :
              ( ( member_a @ Xa3 @ ( set_a2 @ Ys ) )
             => ( less @ ( F @ X3 ) @ ( F @ Xa3 ) ) ) )
     => ( ( sort_key_a_a @ less_eq @ F @ ( append_a @ Xs @ Ys ) )
        = ( append_a @ ( sort_key_a_a @ less_eq @ F @ Xs ) @ ( sort_key_a_a @ less_eq @ F @ Ys ) ) ) ) ).

% local.sort_key_append
thf(fact_1239_local_Obdd__above_Opreordering__bdd__axioms,axiom,
    condit4103000493307248661_bdd_a @ less_eq @ less ).

% local.bdd_above.preordering_bdd_axioms
thf(fact_1240_local_Osort__key__simps_I1_J,axiom,
    ! [F: a > a] :
      ( ( sort_key_a_a @ less_eq @ F @ nil_a )
      = nil_a ) ).

% local.sort_key_simps(1)
thf(fact_1241_local_Oset__sort,axiom,
    ! [F: product_prod_a_b > a,Xs: list_P1467974523212632366od_a_b] :
      ( ( set_Product_prod_a_b2 @ ( sort_k678281911440829620od_a_b @ less_eq @ F @ Xs ) )
      = ( set_Product_prod_a_b2 @ Xs ) ) ).

% local.set_sort
thf(fact_1242_local_Oset__sort,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( set_a2 @ ( sort_key_a_a @ less_eq @ F @ Xs ) )
      = ( set_a2 @ Xs ) ) ).

% local.set_sort
thf(fact_1243_local_Odistinct__sort,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( distinct_a @ ( sort_key_a_a @ less_eq @ F @ Xs ) )
      = ( distinct_a @ Xs ) ) ).

% local.distinct_sort
thf(fact_1244_local_Osorted__list__of__set__remove,axiom,
    ! [A3: set_a,X: a] :
      ( ( finite_finite_a @ A3 )
     => ( ( sorted_list_of_set_a @ less_eq @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) )
        = ( remove1_a @ X @ ( sorted_list_of_set_a @ less_eq @ A3 ) ) ) ) ).

% local.sorted_list_of_set_remove
thf(fact_1245_local_OCompl__set__less__Compl,axiom,
    ! [A3: set_a,B3: set_a] :
      ( ( set_set_less_a @ less_eq @ ( uminus_uminus_set_a @ A3 ) @ ( uminus_uminus_set_a @ B3 ) )
      = ( set_set_less_a @ less_eq @ B3 @ A3 ) ) ).

% local.Compl_set_less_Compl
thf(fact_1246_local_Osorted__sorted__list__of__set,axiom,
    ! [A3: set_a] : ( sorted_wrt_a @ less_eq @ ( sorted_list_of_set_a @ less_eq @ A3 ) ) ).

% local.sorted_sorted_list_of_set
thf(fact_1247_local_Osorted__list__of__set__inject,axiom,
    ! [A3: set_a,B3: set_a] :
      ( ( ( sorted_list_of_set_a @ less_eq @ A3 )
        = ( sorted_list_of_set_a @ less_eq @ B3 ) )
     => ( ( finite_finite_a @ A3 )
       => ( ( finite_finite_a @ B3 )
         => ( A3 = B3 ) ) ) ) ).

% local.sorted_list_of_set_inject
thf(fact_1248_local_Odistinct__sorted__list__of__set,axiom,
    ! [A3: set_a] : ( distinct_a @ ( sorted_list_of_set_a @ less_eq @ A3 ) ) ).

% local.distinct_sorted_list_of_set
thf(fact_1249_local_Onot__set__less__empty,axiom,
    ! [A3: set_a] :
      ~ ( set_set_less_a @ less_eq @ A3 @ bot_bot_set_a ) ).

% local.not_set_less_empty
thf(fact_1250_local_Ostrict__sorted__list__of__set,axiom,
    ! [A3: set_a] : ( sorted_wrt_a @ less @ ( sorted_list_of_set_a @ less_eq @ A3 ) ) ).

% local.strict_sorted_list_of_set
thf(fact_1251_local_Osorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_a] :
      ( ( sorted_wrt_a @ less_eq @ Xs )
     => ( ( distinct_a @ Xs )
       => ( ( sorted_list_of_set_a @ less_eq @ ( set_a2 @ Xs ) )
          = Xs ) ) ) ).

% local.sorted_list_of_set.idem_if_sorted_distinct
thf(fact_1252_local_Oempty__set__less__iff,axiom,
    ! [A3: set_a] :
      ( ( set_set_less_a @ less_eq @ bot_bot_set_a @ A3 )
      = ( A3 != bot_bot_set_a ) ) ).

% local.empty_set_less_iff
thf(fact_1253_local_Osorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A3: set_a] :
      ( ~ ( finite_finite_a @ A3 )
     => ( ( sorted_list_of_set_a @ less_eq @ A3 )
        = nil_a ) ) ).

% local.sorted_list_of_set.fold_insort_key.infinite

% Helper facts (3)
thf(help_If_3_1_If_001tf__b_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001tf__b_T,axiom,
    ! [X: b,Y: b] :
      ( ( if_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001tf__b_T,axiom,
    ! [X: b,Y: b] :
      ( ( if_b @ $true @ X @ Y )
      = X ) ).

% Conjectures (3)
thf(conj_0,hypothesis,
    rBT_is_rbt_a_b @ less @ t ).

thf(conj_1,hypothesis,
    ( ( rBT_rbt_lookup_a_b @ less @ t @ k )
    = ( some_b @ v2 ) ) ).

thf(conj_2,conjecture,
    ( ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ ( rBT_rbt_insert_a_b @ less @ k @ v @ t ) ) )
    = ( insert4534936386344385144od_a_b @ ( product_Pair_a_b @ k @ v ) @ ( minus_6888070959853530735od_a_b @ ( set_Product_prod_a_b2 @ ( rBT_entries_a_b @ t ) ) @ ( insert4534936386344385144od_a_b @ ( product_Pair_a_b @ k @ v2 ) @ bot_bo3428410327501040244od_a_b ) ) ) ) ).

%------------------------------------------------------------------------------