%------------------------------------------------------------------------------
% File     : ITP066^1 : TPTP v8.2.0. Released v7.5.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer HeapImperative problem prob_1195__5351920_1
% Version  : Especial.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source   : [Des21]
% Names    : HeapImperative/prob_1195__5351920_1 [Des21]

% Status   : Theorem
% Rating   : 0.40 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0
% Syntax   : Number of formulae    :  507 ( 186 unt; 150 typ;   0 def)
%            Number of atoms       :  909 ( 536 equ;   0 cnn)
%            Maximal formula atoms :   11 (   2 avg)
%            Number of connectives : 3667 ( 129   ~;  18   |;  95   &;3044   @)
%                                         (   0 <=>; 381  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (   7 avg)
%            Number of types       :   33 (  32 usr)
%            Number of type conns  :  369 ( 369   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  121 ( 118 usr;  20 con; 0-6 aty)
%            Number of variables   : 1283 (  90   ^;1146   !;  47   ?;1283   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Sledgehammer 2021-02-23 15:32:09.933
%------------------------------------------------------------------------------
% Could-be-implicit typings (32)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J,type,
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thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
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thf(ty_n_t__List__Olist_Itf__a_J,type,
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thf(ty_n_t__Heap__OTree_Itf__a_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (118)
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thf(sy_c_HOL_ONO__MATCH_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
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thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Ohs__is__empty_001tf__a,type,
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thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Ohs__of__list_001tf__a,type,
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thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Ohs__remove__max_001tf__a,type,
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thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oleft_001tf__a,type,
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thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oof__list__tree_001tf__a,type,
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thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_OremoveLeaf_001tf__a,type,
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thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oright_001tf__a,type,
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thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_OsiftDown_001tf__a,type,
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thf(sy_c_Heap_OHeap_001t__Heap__OTree_Itf__a_J_001tf__a,type,
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thf(sy_c_Heap_OHeap_001tf__a_001tf__a,type,
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thf(sy_c_Heap_OHeap__axioms_001t__Heap__OTree_Itf__a_J_001tf__a,type,
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thf(sy_c_Heap_OHeap__axioms_001tf__a_001tf__a,type,
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thf(sy_c_Heap_OTree_OE_001tf__a,type,
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thf(sy_c_Heap_OTree_OT_001tf__a,type,
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thf(sy_c_Heap_Oin__tree_001tf__a,type,
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thf(sy_c_Heap_Ois__heap_001tf__a,type,
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thf(sy_c_Heap_Omultiset_001tf__a,type,
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thf(sy_c_Heap_Oval_001tf__a,type,
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thf(sy_c_Lattices__Big_Olinorder__class_OMax_001tf__a,type,
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thf(sy_c_Multiset_Oadd__mset_001t__Multiset__Omultiset_Itf__a_J,type,
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thf(sy_c_Multiset_Oadd__mset_001tf__a,type,
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thf(sy_c_Multiset_Ocomm__monoid__add_Osum__mset_001t__Multiset__Omultiset_Itf__a_J,type,
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thf(sy_c_Multiset_Omult1_001t__Multiset__Omultiset_Itf__a_J,type,
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thf(sy_c_Product__Type_OPair_001t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_001t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
    produc1444952343iset_a: multiset_multiset_a > multiset_multiset_a > produc1293660967iset_a ).

thf(sy_c_Product__Type_OPair_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
    produc1226985431iset_a: multis782275565iset_a > multis782275565iset_a > produc741964647iset_a ).

thf(sy_c_Product__Type_OPair_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J,type,
    produc1608222167Tree_a: multis2082063201Tree_a > multis2082063201Tree_a > produc1136638567Tree_a ).

thf(sy_c_Product__Type_OPair_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    produc1859144151od_a_a: multis599418605od_a_a > multis599418605od_a_a > produc2061001575od_a_a ).

thf(sy_c_Product__Type_OPair_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
    produc2037245207iset_a: multiset_a > multiset_a > produc1127127335iset_a ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
    produc1898392407iset_a: produc1127127335iset_a > produc1127127335iset_a > produc558600423iset_a ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J,type,
    produc1002111575Tree_a: produc143150363Tree_a > produc143150363Tree_a > produc1254371559Tree_a ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a ).

thf(sy_c_Product__Type_OPair_001tf__a_001t__Heap__OTree_Itf__a_J,type,
    produc686083979Tree_a: a > tree_a > produc143150363Tree_a ).

thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
    product_Pair_a_a: a > a > product_prod_a_a ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Heap__OTree_Itf__a_J_001tf__a,type,
    product_fst_Tree_a_a: produc981471411ee_a_a > tree_a ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
    produc653373187iset_a: produc1127127335iset_a > multiset_a ).

thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001t__Heap__OTree_Itf__a_J,type,
    product_fst_a_Tree_a: produc143150363Tree_a > a ).

thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001tf__a,type,
    product_fst_a_a: product_prod_a_a > a ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Heap__OTree_Itf__a_J_001tf__a,type,
    product_snd_Tree_a_a: produc981471411ee_a_a > a ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
    produc148952133iset_a: produc1127127335iset_a > multiset_a ).

thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001t__Heap__OTree_Itf__a_J,type,
    product_snd_a_Tree_a: produc143150363Tree_a > tree_a ).

thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001tf__a,type,
    product_snd_a_a: product_prod_a_a > a ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Heap__OTree_Itf__a_J_001tf__a,type,
    produc1478296771ee_a_a: produc981471411ee_a_a > produc143150363Tree_a ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
    produc506430135iset_a: produc1127127335iset_a > produc1127127335iset_a ).

thf(sy_c_Product__Type_Oprod_Oswap_001tf__a_001t__Heap__OTree_Itf__a_J,type,
    produc1515207979Tree_a: produc143150363Tree_a > produc981471411ee_a_a ).

thf(sy_c_Product__Type_Oprod_Oswap_001tf__a_001tf__a,type,
    product_swap_a_a: product_prod_a_a > product_prod_a_a ).

thf(sy_c_Relation_Oirrefl_001tf__a,type,
    irrefl_a: set_Product_prod_a_a > $o ).

thf(sy_c_Relation_Otrans_001t__Multiset__Omultiset_Itf__a_J,type,
    trans_multiset_a: set_Pr158363655iset_a > $o ).

thf(sy_c_Relation_Otrans_001tf__a,type,
    trans_a: set_Product_prod_a_a > $o ).

thf(sy_c_RemoveMax_OCollection_001t__Heap__OTree_Itf__a_J_001tf__a,type,
    collection_Tree_a_a: tree_a > ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > $o ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
    collec1104713362iset_a: ( produc1127127335iset_a > $o ) > set_Pr158363655iset_a ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J,type,
    collec486369286Tree_a: ( produc143150363Tree_a > $o ) > set_Pr1070816379Tree_a ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).

thf(sy_c_Set_OCollect_001tf__a,type,
    collect_a: ( a > $o ) > set_a ).

thf(sy_c_member_001t__Multiset__Omultiset_Itf__a_J,type,
    member_multiset_a: multiset_a > set_multiset_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Heap__OTree_Itf__a_J_Mtf__a_J,type,
    member1092636252ee_a_a: produc981471411ee_a_a > set_Pr921883667ee_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_Mt__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_J,type,
    member104214352iset_a: produc1293660967iset_a > set_Pr720484615iset_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_J,type,
    member219455632iset_a: produc741964647iset_a > set_Pr1066112583iset_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J_J,type,
    member1298282640Tree_a: produc1136638567Tree_a > set_Pr295926343Tree_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
    member1349055120od_a_a: produc2061001575od_a_a > set_Pr1821581383od_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
    member340150864iset_a: produc1127127335iset_a > set_Pr158363655iset_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
    member567864080iset_a: produc558600423iset_a > set_Pr146455751iset_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J,type,
    member146662416Tree_a: produc1254371559Tree_a > set_Pr220561863Tree_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J,type,
    member254315204Tree_a: produc143150363Tree_a > set_Pr1070816379Tree_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

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

thf(sy_v_l1____,type,
    l1: tree_a ).

thf(sy_v_l2____,type,
    l2: tree_a ).

thf(sy_v_r1____,type,
    r1: tree_a ).

thf(sy_v_r2____,type,
    r2: tree_a ).

thf(sy_v_t,type,
    t: tree_a ).

thf(sy_v_t_H,type,
    t2: tree_a ).

thf(sy_v_t_Ha____,type,
    t_a2: tree_a ).

thf(sy_v_v1____,type,
    v1: a ).

thf(sy_v_v2____,type,
    v2: a ).

thf(sy_v_v_H,type,
    v: a ).

thf(sy_v_v_Ha____,type,
    v_a: a ).

thf(sy_v_v____,type,
    v3: a ).

% Relevant facts (353)
thf(fact_0__C4__2_Oprems_C_I1_J,axiom,
    ( ( produc686083979Tree_a @ v_a @ t_a2 )
    = ( heapIm837449470Leaf_a @ ( t_a @ v3 @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ) ).

% "4_2.prems"(1)
thf(fact_1_assms_I1_J,axiom,
    ( ( produc686083979Tree_a @ v @ t2 )
    = ( heapIm837449470Leaf_a @ t ) ) ).

% assms(1)
thf(fact_2__C4__2_Oprems_C_I2_J,axiom,
    ( ( t_a @ v3 @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) )
   != e_a ) ).

% "4_2.prems"(2)
thf(fact_3__092_060open_062t_H_A_061_AT_Av_A_Isnd_A_IremoveLeaf_A_IT_Av1_Al1_Ar1_J_J_J_A_IT_Av2_Al2_Ar2_J_092_060close_062,axiom,
    ( t_a2
    = ( t_a @ v3 @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ).

% \<open>t' = T v (snd (removeLeaf (T v1 l1 r1))) (T v2 l2 r2)\<close>
thf(fact_4_Tree_Oinject,axiom,
    ! [X21: a,X22: tree_a,X23: tree_a,Y21: a,Y22: tree_a,Y23: tree_a] :
      ( ( ( t_a @ X21 @ X22 @ X23 )
        = ( t_a @ Y21 @ Y22 @ Y23 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 )
        & ( X23 = Y23 ) ) ) ).

% Tree.inject
thf(fact_5_removeLeaf_Osimps_I5_J,axiom,
    ! [V: a,Vd: a,Ve: tree_a,Vf: tree_a,Va: a,Vb: tree_a,Vc: tree_a] :
      ( ( heapIm837449470Leaf_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) @ ( t_a @ V @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).

% removeLeaf.simps(5)
thf(fact_6_removeLeaf_Osimps_I4_J,axiom,
    ! [V: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
      ( ( heapIm837449470Leaf_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) @ ( t_a @ V @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ).

% removeLeaf.simps(4)
thf(fact_7_left_Osimps,axiom,
    ! [V: a,L: tree_a,R: tree_a] :
      ( ( heapIm1140443833left_a @ ( t_a @ V @ L @ R ) )
      = L ) ).

% left.simps
thf(fact_8_right_Osimps,axiom,
    ! [V: a,L: tree_a,R: tree_a] :
      ( ( heapIm1257206334ight_a @ ( t_a @ V @ L @ R ) )
      = R ) ).

% right.simps
thf(fact_9__C4__2_Ohyps_C_I1_J,axiom,
    ! [V2: a,T: tree_a] :
      ( ( ( produc686083979Tree_a @ V2 @ T )
        = ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
     => ( ( ( t_a @ v1 @ l1 @ r1 )
         != e_a )
       => ( ( plus_plus_multiset_a @ ( add_mset_a @ V2 @ zero_zero_multiset_a ) @ ( multiset_a2 @ T ) )
          = ( multiset_a2 @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) ) ).

% "4_2.hyps"(1)
thf(fact_10_removeLeaf_Osimps_I1_J,axiom,
    ! [V: a] :
      ( ( heapIm837449470Leaf_a @ ( t_a @ V @ e_a @ e_a ) )
      = ( produc686083979Tree_a @ V @ e_a ) ) ).

% removeLeaf.simps(1)
thf(fact_11_fstI,axiom,
    ! [X: produc1127127335iset_a,Y: multiset_a,Z: multiset_a] :
      ( ( X
        = ( produc2037245207iset_a @ Y @ Z ) )
     => ( ( produc653373187iset_a @ X )
        = Y ) ) ).

% fstI
thf(fact_12_fstI,axiom,
    ! [X: product_prod_a_a,Y: a,Z: a] :
      ( ( X
        = ( product_Pair_a_a @ Y @ Z ) )
     => ( ( product_fst_a_a @ X )
        = Y ) ) ).

% fstI
thf(fact_13_fstI,axiom,
    ! [X: produc981471411ee_a_a,Y: tree_a,Z: a] :
      ( ( X
        = ( produc649172771ee_a_a @ Y @ Z ) )
     => ( ( product_fst_Tree_a_a @ X )
        = Y ) ) ).

% fstI
thf(fact_14_fstI,axiom,
    ! [X: produc143150363Tree_a,Y: a,Z: tree_a] :
      ( ( X
        = ( produc686083979Tree_a @ Y @ Z ) )
     => ( ( product_fst_a_Tree_a @ X )
        = Y ) ) ).

% fstI
thf(fact_15_fst__eqD,axiom,
    ! [X: multiset_a,Y: multiset_a,A: multiset_a] :
      ( ( ( produc653373187iset_a @ ( produc2037245207iset_a @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_16_fst__eqD,axiom,
    ! [X: a,Y: a,A: a] :
      ( ( ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_17_fst__eqD,axiom,
    ! [X: tree_a,Y: a,A: tree_a] :
      ( ( ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_18_fst__eqD,axiom,
    ! [X: a,Y: tree_a,A: a] :
      ( ( ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_19_fst__conv,axiom,
    ! [X1: multiset_a,X2: multiset_a] :
      ( ( produc653373187iset_a @ ( produc2037245207iset_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_20_fst__conv,axiom,
    ! [X1: a,X2: a] :
      ( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_21_fst__conv,axiom,
    ! [X1: tree_a,X2: a] :
      ( ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_22_fst__conv,axiom,
    ! [X1: a,X2: tree_a] :
      ( ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_23_assms_I2_J,axiom,
    t != e_a ).

% assms(2)
thf(fact_24_prod_Oinject,axiom,
    ! [X1: tree_a,X2: a,Y1: tree_a,Y2: a] :
      ( ( ( produc649172771ee_a_a @ X1 @ X2 )
        = ( produc649172771ee_a_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_25_prod_Oinject,axiom,
    ! [X1: multiset_a,X2: multiset_a,Y1: multiset_a,Y2: multiset_a] :
      ( ( ( produc2037245207iset_a @ X1 @ X2 )
        = ( produc2037245207iset_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_26_prod_Oinject,axiom,
    ! [X1: a,X2: a,Y1: a,Y2: a] :
      ( ( ( product_Pair_a_a @ X1 @ X2 )
        = ( product_Pair_a_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_27_prod_Oinject,axiom,
    ! [X1: a,X2: tree_a,Y1: a,Y2: tree_a] :
      ( ( ( produc686083979Tree_a @ X1 @ X2 )
        = ( produc686083979Tree_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_28_old_Oprod_Oinject,axiom,
    ! [A: tree_a,B: a,A2: tree_a,B2: a] :
      ( ( ( produc649172771ee_a_a @ A @ B )
        = ( produc649172771ee_a_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_29_old_Oprod_Oinject,axiom,
    ! [A: multiset_a,B: multiset_a,A2: multiset_a,B2: multiset_a] :
      ( ( ( produc2037245207iset_a @ A @ B )
        = ( produc2037245207iset_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_30_old_Oprod_Oinject,axiom,
    ! [A: a,B: a,A2: a,B2: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_31_old_Oprod_Oinject,axiom,
    ! [A: a,B: tree_a,A2: a,B2: tree_a] :
      ( ( ( produc686083979Tree_a @ A @ B )
        = ( produc686083979Tree_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_32_prod_Ocollapse,axiom,
    ! [Prod: produc1127127335iset_a] :
      ( ( produc2037245207iset_a @ ( produc653373187iset_a @ Prod ) @ ( produc148952133iset_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_33_prod_Ocollapse,axiom,
    ! [Prod: product_prod_a_a] :
      ( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_34_prod_Ocollapse,axiom,
    ! [Prod: produc981471411ee_a_a] :
      ( ( produc649172771ee_a_a @ ( product_fst_Tree_a_a @ Prod ) @ ( product_snd_Tree_a_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_35_prod_Ocollapse,axiom,
    ! [Prod: produc143150363Tree_a] :
      ( ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ Prod ) @ ( product_snd_a_Tree_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_36_multiset_Osimps_I2_J,axiom,
    ! [V: a,L: tree_a,R: tree_a] :
      ( ( multiset_a2 @ ( t_a @ V @ L @ R ) )
      = ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ ( multiset_a2 @ L ) @ ( add_mset_a @ V @ zero_zero_multiset_a ) ) @ ( multiset_a2 @ R ) ) ) ).

% multiset.simps(2)
thf(fact_37_multiset_Osimps_I1_J,axiom,
    ( ( multiset_a2 @ e_a )
    = zero_zero_multiset_a ) ).

% multiset.simps(1)
thf(fact_38_sndI,axiom,
    ! [X: produc1127127335iset_a,Y: multiset_a,Z: multiset_a] :
      ( ( X
        = ( produc2037245207iset_a @ Y @ Z ) )
     => ( ( produc148952133iset_a @ X )
        = Z ) ) ).

% sndI
thf(fact_39_sndI,axiom,
    ! [X: product_prod_a_a,Y: a,Z: a] :
      ( ( X
        = ( product_Pair_a_a @ Y @ Z ) )
     => ( ( product_snd_a_a @ X )
        = Z ) ) ).

% sndI
thf(fact_40_sndI,axiom,
    ! [X: produc981471411ee_a_a,Y: tree_a,Z: a] :
      ( ( X
        = ( produc649172771ee_a_a @ Y @ Z ) )
     => ( ( product_snd_Tree_a_a @ X )
        = Z ) ) ).

% sndI
thf(fact_41_sndI,axiom,
    ! [X: produc143150363Tree_a,Y: a,Z: tree_a] :
      ( ( X
        = ( produc686083979Tree_a @ Y @ Z ) )
     => ( ( product_snd_a_Tree_a @ X )
        = Z ) ) ).

% sndI
thf(fact_42_snd__eqD,axiom,
    ! [X: multiset_a,Y: multiset_a,A: multiset_a] :
      ( ( ( produc148952133iset_a @ ( produc2037245207iset_a @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_43_snd__eqD,axiom,
    ! [X: a,Y: a,A: a] :
      ( ( ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_44_snd__eqD,axiom,
    ! [X: tree_a,Y: a,A: a] :
      ( ( ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_45_snd__eqD,axiom,
    ! [X: a,Y: tree_a,A: tree_a] :
      ( ( ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_46_snd__conv,axiom,
    ! [X1: multiset_a,X2: multiset_a] :
      ( ( produc148952133iset_a @ ( produc2037245207iset_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_47_snd__conv,axiom,
    ! [X1: a,X2: a] :
      ( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_48_snd__conv,axiom,
    ! [X1: tree_a,X2: a] :
      ( ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_49_snd__conv,axiom,
    ! [X1: a,X2: tree_a] :
      ( ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_50_surj__pair,axiom,
    ! [P: produc981471411ee_a_a] :
    ? [X3: tree_a,Y3: a] :
      ( P
      = ( produc649172771ee_a_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_51_surj__pair,axiom,
    ! [P: produc1127127335iset_a] :
    ? [X3: multiset_a,Y3: multiset_a] :
      ( P
      = ( produc2037245207iset_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_52_surj__pair,axiom,
    ! [P: product_prod_a_a] :
    ? [X3: a,Y3: a] :
      ( P
      = ( product_Pair_a_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_53_surj__pair,axiom,
    ! [P: produc143150363Tree_a] :
    ? [X3: a,Y3: tree_a] :
      ( P
      = ( produc686083979Tree_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_54_prod__cases,axiom,
    ! [P2: produc981471411ee_a_a > $o,P: produc981471411ee_a_a] :
      ( ! [A3: tree_a,B3: a] : ( P2 @ ( produc649172771ee_a_a @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_55_prod__cases,axiom,
    ! [P2: produc1127127335iset_a > $o,P: produc1127127335iset_a] :
      ( ! [A3: multiset_a,B3: multiset_a] : ( P2 @ ( produc2037245207iset_a @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_56_prod__cases,axiom,
    ! [P2: product_prod_a_a > $o,P: product_prod_a_a] :
      ( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_57_prod__cases,axiom,
    ! [P2: produc143150363Tree_a > $o,P: produc143150363Tree_a] :
      ( ! [A3: a,B3: tree_a] : ( P2 @ ( produc686083979Tree_a @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_58_Pair__inject,axiom,
    ! [A: tree_a,B: a,A2: tree_a,B2: a] :
      ( ( ( produc649172771ee_a_a @ A @ B )
        = ( produc649172771ee_a_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_59_Pair__inject,axiom,
    ! [A: multiset_a,B: multiset_a,A2: multiset_a,B2: multiset_a] :
      ( ( ( produc2037245207iset_a @ A @ B )
        = ( produc2037245207iset_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_60_Pair__inject,axiom,
    ! [A: a,B: a,A2: a,B2: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_61_Pair__inject,axiom,
    ! [A: a,B: tree_a,A2: a,B2: tree_a] :
      ( ( ( produc686083979Tree_a @ A @ B )
        = ( produc686083979Tree_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_62_old_Oprod_Oexhaust,axiom,
    ! [Y: produc981471411ee_a_a] :
      ~ ! [A3: tree_a,B3: a] :
          ( Y
         != ( produc649172771ee_a_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_63_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1127127335iset_a] :
      ~ ! [A3: multiset_a,B3: multiset_a] :
          ( Y
         != ( produc2037245207iset_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_64_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_a_a] :
      ~ ! [A3: a,B3: a] :
          ( Y
         != ( product_Pair_a_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_65_old_Oprod_Oexhaust,axiom,
    ! [Y: produc143150363Tree_a] :
      ~ ! [A3: a,B3: tree_a] :
          ( Y
         != ( produc686083979Tree_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_66_old_Oprod_Oinducts,axiom,
    ! [P2: produc981471411ee_a_a > $o,Prod: produc981471411ee_a_a] :
      ( ! [A3: tree_a,B3: a] : ( P2 @ ( produc649172771ee_a_a @ A3 @ B3 ) )
     => ( P2 @ Prod ) ) ).

% old.prod.inducts
thf(fact_67_old_Oprod_Oinducts,axiom,
    ! [P2: produc1127127335iset_a > $o,Prod: produc1127127335iset_a] :
      ( ! [A3: multiset_a,B3: multiset_a] : ( P2 @ ( produc2037245207iset_a @ A3 @ B3 ) )
     => ( P2 @ Prod ) ) ).

% old.prod.inducts
thf(fact_68_old_Oprod_Oinducts,axiom,
    ! [P2: product_prod_a_a > $o,Prod: product_prod_a_a] :
      ( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
     => ( P2 @ Prod ) ) ).

% old.prod.inducts
thf(fact_69_old_Oprod_Oinducts,axiom,
    ! [P2: produc143150363Tree_a > $o,Prod: produc143150363Tree_a] :
      ( ! [A3: a,B3: tree_a] : ( P2 @ ( produc686083979Tree_a @ A3 @ B3 ) )
     => ( P2 @ Prod ) ) ).

% old.prod.inducts
thf(fact_70_HS_Omultiset__empty,axiom,
    ( ( multiset_a2 @ e_a )
    = zero_zero_multiset_a ) ).

% HS.multiset_empty
thf(fact_71_surjective__pairing,axiom,
    ! [T2: produc1127127335iset_a] :
      ( T2
      = ( produc2037245207iset_a @ ( produc653373187iset_a @ T2 ) @ ( produc148952133iset_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_72_surjective__pairing,axiom,
    ! [T2: product_prod_a_a] :
      ( T2
      = ( product_Pair_a_a @ ( product_fst_a_a @ T2 ) @ ( product_snd_a_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_73_surjective__pairing,axiom,
    ! [T2: produc981471411ee_a_a] :
      ( T2
      = ( produc649172771ee_a_a @ ( product_fst_Tree_a_a @ T2 ) @ ( product_snd_Tree_a_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_74_surjective__pairing,axiom,
    ! [T2: produc143150363Tree_a] :
      ( T2
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ T2 ) @ ( product_snd_a_Tree_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_75_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1127127335iset_a] :
      ( Prod
      = ( produc2037245207iset_a @ ( produc653373187iset_a @ Prod ) @ ( produc148952133iset_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_76_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_a_a] :
      ( Prod
      = ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_77_prod_Oexhaust__sel,axiom,
    ! [Prod: produc981471411ee_a_a] :
      ( Prod
      = ( produc649172771ee_a_a @ ( product_fst_Tree_a_a @ Prod ) @ ( product_snd_Tree_a_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_78_prod_Oexhaust__sel,axiom,
    ! [Prod: produc143150363Tree_a] :
      ( Prod
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ Prod ) @ ( product_snd_a_Tree_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_79_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc981471411ee_a_a,Z2: produc981471411ee_a_a] : Y4 = Z2 )
    = ( ^ [S: produc981471411ee_a_a,T3: produc981471411ee_a_a] :
          ( ( ( product_fst_Tree_a_a @ S )
            = ( product_fst_Tree_a_a @ T3 ) )
          & ( ( product_snd_Tree_a_a @ S )
            = ( product_snd_Tree_a_a @ T3 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_80_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc143150363Tree_a,Z2: produc143150363Tree_a] : Y4 = Z2 )
    = ( ^ [S: produc143150363Tree_a,T3: produc143150363Tree_a] :
          ( ( ( product_fst_a_Tree_a @ S )
            = ( product_fst_a_Tree_a @ T3 ) )
          & ( ( product_snd_a_Tree_a @ S )
            = ( product_snd_a_Tree_a @ T3 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_81_prod_Oexpand,axiom,
    ! [Prod: produc981471411ee_a_a,Prod2: produc981471411ee_a_a] :
      ( ( ( ( product_fst_Tree_a_a @ Prod )
          = ( product_fst_Tree_a_a @ Prod2 ) )
        & ( ( product_snd_Tree_a_a @ Prod )
          = ( product_snd_Tree_a_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_82_prod_Oexpand,axiom,
    ! [Prod: produc143150363Tree_a,Prod2: produc143150363Tree_a] :
      ( ( ( ( product_fst_a_Tree_a @ Prod )
          = ( product_fst_a_Tree_a @ Prod2 ) )
        & ( ( product_snd_a_Tree_a @ Prod )
          = ( product_snd_a_Tree_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_83_prod__eqI,axiom,
    ! [P: produc981471411ee_a_a,Q: produc981471411ee_a_a] :
      ( ( ( product_fst_Tree_a_a @ P )
        = ( product_fst_Tree_a_a @ Q ) )
     => ( ( ( product_snd_Tree_a_a @ P )
          = ( product_snd_Tree_a_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_84_prod__eqI,axiom,
    ! [P: produc143150363Tree_a,Q: produc143150363Tree_a] :
      ( ( ( product_fst_a_Tree_a @ P )
        = ( product_fst_a_Tree_a @ Q ) )
     => ( ( ( product_snd_a_Tree_a @ P )
          = ( product_snd_a_Tree_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_85_is__heap_Ocases,axiom,
    ! [X: tree_a] :
      ( ( X != e_a )
     => ( ! [V3: a] :
            ( X
           != ( t_a @ V3 @ e_a @ e_a ) )
       => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
              ( X
             != ( t_a @ V3 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
         => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
                ( X
               != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
           => ~ ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
                  ( X
                 != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ).

% is_heap.cases
thf(fact_86_mem__Collect__eq,axiom,
    ! [A: produc143150363Tree_a,P2: produc143150363Tree_a > $o] :
      ( ( member254315204Tree_a @ A @ ( collec486369286Tree_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_87_mem__Collect__eq,axiom,
    ! [A: a,P2: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_88_mem__Collect__eq,axiom,
    ! [A: produc1127127335iset_a,P2: produc1127127335iset_a > $o] :
      ( ( member340150864iset_a @ A @ ( collec1104713362iset_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_89_mem__Collect__eq,axiom,
    ! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
      ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_90_Collect__mem__eq,axiom,
    ! [A4: set_Pr1070816379Tree_a] :
      ( ( collec486369286Tree_a
        @ ^ [X4: produc143150363Tree_a] : ( member254315204Tree_a @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_91_Collect__mem__eq,axiom,
    ! [A4: set_a] :
      ( ( collect_a
        @ ^ [X4: a] : ( member_a @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_92_Collect__mem__eq,axiom,
    ! [A4: set_Pr158363655iset_a] :
      ( ( collec1104713362iset_a
        @ ^ [X4: produc1127127335iset_a] : ( member340150864iset_a @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_93_Collect__mem__eq,axiom,
    ! [A4: set_Product_prod_a_a] :
      ( ( collec645855634od_a_a
        @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_94_Tree_Oexhaust,axiom,
    ! [Y: tree_a] :
      ( ( Y != e_a )
     => ~ ! [X212: a,X222: tree_a,X232: tree_a] :
            ( Y
           != ( t_a @ X212 @ X222 @ X232 ) ) ) ).

% Tree.exhaust
thf(fact_95_Tree_Oinduct,axiom,
    ! [P2: tree_a > $o,Tree: tree_a] :
      ( ( P2 @ e_a )
     => ( ! [X12: a,X24: tree_a,X32: tree_a] :
            ( ( P2 @ X24 )
           => ( ( P2 @ X32 )
             => ( P2 @ ( t_a @ X12 @ X24 @ X32 ) ) ) )
       => ( P2 @ Tree ) ) ) ).

% Tree.induct
thf(fact_96_Tree_Odistinct_I1_J,axiom,
    ! [X21: a,X22: tree_a,X23: tree_a] :
      ( e_a
     != ( t_a @ X21 @ X22 @ X23 ) ) ).

% Tree.distinct(1)
thf(fact_97_removeLeaf_Oinduct,axiom,
    ! [P2: tree_a > $o,A0: tree_a] :
      ( ! [V3: a] : ( P2 @ ( t_a @ V3 @ e_a @ e_a ) )
     => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
            ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
           => ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
             => ( P2 @ ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) ) ) )
       => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
              ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
             => ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
               => ( P2 @ ( t_a @ V3 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
         => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
                ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
               => ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
                 => ( P2 @ ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) )
           => ( ! [V3: a,Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
                  ( ( P2 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) )
                 => ( ( P2 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) )
                   => ( P2 @ ( t_a @ V3 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
             => ( ( P2 @ e_a )
               => ( P2 @ A0 ) ) ) ) ) ) ) ).

% removeLeaf.induct
thf(fact_98_removeLeaf_Ocases,axiom,
    ! [X: tree_a] :
      ( ! [V3: a] :
          ( X
         != ( t_a @ V3 @ e_a @ e_a ) )
     => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
            ( X
           != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
       => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
              ( X
             != ( t_a @ V3 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
         => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
                ( X
               != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
           => ( ! [V3: a,Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
                  ( X
                 != ( t_a @ V3 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
             => ( X = e_a ) ) ) ) ) ) ).

% removeLeaf.cases
thf(fact_99_siftDown_Ocases,axiom,
    ! [X: tree_a] :
      ( ( X != e_a )
     => ( ! [V3: a] :
            ( X
           != ( t_a @ V3 @ e_a @ e_a ) )
       => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
              ( X
             != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
         => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
                ( X
               != ( t_a @ V3 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
           => ~ ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
                  ( X
                 != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ).

% siftDown.cases
thf(fact_100_removeLeaf_Osimps_I2_J,axiom,
    ! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
      ( ( heapIm837449470Leaf_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) @ ( t_a @ V @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) @ e_a ) ) ) ).

% removeLeaf.simps(2)
thf(fact_101_removeLeaf_Osimps_I3_J,axiom,
    ! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
      ( ( heapIm837449470Leaf_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) @ ( t_a @ V @ e_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ) ) ).

% removeLeaf.simps(3)
thf(fact_102_union__mset__add__mset__left,axiom,
    ! [A: a,A4: multiset_a,B4: multiset_a] :
      ( ( plus_plus_multiset_a @ ( add_mset_a @ A @ A4 ) @ B4 )
      = ( add_mset_a @ A @ ( plus_plus_multiset_a @ A4 @ B4 ) ) ) ).

% union_mset_add_mset_left
thf(fact_103_union__mset__add__mset__right,axiom,
    ! [A4: multiset_a,A: a,B4: multiset_a] :
      ( ( plus_plus_multiset_a @ A4 @ ( add_mset_a @ A @ B4 ) )
      = ( add_mset_a @ A @ ( plus_plus_multiset_a @ A4 @ B4 ) ) ) ).

% union_mset_add_mset_right
thf(fact_104_single__eq__single,axiom,
    ! [A: a,B: a] :
      ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
        = ( add_mset_a @ B @ zero_zero_multiset_a ) )
      = ( A = B ) ) ).

% single_eq_single
thf(fact_105_add__mset__eq__single,axiom,
    ! [B: a,M: multiset_a,A: a] :
      ( ( ( add_mset_a @ B @ M )
        = ( add_mset_a @ A @ zero_zero_multiset_a ) )
      = ( ( B = A )
        & ( M = zero_zero_multiset_a ) ) ) ).

% add_mset_eq_single
thf(fact_106_single__eq__add__mset,axiom,
    ! [A: a,B: a,M: multiset_a] :
      ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
        = ( add_mset_a @ B @ M ) )
      = ( ( B = A )
        & ( M = zero_zero_multiset_a ) ) ) ).

% single_eq_add_mset
thf(fact_107_add__mset__eq__singleton__iff,axiom,
    ! [X: a,M: multiset_a,Y: a] :
      ( ( ( add_mset_a @ X @ M )
        = ( add_mset_a @ Y @ zero_zero_multiset_a ) )
      = ( ( M = zero_zero_multiset_a )
        & ( X = Y ) ) ) ).

% add_mset_eq_singleton_iff
thf(fact_108_empty__eq__union,axiom,
    ! [M: multiset_a,N: multiset_a] :
      ( ( zero_zero_multiset_a
        = ( plus_plus_multiset_a @ M @ N ) )
      = ( ( M = zero_zero_multiset_a )
        & ( N = zero_zero_multiset_a ) ) ) ).

% empty_eq_union
thf(fact_109_union__eq__empty,axiom,
    ! [M: multiset_a,N: multiset_a] :
      ( ( ( plus_plus_multiset_a @ M @ N )
        = zero_zero_multiset_a )
      = ( ( M = zero_zero_multiset_a )
        & ( N = zero_zero_multiset_a ) ) ) ).

% union_eq_empty
thf(fact_110_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [X: multiset_a,Y: multiset_a] :
      ( ( ( plus_plus_multiset_a @ X @ Y )
        = zero_zero_multiset_a )
      = ( ( X = zero_zero_multiset_a )
        & ( Y = zero_zero_multiset_a ) ) ) ).

% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_111_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
    ! [X: multiset_a,Y: multiset_a] :
      ( ( zero_zero_multiset_a
        = ( plus_plus_multiset_a @ X @ Y ) )
      = ( ( X = zero_zero_multiset_a )
        & ( Y = zero_zero_multiset_a ) ) ) ).

% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_112_add_Oleft__neutral,axiom,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
      = A ) ).

% add.left_neutral
thf(fact_113_add__right__cancel,axiom,
    ! [B: multiset_a,A: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ B @ A )
        = ( plus_plus_multiset_a @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_114_add__left__cancel,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( plus_plus_multiset_a @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_115_multi__self__add__other__not__self,axiom,
    ! [M: multiset_a,X: a] :
      ( M
     != ( add_mset_a @ X @ M ) ) ).

% multi_self_add_other_not_self
thf(fact_116_add__mset__add__mset__same__iff,axiom,
    ! [A: a,A4: multiset_a,B4: multiset_a] :
      ( ( ( add_mset_a @ A @ A4 )
        = ( add_mset_a @ A @ B4 ) )
      = ( A4 = B4 ) ) ).

% add_mset_add_mset_same_iff
thf(fact_117_add__cancel__right__right,axiom,
    ! [A: multiset_a,B: multiset_a] :
      ( ( A
        = ( plus_plus_multiset_a @ A @ B ) )
      = ( B = zero_zero_multiset_a ) ) ).

% add_cancel_right_right
thf(fact_118_add__cancel__right__left,axiom,
    ! [A: multiset_a,B: multiset_a] :
      ( ( A
        = ( plus_plus_multiset_a @ B @ A ) )
      = ( B = zero_zero_multiset_a ) ) ).

% add_cancel_right_left
thf(fact_119_add__cancel__left__right,axiom,
    ! [A: multiset_a,B: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = A )
      = ( B = zero_zero_multiset_a ) ) ).

% add_cancel_left_right
thf(fact_120_add__cancel__left__left,axiom,
    ! [B: multiset_a,A: multiset_a] :
      ( ( ( plus_plus_multiset_a @ B @ A )
        = A )
      = ( B = zero_zero_multiset_a ) ) ).

% add_cancel_left_left
thf(fact_121_add_Oright__neutral,axiom,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
      = A ) ).

% add.right_neutral
thf(fact_122_zero__reorient,axiom,
    ! [X: multiset_a] :
      ( ( zero_zero_multiset_a = X )
      = ( X = zero_zero_multiset_a ) ) ).

% zero_reorient
thf(fact_123_add__right__imp__eq,axiom,
    ! [B: multiset_a,A: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ B @ A )
        = ( plus_plus_multiset_a @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_124_add__left__imp__eq,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( plus_plus_multiset_a @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_125_add_Oleft__commute,axiom,
    ! [B: multiset_a,A: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ B @ ( plus_plus_multiset_a @ A @ C ) )
      = ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ) ).

% add.left_commute
thf(fact_126_add_Ocommute,axiom,
    ( plus_plus_multiset_a
    = ( ^ [A5: multiset_a,B5: multiset_a] : ( plus_plus_multiset_a @ B5 @ A5 ) ) ) ).

% add.commute
thf(fact_127_add_Oassoc,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C )
      = ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ) ).

% add.assoc
thf(fact_128_group__cancel_Oadd2,axiom,
    ! [B4: multiset_a,K: multiset_a,B: multiset_a,A: multiset_a] :
      ( ( B4
        = ( plus_plus_multiset_a @ K @ B ) )
     => ( ( plus_plus_multiset_a @ A @ B4 )
        = ( plus_plus_multiset_a @ K @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_129_group__cancel_Oadd1,axiom,
    ! [A4: multiset_a,K: multiset_a,A: multiset_a,B: multiset_a] :
      ( ( A4
        = ( plus_plus_multiset_a @ K @ A ) )
     => ( ( plus_plus_multiset_a @ A4 @ B )
        = ( plus_plus_multiset_a @ K @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_130_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_multiset_a @ I @ K )
        = ( plus_plus_multiset_a @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_131_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C )
      = ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_132_multi__union__self__other__eq,axiom,
    ! [A4: multiset_a,X5: multiset_a,Y5: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A4 @ X5 )
        = ( plus_plus_multiset_a @ A4 @ Y5 ) )
     => ( X5 = Y5 ) ) ).

% multi_union_self_other_eq
thf(fact_133_union__right__cancel,axiom,
    ! [M: multiset_a,K2: multiset_a,N: multiset_a] :
      ( ( ( plus_plus_multiset_a @ M @ K2 )
        = ( plus_plus_multiset_a @ N @ K2 ) )
      = ( M = N ) ) ).

% union_right_cancel
thf(fact_134_union__left__cancel,axiom,
    ! [K2: multiset_a,M: multiset_a,N: multiset_a] :
      ( ( ( plus_plus_multiset_a @ K2 @ M )
        = ( plus_plus_multiset_a @ K2 @ N ) )
      = ( M = N ) ) ).

% union_left_cancel
thf(fact_135_union__commute,axiom,
    ( plus_plus_multiset_a
    = ( ^ [M2: multiset_a,N2: multiset_a] : ( plus_plus_multiset_a @ N2 @ M2 ) ) ) ).

% union_commute
thf(fact_136_union__lcomm,axiom,
    ! [M: multiset_a,N: multiset_a,K2: multiset_a] :
      ( ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N @ K2 ) )
      = ( plus_plus_multiset_a @ N @ ( plus_plus_multiset_a @ M @ K2 ) ) ) ).

% union_lcomm
thf(fact_137_union__assoc,axiom,
    ! [M: multiset_a,N: multiset_a,K2: multiset_a] :
      ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ M @ N ) @ K2 )
      = ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N @ K2 ) ) ) ).

% union_assoc
thf(fact_138_add__mset__commute,axiom,
    ! [X: a,Y: a,M: multiset_a] :
      ( ( add_mset_a @ X @ ( add_mset_a @ Y @ M ) )
      = ( add_mset_a @ Y @ ( add_mset_a @ X @ M ) ) ) ).

% add_mset_commute
thf(fact_139_add__eq__conv__ex,axiom,
    ! [A: a,M: multiset_a,B: a,N: multiset_a] :
      ( ( ( add_mset_a @ A @ M )
        = ( add_mset_a @ B @ N ) )
      = ( ( ( M = N )
          & ( A = B ) )
        | ? [K3: multiset_a] :
            ( ( M
              = ( add_mset_a @ B @ K3 ) )
            & ( N
              = ( add_mset_a @ A @ K3 ) ) ) ) ) ).

% add_eq_conv_ex
thf(fact_140_add_Ocomm__neutral,axiom,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
      = A ) ).

% add.comm_neutral
thf(fact_141_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_142_empty__neutral_I1_J,axiom,
    ! [X: multiset_a] :
      ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X )
      = X ) ).

% empty_neutral(1)
thf(fact_143_empty__neutral_I2_J,axiom,
    ! [X: multiset_a] :
      ( ( plus_plus_multiset_a @ X @ zero_zero_multiset_a )
      = X ) ).

% empty_neutral(2)
thf(fact_144_multi__nonempty__split,axiom,
    ! [M: multiset_a] :
      ( ( M != zero_zero_multiset_a )
     => ? [A6: multiset_a,A3: a] :
          ( M
          = ( add_mset_a @ A3 @ A6 ) ) ) ).

% multi_nonempty_split
thf(fact_145_empty__not__add__mset,axiom,
    ! [A: a,A4: multiset_a] :
      ( zero_zero_multiset_a
     != ( add_mset_a @ A @ A4 ) ) ).

% empty_not_add_mset
thf(fact_146_multiset__induct2,axiom,
    ! [P2: multiset_a > multiset_a > $o,M: multiset_a,N: multiset_a] :
      ( ( P2 @ zero_zero_multiset_a @ zero_zero_multiset_a )
     => ( ! [A3: a,M3: multiset_a,N3: multiset_a] :
            ( ( P2 @ M3 @ N3 )
           => ( P2 @ ( add_mset_a @ A3 @ M3 ) @ N3 ) )
       => ( ! [A3: a,M3: multiset_a,N3: multiset_a] :
              ( ( P2 @ M3 @ N3 )
             => ( P2 @ M3 @ ( add_mset_a @ A3 @ N3 ) ) )
         => ( P2 @ M @ N ) ) ) ) ).

% multiset_induct2
thf(fact_147_multiset__induct,axiom,
    ! [P2: multiset_a > $o,M: multiset_a] :
      ( ( P2 @ zero_zero_multiset_a )
     => ( ! [X3: a,M3: multiset_a] :
            ( ( P2 @ M3 )
           => ( P2 @ ( add_mset_a @ X3 @ M3 ) ) )
       => ( P2 @ M ) ) ) ).

% multiset_induct
thf(fact_148_multiset__cases,axiom,
    ! [M: multiset_a] :
      ( ( M != zero_zero_multiset_a )
     => ~ ! [X3: a,N3: multiset_a] :
            ( M
           != ( add_mset_a @ X3 @ N3 ) ) ) ).

% multiset_cases
thf(fact_149_add__mset__add__single,axiom,
    ( add_mset_a
    = ( ^ [A5: a,A7: multiset_a] : ( plus_plus_multiset_a @ A7 @ ( add_mset_a @ A5 @ zero_zero_multiset_a ) ) ) ) ).

% add_mset_add_single
thf(fact_150_union__is__single,axiom,
    ! [M: multiset_a,N: multiset_a,A: a] :
      ( ( ( plus_plus_multiset_a @ M @ N )
        = ( add_mset_a @ A @ zero_zero_multiset_a ) )
      = ( ( ( M
            = ( add_mset_a @ A @ zero_zero_multiset_a ) )
          & ( N = zero_zero_multiset_a ) )
        | ( ( M = zero_zero_multiset_a )
          & ( N
            = ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).

% union_is_single
thf(fact_151_single__is__union,axiom,
    ! [A: a,M: multiset_a,N: multiset_a] :
      ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
        = ( plus_plus_multiset_a @ M @ N ) )
      = ( ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
            = M )
          & ( N = zero_zero_multiset_a ) )
        | ( ( M = zero_zero_multiset_a )
          & ( ( add_mset_a @ A @ zero_zero_multiset_a )
            = N ) ) ) ) ).

% single_is_union
thf(fact_152_exI__realizer,axiom,
    ! [P2: multiset_a > multiset_a > $o,Y: multiset_a,X: multiset_a] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc148952133iset_a @ ( produc2037245207iset_a @ X @ Y ) ) @ ( produc653373187iset_a @ ( produc2037245207iset_a @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_153_exI__realizer,axiom,
    ! [P2: a > a > $o,Y: a,X: a] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_154_exI__realizer,axiom,
    ! [P2: a > tree_a > $o,Y: a,X: tree_a] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ X @ Y ) ) @ ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_155_exI__realizer,axiom,
    ! [P2: tree_a > a > $o,Y: tree_a,X: a] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ X @ Y ) ) @ ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_156_conjI__realizer,axiom,
    ! [P2: multiset_a > $o,P: multiset_a,Q2: multiset_a > $o,Q: multiset_a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc653373187iset_a @ ( produc2037245207iset_a @ P @ Q ) ) )
          & ( Q2 @ ( produc148952133iset_a @ ( produc2037245207iset_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_157_conjI__realizer,axiom,
    ! [P2: a > $o,P: a,Q2: a > $o,Q: a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_a_a @ ( product_Pair_a_a @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_a_a @ ( product_Pair_a_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_158_conjI__realizer,axiom,
    ! [P2: tree_a > $o,P: tree_a,Q2: a > $o,Q: a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_159_conjI__realizer,axiom,
    ! [P2: a > $o,P: a,Q2: tree_a > $o,Q: tree_a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_160_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: multiset_a > multiset_a > $o,X: multiset_a,Y: multiset_a,A: produc1127127335iset_a] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc2037245207iset_a @ X @ Y ) )
       => ( P2 @ ( produc653373187iset_a @ A ) @ ( produc148952133iset_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_161_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: a > a > $o,X: a,Y: a,A: product_prod_a_a] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( product_Pair_a_a @ X @ Y ) )
       => ( P2 @ ( product_fst_a_a @ A ) @ ( product_snd_a_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_162_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: tree_a > a > $o,X: tree_a,Y: a,A: produc981471411ee_a_a] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc649172771ee_a_a @ X @ Y ) )
       => ( P2 @ ( product_fst_Tree_a_a @ A ) @ ( product_snd_Tree_a_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_163_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: a > tree_a > $o,X: a,Y: tree_a,A: produc143150363Tree_a] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc686083979Tree_a @ X @ Y ) )
       => ( P2 @ ( product_fst_a_Tree_a @ A ) @ ( product_snd_a_Tree_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_164_Multiset_Ois__empty__def,axiom,
    ( is_empty_a
    = ( ^ [A7: multiset_a] : A7 = zero_zero_multiset_a ) ) ).

% Multiset.is_empty_def
thf(fact_165_exE__realizer_H,axiom,
    ! [P2: a > tree_a > $o,P: produc981471411ee_a_a] :
      ( ( P2 @ ( product_snd_Tree_a_a @ P ) @ ( product_fst_Tree_a_a @ P ) )
     => ~ ! [X3: tree_a,Y3: a] :
            ~ ( P2 @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_166_exE__realizer_H,axiom,
    ! [P2: tree_a > a > $o,P: produc143150363Tree_a] :
      ( ( P2 @ ( product_snd_a_Tree_a @ P ) @ ( product_fst_a_Tree_a @ P ) )
     => ~ ! [X3: a,Y3: tree_a] :
            ~ ( P2 @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_167_eq__snd__iff,axiom,
    ! [B: multiset_a,P: produc1127127335iset_a] :
      ( ( B
        = ( produc148952133iset_a @ P ) )
      = ( ? [A5: multiset_a] :
            ( P
            = ( produc2037245207iset_a @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_168_eq__snd__iff,axiom,
    ! [B: a,P: product_prod_a_a] :
      ( ( B
        = ( product_snd_a_a @ P ) )
      = ( ? [A5: a] :
            ( P
            = ( product_Pair_a_a @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_169_eq__snd__iff,axiom,
    ! [B: a,P: produc981471411ee_a_a] :
      ( ( B
        = ( product_snd_Tree_a_a @ P ) )
      = ( ? [A5: tree_a] :
            ( P
            = ( produc649172771ee_a_a @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_170_eq__snd__iff,axiom,
    ! [B: tree_a,P: produc143150363Tree_a] :
      ( ( B
        = ( product_snd_a_Tree_a @ P ) )
      = ( ? [A5: a] :
            ( P
            = ( produc686083979Tree_a @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_171_eq__fst__iff,axiom,
    ! [A: multiset_a,P: produc1127127335iset_a] :
      ( ( A
        = ( produc653373187iset_a @ P ) )
      = ( ? [B5: multiset_a] :
            ( P
            = ( produc2037245207iset_a @ A @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_172_eq__fst__iff,axiom,
    ! [A: a,P: product_prod_a_a] :
      ( ( A
        = ( product_fst_a_a @ P ) )
      = ( ? [B5: a] :
            ( P
            = ( product_Pair_a_a @ A @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_173_eq__fst__iff,axiom,
    ! [A: tree_a,P: produc981471411ee_a_a] :
      ( ( A
        = ( product_fst_Tree_a_a @ P ) )
      = ( ? [B5: a] :
            ( P
            = ( produc649172771ee_a_a @ A @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_174_eq__fst__iff,axiom,
    ! [A: a,P: produc143150363Tree_a] :
      ( ( A
        = ( product_fst_a_Tree_a @ P ) )
      = ( ? [B5: tree_a] :
            ( P
            = ( produc686083979Tree_a @ A @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_175_verit__sum__simplify,axiom,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
      = A ) ).

% verit_sum_simplify
thf(fact_176_HS_Ois__empty__as__list,axiom,
    ! [E: tree_a] :
      ( ( heapIm229596386mpty_a @ E )
     => ( ( multiset_a2 @ E )
        = zero_zero_multiset_a ) ) ).

% HS.is_empty_as_list
thf(fact_177_add__mset__replicate__mset__safe,axiom,
    ! [M: multiset_a,A: a] :
      ( ( nO_MAT1617603563iset_a @ zero_zero_multiset_a @ M )
     => ( ( add_mset_a @ A @ M )
        = ( plus_plus_multiset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M ) ) ) ).

% add_mset_replicate_mset_safe
thf(fact_178_HS_Ois__empty__empty,axiom,
    heapIm229596386mpty_a @ e_a ).

% HS.is_empty_empty
thf(fact_179_hs__is__empty__def,axiom,
    ( heapIm229596386mpty_a
    = ( ^ [T3: tree_a] : T3 = e_a ) ) ).

% hs_is_empty_def
thf(fact_180_HS_Ois__empty__inj,axiom,
    ! [E: tree_a] :
      ( ( heapIm229596386mpty_a @ E )
     => ( E = e_a ) ) ).

% HS.is_empty_inj
thf(fact_181_prod_Oswap__def,axiom,
    ( produc506430135iset_a
    = ( ^ [P3: produc1127127335iset_a] : ( produc2037245207iset_a @ ( produc148952133iset_a @ P3 ) @ ( produc653373187iset_a @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_182_prod_Oswap__def,axiom,
    ( product_swap_a_a
    = ( ^ [P3: product_prod_a_a] : ( product_Pair_a_a @ ( product_snd_a_a @ P3 ) @ ( product_fst_a_a @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_183_prod_Oswap__def,axiom,
    ( produc1478296771ee_a_a
    = ( ^ [P3: produc981471411ee_a_a] : ( produc686083979Tree_a @ ( product_snd_Tree_a_a @ P3 ) @ ( product_fst_Tree_a_a @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_184_prod_Oswap__def,axiom,
    ( produc1515207979Tree_a
    = ( ^ [P3: produc143150363Tree_a] : ( produc649172771ee_a_a @ ( product_snd_a_Tree_a @ P3 ) @ ( product_fst_a_Tree_a @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_185_removeLeaf__heap__is__heap,axiom,
    ! [T2: tree_a] :
      ( ( is_heap_a @ T2 )
     => ( ( T2 != e_a )
       => ( is_heap_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T2 ) ) ) ) ) ).

% removeLeaf_heap_is_heap
thf(fact_186_NO__MATCH__cong,axiom,
    nO_MAT1617603563iset_a = nO_MAT1617603563iset_a ).

% NO_MATCH_cong
thf(fact_187_NO__MATCH__def,axiom,
    ( nO_MAT1617603563iset_a
    = ( ^ [Pat: multiset_a,Val: multiset_a] : $true ) ) ).

% NO_MATCH_def
thf(fact_188_swap__swap,axiom,
    ! [P: produc981471411ee_a_a] :
      ( ( produc1515207979Tree_a @ ( produc1478296771ee_a_a @ P ) )
      = P ) ).

% swap_swap
thf(fact_189_swap__swap,axiom,
    ! [P: produc143150363Tree_a] :
      ( ( produc1478296771ee_a_a @ ( produc1515207979Tree_a @ P ) )
      = P ) ).

% swap_swap
thf(fact_190_swap__simp,axiom,
    ! [X: multiset_a,Y: multiset_a] :
      ( ( produc506430135iset_a @ ( produc2037245207iset_a @ X @ Y ) )
      = ( produc2037245207iset_a @ Y @ X ) ) ).

% swap_simp
thf(fact_191_swap__simp,axiom,
    ! [X: a,Y: a] :
      ( ( product_swap_a_a @ ( product_Pair_a_a @ X @ Y ) )
      = ( product_Pair_a_a @ Y @ X ) ) ).

% swap_simp
thf(fact_192_swap__simp,axiom,
    ! [X: tree_a,Y: a] :
      ( ( produc1478296771ee_a_a @ ( produc649172771ee_a_a @ X @ Y ) )
      = ( produc686083979Tree_a @ Y @ X ) ) ).

% swap_simp
thf(fact_193_swap__simp,axiom,
    ! [X: a,Y: tree_a] :
      ( ( produc1515207979Tree_a @ ( produc686083979Tree_a @ X @ Y ) )
      = ( produc649172771ee_a_a @ Y @ X ) ) ).

% swap_simp
thf(fact_194_snd__swap,axiom,
    ! [X: produc143150363Tree_a] :
      ( ( product_snd_Tree_a_a @ ( produc1515207979Tree_a @ X ) )
      = ( product_fst_a_Tree_a @ X ) ) ).

% snd_swap
thf(fact_195_snd__swap,axiom,
    ! [X: produc981471411ee_a_a] :
      ( ( product_snd_a_Tree_a @ ( produc1478296771ee_a_a @ X ) )
      = ( product_fst_Tree_a_a @ X ) ) ).

% snd_swap
thf(fact_196_fst__swap,axiom,
    ! [X: produc981471411ee_a_a] :
      ( ( product_fst_a_Tree_a @ ( produc1478296771ee_a_a @ X ) )
      = ( product_snd_Tree_a_a @ X ) ) ).

% fst_swap
thf(fact_197_fst__swap,axiom,
    ! [X: produc143150363Tree_a] :
      ( ( product_fst_Tree_a_a @ ( produc1515207979Tree_a @ X ) )
      = ( product_snd_a_Tree_a @ X ) ) ).

% fst_swap
thf(fact_198_is__heap_Osimps_I1_J,axiom,
    is_heap_a @ e_a ).

% is_heap.simps(1)
thf(fact_199_is__heap_Osimps_I2_J,axiom,
    ! [V: a] : ( is_heap_a @ ( t_a @ V @ e_a @ e_a ) ) ).

% is_heap.simps(2)
thf(fact_200_removeLeaf__val__val,axiom,
    ! [T2: tree_a] :
      ( ( ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T2 ) )
       != e_a )
     => ( ( T2 != e_a )
       => ( ( val_a @ T2 )
          = ( val_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T2 ) ) ) ) ) ) ).

% removeLeaf_val_val
thf(fact_201_Heap__axioms__def,axiom,
    ( heap_axioms_a_a
    = ( ^ [Is_empty: a > $o,Of_list: list_a > a,Multiset: a > multiset_a,As_tree: a > tree_a,Remove_max: a > product_prod_a_a] :
          ( ! [L2: a] :
              ( ( Multiset @ L2 )
              = ( multiset_a2 @ ( As_tree @ L2 ) ) )
          & ! [I2: list_a] : ( is_heap_a @ ( As_tree @ ( Of_list @ I2 ) ) )
          & ! [T3: a] :
              ( ( ( As_tree @ T3 )
                = e_a )
              = ( Is_empty @ T3 ) )
          & ! [L2: a,M4: a,L3: a] :
              ( ~ ( Is_empty @ L2 )
             => ( ( ( product_Pair_a_a @ M4 @ L3 )
                  = ( Remove_max @ L2 ) )
               => ( ( add_mset_a @ M4 @ ( Multiset @ L3 ) )
                  = ( Multiset @ L2 ) ) ) )
          & ! [L2: a,M4: a,L3: a] :
              ( ~ ( Is_empty @ L2 )
             => ( ( is_heap_a @ ( As_tree @ L2 ) )
               => ( ( ( product_Pair_a_a @ M4 @ L3 )
                    = ( Remove_max @ L2 ) )
                 => ( is_heap_a @ ( As_tree @ L3 ) ) ) ) )
          & ! [T3: a,M4: a,T4: a] :
              ( ~ ( Is_empty @ T3 )
             => ( ( ( product_Pair_a_a @ M4 @ T4 )
                  = ( Remove_max @ T3 ) )
               => ( M4
                  = ( val_a @ ( As_tree @ T3 ) ) ) ) ) ) ) ) ).

% Heap_axioms_def
thf(fact_202_Heap__axioms__def,axiom,
    ( heap_axioms_Tree_a_a
    = ( ^ [Is_empty: tree_a > $o,Of_list: list_a > tree_a,Multiset: tree_a > multiset_a,As_tree: tree_a > tree_a,Remove_max: tree_a > produc143150363Tree_a] :
          ( ! [L2: tree_a] :
              ( ( Multiset @ L2 )
              = ( multiset_a2 @ ( As_tree @ L2 ) ) )
          & ! [I2: list_a] : ( is_heap_a @ ( As_tree @ ( Of_list @ I2 ) ) )
          & ! [T3: tree_a] :
              ( ( ( As_tree @ T3 )
                = e_a )
              = ( Is_empty @ T3 ) )
          & ! [L2: tree_a,M4: a,L3: tree_a] :
              ( ~ ( Is_empty @ L2 )
             => ( ( ( produc686083979Tree_a @ M4 @ L3 )
                  = ( Remove_max @ L2 ) )
               => ( ( add_mset_a @ M4 @ ( Multiset @ L3 ) )
                  = ( Multiset @ L2 ) ) ) )
          & ! [L2: tree_a,M4: a,L3: tree_a] :
              ( ~ ( Is_empty @ L2 )
             => ( ( is_heap_a @ ( As_tree @ L2 ) )
               => ( ( ( produc686083979Tree_a @ M4 @ L3 )
                    = ( Remove_max @ L2 ) )
                 => ( is_heap_a @ ( As_tree @ L3 ) ) ) ) )
          & ! [T3: tree_a,M4: a,T4: tree_a] :
              ( ~ ( Is_empty @ T3 )
             => ( ( ( produc686083979Tree_a @ M4 @ T4 )
                  = ( Remove_max @ T3 ) )
               => ( M4
                  = ( val_a @ ( As_tree @ T3 ) ) ) ) ) ) ) ) ).

% Heap_axioms_def
thf(fact_203_Heap__axioms_Ointro,axiom,
    ! [Multiset2: a > multiset_a,As_tree2: a > tree_a,Of_list2: list_a > a,Is_empty2: a > $o,Remove_max2: a > product_prod_a_a] :
      ( ! [L4: a] :
          ( ( Multiset2 @ L4 )
          = ( multiset_a2 @ ( As_tree2 @ L4 ) ) )
     => ( ! [I3: list_a] : ( is_heap_a @ ( As_tree2 @ ( Of_list2 @ I3 ) ) )
       => ( ! [T5: a] :
              ( ( ( As_tree2 @ T5 )
                = e_a )
              = ( Is_empty2 @ T5 ) )
         => ( ! [L4: a,M5: a,L5: a] :
                ( ~ ( Is_empty2 @ L4 )
               => ( ( ( product_Pair_a_a @ M5 @ L5 )
                    = ( Remove_max2 @ L4 ) )
                 => ( ( add_mset_a @ M5 @ ( Multiset2 @ L5 ) )
                    = ( Multiset2 @ L4 ) ) ) )
           => ( ! [L4: a,M5: a,L5: a] :
                  ( ~ ( Is_empty2 @ L4 )
                 => ( ( is_heap_a @ ( As_tree2 @ L4 ) )
                   => ( ( ( product_Pair_a_a @ M5 @ L5 )
                        = ( Remove_max2 @ L4 ) )
                     => ( is_heap_a @ ( As_tree2 @ L5 ) ) ) ) )
             => ( ! [T5: a,M5: a,T6: a] :
                    ( ~ ( Is_empty2 @ T5 )
                   => ( ( ( product_Pair_a_a @ M5 @ T6 )
                        = ( Remove_max2 @ T5 ) )
                     => ( M5
                        = ( val_a @ ( As_tree2 @ T5 ) ) ) ) )
               => ( heap_axioms_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ) ) ) ) ).

% Heap_axioms.intro
thf(fact_204_Heap__axioms_Ointro,axiom,
    ! [Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Of_list2: list_a > tree_a,Is_empty2: tree_a > $o,Remove_max2: tree_a > produc143150363Tree_a] :
      ( ! [L4: tree_a] :
          ( ( Multiset2 @ L4 )
          = ( multiset_a2 @ ( As_tree2 @ L4 ) ) )
     => ( ! [I3: list_a] : ( is_heap_a @ ( As_tree2 @ ( Of_list2 @ I3 ) ) )
       => ( ! [T5: tree_a] :
              ( ( ( As_tree2 @ T5 )
                = e_a )
              = ( Is_empty2 @ T5 ) )
         => ( ! [L4: tree_a,M5: a,L5: tree_a] :
                ( ~ ( Is_empty2 @ L4 )
               => ( ( ( produc686083979Tree_a @ M5 @ L5 )
                    = ( Remove_max2 @ L4 ) )
                 => ( ( add_mset_a @ M5 @ ( Multiset2 @ L5 ) )
                    = ( Multiset2 @ L4 ) ) ) )
           => ( ! [L4: tree_a,M5: a,L5: tree_a] :
                  ( ~ ( Is_empty2 @ L4 )
                 => ( ( is_heap_a @ ( As_tree2 @ L4 ) )
                   => ( ( ( produc686083979Tree_a @ M5 @ L5 )
                        = ( Remove_max2 @ L4 ) )
                     => ( is_heap_a @ ( As_tree2 @ L5 ) ) ) ) )
             => ( ! [T5: tree_a,M5: a,T6: tree_a] :
                    ( ~ ( Is_empty2 @ T5 )
                   => ( ( ( produc686083979Tree_a @ M5 @ T6 )
                        = ( Remove_max2 @ T5 ) )
                     => ( M5
                        = ( val_a @ ( As_tree2 @ T5 ) ) ) ) )
               => ( heap_axioms_Tree_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ) ) ) ) ).

% Heap_axioms.intro
thf(fact_205_val_Osimps,axiom,
    ! [V: a,Uu: tree_a,Uv: tree_a] :
      ( ( val_a @ ( t_a @ V @ Uu @ Uv ) )
      = V ) ).

% val.simps
thf(fact_206_is__heap_Osimps_I4_J,axiom,
    ! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
      ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
      = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
        & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).

% is_heap.simps(4)
thf(fact_207_is__heap_Osimps_I3_J,axiom,
    ! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
      ( ( is_heap_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
      = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
        & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).

% is_heap.simps(3)
thf(fact_208_Heap_Oremove__max__is__heap,axiom,
    ! [Empty: a,Is_empty2: a > $o,Of_list2: list_a > a,Multiset2: a > multiset_a,As_tree2: a > tree_a,Remove_max2: a > product_prod_a_a,L: a,M6: a,L6: a] :
      ( ( heap_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
     => ( ~ ( Is_empty2 @ L )
       => ( ( is_heap_a @ ( As_tree2 @ L ) )
         => ( ( ( product_Pair_a_a @ M6 @ L6 )
              = ( Remove_max2 @ L ) )
           => ( is_heap_a @ ( As_tree2 @ L6 ) ) ) ) ) ) ).

% Heap.remove_max_is_heap
thf(fact_209_Heap_Oremove__max__is__heap,axiom,
    ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,L: tree_a,M6: a,L6: tree_a] :
      ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
     => ( ~ ( Is_empty2 @ L )
       => ( ( is_heap_a @ ( As_tree2 @ L ) )
         => ( ( ( produc686083979Tree_a @ M6 @ L6 )
              = ( Remove_max2 @ L ) )
           => ( is_heap_a @ ( As_tree2 @ L6 ) ) ) ) ) ) ).

% Heap.remove_max_is_heap
thf(fact_210_verit__la__disequality,axiom,
    ! [A: a,B: a] :
      ( ( A = B )
      | ~ ( ord_less_eq_a @ A @ B )
      | ~ ( ord_less_eq_a @ B @ A ) ) ).

% verit_la_disequality
thf(fact_211_add__right__mono,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ord_le1199012836iset_a @ A @ B )
     => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).

% add_right_mono
thf(fact_212_add__left__mono,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ord_le1199012836iset_a @ A @ B )
     => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).

% add_left_mono
thf(fact_213_add__mono,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
      ( ( ord_le1199012836iset_a @ A @ B )
     => ( ( ord_le1199012836iset_a @ C @ D )
       => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).

% add_mono
thf(fact_214_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
      ( ( ( ord_le1199012836iset_a @ I @ J )
        & ( ord_le1199012836iset_a @ K @ L ) )
     => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_215_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
      ( ( ( I = J )
        & ( ord_le1199012836iset_a @ K @ L ) )
     => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_216_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
      ( ( ( ord_le1199012836iset_a @ I @ J )
        & ( K = L ) )
     => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_217_Heap_Oas__tree__empty,axiom,
    ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,T2: tree_a] :
      ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
     => ( ( ( As_tree2 @ T2 )
          = e_a )
        = ( Is_empty2 @ T2 ) ) ) ).

% Heap.as_tree_empty
thf(fact_218_Heap_Ois__heap__of__list,axiom,
    ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,I: list_a] :
      ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
     => ( is_heap_a @ ( As_tree2 @ ( Of_list2 @ I ) ) ) ) ).

% Heap.is_heap_of_list
thf(fact_219_Heap_Omultiset,axiom,
    ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,L: tree_a] :
      ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
     => ( ( Multiset2 @ L )
        = ( multiset_a2 @ ( As_tree2 @ L ) ) ) ) ).

% Heap.multiset
thf(fact_220_Heap_Oaxioms_I2_J,axiom,
    ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a] :
      ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
     => ( heap_axioms_Tree_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ).

% Heap.axioms(2)
thf(fact_221_is__heap_Osimps_I6_J,axiom,
    ! [V: a,Vd: a,Ve: tree_a,Vf: tree_a,Va: a,Vb: tree_a,Vc: tree_a] :
      ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
      = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
        & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) )
        & ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
        & ( is_heap_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ).

% is_heap.simps(6)
thf(fact_222_is__heap_Osimps_I5_J,axiom,
    ! [V: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
      ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
      = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
        & ( is_heap_a @ ( t_a @ Vd @ Ve @ Vf ) )
        & ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
        & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).

% is_heap.simps(5)
thf(fact_223_Heap_Oremove__max__multiset_H,axiom,
    ! [Empty: a,Is_empty2: a > $o,Of_list2: list_a > a,Multiset2: a > multiset_a,As_tree2: a > tree_a,Remove_max2: a > product_prod_a_a,L: a,M6: a,L6: a] :
      ( ( heap_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
     => ( ~ ( Is_empty2 @ L )
       => ( ( ( product_Pair_a_a @ M6 @ L6 )
            = ( Remove_max2 @ L ) )
         => ( ( add_mset_a @ M6 @ ( Multiset2 @ L6 ) )
            = ( Multiset2 @ L ) ) ) ) ) ).

% Heap.remove_max_multiset'
thf(fact_224_Heap_Oremove__max__multiset_H,axiom,
    ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,L: tree_a,M6: a,L6: tree_a] :
      ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
     => ( ~ ( Is_empty2 @ L )
       => ( ( ( produc686083979Tree_a @ M6 @ L6 )
            = ( Remove_max2 @ L ) )
         => ( ( add_mset_a @ M6 @ ( Multiset2 @ L6 ) )
            = ( Multiset2 @ L ) ) ) ) ) ).

% Heap.remove_max_multiset'
thf(fact_225_Heap_Oremove__max__val,axiom,
    ! [Empty: a,Is_empty2: a > $o,Of_list2: list_a > a,Multiset2: a > multiset_a,As_tree2: a > tree_a,Remove_max2: a > product_prod_a_a,T2: a,M6: a,T: a] :
      ( ( heap_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
     => ( ~ ( Is_empty2 @ T2 )
       => ( ( ( product_Pair_a_a @ M6 @ T )
            = ( Remove_max2 @ T2 ) )
         => ( M6
            = ( val_a @ ( As_tree2 @ T2 ) ) ) ) ) ) ).

% Heap.remove_max_val
thf(fact_226_Heap_Oremove__max__val,axiom,
    ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,T2: tree_a,M6: a,T: tree_a] :
      ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
     => ( ~ ( Is_empty2 @ T2 )
       => ( ( ( produc686083979Tree_a @ M6 @ T )
            = ( Remove_max2 @ T2 ) )
         => ( M6
            = ( val_a @ ( As_tree2 @ T2 ) ) ) ) ) ) ).

% Heap.remove_max_val
thf(fact_227_siftDown_Osimps_I4_J,axiom,
    ! [Va: a,Vb: tree_a,Vc: tree_a,V: a] :
      ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
       => ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
          = ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) ) )
      & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
       => ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
          = ( t_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ e_a @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ) ) ) ) ).

% siftDown.simps(4)
thf(fact_228_siftDown_Osimps_I3_J,axiom,
    ! [Va: a,Vb: tree_a,Vc: tree_a,V: a] :
      ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
       => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
          = ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) ) )
      & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
       => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
          = ( t_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) @ e_a ) ) ) ) ).

% siftDown.simps(3)
thf(fact_229_siftDown_Osimps_I5_J,axiom,
    ! [Vd: a,Ve: tree_a,Vf: tree_a,Va: a,Vb: tree_a,Vc: tree_a,V: a] :
      ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) )
       => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
           => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
              = ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) )
          & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
           => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
              = ( t_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ) )
      & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) )
       => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
           => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
              = ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) )
          & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
           => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
              = ( t_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( t_a @ Va @ Vb @ Vc ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ) ) ) ).

% siftDown.simps(5)
thf(fact_230_siftDown_Osimps_I1_J,axiom,
    ( ( heapIm1091024090Down_a @ e_a )
    = e_a ) ).

% siftDown.simps(1)
thf(fact_231_siftDown__multiset,axiom,
    ! [T2: tree_a] :
      ( ( multiset_a2 @ ( heapIm1091024090Down_a @ T2 ) )
      = ( multiset_a2 @ T2 ) ) ).

% siftDown_multiset
thf(fact_232_siftDown__Node,axiom,
    ! [T2: tree_a,V: a,L: tree_a,R: tree_a] :
      ( ( T2
        = ( t_a @ V @ L @ R ) )
     => ? [L5: tree_a,V4: a,R2: tree_a] :
          ( ( ( heapIm1091024090Down_a @ T2 )
            = ( t_a @ V4 @ L5 @ R2 ) )
          & ( ord_less_eq_a @ V @ V4 ) ) ) ).

% siftDown_Node
thf(fact_233_siftDown_Osimps_I2_J,axiom,
    ! [V: a] :
      ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ e_a ) )
      = ( t_a @ V @ e_a @ e_a ) ) ).

% siftDown.simps(2)
thf(fact_234_siftDown__heap__is__heap,axiom,
    ! [L: tree_a,R: tree_a,T2: tree_a,V: a] :
      ( ( is_heap_a @ L )
     => ( ( is_heap_a @ R )
       => ( ( T2
            = ( t_a @ V @ L @ R ) )
         => ( is_heap_a @ ( heapIm1091024090Down_a @ T2 ) ) ) ) ) ).

% siftDown_heap_is_heap
thf(fact_235_siftDown_Osimps_I6_J,axiom,
    ! [Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a,V: a] :
      ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) )
       => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
           => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
              = ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) ) )
          & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
           => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
              = ( t_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ) )
      & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) )
       => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
           => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
              = ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) ) )
          & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
           => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
              = ( t_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( t_a @ Vd @ Ve @ Vf ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ) ) ) ) ) ) ).

% siftDown.simps(6)
thf(fact_236_siftDown__in__tree,axiom,
    ! [T2: tree_a] :
      ( ( T2 != e_a )
     => ( in_tree_a @ ( val_a @ ( heapIm1091024090Down_a @ T2 ) ) @ T2 ) ) ).

% siftDown_in_tree
thf(fact_237_is__heap__max,axiom,
    ! [V: a,T2: tree_a] :
      ( ( in_tree_a @ V @ T2 )
     => ( ( is_heap_a @ T2 )
       => ( ord_less_eq_a @ V @ ( val_a @ T2 ) ) ) ) ).

% is_heap_max
thf(fact_238_in__tree_Osimps_I2_J,axiom,
    ! [V: a,V2: a,L: tree_a,R: tree_a] :
      ( ( in_tree_a @ V @ ( t_a @ V2 @ L @ R ) )
      = ( ( V = V2 )
        | ( in_tree_a @ V @ L )
        | ( in_tree_a @ V @ R ) ) ) ).

% in_tree.simps(2)
thf(fact_239_in__tree_Osimps_I1_J,axiom,
    ! [V: a] :
      ~ ( in_tree_a @ V @ e_a ) ).

% in_tree.simps(1)
thf(fact_240_siftDown__in__tree__set,axiom,
    ( in_tree_a
    = ( ^ [V5: a,T3: tree_a] : ( in_tree_a @ V5 @ ( heapIm1091024090Down_a @ T3 ) ) ) ) ).

% siftDown_in_tree_set
thf(fact_241_hs__remove__max__def,axiom,
    ( heapIm915766629_max_a
    = ( ^ [T3: tree_a] :
          ( if_Pro1144176865Tree_a
          @ ( ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T3 ) )
            = e_a )
          @ ( produc686083979Tree_a @ ( val_a @ T3 ) @ e_a )
          @ ( produc686083979Tree_a @ ( val_a @ T3 ) @ ( heapIm1091024090Down_a @ ( t_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ T3 ) ) @ ( heapIm1140443833left_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T3 ) ) ) @ ( heapIm1257206334ight_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T3 ) ) ) ) ) ) ) ) ) ).

% hs_remove_max_def
thf(fact_242_heapify_Osimps_I2_J,axiom,
    ! [V: a,L: tree_a,R: tree_a] :
      ( ( heapIm970322378pify_a @ ( t_a @ V @ L @ R ) )
      = ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm970322378pify_a @ L ) @ ( heapIm970322378pify_a @ R ) ) ) ) ).

% heapify.simps(2)
thf(fact_243_heapify_Osimps_I1_J,axiom,
    ( ( heapIm970322378pify_a @ e_a )
    = e_a ) ).

% heapify.simps(1)
thf(fact_244_heapify__heap__is__heap,axiom,
    ! [T2: tree_a] : ( is_heap_a @ ( heapIm970322378pify_a @ T2 ) ) ).

% heapify_heap_is_heap
thf(fact_245_multiset__heapify,axiom,
    ! [T2: tree_a] :
      ( ( multiset_a2 @ ( heapIm970322378pify_a @ T2 ) )
      = ( multiset_a2 @ T2 ) ) ).

% multiset_heapify
thf(fact_246_Heap__def,axiom,
    ( heap_Tree_a_a
    = ( ^ [Empty2: tree_a,Is_empty: tree_a > $o,Of_list: list_a > tree_a,Multiset: tree_a > multiset_a,As_tree: tree_a > tree_a,Remove_max: tree_a > produc143150363Tree_a] :
          ( ( collection_Tree_a_a @ Empty2 @ Is_empty @ Of_list @ Multiset )
          & ( heap_axioms_Tree_a_a @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ) ).

% Heap_def
thf(fact_247_Heap_Ointro,axiom,
    ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a] :
      ( ( collection_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 )
     => ( ( heap_axioms_Tree_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
       => ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ).

% Heap.intro
thf(fact_248_Heap_Oaxioms_I1_J,axiom,
    ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a] :
      ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
     => ( collection_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 ) ) ).

% Heap.axioms(1)
thf(fact_249_HS_OCollection__axioms,axiom,
    collection_Tree_a_a @ e_a @ heapIm229596386mpty_a @ heapIm1057938560list_a @ multiset_a2 ).

% HS.Collection_axioms
thf(fact_250_Collection_Omultiset__empty,axiom,
    ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a] :
      ( ( collection_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 )
     => ( ( Multiset2 @ Empty )
        = zero_zero_multiset_a ) ) ).

% Collection.multiset_empty
thf(fact_251_Collection_Ois__empty__as__list,axiom,
    ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,E: tree_a] :
      ( ( collection_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 )
     => ( ( Is_empty2 @ E )
       => ( ( Multiset2 @ E )
          = zero_zero_multiset_a ) ) ) ).

% Collection.is_empty_as_list
thf(fact_252_hs__of__list__def,axiom,
    ( heapIm1057938560list_a
    = ( ^ [L2: list_a] : ( heapIm970322378pify_a @ ( heapIm1637418125tree_a @ L2 ) ) ) ) ).

% hs_of_list_def
thf(fact_253_pred__equals__eq2,axiom,
    ! [R3: set_Pr921883667ee_a_a,S2: set_Pr921883667ee_a_a] :
      ( ( ( ^ [X4: tree_a,Y6: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X4 @ Y6 ) @ R3 ) )
        = ( ^ [X4: tree_a,Y6: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X4 @ Y6 ) @ S2 ) ) )
      = ( R3 = S2 ) ) ).

% pred_equals_eq2
thf(fact_254_pred__equals__eq2,axiom,
    ! [R3: set_Pr158363655iset_a,S2: set_Pr158363655iset_a] :
      ( ( ( ^ [X4: multiset_a,Y6: multiset_a] : ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y6 ) @ R3 ) )
        = ( ^ [X4: multiset_a,Y6: multiset_a] : ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y6 ) @ S2 ) ) )
      = ( R3 = S2 ) ) ).

% pred_equals_eq2
thf(fact_255_pred__equals__eq2,axiom,
    ! [R3: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
      ( ( ( ^ [X4: a,Y6: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ R3 ) )
        = ( ^ [X4: a,Y6: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ S2 ) ) )
      = ( R3 = S2 ) ) ).

% pred_equals_eq2
thf(fact_256_pred__equals__eq2,axiom,
    ! [R3: set_Pr1070816379Tree_a,S2: set_Pr1070816379Tree_a] :
      ( ( ( ^ [X4: a,Y6: tree_a] : ( member254315204Tree_a @ ( produc686083979Tree_a @ X4 @ Y6 ) @ R3 ) )
        = ( ^ [X4: a,Y6: tree_a] : ( member254315204Tree_a @ ( produc686083979Tree_a @ X4 @ Y6 ) @ S2 ) ) )
      = ( R3 = S2 ) ) ).

% pred_equals_eq2
thf(fact_257_subrelI,axiom,
    ! [R: set_Pr921883667ee_a_a,S3: set_Pr921883667ee_a_a] :
      ( ! [X3: tree_a,Y3: a] :
          ( ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X3 @ Y3 ) @ R )
         => ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le864074675ee_a_a @ R @ S3 ) ) ).

% subrelI
thf(fact_258_subrelI,axiom,
    ! [R: set_Pr158363655iset_a,S3: set_Pr158363655iset_a] :
      ( ! [X3: multiset_a,Y3: multiset_a] :
          ( ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ R )
         => ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le1978107815iset_a @ R @ S3 ) ) ).

% subrelI
thf(fact_259_subrelI,axiom,
    ! [R: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
      ( ! [X3: a,Y3: a] :
          ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
         => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le1824328871od_a_a @ R @ S3 ) ) ).

% subrelI
thf(fact_260_subrelI,axiom,
    ! [R: set_Pr1070816379Tree_a,S3: set_Pr1070816379Tree_a] :
      ( ! [X3: a,Y3: tree_a] :
          ( ( member254315204Tree_a @ ( produc686083979Tree_a @ X3 @ Y3 ) @ R )
         => ( member254315204Tree_a @ ( produc686083979Tree_a @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le1013007387Tree_a @ R @ S3 ) ) ).

% subrelI
thf(fact_261_pred__subset__eq2,axiom,
    ! [R3: set_Pr921883667ee_a_a,S2: set_Pr921883667ee_a_a] :
      ( ( ord_le1748873858_a_a_o
        @ ^ [X4: tree_a,Y6: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X4 @ Y6 ) @ R3 )
        @ ^ [X4: tree_a,Y6: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X4 @ Y6 ) @ S2 ) )
      = ( ord_le864074675ee_a_a @ R3 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_262_pred__subset__eq2,axiom,
    ! [R3: set_Pr158363655iset_a,S2: set_Pr158363655iset_a] :
      ( ( ord_le51495822et_a_o
        @ ^ [X4: multiset_a,Y6: multiset_a] : ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y6 ) @ R3 )
        @ ^ [X4: multiset_a,Y6: multiset_a] : ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y6 ) @ S2 ) )
      = ( ord_le1978107815iset_a @ R3 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_263_pred__subset__eq2,axiom,
    ! [R3: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
      ( ( ord_less_eq_a_a_o
        @ ^ [X4: a,Y6: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ R3 )
        @ ^ [X4: a,Y6: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ S2 ) )
      = ( ord_le1824328871od_a_a @ R3 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_264_pred__subset__eq2,axiom,
    ! [R3: set_Pr1070816379Tree_a,S2: set_Pr1070816379Tree_a] :
      ( ( ord_le1327026842ee_a_o
        @ ^ [X4: a,Y6: tree_a] : ( member254315204Tree_a @ ( produc686083979Tree_a @ X4 @ Y6 ) @ R3 )
        @ ^ [X4: a,Y6: tree_a] : ( member254315204Tree_a @ ( produc686083979Tree_a @ X4 @ Y6 ) @ S2 ) )
      = ( ord_le1013007387Tree_a @ R3 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_265_heap__top__geq,axiom,
    ! [A: a,T2: tree_a] :
      ( ( member_a @ A @ ( set_mset_a @ ( multiset_a2 @ T2 ) ) )
     => ( ( is_heap_a @ T2 )
       => ( ord_less_eq_a @ A @ ( val_a @ T2 ) ) ) ) ).

% heap_top_geq
thf(fact_266_multi__member__last,axiom,
    ! [X: produc143150363Tree_a] : ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ ( add_ms205941497Tree_a @ X @ zero_z2144236696Tree_a ) ) ) ).

% multi_member_last
thf(fact_267_multi__member__last,axiom,
    ! [X: produc1127127335iset_a] : ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ ( add_ms1353365509iset_a @ X @ zero_z1941791140iset_a ) ) ) ).

% multi_member_last
thf(fact_268_multi__member__last,axiom,
    ! [X: product_prod_a_a] : ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) ) ) ).

% multi_member_last
thf(fact_269_multi__member__last,axiom,
    ! [X: multiset_a] : ( member_multiset_a @ X @ ( set_mset_multiset_a @ ( add_mset_multiset_a @ X @ zero_z1580389697iset_a ) ) ) ).

% multi_member_last
thf(fact_270_multi__member__last,axiom,
    ! [X: a] : ( member_a @ X @ ( set_mset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ).

% multi_member_last
thf(fact_271_union__single__eq__member,axiom,
    ! [X: produc143150363Tree_a,M: multis2082063201Tree_a,N: multis2082063201Tree_a] :
      ( ( ( add_ms205941497Tree_a @ X @ M )
        = N )
     => ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ N ) ) ) ).

% union_single_eq_member
thf(fact_272_union__single__eq__member,axiom,
    ! [X: produc1127127335iset_a,M: multis782275565iset_a,N: multis782275565iset_a] :
      ( ( ( add_ms1353365509iset_a @ X @ M )
        = N )
     => ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ N ) ) ) ).

% union_single_eq_member
thf(fact_273_union__single__eq__member,axiom,
    ! [X: product_prod_a_a,M: multis599418605od_a_a,N: multis599418605od_a_a] :
      ( ( ( add_ms1725977093od_a_a @ X @ M )
        = N )
     => ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ N ) ) ) ).

% union_single_eq_member
thf(fact_274_union__single__eq__member,axiom,
    ! [X: multiset_a,M: multiset_multiset_a,N: multiset_multiset_a] :
      ( ( ( add_mset_multiset_a @ X @ M )
        = N )
     => ( member_multiset_a @ X @ ( set_mset_multiset_a @ N ) ) ) ).

% union_single_eq_member
thf(fact_275_union__single__eq__member,axiom,
    ! [X: a,M: multiset_a,N: multiset_a] :
      ( ( ( add_mset_a @ X @ M )
        = N )
     => ( member_a @ X @ ( set_mset_a @ N ) ) ) ).

% union_single_eq_member
thf(fact_276_insert__noteq__member,axiom,
    ! [B: produc143150363Tree_a,B4: multis2082063201Tree_a,C: produc143150363Tree_a,C2: multis2082063201Tree_a] :
      ( ( ( add_ms205941497Tree_a @ B @ B4 )
        = ( add_ms205941497Tree_a @ C @ C2 ) )
     => ( ( B != C )
       => ( member254315204Tree_a @ C @ ( set_ms1129970904Tree_a @ B4 ) ) ) ) ).

% insert_noteq_member
thf(fact_277_insert__noteq__member,axiom,
    ! [B: produc1127127335iset_a,B4: multis782275565iset_a,C: produc1127127335iset_a,C2: multis782275565iset_a] :
      ( ( ( add_ms1353365509iset_a @ B @ B4 )
        = ( add_ms1353365509iset_a @ C @ C2 ) )
     => ( ( B != C )
       => ( member340150864iset_a @ C @ ( set_ms1768116836iset_a @ B4 ) ) ) ) ).

% insert_noteq_member
thf(fact_278_insert__noteq__member,axiom,
    ! [B: product_prod_a_a,B4: multis599418605od_a_a,C: product_prod_a_a,C2: multis599418605od_a_a] :
      ( ( ( add_ms1725977093od_a_a @ B @ B4 )
        = ( add_ms1725977093od_a_a @ C @ C2 ) )
     => ( ( B != C )
       => ( member449909584od_a_a @ C @ ( set_ms1818337124od_a_a @ B4 ) ) ) ) ).

% insert_noteq_member
thf(fact_279_insert__noteq__member,axiom,
    ! [B: multiset_a,B4: multiset_multiset_a,C: multiset_a,C2: multiset_multiset_a] :
      ( ( ( add_mset_multiset_a @ B @ B4 )
        = ( add_mset_multiset_a @ C @ C2 ) )
     => ( ( B != C )
       => ( member_multiset_a @ C @ ( set_mset_multiset_a @ B4 ) ) ) ) ).

% insert_noteq_member
thf(fact_280_insert__noteq__member,axiom,
    ! [B: a,B4: multiset_a,C: a,C2: multiset_a] :
      ( ( ( add_mset_a @ B @ B4 )
        = ( add_mset_a @ C @ C2 ) )
     => ( ( B != C )
       => ( member_a @ C @ ( set_mset_a @ B4 ) ) ) ) ).

% insert_noteq_member
thf(fact_281_multi__member__split,axiom,
    ! [X: produc143150363Tree_a,M: multis2082063201Tree_a] :
      ( ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ M ) )
     => ? [A6: multis2082063201Tree_a] :
          ( M
          = ( add_ms205941497Tree_a @ X @ A6 ) ) ) ).

% multi_member_split
thf(fact_282_multi__member__split,axiom,
    ! [X: produc1127127335iset_a,M: multis782275565iset_a] :
      ( ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ M ) )
     => ? [A6: multis782275565iset_a] :
          ( M
          = ( add_ms1353365509iset_a @ X @ A6 ) ) ) ).

% multi_member_split
thf(fact_283_multi__member__split,axiom,
    ! [X: product_prod_a_a,M: multis599418605od_a_a] :
      ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
     => ? [A6: multis599418605od_a_a] :
          ( M
          = ( add_ms1725977093od_a_a @ X @ A6 ) ) ) ).

% multi_member_split
thf(fact_284_multi__member__split,axiom,
    ! [X: multiset_a,M: multiset_multiset_a] :
      ( ( member_multiset_a @ X @ ( set_mset_multiset_a @ M ) )
     => ? [A6: multiset_multiset_a] :
          ( M
          = ( add_mset_multiset_a @ X @ A6 ) ) ) ).

% multi_member_split
thf(fact_285_multi__member__split,axiom,
    ! [X: a,M: multiset_a] :
      ( ( member_a @ X @ ( set_mset_a @ M ) )
     => ? [A6: multiset_a] :
          ( M
          = ( add_mset_a @ X @ A6 ) ) ) ).

% multi_member_split
thf(fact_286_mset__add,axiom,
    ! [A: produc143150363Tree_a,A4: multis2082063201Tree_a] :
      ( ( member254315204Tree_a @ A @ ( set_ms1129970904Tree_a @ A4 ) )
     => ~ ! [B6: multis2082063201Tree_a] :
            ( A4
           != ( add_ms205941497Tree_a @ A @ B6 ) ) ) ).

% mset_add
thf(fact_287_mset__add,axiom,
    ! [A: produc1127127335iset_a,A4: multis782275565iset_a] :
      ( ( member340150864iset_a @ A @ ( set_ms1768116836iset_a @ A4 ) )
     => ~ ! [B6: multis782275565iset_a] :
            ( A4
           != ( add_ms1353365509iset_a @ A @ B6 ) ) ) ).

% mset_add
thf(fact_288_mset__add,axiom,
    ! [A: product_prod_a_a,A4: multis599418605od_a_a] :
      ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ A4 ) )
     => ~ ! [B6: multis599418605od_a_a] :
            ( A4
           != ( add_ms1725977093od_a_a @ A @ B6 ) ) ) ).

% mset_add
thf(fact_289_mset__add,axiom,
    ! [A: multiset_a,A4: multiset_multiset_a] :
      ( ( member_multiset_a @ A @ ( set_mset_multiset_a @ A4 ) )
     => ~ ! [B6: multiset_multiset_a] :
            ( A4
           != ( add_mset_multiset_a @ A @ B6 ) ) ) ).

% mset_add
thf(fact_290_mset__add,axiom,
    ! [A: a,A4: multiset_a] :
      ( ( member_a @ A @ ( set_mset_a @ A4 ) )
     => ~ ! [B6: multiset_a] :
            ( A4
           != ( add_mset_a @ A @ B6 ) ) ) ).

% mset_add
thf(fact_291_union__iff,axiom,
    ! [A: produc143150363Tree_a,A4: multis2082063201Tree_a,B4: multis2082063201Tree_a] :
      ( ( member254315204Tree_a @ A @ ( set_ms1129970904Tree_a @ ( plus_p1065544216Tree_a @ A4 @ B4 ) ) )
      = ( ( member254315204Tree_a @ A @ ( set_ms1129970904Tree_a @ A4 ) )
        | ( member254315204Tree_a @ A @ ( set_ms1129970904Tree_a @ B4 ) ) ) ) ).

% union_iff
thf(fact_292_union__iff,axiom,
    ! [A: produc1127127335iset_a,A4: multis782275565iset_a,B4: multis782275565iset_a] :
      ( ( member340150864iset_a @ A @ ( set_ms1768116836iset_a @ ( plus_p1919096612iset_a @ A4 @ B4 ) ) )
      = ( ( member340150864iset_a @ A @ ( set_ms1768116836iset_a @ A4 ) )
        | ( member340150864iset_a @ A @ ( set_ms1768116836iset_a @ B4 ) ) ) ) ).

% union_iff
thf(fact_293_union__iff,axiom,
    ! [A: product_prod_a_a,A4: multis599418605od_a_a,B4: multis599418605od_a_a] :
      ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ A4 @ B4 ) ) )
      = ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ A4 ) )
        | ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ B4 ) ) ) ) ).

% union_iff
thf(fact_294_union__iff,axiom,
    ! [A: multiset_a,A4: multiset_multiset_a,B4: multiset_multiset_a] :
      ( ( member_multiset_a @ A @ ( set_mset_multiset_a @ ( plus_p1957546689iset_a @ A4 @ B4 ) ) )
      = ( ( member_multiset_a @ A @ ( set_mset_multiset_a @ A4 ) )
        | ( member_multiset_a @ A @ ( set_mset_multiset_a @ B4 ) ) ) ) ).

% union_iff
thf(fact_295_union__iff,axiom,
    ! [A: a,A4: multiset_a,B4: multiset_a] :
      ( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A4 @ B4 ) ) )
      = ( ( member_a @ A @ ( set_mset_a @ A4 ) )
        | ( member_a @ A @ ( set_mset_a @ B4 ) ) ) ) ).

% union_iff
thf(fact_296_multiset__nonemptyE,axiom,
    ! [A4: multis2082063201Tree_a] :
      ( ( A4 != zero_z2144236696Tree_a )
     => ~ ! [X3: produc143150363Tree_a] :
            ~ ( member254315204Tree_a @ X3 @ ( set_ms1129970904Tree_a @ A4 ) ) ) ).

% multiset_nonemptyE
thf(fact_297_multiset__nonemptyE,axiom,
    ! [A4: multis782275565iset_a] :
      ( ( A4 != zero_z1941791140iset_a )
     => ~ ! [X3: produc1127127335iset_a] :
            ~ ( member340150864iset_a @ X3 @ ( set_ms1768116836iset_a @ A4 ) ) ) ).

% multiset_nonemptyE
thf(fact_298_multiset__nonemptyE,axiom,
    ! [A4: multis599418605od_a_a] :
      ( ( A4 != zero_z160849828od_a_a )
     => ~ ! [X3: product_prod_a_a] :
            ~ ( member449909584od_a_a @ X3 @ ( set_ms1818337124od_a_a @ A4 ) ) ) ).

% multiset_nonemptyE
thf(fact_299_multiset__nonemptyE,axiom,
    ! [A4: multiset_multiset_a] :
      ( ( A4 != zero_z1580389697iset_a )
     => ~ ! [X3: multiset_a] :
            ~ ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ A4 ) ) ) ).

% multiset_nonemptyE
thf(fact_300_multiset__nonemptyE,axiom,
    ! [A4: multiset_a] :
      ( ( A4 != zero_zero_multiset_a )
     => ~ ! [X3: a] :
            ~ ( member_a @ X3 @ ( set_mset_a @ A4 ) ) ) ).

% multiset_nonemptyE
thf(fact_301_subset__CollectI,axiom,
    ! [B4: set_a,A4: set_a,Q2: a > $o,P2: a > $o] :
      ( ( ord_less_eq_set_a @ B4 @ A4 )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ B4 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_less_eq_set_a
          @ ( collect_a
            @ ^ [X4: a] :
                ( ( member_a @ X4 @ B4 )
                & ( Q2 @ X4 ) ) )
          @ ( collect_a
            @ ^ [X4: a] :
                ( ( member_a @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_302_subset__CollectI,axiom,
    ! [B4: set_Pr158363655iset_a,A4: set_Pr158363655iset_a,Q2: produc1127127335iset_a > $o,P2: produc1127127335iset_a > $o] :
      ( ( ord_le1978107815iset_a @ B4 @ A4 )
     => ( ! [X3: produc1127127335iset_a] :
            ( ( member340150864iset_a @ X3 @ B4 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le1978107815iset_a
          @ ( collec1104713362iset_a
            @ ^ [X4: produc1127127335iset_a] :
                ( ( member340150864iset_a @ X4 @ B4 )
                & ( Q2 @ X4 ) ) )
          @ ( collec1104713362iset_a
            @ ^ [X4: produc1127127335iset_a] :
                ( ( member340150864iset_a @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_303_subset__CollectI,axiom,
    ! [B4: set_Product_prod_a_a,A4: set_Product_prod_a_a,Q2: product_prod_a_a > $o,P2: product_prod_a_a > $o] :
      ( ( ord_le1824328871od_a_a @ B4 @ A4 )
     => ( ! [X3: product_prod_a_a] :
            ( ( member449909584od_a_a @ X3 @ B4 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le1824328871od_a_a
          @ ( collec645855634od_a_a
            @ ^ [X4: product_prod_a_a] :
                ( ( member449909584od_a_a @ X4 @ B4 )
                & ( Q2 @ X4 ) ) )
          @ ( collec645855634od_a_a
            @ ^ [X4: product_prod_a_a] :
                ( ( member449909584od_a_a @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_304_subset__CollectI,axiom,
    ! [B4: set_Pr1070816379Tree_a,A4: set_Pr1070816379Tree_a,Q2: produc143150363Tree_a > $o,P2: produc143150363Tree_a > $o] :
      ( ( ord_le1013007387Tree_a @ B4 @ A4 )
     => ( ! [X3: produc143150363Tree_a] :
            ( ( member254315204Tree_a @ X3 @ B4 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le1013007387Tree_a
          @ ( collec486369286Tree_a
            @ ^ [X4: produc143150363Tree_a] :
                ( ( member254315204Tree_a @ X4 @ B4 )
                & ( Q2 @ X4 ) ) )
          @ ( collec486369286Tree_a
            @ ^ [X4: produc143150363Tree_a] :
                ( ( member254315204Tree_a @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_305_subset__Collect__iff,axiom,
    ! [B4: set_a,A4: set_a,P2: a > $o] :
      ( ( ord_less_eq_set_a @ B4 @ A4 )
     => ( ( ord_less_eq_set_a @ B4
          @ ( collect_a
            @ ^ [X4: a] :
                ( ( member_a @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: a] :
              ( ( member_a @ X4 @ B4 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_306_subset__Collect__iff,axiom,
    ! [B4: set_Pr158363655iset_a,A4: set_Pr158363655iset_a,P2: produc1127127335iset_a > $o] :
      ( ( ord_le1978107815iset_a @ B4 @ A4 )
     => ( ( ord_le1978107815iset_a @ B4
          @ ( collec1104713362iset_a
            @ ^ [X4: produc1127127335iset_a] :
                ( ( member340150864iset_a @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: produc1127127335iset_a] :
              ( ( member340150864iset_a @ X4 @ B4 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_307_subset__Collect__iff,axiom,
    ! [B4: set_Product_prod_a_a,A4: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
      ( ( ord_le1824328871od_a_a @ B4 @ A4 )
     => ( ( ord_le1824328871od_a_a @ B4
          @ ( collec645855634od_a_a
            @ ^ [X4: product_prod_a_a] :
                ( ( member449909584od_a_a @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: product_prod_a_a] :
              ( ( member449909584od_a_a @ X4 @ B4 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_308_subset__Collect__iff,axiom,
    ! [B4: set_Pr1070816379Tree_a,A4: set_Pr1070816379Tree_a,P2: produc143150363Tree_a > $o] :
      ( ( ord_le1013007387Tree_a @ B4 @ A4 )
     => ( ( ord_le1013007387Tree_a @ B4
          @ ( collec486369286Tree_a
            @ ^ [X4: produc143150363Tree_a] :
                ( ( member254315204Tree_a @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: produc143150363Tree_a] :
              ( ( member254315204Tree_a @ X4 @ B4 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_309_multiset__induct__min,axiom,
    ! [P2: multiset_a > $o,M: multiset_a] :
      ( ( P2 @ zero_zero_multiset_a )
     => ( ! [X3: a,M3: multiset_a] :
            ( ( P2 @ M3 )
           => ( ! [Xa: a] :
                  ( ( member_a @ Xa @ ( set_mset_a @ M3 ) )
                 => ( ord_less_eq_a @ X3 @ Xa ) )
             => ( P2 @ ( add_mset_a @ X3 @ M3 ) ) ) )
       => ( P2 @ M ) ) ) ).

% multiset_induct_min
thf(fact_310_multiset__induct__max,axiom,
    ! [P2: multiset_a > $o,M: multiset_a] :
      ( ( P2 @ zero_zero_multiset_a )
     => ( ! [X3: a,M3: multiset_a] :
            ( ( P2 @ M3 )
           => ( ! [Xa: a] :
                  ( ( member_a @ Xa @ ( set_mset_a @ M3 ) )
                 => ( ord_less_eq_a @ Xa @ X3 ) )
             => ( P2 @ ( add_mset_a @ X3 @ M3 ) ) ) )
       => ( P2 @ M ) ) ) ).

% multiset_induct_max
thf(fact_311_multi__member__skip,axiom,
    ! [X: produc143150363Tree_a,XS: multis2082063201Tree_a,Y: produc143150363Tree_a] :
      ( ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ XS ) )
     => ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ ( plus_p1065544216Tree_a @ ( add_ms205941497Tree_a @ Y @ zero_z2144236696Tree_a ) @ XS ) ) ) ) ).

% multi_member_skip
thf(fact_312_multi__member__skip,axiom,
    ! [X: produc1127127335iset_a,XS: multis782275565iset_a,Y: produc1127127335iset_a] :
      ( ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ XS ) )
     => ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ ( plus_p1919096612iset_a @ ( add_ms1353365509iset_a @ Y @ zero_z1941791140iset_a ) @ XS ) ) ) ) ).

% multi_member_skip
thf(fact_313_multi__member__skip,axiom,
    ! [X: product_prod_a_a,XS: multis599418605od_a_a,Y: product_prod_a_a] :
      ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ XS ) )
     => ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ ( add_ms1725977093od_a_a @ Y @ zero_z160849828od_a_a ) @ XS ) ) ) ) ).

% multi_member_skip
thf(fact_314_multi__member__skip,axiom,
    ! [X: multiset_a,XS: multiset_multiset_a,Y: multiset_a] :
      ( ( member_multiset_a @ X @ ( set_mset_multiset_a @ XS ) )
     => ( member_multiset_a @ X @ ( set_mset_multiset_a @ ( plus_p1957546689iset_a @ ( add_mset_multiset_a @ Y @ zero_z1580389697iset_a ) @ XS ) ) ) ) ).

% multi_member_skip
thf(fact_315_multi__member__skip,axiom,
    ! [X: a,XS: multiset_a,Y: a] :
      ( ( member_a @ X @ ( set_mset_a @ XS ) )
     => ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ Y @ zero_zero_multiset_a ) @ XS ) ) ) ) ).

% multi_member_skip
thf(fact_316_multi__member__this,axiom,
    ! [X: produc143150363Tree_a,XS: multis2082063201Tree_a] : ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ ( plus_p1065544216Tree_a @ ( add_ms205941497Tree_a @ X @ zero_z2144236696Tree_a ) @ XS ) ) ) ).

% multi_member_this
thf(fact_317_multi__member__this,axiom,
    ! [X: produc1127127335iset_a,XS: multis782275565iset_a] : ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ ( plus_p1919096612iset_a @ ( add_ms1353365509iset_a @ X @ zero_z1941791140iset_a ) @ XS ) ) ) ).

% multi_member_this
thf(fact_318_multi__member__this,axiom,
    ! [X: product_prod_a_a,XS: multis599418605od_a_a] : ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) @ XS ) ) ) ).

% multi_member_this
thf(fact_319_multi__member__this,axiom,
    ! [X: multiset_a,XS: multiset_multiset_a] : ( member_multiset_a @ X @ ( set_mset_multiset_a @ ( plus_p1957546689iset_a @ ( add_mset_multiset_a @ X @ zero_z1580389697iset_a ) @ XS ) ) ) ).

% multi_member_this
thf(fact_320_multi__member__this,axiom,
    ! [X: a,XS: multiset_a] : ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) @ XS ) ) ) ).

% multi_member_this
thf(fact_321_less__add,axiom,
    ! [N: multis2082063201Tree_a,A: produc143150363Tree_a,M0: multis2082063201Tree_a,R: set_Pr220561863Tree_a] :
      ( ( member1298282640Tree_a @ ( produc1608222167Tree_a @ N @ ( add_ms205941497Tree_a @ A @ M0 ) ) @ ( mult1_470979598Tree_a @ R ) )
     => ( ? [M3: multis2082063201Tree_a] :
            ( ( member1298282640Tree_a @ ( produc1608222167Tree_a @ M3 @ M0 ) @ ( mult1_470979598Tree_a @ R ) )
            & ( N
              = ( add_ms205941497Tree_a @ A @ M3 ) ) )
        | ? [K4: multis2082063201Tree_a] :
            ( ! [B7: produc143150363Tree_a] :
                ( ( member254315204Tree_a @ B7 @ ( set_ms1129970904Tree_a @ K4 ) )
               => ( member146662416Tree_a @ ( produc1002111575Tree_a @ B7 @ A ) @ R ) )
            & ( N
              = ( plus_p1065544216Tree_a @ M0 @ K4 ) ) ) ) ) ).

% less_add
thf(fact_322_less__add,axiom,
    ! [N: multis782275565iset_a,A: produc1127127335iset_a,M0: multis782275565iset_a,R: set_Pr146455751iset_a] :
      ( ( member219455632iset_a @ ( produc1226985431iset_a @ N @ ( add_ms1353365509iset_a @ A @ M0 ) ) @ ( mult1_599785114iset_a @ R ) )
     => ( ? [M3: multis782275565iset_a] :
            ( ( member219455632iset_a @ ( produc1226985431iset_a @ M3 @ M0 ) @ ( mult1_599785114iset_a @ R ) )
            & ( N
              = ( add_ms1353365509iset_a @ A @ M3 ) ) )
        | ? [K4: multis782275565iset_a] :
            ( ! [B7: produc1127127335iset_a] :
                ( ( member340150864iset_a @ B7 @ ( set_ms1768116836iset_a @ K4 ) )
               => ( member567864080iset_a @ ( produc1898392407iset_a @ B7 @ A ) @ R ) )
            & ( N
              = ( plus_p1919096612iset_a @ M0 @ K4 ) ) ) ) ) ).

% less_add
thf(fact_323_less__add,axiom,
    ! [N: multis599418605od_a_a,A: product_prod_a_a,M0: multis599418605od_a_a,R: set_Pr1948701895od_a_a] :
      ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ ( add_ms1725977093od_a_a @ A @ M0 ) ) @ ( mult1_1719650714od_a_a @ R ) )
     => ( ? [M3: multis599418605od_a_a] :
            ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ M3 @ M0 ) @ ( mult1_1719650714od_a_a @ R ) )
            & ( N
              = ( add_ms1725977093od_a_a @ A @ M3 ) ) )
        | ? [K4: multis599418605od_a_a] :
            ( ! [B7: product_prod_a_a] :
                ( ( member449909584od_a_a @ B7 @ ( set_ms1818337124od_a_a @ K4 ) )
               => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B7 @ A ) @ R ) )
            & ( N
              = ( plus_p404985124od_a_a @ M0 @ K4 ) ) ) ) ) ).

% less_add
thf(fact_324_less__add,axiom,
    ! [N: multiset_multiset_a,A: multiset_a,M0: multiset_multiset_a,R: set_Pr158363655iset_a] :
      ( ( member104214352iset_a @ ( produc1444952343iset_a @ N @ ( add_mset_multiset_a @ A @ M0 ) ) @ ( mult1_multiset_a @ R ) )
     => ( ? [M3: multiset_multiset_a] :
            ( ( member104214352iset_a @ ( produc1444952343iset_a @ M3 @ M0 ) @ ( mult1_multiset_a @ R ) )
            & ( N
              = ( add_mset_multiset_a @ A @ M3 ) ) )
        | ? [K4: multiset_multiset_a] :
            ( ! [B7: multiset_a] :
                ( ( member_multiset_a @ B7 @ ( set_mset_multiset_a @ K4 ) )
               => ( member340150864iset_a @ ( produc2037245207iset_a @ B7 @ A ) @ R ) )
            & ( N
              = ( plus_p1957546689iset_a @ M0 @ K4 ) ) ) ) ) ).

% less_add
thf(fact_325_less__add,axiom,
    ! [N: multiset_a,A: a,M0: multiset_a,R: set_Product_prod_a_a] :
      ( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ ( add_mset_a @ A @ M0 ) ) @ ( mult1_a @ R ) )
     => ( ? [M3: multiset_a] :
            ( ( member340150864iset_a @ ( produc2037245207iset_a @ M3 @ M0 ) @ ( mult1_a @ R ) )
            & ( N
              = ( add_mset_a @ A @ M3 ) ) )
        | ? [K4: multiset_a] :
            ( ! [B7: a] :
                ( ( member_a @ B7 @ ( set_mset_a @ K4 ) )
               => ( member449909584od_a_a @ ( product_Pair_a_a @ B7 @ A ) @ R ) )
            & ( N
              = ( plus_plus_multiset_a @ M0 @ K4 ) ) ) ) ) ).

% less_add
thf(fact_326_mult1I,axiom,
    ! [M: multis2082063201Tree_a,A: produc143150363Tree_a,M0: multis2082063201Tree_a,N: multis2082063201Tree_a,K2: multis2082063201Tree_a,R: set_Pr220561863Tree_a] :
      ( ( M
        = ( add_ms205941497Tree_a @ A @ M0 ) )
     => ( ( N
          = ( plus_p1065544216Tree_a @ M0 @ K2 ) )
       => ( ! [B3: produc143150363Tree_a] :
              ( ( member254315204Tree_a @ B3 @ ( set_ms1129970904Tree_a @ K2 ) )
             => ( member146662416Tree_a @ ( produc1002111575Tree_a @ B3 @ A ) @ R ) )
         => ( member1298282640Tree_a @ ( produc1608222167Tree_a @ N @ M ) @ ( mult1_470979598Tree_a @ R ) ) ) ) ) ).

% mult1I
thf(fact_327_mult1I,axiom,
    ! [M: multis782275565iset_a,A: produc1127127335iset_a,M0: multis782275565iset_a,N: multis782275565iset_a,K2: multis782275565iset_a,R: set_Pr146455751iset_a] :
      ( ( M
        = ( add_ms1353365509iset_a @ A @ M0 ) )
     => ( ( N
          = ( plus_p1919096612iset_a @ M0 @ K2 ) )
       => ( ! [B3: produc1127127335iset_a] :
              ( ( member340150864iset_a @ B3 @ ( set_ms1768116836iset_a @ K2 ) )
             => ( member567864080iset_a @ ( produc1898392407iset_a @ B3 @ A ) @ R ) )
         => ( member219455632iset_a @ ( produc1226985431iset_a @ N @ M ) @ ( mult1_599785114iset_a @ R ) ) ) ) ) ).

% mult1I
thf(fact_328_mult1I,axiom,
    ! [M: multis599418605od_a_a,A: product_prod_a_a,M0: multis599418605od_a_a,N: multis599418605od_a_a,K2: multis599418605od_a_a,R: set_Pr1948701895od_a_a] :
      ( ( M
        = ( add_ms1725977093od_a_a @ A @ M0 ) )
     => ( ( N
          = ( plus_p404985124od_a_a @ M0 @ K2 ) )
       => ( ! [B3: product_prod_a_a] :
              ( ( member449909584od_a_a @ B3 @ ( set_ms1818337124od_a_a @ K2 ) )
             => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B3 @ A ) @ R ) )
         => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ M ) @ ( mult1_1719650714od_a_a @ R ) ) ) ) ) ).

% mult1I
thf(fact_329_mult1I,axiom,
    ! [M: multiset_multiset_a,A: multiset_a,M0: multiset_multiset_a,N: multiset_multiset_a,K2: multiset_multiset_a,R: set_Pr158363655iset_a] :
      ( ( M
        = ( add_mset_multiset_a @ A @ M0 ) )
     => ( ( N
          = ( plus_p1957546689iset_a @ M0 @ K2 ) )
       => ( ! [B3: multiset_a] :
              ( ( member_multiset_a @ B3 @ ( set_mset_multiset_a @ K2 ) )
             => ( member340150864iset_a @ ( produc2037245207iset_a @ B3 @ A ) @ R ) )
         => ( member104214352iset_a @ ( produc1444952343iset_a @ N @ M ) @ ( mult1_multiset_a @ R ) ) ) ) ) ).

% mult1I
thf(fact_330_mult1I,axiom,
    ! [M: multiset_a,A: a,M0: multiset_a,N: multiset_a,K2: multiset_a,R: set_Product_prod_a_a] :
      ( ( M
        = ( add_mset_a @ A @ M0 ) )
     => ( ( N
          = ( plus_plus_multiset_a @ M0 @ K2 ) )
       => ( ! [B3: a] :
              ( ( member_a @ B3 @ ( set_mset_a @ K2 ) )
             => ( member449909584od_a_a @ ( product_Pair_a_a @ B3 @ A ) @ R ) )
         => ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R ) ) ) ) ) ).

% mult1I
thf(fact_331_mono__mult1,axiom,
    ! [R: set_Product_prod_a_a,R4: set_Product_prod_a_a] :
      ( ( ord_le1824328871od_a_a @ R @ R4 )
     => ( ord_le1978107815iset_a @ ( mult1_a @ R ) @ ( mult1_a @ R4 ) ) ) ).

% mono_mult1
thf(fact_332_mult1__union,axiom,
    ! [B4: multiset_a,D2: multiset_a,R: set_Product_prod_a_a,C2: multiset_a] :
      ( ( member340150864iset_a @ ( produc2037245207iset_a @ B4 @ D2 ) @ ( mult1_a @ R ) )
     => ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ C2 @ B4 ) @ ( plus_plus_multiset_a @ C2 @ D2 ) ) @ ( mult1_a @ R ) ) ) ).

% mult1_union
thf(fact_333_not__less__empty,axiom,
    ! [M: multiset_a,R: set_Product_prod_a_a] :
      ~ ( member340150864iset_a @ ( produc2037245207iset_a @ M @ zero_zero_multiset_a ) @ ( mult1_a @ R ) ) ).

% not_less_empty
thf(fact_334_mult1E,axiom,
    ! [N: multis2082063201Tree_a,M: multis2082063201Tree_a,R: set_Pr220561863Tree_a] :
      ( ( member1298282640Tree_a @ ( produc1608222167Tree_a @ N @ M ) @ ( mult1_470979598Tree_a @ R ) )
     => ~ ! [A3: produc143150363Tree_a,M02: multis2082063201Tree_a] :
            ( ( M
              = ( add_ms205941497Tree_a @ A3 @ M02 ) )
           => ! [K4: multis2082063201Tree_a] :
                ( ( N
                  = ( plus_p1065544216Tree_a @ M02 @ K4 ) )
               => ~ ! [B7: produc143150363Tree_a] :
                      ( ( member254315204Tree_a @ B7 @ ( set_ms1129970904Tree_a @ K4 ) )
                     => ( member146662416Tree_a @ ( produc1002111575Tree_a @ B7 @ A3 ) @ R ) ) ) ) ) ).

% mult1E
thf(fact_335_mult1E,axiom,
    ! [N: multis782275565iset_a,M: multis782275565iset_a,R: set_Pr146455751iset_a] :
      ( ( member219455632iset_a @ ( produc1226985431iset_a @ N @ M ) @ ( mult1_599785114iset_a @ R ) )
     => ~ ! [A3: produc1127127335iset_a,M02: multis782275565iset_a] :
            ( ( M
              = ( add_ms1353365509iset_a @ A3 @ M02 ) )
           => ! [K4: multis782275565iset_a] :
                ( ( N
                  = ( plus_p1919096612iset_a @ M02 @ K4 ) )
               => ~ ! [B7: produc1127127335iset_a] :
                      ( ( member340150864iset_a @ B7 @ ( set_ms1768116836iset_a @ K4 ) )
                     => ( member567864080iset_a @ ( produc1898392407iset_a @ B7 @ A3 ) @ R ) ) ) ) ) ).

% mult1E
thf(fact_336_mult1E,axiom,
    ! [N: multis599418605od_a_a,M: multis599418605od_a_a,R: set_Pr1948701895od_a_a] :
      ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ M ) @ ( mult1_1719650714od_a_a @ R ) )
     => ~ ! [A3: product_prod_a_a,M02: multis599418605od_a_a] :
            ( ( M
              = ( add_ms1725977093od_a_a @ A3 @ M02 ) )
           => ! [K4: multis599418605od_a_a] :
                ( ( N
                  = ( plus_p404985124od_a_a @ M02 @ K4 ) )
               => ~ ! [B7: product_prod_a_a] :
                      ( ( member449909584od_a_a @ B7 @ ( set_ms1818337124od_a_a @ K4 ) )
                     => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B7 @ A3 ) @ R ) ) ) ) ) ).

% mult1E
thf(fact_337_mult1E,axiom,
    ! [N: multiset_multiset_a,M: multiset_multiset_a,R: set_Pr158363655iset_a] :
      ( ( member104214352iset_a @ ( produc1444952343iset_a @ N @ M ) @ ( mult1_multiset_a @ R ) )
     => ~ ! [A3: multiset_a,M02: multiset_multiset_a] :
            ( ( M
              = ( add_mset_multiset_a @ A3 @ M02 ) )
           => ! [K4: multiset_multiset_a] :
                ( ( N
                  = ( plus_p1957546689iset_a @ M02 @ K4 ) )
               => ~ ! [B7: multiset_a] :
                      ( ( member_multiset_a @ B7 @ ( set_mset_multiset_a @ K4 ) )
                     => ( member340150864iset_a @ ( produc2037245207iset_a @ B7 @ A3 ) @ R ) ) ) ) ) ).

% mult1E
thf(fact_338_mult1E,axiom,
    ! [N: multiset_a,M: multiset_a,R: set_Product_prod_a_a] :
      ( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R ) )
     => ~ ! [A3: a,M02: multiset_a] :
            ( ( M
              = ( add_mset_a @ A3 @ M02 ) )
           => ! [K4: multiset_a] :
                ( ( N
                  = ( plus_plus_multiset_a @ M02 @ K4 ) )
               => ~ ! [B7: a] :
                      ( ( member_a @ B7 @ ( set_mset_a @ K4 ) )
                     => ( member449909584od_a_a @ ( product_Pair_a_a @ B7 @ A3 ) @ R ) ) ) ) ) ).

% mult1E
thf(fact_339_one__step__implies__mult,axiom,
    ! [J2: multiset_multiset_a,K2: multiset_multiset_a,R: set_Pr158363655iset_a,I4: multiset_multiset_a] :
      ( ( J2 != zero_z1580389697iset_a )
     => ( ! [X3: multiset_a] :
            ( ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ K2 ) )
           => ? [Xa: multiset_a] :
                ( ( member_multiset_a @ Xa @ ( set_mset_multiset_a @ J2 ) )
                & ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Xa ) @ R ) ) )
       => ( member104214352iset_a @ ( produc1444952343iset_a @ ( plus_p1957546689iset_a @ I4 @ K2 ) @ ( plus_p1957546689iset_a @ I4 @ J2 ) ) @ ( mult_multiset_a @ R ) ) ) ) ).

% one_step_implies_mult
thf(fact_340_one__step__implies__mult,axiom,
    ! [J2: multiset_a,K2: multiset_a,R: set_Product_prod_a_a,I4: multiset_a] :
      ( ( J2 != zero_zero_multiset_a )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_mset_a @ K2 ) )
           => ? [Xa: a] :
                ( ( member_a @ Xa @ ( set_mset_a @ J2 ) )
                & ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Xa ) @ R ) ) )
       => ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ I4 @ K2 ) @ ( plus_plus_multiset_a @ I4 @ J2 ) ) @ ( mult_a @ R ) ) ) ) ).

% one_step_implies_mult
thf(fact_341_heap__top__max,axiom,
    ! [T2: tree_a] :
      ( ( T2 != e_a )
     => ( ( is_heap_a @ T2 )
       => ( ( val_a @ T2 )
          = ( lattic146396397_Max_a @ ( set_mset_a @ ( multiset_a2 @ T2 ) ) ) ) ) ) ).

% heap_top_max
thf(fact_342_mono__mult,axiom,
    ! [R: set_Product_prod_a_a,R4: set_Product_prod_a_a] :
      ( ( ord_le1824328871od_a_a @ R @ R4 )
     => ( ord_le1978107815iset_a @ ( mult_a @ R ) @ ( mult_a @ R4 ) ) ) ).

% mono_mult
thf(fact_343_mult__implies__one__step,axiom,
    ! [R: set_Pr158363655iset_a,M: multiset_multiset_a,N: multiset_multiset_a] :
      ( ( trans_multiset_a @ R )
     => ( ( member104214352iset_a @ ( produc1444952343iset_a @ M @ N ) @ ( mult_multiset_a @ R ) )
       => ? [I5: multiset_multiset_a,J3: multiset_multiset_a] :
            ( ( N
              = ( plus_p1957546689iset_a @ I5 @ J3 ) )
            & ? [K4: multiset_multiset_a] :
                ( ( M
                  = ( plus_p1957546689iset_a @ I5 @ K4 ) )
                & ( J3 != zero_z1580389697iset_a )
                & ! [X6: multiset_a] :
                    ( ( member_multiset_a @ X6 @ ( set_mset_multiset_a @ K4 ) )
                   => ? [Xa2: multiset_a] :
                        ( ( member_multiset_a @ Xa2 @ ( set_mset_multiset_a @ J3 ) )
                        & ( member340150864iset_a @ ( produc2037245207iset_a @ X6 @ Xa2 ) @ R ) ) ) ) ) ) ) ).

% mult_implies_one_step
thf(fact_344_mult__implies__one__step,axiom,
    ! [R: set_Product_prod_a_a,M: multiset_a,N: multiset_a] :
      ( ( trans_a @ R )
     => ( ( member340150864iset_a @ ( produc2037245207iset_a @ M @ N ) @ ( mult_a @ R ) )
       => ? [I5: multiset_a,J3: multiset_a] :
            ( ( N
              = ( plus_plus_multiset_a @ I5 @ J3 ) )
            & ? [K4: multiset_a] :
                ( ( M
                  = ( plus_plus_multiset_a @ I5 @ K4 ) )
                & ( J3 != zero_zero_multiset_a )
                & ! [X6: a] :
                    ( ( member_a @ X6 @ ( set_mset_a @ K4 ) )
                   => ? [Xa2: a] :
                        ( ( member_a @ Xa2 @ ( set_mset_a @ J3 ) )
                        & ( member449909584od_a_a @ ( product_Pair_a_a @ X6 @ Xa2 ) @ R ) ) ) ) ) ) ) ).

% mult_implies_one_step
thf(fact_345_subset__mset_Osum__mset__0__iff,axiom,
    ! [M: multiset_multiset_a] :
      ( ( ( comm_m315775925iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ M )
        = zero_zero_multiset_a )
      = ( ! [X4: multiset_a] :
            ( ( member_multiset_a @ X4 @ ( set_mset_multiset_a @ M ) )
           => ( X4 = zero_zero_multiset_a ) ) ) ) ).

% subset_mset.sum_mset_0_iff
thf(fact_346_comm__monoid__add_Osum__mset_Ocong,axiom,
    comm_m315775925iset_a = comm_m315775925iset_a ).

% comm_monoid_add.sum_mset.cong
thf(fact_347_transD,axiom,
    ! [R: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
      ( ( trans_multiset_a @ R )
     => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R )
       => ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ R )
         => ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ R ) ) ) ) ).

% transD
thf(fact_348_transD,axiom,
    ! [R: set_Product_prod_a_a,X: a,Y: a,Z: a] :
      ( ( trans_a @ R )
     => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
       => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
         => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R ) ) ) ) ).

% transD
thf(fact_349_transE,axiom,
    ! [R: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
      ( ( trans_multiset_a @ R )
     => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R )
       => ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ R )
         => ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ R ) ) ) ) ).

% transE
thf(fact_350_transE,axiom,
    ! [R: set_Product_prod_a_a,X: a,Y: a,Z: a] :
      ( ( trans_a @ R )
     => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
       => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
         => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R ) ) ) ) ).

% transE
thf(fact_351_transI,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ! [X3: a,Y3: a,Z3: a] :
          ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
         => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Z3 ) @ R )
           => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Z3 ) @ R ) ) )
     => ( trans_a @ R ) ) ).

% transI
thf(fact_352_mult__cancel__add__mset,axiom,
    ! [S3: set_Product_prod_a_a,Uu: a,X5: multiset_a,Y5: multiset_a] :
      ( ( trans_a @ S3 )
     => ( ( irrefl_a @ S3 )
       => ( ( member340150864iset_a @ ( produc2037245207iset_a @ ( add_mset_a @ Uu @ X5 ) @ ( add_mset_a @ Uu @ Y5 ) ) @ ( mult_a @ S3 ) )
          = ( member340150864iset_a @ ( produc2037245207iset_a @ X5 @ Y5 ) @ ( mult_a @ S3 ) ) ) ) ) ).

% mult_cancel_add_mset

% Helper facts (3)
thf(help_If_3_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_T,axiom,
    ! [X: produc143150363Tree_a,Y: produc143150363Tree_a] :
      ( ( if_Pro1144176865Tree_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_T,axiom,
    ! [X: produc143150363Tree_a,Y: produc143150363Tree_a] :
      ( ( if_Pro1144176865Tree_a @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( v_a
    = ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) ).

%------------------------------------------------------------------------------