TPTP Problem File: ITP179^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP179^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer StandardRules problem prob_555__5392816_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 : StandardRules/prob_555__5392816_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 554 ( 231 unt; 201 typ; 0 def)
% Number of atoms : 847 ( 345 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 2464 ( 112 ~; 9 |; 56 &;2064 @)
% ( 0 <=>; 223 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Number of types : 55 ( 54 usr)
% Number of type conns : 262 ( 262 >; 0 *; 0 +; 0 <<)
% Number of symbols : 148 ( 147 usr; 27 con; 0-3 aty)
% Number of variables : 835 ( 20 ^; 788 !; 27 ?; 835 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:38:59.416
%------------------------------------------------------------------------------
% Could-be-implicit typings (54)
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% Explicit typings (147)
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image_nat_nat: set_Pr1986765409at_nat > set_nat > set_nat ).
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image_489788268at_nat: set_Pr1134469976at_nat > set_nat > set_Pr1647387645at_nat ).
thf(sy_c_Relation_OImage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
image_1228479730um_a_b: set_Pr1618094130um_a_b > set_nat > set_Pr1174980151um_a_b ).
thf(sy_c_Relation_OImage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_Relation_OImage_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
image_256773707um_a_b: set_Pr1174980151um_a_b > set_nat > set_Sum_sum_a_b ).
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image_1774897262um_a_b: set_Pr1121389018um_a_b > set_Pr409224873um_a_b > set_Sum_sum_a_b ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
image_320480233um_a_b: set_Pr386468381um_a_b > set_Pr1986765409at_nat > set_Pr1174980151um_a_b ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
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thf(sy_c_Relation_OImage_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
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image_1482984686um_a_b: set_Pr1558970842um_a_b > set_Sum_sum_a_b > set_Pr409224873um_a_b ).
thf(sy_c_Relation_OImage_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
image_1675391388um_a_b: set_Pr1783309276um_a_b > set_Sum_sum_a_b > set_Pr1174980151um_a_b ).
thf(sy_c_Relation_OImage_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
image_1918588449um_a_b: set_Pr1916610317um_a_b > set_Sum_sum_a_b > set_Sum_sum_a_b ).
thf(sy_c_RulesAndChains_Omaintained_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
mainta522127984um_a_b: produc1871334759_a_nat > labele431970251um_a_b > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collec52779676at_nat: ( produc1032616263at_nat > $o ) > set_Pr1647387645at_nat ).
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collec1254333256um_a_b: ( produc1963079155um_a_b > $o ) > set_Pr409224873um_a_b ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
collec492052930um_a_b: ( produc1124793815um_a_b > $o ) > set_Pr1174980151um_a_b ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
collec1769032088um_a_b: ( produc1548871597um_a_b > $o ) > set_Pr1916610317um_a_b ).
thf(sy_c_Set_OCollect_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
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thf(sy_c_Set_Oinsert_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
insert1909710879tant_a: standard_Constant_a > set_St761939237tant_a > set_St761939237tant_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
insert1625259895at_nat: produc1032616263at_nat > set_Pr1647387645at_nat > set_Pr1647387645at_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J,type,
insert323157027um_a_b: produc1963079155um_a_b > set_Pr409224873um_a_b > set_Pr409224873um_a_b ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert271595217at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > set_Pr1986765409at_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
insert983991207um_a_b: produc1124793815um_a_b > set_Pr1174980151um_a_b > set_Pr1174980151um_a_b ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
insert1435405693um_a_b: produc1548871597um_a_b > set_Pr1916610317um_a_b > set_Pr1916610317um_a_b ).
thf(sy_c_Set_Oinsert_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
insert_Sum_sum_a_b: sum_sum_a_b > set_Sum_sum_a_b > set_Sum_sum_a_b ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Ocongruence__rule_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
standa1343274079tant_a: standard_Constant_a > standard_Constant_a > produc1871334759_a_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Ononempty__rule_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
standa1410829644tant_a: produc1871334759_a_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Oreflexivity__rule_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
standa63370785tant_a: standard_Constant_a > produc1871334759_a_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Osymmetry__rule_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
standa997693288tant_a: standard_Constant_a > produc1871334759_a_nat ).
thf(sy_c_StandardRules__Mirabelle__iljqcenreq_Otransitive__rule_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
standa1795879409tant_a: standard_Constant_a > produc1871334759_a_nat ).
thf(sy_c_member_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
member1632892294tant_a: standard_Constant_a > set_St761939237tant_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member1696759390at_nat: produc1032616263at_nat > set_Pr1647387645at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J,type,
member1502344244um_a_b: produc1003469085um_a_b > set_Pr326391507um_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J,type,
member1998628618um_a_b: produc1963079155um_a_b > set_Pr409224873um_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
member1641046175um_a_b: produc1394389750um_a_b > set_Pr1994339542um_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J,type,
member1139034387um_a_b: produc204140796um_a_b > set_Pr1618094130um_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
member1294585472um_a_b: produc1124793815um_a_b > set_Pr1174980151um_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
member1590479267um_a_b: produc1681803642um_a_b > set_Pr1121389018um_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J,type,
member834135910um_a_b: produc1546082365um_a_b > set_Pr386468381um_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
member585880557um_a_b: produc1798470614um_a_b > set_Pr195945996um_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J,type,
member429827856um_a_b: produc1918750183um_a_b > set_Pr45173959um_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
member748122307um_a_b: produc585753260um_a_b > set_Pr653770722um_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
member1249152_b_nat: produc1978941143_b_nat > set_Pr375490359_b_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J_J,type,
member1641922979um_a_b: produc1733247354um_a_b > set_Pr1558970842um_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J,type,
member485470653um_a_b: produc323101606um_a_b > set_Pr1783309276um_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
member947389014um_a_b: produc1548871597um_a_b > set_Pr1916610317um_a_b > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
member_Sum_sum_a_b: sum_sum_a_b > set_Sum_sum_a_b > $o ).
thf(sy_v_G,type,
g: labele431970251um_a_b ).
thf(sy_v_f____,type,
f: set_Pr1174980151um_a_b ).
thf(sy_v_l,type,
l: standard_Constant_a ).
thf(sy_v_v____,type,
v: sum_sum_a_b ).
% Relevant facts (352)
thf(fact_0_g,axiom,
( g
= ( restri1162247455um_a_b @ g ) ) ).
% g
thf(fact_1_v,axiom,
member_Sum_sum_a_b @ v @ ( labele577278695um_a_b @ g ) ).
% v
thf(fact_2__092_060open_062edge__preserving_A_123_I0_058_058_063_Hc1_M_Av_J_125_A_123_IS__Idt_M_A0_058_058_063_Hc1_M_A0_058_058_063_Hc1_J_125_A_Iedges_AG_J_092_060close_062,axiom,
edge_p1382426714tant_a @ ( insert983991207um_a_b @ ( produc1808556047um_a_b @ zero_zero_nat @ v ) @ bot_bo575978147um_a_b ) @ ( insert1625259895at_nat @ ( produc407553657at_nat @ standard_S_Idt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) @ bot_bo810816657at_nat ) @ ( labele1939049654um_a_b @ g ) ).
% \<open>edge_preserving {(0::?'c1, v)} {(S_Idt, 0::?'c1, 0::?'c1)} (edges G)\<close>
thf(fact_3_r,axiom,
ord_le192794300um_a_b @ ( image_256773707um_a_b @ f @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) @ ( labele577278695um_a_b @ g ) ).
% r
thf(fact_4_f,axiom,
( f
= ( insert983991207um_a_b @ ( produc1808556047um_a_b @ zero_zero_nat @ v ) @ bot_bo575978147um_a_b ) ) ).
% f
thf(fact_5_d,axiom,
( ( domain1368163076um_a_b @ f )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% d
thf(fact_6_graph__single,axiom,
! [A: standard_Constant_a,B: sum_sum_a_b,C: sum_sum_a_b] :
( ( labele1729654377um_a_b @ ( insert323157027um_a_b @ ( produc1697725733um_a_b @ A @ ( produc176426981um_a_b @ B @ C ) ) @ bot_bo1262634813um_a_b ) @ ( insert_Sum_sum_a_b @ B @ ( insert_Sum_sum_a_b @ C @ bot_bo1491243248um_a_b ) ) )
= ( restri1162247455um_a_b @ ( labele1729654377um_a_b @ ( insert323157027um_a_b @ ( produc1697725733um_a_b @ A @ ( produc176426981um_a_b @ B @ C ) ) @ bot_bo1262634813um_a_b ) @ ( insert_Sum_sum_a_b @ B @ ( insert_Sum_sum_a_b @ C @ bot_bo1491243248um_a_b ) ) ) ) ) ).
% graph_single
thf(fact_7_graph__single,axiom,
! [A: standard_Constant_a,B: nat,C: nat] :
( ( labele16114835_a_nat @ ( insert1625259895at_nat @ ( produc407553657at_nat @ A @ ( product_Pair_nat_nat @ B @ C ) ) @ bot_bo810816657at_nat ) @ ( insert_nat @ B @ ( insert_nat @ C @ bot_bot_set_nat ) ) )
= ( restri572569417_a_nat @ ( labele16114835_a_nat @ ( insert1625259895at_nat @ ( produc407553657at_nat @ A @ ( product_Pair_nat_nat @ B @ C ) ) @ bot_bo810816657at_nat ) @ ( insert_nat @ B @ ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ) ).
% graph_single
thf(fact_8__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062v_O_A_I0_M_Av_J_A_092_060in_062_Af_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [V: sum_sum_a_b] :
~ ( member1294585472um_a_b @ ( produc1808556047um_a_b @ zero_zero_nat @ V ) @ f ) ).
% \<open>\<And>thesis. (\<And>v. (0, v) \<in> f \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_9_u,axiom,
unival2092813468um_a_b @ f ).
% u
thf(fact_10_Domain__insert,axiom,
! [A: standard_Constant_a,B: product_prod_nat_nat,R: set_Pr1647387645at_nat] :
( ( domain1060562500at_nat @ ( insert1625259895at_nat @ ( produc407553657at_nat @ A @ B ) @ R ) )
= ( insert1909710879tant_a @ A @ ( domain1060562500at_nat @ R ) ) ) ).
% Domain_insert
thf(fact_11_Domain__insert,axiom,
! [A: nat,B: nat,R: set_Pr1986765409at_nat] :
( ( domain_nat_nat @ ( insert271595217at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) )
= ( insert_nat @ A @ ( domain_nat_nat @ R ) ) ) ).
% Domain_insert
thf(fact_12_Domain__insert,axiom,
! [A: nat,B: sum_sum_a_b,R: set_Pr1174980151um_a_b] :
( ( domain1368163076um_a_b @ ( insert983991207um_a_b @ ( produc1808556047um_a_b @ A @ B ) @ R ) )
= ( insert_nat @ A @ ( domain1368163076um_a_b @ R ) ) ) ).
% Domain_insert
thf(fact_13_Domain__insert,axiom,
! [A: standard_Constant_a,B: produc1548871597um_a_b,R: set_Pr409224873um_a_b] :
( ( domain1362581744um_a_b @ ( insert323157027um_a_b @ ( produc1697725733um_a_b @ A @ B ) @ R ) )
= ( insert1909710879tant_a @ A @ ( domain1362581744um_a_b @ R ) ) ) ).
% Domain_insert
thf(fact_14_Domain__insert,axiom,
! [A: sum_sum_a_b,B: sum_sum_a_b,R: set_Pr1916610317um_a_b] :
( ( domain2069673178um_a_b @ ( insert1435405693um_a_b @ ( produc176426981um_a_b @ A @ B ) @ R ) )
= ( insert_Sum_sum_a_b @ A @ ( domain2069673178um_a_b @ R ) ) ) ).
% Domain_insert
thf(fact_15_Image__singleton__iff,axiom,
! [B: nat,R: set_Pr1986765409at_nat,A: nat] :
( ( member_nat @ B @ ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_16_Image__singleton__iff,axiom,
! [B: sum_sum_a_b,R: set_Pr1174980151um_a_b,A: nat] :
( ( member_Sum_sum_a_b @ B @ ( image_256773707um_a_b @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( member1294585472um_a_b @ ( produc1808556047um_a_b @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_17_Image__singleton__iff,axiom,
! [B: sum_sum_a_b,R: set_Pr1994339542um_a_b,A: standard_Constant_a] :
( ( member_Sum_sum_a_b @ B @ ( image_2129435242um_a_b @ R @ ( insert1909710879tant_a @ A @ bot_bo1160111033tant_a ) ) )
= ( member1641046175um_a_b @ ( produc584843310um_a_b @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_18_Image__singleton__iff,axiom,
! [B: product_prod_nat_nat,R: set_Pr1647387645at_nat,A: standard_Constant_a] :
( ( member701585322at_nat @ B @ ( image_127502653at_nat @ R @ ( insert1909710879tant_a @ A @ bot_bo1160111033tant_a ) ) )
= ( member1696759390at_nat @ ( produc407553657at_nat @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_19_Image__singleton__iff,axiom,
! [B: sum_sum_a_b,R: set_Pr195945996um_a_b,A: product_prod_nat_nat] :
( ( member_Sum_sum_a_b @ B @ ( image_1754351636um_a_b @ R @ ( insert271595217at_nat @ A @ bot_bo2130386637at_nat ) ) )
= ( member585880557um_a_b @ ( produc1872245072um_a_b @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_20_Image__singleton__iff,axiom,
! [B: produc1124793815um_a_b,R: set_Pr1618094130um_a_b,A: nat] :
( ( member1294585472um_a_b @ B @ ( image_1228479730um_a_b @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( member1139034387um_a_b @ ( produc1346373166um_a_b @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_21_Image__singleton__iff,axiom,
! [B: sum_sum_a_b,R: set_Pr1916610317um_a_b,A: sum_sum_a_b] :
( ( member_Sum_sum_a_b @ B @ ( image_1918588449um_a_b @ R @ ( insert_Sum_sum_a_b @ A @ bot_bo1491243248um_a_b ) ) )
= ( member947389014um_a_b @ ( produc176426981um_a_b @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_22_Image__singleton__iff,axiom,
! [B: produc1124793815um_a_b,R: set_Pr326391507um_a_b,A: standard_Constant_a] :
( ( member1294585472um_a_b @ B @ ( image_1179143699um_a_b @ R @ ( insert1909710879tant_a @ A @ bot_bo1160111033tant_a ) ) )
= ( member1502344244um_a_b @ ( produc414524239um_a_b @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_23_Image__singleton__iff,axiom,
! [B: produc1124793815um_a_b,R: set_Pr386468381um_a_b,A: product_prod_nat_nat] :
( ( member1294585472um_a_b @ B @ ( image_320480233um_a_b @ R @ ( insert271595217at_nat @ A @ bot_bo2130386637at_nat ) ) )
= ( member834135910um_a_b @ ( produc335036333um_a_b @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_24_Image__singleton__iff,axiom,
! [B: sum_sum_a_b,R: set_Pr653770722um_a_b,A: produc1124793815um_a_b] :
( ( member_Sum_sum_a_b @ B @ ( image_869585130um_a_b @ R @ ( insert983991207um_a_b @ A @ bot_bo575978147um_a_b ) ) )
= ( member748122307um_a_b @ ( produc831476262um_a_b @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_25_graph__empty__e,axiom,
! [V2: set_nat] :
( ( labele16114835_a_nat @ bot_bo810816657at_nat @ V2 )
= ( restri572569417_a_nat @ ( labele16114835_a_nat @ bot_bo810816657at_nat @ V2 ) ) ) ).
% graph_empty_e
thf(fact_26_graph__empty__e,axiom,
! [V2: set_Sum_sum_a_b] :
( ( labele1729654377um_a_b @ bot_bo1262634813um_a_b @ V2 )
= ( restri1162247455um_a_b @ ( labele1729654377um_a_b @ bot_bo1262634813um_a_b @ V2 ) ) ) ).
% graph_empty_e
thf(fact_27_labeled__graph_Ocollapse,axiom,
! [Labeled_graph: labele935650037_a_nat] :
( ( labele16114835_a_nat @ ( labele195203296_a_nat @ Labeled_graph ) @ ( labele1810595089_a_nat @ Labeled_graph ) )
= Labeled_graph ) ).
% labeled_graph.collapse
thf(fact_28_labeled__graph_Ocollapse,axiom,
! [Labeled_graph: labele431970251um_a_b] :
( ( labele1729654377um_a_b @ ( labele1939049654um_a_b @ Labeled_graph ) @ ( labele577278695um_a_b @ Labeled_graph ) )
= Labeled_graph ) ).
% labeled_graph.collapse
thf(fact_29_Domain__empty,axiom,
( ( domain_nat_nat @ bot_bo2130386637at_nat )
= bot_bot_set_nat ) ).
% Domain_empty
thf(fact_30_Domain__empty,axiom,
( ( domain1060562500at_nat @ bot_bo810816657at_nat )
= bot_bo1160111033tant_a ) ).
% Domain_empty
thf(fact_31_Domain__empty,axiom,
( ( domain1368163076um_a_b @ bot_bo575978147um_a_b )
= bot_bot_set_nat ) ).
% Domain_empty
thf(fact_32_Domain__empty,axiom,
( ( domain2069673178um_a_b @ bot_bo225809273um_a_b )
= bot_bo1491243248um_a_b ) ).
% Domain_empty
thf(fact_33_Domain__empty,axiom,
( ( domain1362581744um_a_b @ bot_bo1262634813um_a_b )
= bot_bo1160111033tant_a ) ).
% Domain_empty
thf(fact_34_Image__empty1,axiom,
! [X: set_St761939237tant_a] :
( ( image_127502653at_nat @ bot_bo810816657at_nat @ X )
= bot_bo2130386637at_nat ) ).
% Image_empty1
thf(fact_35_Image__empty1,axiom,
! [X: set_nat] :
( ( image_256773707um_a_b @ bot_bo575978147um_a_b @ X )
= bot_bo1491243248um_a_b ) ).
% Image_empty1
thf(fact_36_Image__empty1,axiom,
! [X: set_Sum_sum_a_b] :
( ( image_1918588449um_a_b @ bot_bo225809273um_a_b @ X )
= bot_bo1491243248um_a_b ) ).
% Image_empty1
thf(fact_37_Image__empty1,axiom,
! [X: set_St761939237tant_a] :
( ( image_2023280617um_a_b @ bot_bo1262634813um_a_b @ X )
= bot_bo225809273um_a_b ) ).
% Image_empty1
thf(fact_38_restrict__idemp,axiom,
! [X2: labele935650037_a_nat] :
( ( restri572569417_a_nat @ ( restri572569417_a_nat @ X2 ) )
= ( restri572569417_a_nat @ X2 ) ) ).
% restrict_idemp
thf(fact_39_restrict__idemp,axiom,
! [X2: labele431970251um_a_b] :
( ( restri1162247455um_a_b @ ( restri1162247455um_a_b @ X2 ) )
= ( restri1162247455um_a_b @ X2 ) ) ).
% restrict_idemp
thf(fact_40_labeled__graph_Oinject,axiom,
! [X1: set_Pr1647387645at_nat,X22: set_nat,Y1: set_Pr1647387645at_nat,Y2: set_nat] :
( ( ( labele16114835_a_nat @ X1 @ X22 )
= ( labele16114835_a_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% labeled_graph.inject
thf(fact_41_labeled__graph_Oinject,axiom,
! [X1: set_Pr409224873um_a_b,X22: set_Sum_sum_a_b,Y1: set_Pr409224873um_a_b,Y2: set_Sum_sum_a_b] :
( ( ( labele1729654377um_a_b @ X1 @ X22 )
= ( labele1729654377um_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% labeled_graph.inject
thf(fact_42_v__gr,axiom,
! [A2: sum_sum_a_b,B2: sum_sum_a_b] :
( ( member1998628618um_a_b @ ( produc1697725733um_a_b @ standard_S_Idt_a @ ( produc176426981um_a_b @ A2 @ B2 ) ) @ ( labele1939049654um_a_b @ g ) )
= ( ( member_Sum_sum_a_b @ A2 @ ( labele577278695um_a_b @ g ) )
& ( B2 = A2 ) ) ) ).
% v_gr
thf(fact_43__092_060open_062_IS__Idt_M_Av_M_Av_J_A_092_060in_062_Aedges_AG_092_060close_062,axiom,
member1998628618um_a_b @ ( produc1697725733um_a_b @ standard_S_Idt_a @ ( produc176426981um_a_b @ v @ v ) ) @ ( labele1939049654um_a_b @ g ) ).
% \<open>(S_Idt, v, v) \<in> edges G\<close>
thf(fact_44_ImageI,axiom,
! [A: nat,B: nat,R: set_Pr1986765409at_nat,A3: set_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( member_nat @ A @ A3 )
=> ( member_nat @ B @ ( image_nat_nat @ R @ A3 ) ) ) ) ).
% ImageI
thf(fact_45_ImageI,axiom,
! [A: nat,B: sum_sum_a_b,R: set_Pr1174980151um_a_b,A3: set_nat] :
( ( member1294585472um_a_b @ ( produc1808556047um_a_b @ A @ B ) @ R )
=> ( ( member_nat @ A @ A3 )
=> ( member_Sum_sum_a_b @ B @ ( image_256773707um_a_b @ R @ A3 ) ) ) ) ).
% ImageI
thf(fact_46_ImageI,axiom,
! [A: standard_Constant_a,B: product_prod_nat_nat,R: set_Pr1647387645at_nat,A3: set_St761939237tant_a] :
( ( member1696759390at_nat @ ( produc407553657at_nat @ A @ B ) @ R )
=> ( ( member1632892294tant_a @ A @ A3 )
=> ( member701585322at_nat @ B @ ( image_127502653at_nat @ R @ A3 ) ) ) ) ).
% ImageI
thf(fact_47_ImageI,axiom,
! [A: sum_sum_a_b,B: sum_sum_a_b,R: set_Pr1916610317um_a_b,A3: set_Sum_sum_a_b] :
( ( member947389014um_a_b @ ( produc176426981um_a_b @ A @ B ) @ R )
=> ( ( member_Sum_sum_a_b @ A @ A3 )
=> ( member_Sum_sum_a_b @ B @ ( image_1918588449um_a_b @ R @ A3 ) ) ) ) ).
% ImageI
thf(fact_48_ImageI,axiom,
! [A: sum_sum_a_b,B: produc1124793815um_a_b,R: set_Pr1783309276um_a_b,A3: set_Sum_sum_a_b] :
( ( member485470653um_a_b @ ( produc1637282520um_a_b @ A @ B ) @ R )
=> ( ( member_Sum_sum_a_b @ A @ A3 )
=> ( member1294585472um_a_b @ B @ ( image_1675391388um_a_b @ R @ A3 ) ) ) ) ).
% ImageI
thf(fact_49_ImageI,axiom,
! [A: produc1124793815um_a_b,B: sum_sum_a_b,R: set_Pr653770722um_a_b,A3: set_Pr1174980151um_a_b] :
( ( member748122307um_a_b @ ( produc831476262um_a_b @ A @ B ) @ R )
=> ( ( member1294585472um_a_b @ A @ A3 )
=> ( member_Sum_sum_a_b @ B @ ( image_869585130um_a_b @ R @ A3 ) ) ) ) ).
% ImageI
thf(fact_50_ImageI,axiom,
! [A: standard_Constant_a,B: produc1548871597um_a_b,R: set_Pr409224873um_a_b,A3: set_St761939237tant_a] :
( ( member1998628618um_a_b @ ( produc1697725733um_a_b @ A @ B ) @ R )
=> ( ( member1632892294tant_a @ A @ A3 )
=> ( member947389014um_a_b @ B @ ( image_2023280617um_a_b @ R @ A3 ) ) ) ) ).
% ImageI
thf(fact_51_ImageI,axiom,
! [A: produc1124793815um_a_b,B: produc1124793815um_a_b,R: set_Pr45173959um_a_b,A3: set_Pr1174980151um_a_b] :
( ( member429827856um_a_b @ ( produc27163479um_a_b @ A @ B ) @ R )
=> ( ( member1294585472um_a_b @ A @ A3 )
=> ( member1294585472um_a_b @ B @ ( image_1464989587um_a_b @ R @ A3 ) ) ) ) ).
% ImageI
thf(fact_52_ImageI,axiom,
! [A: sum_sum_a_b,B: produc1963079155um_a_b,R: set_Pr1558970842um_a_b,A3: set_Sum_sum_a_b] :
( ( member1641922979um_a_b @ ( produc230853810um_a_b @ A @ B ) @ R )
=> ( ( member_Sum_sum_a_b @ A @ A3 )
=> ( member1998628618um_a_b @ B @ ( image_1482984686um_a_b @ R @ A3 ) ) ) ) ).
% ImageI
thf(fact_53_ImageI,axiom,
! [A: produc1963079155um_a_b,B: sum_sum_a_b,R: set_Pr1121389018um_a_b,A3: set_Pr409224873um_a_b] :
( ( member1590479267um_a_b @ ( produc522766386um_a_b @ A @ B ) @ R )
=> ( ( member1998628618um_a_b @ A @ A3 )
=> ( member_Sum_sum_a_b @ B @ ( image_1774897262um_a_b @ R @ A3 ) ) ) ) ).
% ImageI
thf(fact_54_Image__empty2,axiom,
! [R2: set_Pr1986765409at_nat] :
( ( image_nat_nat @ R2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Image_empty2
thf(fact_55_Image__empty2,axiom,
! [R2: set_Pr1174980151um_a_b] :
( ( image_256773707um_a_b @ R2 @ bot_bot_set_nat )
= bot_bo1491243248um_a_b ) ).
% Image_empty2
thf(fact_56_Image__empty2,axiom,
! [R2: set_Pr375490359_b_nat] :
( ( image_1117854795_b_nat @ R2 @ bot_bo1491243248um_a_b )
= bot_bot_set_nat ) ).
% Image_empty2
thf(fact_57_Image__empty2,axiom,
! [R2: set_Pr1618094130um_a_b] :
( ( image_1228479730um_a_b @ R2 @ bot_bot_set_nat )
= bot_bo575978147um_a_b ) ).
% Image_empty2
thf(fact_58_Image__empty2,axiom,
! [R2: set_Pr1902304780_b_nat] :
( ( image_128716052_b_nat @ R2 @ bot_bo575978147um_a_b )
= bot_bot_set_nat ) ).
% Image_empty2
thf(fact_59_Image__empty2,axiom,
! [R2: set_Pr1916610317um_a_b] :
( ( image_1918588449um_a_b @ R2 @ bot_bo1491243248um_a_b )
= bot_bo1491243248um_a_b ) ).
% Image_empty2
thf(fact_60_Image__empty2,axiom,
! [R2: set_Pr689416536at_nat] :
( ( image_452296684at_nat @ R2 @ bot_bo810816657at_nat )
= bot_bot_set_nat ) ).
% Image_empty2
thf(fact_61_Image__empty2,axiom,
! [R2: set_Pr1134469976at_nat] :
( ( image_489788268at_nat @ R2 @ bot_bot_set_nat )
= bot_bo810816657at_nat ) ).
% Image_empty2
thf(fact_62_Image__empty2,axiom,
! [R2: set_Pr1155046920um_a_b] :
( ( image_166667464um_a_b @ R2 @ bot_bot_set_nat )
= bot_bo225809273um_a_b ) ).
% Image_empty2
thf(fact_63_Image__empty2,axiom,
! [R2: set_Pr653770722um_a_b] :
( ( image_869585130um_a_b @ R2 @ bot_bo575978147um_a_b )
= bot_bo1491243248um_a_b ) ).
% Image_empty2
thf(fact_64_vertices__restrict,axiom,
! [G: labele935650037_a_nat] :
( ( labele1810595089_a_nat @ ( restri572569417_a_nat @ G ) )
= ( labele1810595089_a_nat @ G ) ) ).
% vertices_restrict
thf(fact_65_vertices__restrict,axiom,
! [G: labele431970251um_a_b] :
( ( labele577278695um_a_b @ ( restri1162247455um_a_b @ G ) )
= ( labele577278695um_a_b @ G ) ) ).
% vertices_restrict
thf(fact_66__092_060open_062graph__homomorphism_A_ILG_A_123_125_A_1230_125_J_AG_Af_092_060close_062,axiom,
graph_1452133198um_a_b @ ( labele16114835_a_nat @ bot_bo810816657at_nat @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) @ g @ f ).
% \<open>graph_homomorphism (LG {} {0}) G f\<close>
thf(fact_67_labeled__graph_Oexhaust,axiom,
! [Y: labele935650037_a_nat] :
~ ! [X12: set_Pr1647387645at_nat,X23: set_nat] :
( Y
!= ( labele16114835_a_nat @ X12 @ X23 ) ) ).
% labeled_graph.exhaust
thf(fact_68_labeled__graph_Oexhaust,axiom,
! [Y: labele431970251um_a_b] :
~ ! [X12: set_Pr409224873um_a_b,X23: set_Sum_sum_a_b] :
( Y
!= ( labele1729654377um_a_b @ X12 @ X23 ) ) ).
% labeled_graph.exhaust
thf(fact_69_labeled__graph_Oinduct,axiom,
! [P: labele935650037_a_nat > $o,Labeled_graph: labele935650037_a_nat] :
( ! [X1a: set_Pr1647387645at_nat,X2a: set_nat] : ( P @ ( labele16114835_a_nat @ X1a @ X2a ) )
=> ( P @ Labeled_graph ) ) ).
% labeled_graph.induct
thf(fact_70_labeled__graph_Oinduct,axiom,
! [P: labele431970251um_a_b > $o,Labeled_graph: labele431970251um_a_b] :
( ! [X1a: set_Pr409224873um_a_b,X2a: set_Sum_sum_a_b] : ( P @ ( labele1729654377um_a_b @ X1a @ X2a ) )
=> ( P @ Labeled_graph ) ) ).
% labeled_graph.induct
thf(fact_71_rev__ImageI,axiom,
! [A: nat,A3: set_nat,B: nat,R: set_Pr1986765409at_nat] :
( ( member_nat @ A @ A3 )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( member_nat @ B @ ( image_nat_nat @ R @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_72_rev__ImageI,axiom,
! [A: nat,A3: set_nat,B: sum_sum_a_b,R: set_Pr1174980151um_a_b] :
( ( member_nat @ A @ A3 )
=> ( ( member1294585472um_a_b @ ( produc1808556047um_a_b @ A @ B ) @ R )
=> ( member_Sum_sum_a_b @ B @ ( image_256773707um_a_b @ R @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_73_rev__ImageI,axiom,
! [A: standard_Constant_a,A3: set_St761939237tant_a,B: product_prod_nat_nat,R: set_Pr1647387645at_nat] :
( ( member1632892294tant_a @ A @ A3 )
=> ( ( member1696759390at_nat @ ( produc407553657at_nat @ A @ B ) @ R )
=> ( member701585322at_nat @ B @ ( image_127502653at_nat @ R @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_74_rev__ImageI,axiom,
! [A: sum_sum_a_b,A3: set_Sum_sum_a_b,B: sum_sum_a_b,R: set_Pr1916610317um_a_b] :
( ( member_Sum_sum_a_b @ A @ A3 )
=> ( ( member947389014um_a_b @ ( produc176426981um_a_b @ A @ B ) @ R )
=> ( member_Sum_sum_a_b @ B @ ( image_1918588449um_a_b @ R @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_75_rev__ImageI,axiom,
! [A: sum_sum_a_b,A3: set_Sum_sum_a_b,B: produc1124793815um_a_b,R: set_Pr1783309276um_a_b] :
( ( member_Sum_sum_a_b @ A @ A3 )
=> ( ( member485470653um_a_b @ ( produc1637282520um_a_b @ A @ B ) @ R )
=> ( member1294585472um_a_b @ B @ ( image_1675391388um_a_b @ R @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_76_rev__ImageI,axiom,
! [A: produc1124793815um_a_b,A3: set_Pr1174980151um_a_b,B: sum_sum_a_b,R: set_Pr653770722um_a_b] :
( ( member1294585472um_a_b @ A @ A3 )
=> ( ( member748122307um_a_b @ ( produc831476262um_a_b @ A @ B ) @ R )
=> ( member_Sum_sum_a_b @ B @ ( image_869585130um_a_b @ R @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_77_rev__ImageI,axiom,
! [A: standard_Constant_a,A3: set_St761939237tant_a,B: produc1548871597um_a_b,R: set_Pr409224873um_a_b] :
( ( member1632892294tant_a @ A @ A3 )
=> ( ( member1998628618um_a_b @ ( produc1697725733um_a_b @ A @ B ) @ R )
=> ( member947389014um_a_b @ B @ ( image_2023280617um_a_b @ R @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_78_rev__ImageI,axiom,
! [A: produc1124793815um_a_b,A3: set_Pr1174980151um_a_b,B: produc1124793815um_a_b,R: set_Pr45173959um_a_b] :
( ( member1294585472um_a_b @ A @ A3 )
=> ( ( member429827856um_a_b @ ( produc27163479um_a_b @ A @ B ) @ R )
=> ( member1294585472um_a_b @ B @ ( image_1464989587um_a_b @ R @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_79_rev__ImageI,axiom,
! [A: sum_sum_a_b,A3: set_Sum_sum_a_b,B: produc1963079155um_a_b,R: set_Pr1558970842um_a_b] :
( ( member_Sum_sum_a_b @ A @ A3 )
=> ( ( member1641922979um_a_b @ ( produc230853810um_a_b @ A @ B ) @ R )
=> ( member1998628618um_a_b @ B @ ( image_1482984686um_a_b @ R @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_80_rev__ImageI,axiom,
! [A: produc1963079155um_a_b,A3: set_Pr409224873um_a_b,B: sum_sum_a_b,R: set_Pr1121389018um_a_b] :
( ( member1998628618um_a_b @ A @ A3 )
=> ( ( member1590479267um_a_b @ ( produc522766386um_a_b @ A @ B ) @ R )
=> ( member_Sum_sum_a_b @ B @ ( image_1774897262um_a_b @ R @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_81_Image__iff,axiom,
! [B: product_prod_nat_nat,R: set_Pr1647387645at_nat,A3: set_St761939237tant_a] :
( ( member701585322at_nat @ B @ ( image_127502653at_nat @ R @ A3 ) )
= ( ? [X3: standard_Constant_a] :
( ( member1632892294tant_a @ X3 @ A3 )
& ( member1696759390at_nat @ ( produc407553657at_nat @ X3 @ B ) @ R ) ) ) ) ).
% Image_iff
thf(fact_82_Image__iff,axiom,
! [B: nat,R: set_Pr1986765409at_nat,A3: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ R @ A3 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ B ) @ R ) ) ) ) ).
% Image_iff
thf(fact_83_Image__iff,axiom,
! [B: sum_sum_a_b,R: set_Pr1174980151um_a_b,A3: set_nat] :
( ( member_Sum_sum_a_b @ B @ ( image_256773707um_a_b @ R @ A3 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( member1294585472um_a_b @ ( produc1808556047um_a_b @ X3 @ B ) @ R ) ) ) ) ).
% Image_iff
thf(fact_84_Image__iff,axiom,
! [B: produc1548871597um_a_b,R: set_Pr409224873um_a_b,A3: set_St761939237tant_a] :
( ( member947389014um_a_b @ B @ ( image_2023280617um_a_b @ R @ A3 ) )
= ( ? [X3: standard_Constant_a] :
( ( member1632892294tant_a @ X3 @ A3 )
& ( member1998628618um_a_b @ ( produc1697725733um_a_b @ X3 @ B ) @ R ) ) ) ) ).
% Image_iff
thf(fact_85_Image__iff,axiom,
! [B: sum_sum_a_b,R: set_Pr1916610317um_a_b,A3: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b @ B @ ( image_1918588449um_a_b @ R @ A3 ) )
= ( ? [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b @ X3 @ A3 )
& ( member947389014um_a_b @ ( produc176426981um_a_b @ X3 @ B ) @ R ) ) ) ) ).
% Image_iff
thf(fact_86_ImageE,axiom,
! [B: nat,R: set_Pr1986765409at_nat,A3: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ R @ A3 ) )
=> ~ ! [X4: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ B ) @ R )
=> ~ ( member_nat @ X4 @ A3 ) ) ) ).
% ImageE
thf(fact_87_ImageE,axiom,
! [B: sum_sum_a_b,R: set_Pr1174980151um_a_b,A3: set_nat] :
( ( member_Sum_sum_a_b @ B @ ( image_256773707um_a_b @ R @ A3 ) )
=> ~ ! [X4: nat] :
( ( member1294585472um_a_b @ ( produc1808556047um_a_b @ X4 @ B ) @ R )
=> ~ ( member_nat @ X4 @ A3 ) ) ) ).
% ImageE
thf(fact_88_ImageE,axiom,
! [B: product_prod_nat_nat,R: set_Pr1647387645at_nat,A3: set_St761939237tant_a] :
( ( member701585322at_nat @ B @ ( image_127502653at_nat @ R @ A3 ) )
=> ~ ! [X4: standard_Constant_a] :
( ( member1696759390at_nat @ ( produc407553657at_nat @ X4 @ B ) @ R )
=> ~ ( member1632892294tant_a @ X4 @ A3 ) ) ) ).
% ImageE
thf(fact_89_ImageE,axiom,
! [B: sum_sum_a_b,R: set_Pr1916610317um_a_b,A3: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b @ B @ ( image_1918588449um_a_b @ R @ A3 ) )
=> ~ ! [X4: sum_sum_a_b] :
( ( member947389014um_a_b @ ( produc176426981um_a_b @ X4 @ B ) @ R )
=> ~ ( member_Sum_sum_a_b @ X4 @ A3 ) ) ) ).
% ImageE
thf(fact_90_ImageE,axiom,
! [B: sum_sum_a_b,R: set_Pr653770722um_a_b,A3: set_Pr1174980151um_a_b] :
( ( member_Sum_sum_a_b @ B @ ( image_869585130um_a_b @ R @ A3 ) )
=> ~ ! [X4: produc1124793815um_a_b] :
( ( member748122307um_a_b @ ( produc831476262um_a_b @ X4 @ B ) @ R )
=> ~ ( member1294585472um_a_b @ X4 @ A3 ) ) ) ).
% ImageE
thf(fact_91_ImageE,axiom,
! [B: produc1124793815um_a_b,R: set_Pr1783309276um_a_b,A3: set_Sum_sum_a_b] :
( ( member1294585472um_a_b @ B @ ( image_1675391388um_a_b @ R @ A3 ) )
=> ~ ! [X4: sum_sum_a_b] :
( ( member485470653um_a_b @ ( produc1637282520um_a_b @ X4 @ B ) @ R )
=> ~ ( member_Sum_sum_a_b @ X4 @ A3 ) ) ) ).
% ImageE
thf(fact_92_ImageE,axiom,
! [B: produc1548871597um_a_b,R: set_Pr409224873um_a_b,A3: set_St761939237tant_a] :
( ( member947389014um_a_b @ B @ ( image_2023280617um_a_b @ R @ A3 ) )
=> ~ ! [X4: standard_Constant_a] :
( ( member1998628618um_a_b @ ( produc1697725733um_a_b @ X4 @ B ) @ R )
=> ~ ( member1632892294tant_a @ X4 @ A3 ) ) ) ).
% ImageE
thf(fact_93_ImageE,axiom,
! [B: produc1124793815um_a_b,R: set_Pr45173959um_a_b,A3: set_Pr1174980151um_a_b] :
( ( member1294585472um_a_b @ B @ ( image_1464989587um_a_b @ R @ A3 ) )
=> ~ ! [X4: produc1124793815um_a_b] :
( ( member429827856um_a_b @ ( produc27163479um_a_b @ X4 @ B ) @ R )
=> ~ ( member1294585472um_a_b @ X4 @ A3 ) ) ) ).
% ImageE
thf(fact_94_ImageE,axiom,
! [B: sum_sum_a_b,R: set_Pr1121389018um_a_b,A3: set_Pr409224873um_a_b] :
( ( member_Sum_sum_a_b @ B @ ( image_1774897262um_a_b @ R @ A3 ) )
=> ~ ! [X4: produc1963079155um_a_b] :
( ( member1590479267um_a_b @ ( produc522766386um_a_b @ X4 @ B ) @ R )
=> ~ ( member1998628618um_a_b @ X4 @ A3 ) ) ) ).
% ImageE
thf(fact_95_ImageE,axiom,
! [B: produc1963079155um_a_b,R: set_Pr1558970842um_a_b,A3: set_Sum_sum_a_b] :
( ( member1998628618um_a_b @ B @ ( image_1482984686um_a_b @ R @ A3 ) )
=> ~ ! [X4: sum_sum_a_b] :
( ( member1641922979um_a_b @ ( produc230853810um_a_b @ X4 @ B ) @ R )
=> ~ ( member_Sum_sum_a_b @ X4 @ A3 ) ) ) ).
% ImageE
thf(fact_96_Image__mono,axiom,
! [R3: set_Pr1174980151um_a_b,R: set_Pr1174980151um_a_b,A4: set_nat,A3: set_nat] :
( ( ord_le823954903um_a_b @ R3 @ R )
=> ( ( ord_less_eq_set_nat @ A4 @ A3 )
=> ( ord_le192794300um_a_b @ ( image_256773707um_a_b @ R3 @ A4 ) @ ( image_256773707um_a_b @ R @ A3 ) ) ) ) ).
% Image_mono
thf(fact_97_Image__mono,axiom,
! [R3: set_Pr1916610317um_a_b,R: set_Pr1916610317um_a_b,A4: set_Sum_sum_a_b,A3: set_Sum_sum_a_b] :
( ( ord_le1059794605um_a_b @ R3 @ R )
=> ( ( ord_le192794300um_a_b @ A4 @ A3 )
=> ( ord_le192794300um_a_b @ ( image_1918588449um_a_b @ R3 @ A4 ) @ ( image_1918588449um_a_b @ R @ A3 ) ) ) ) ).
% Image_mono
thf(fact_98_Domain_Oinducts,axiom,
! [X2: standard_Constant_a,R: set_Pr1647387645at_nat,P: standard_Constant_a > $o] :
( ( member1632892294tant_a @ X2 @ ( domain1060562500at_nat @ R ) )
=> ( ! [A5: standard_Constant_a,B3: product_prod_nat_nat] :
( ( member1696759390at_nat @ ( produc407553657at_nat @ A5 @ B3 ) @ R )
=> ( P @ A5 ) )
=> ( P @ X2 ) ) ) ).
% Domain.inducts
thf(fact_99_Domain_Oinducts,axiom,
! [X2: nat,R: set_Pr1986765409at_nat,P: nat > $o] :
( ( member_nat @ X2 @ ( domain_nat_nat @ R ) )
=> ( ! [A5: nat,B3: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A5 @ B3 ) @ R )
=> ( P @ A5 ) )
=> ( P @ X2 ) ) ) ).
% Domain.inducts
thf(fact_100_Domain_Oinducts,axiom,
! [X2: nat,R: set_Pr1174980151um_a_b,P: nat > $o] :
( ( member_nat @ X2 @ ( domain1368163076um_a_b @ R ) )
=> ( ! [A5: nat,B3: sum_sum_a_b] :
( ( member1294585472um_a_b @ ( produc1808556047um_a_b @ A5 @ B3 ) @ R )
=> ( P @ A5 ) )
=> ( P @ X2 ) ) ) ).
% Domain.inducts
thf(fact_101_Domain_Oinducts,axiom,
! [X2: standard_Constant_a,R: set_Pr409224873um_a_b,P: standard_Constant_a > $o] :
( ( member1632892294tant_a @ X2 @ ( domain1362581744um_a_b @ R ) )
=> ( ! [A5: standard_Constant_a,B3: produc1548871597um_a_b] :
( ( member1998628618um_a_b @ ( produc1697725733um_a_b @ A5 @ B3 ) @ R )
=> ( P @ A5 ) )
=> ( P @ X2 ) ) ) ).
% Domain.inducts
thf(fact_102_Domain_Oinducts,axiom,
! [X2: sum_sum_a_b,R: set_Pr1916610317um_a_b,P: sum_sum_a_b > $o] :
( ( member_Sum_sum_a_b @ X2 @ ( domain2069673178um_a_b @ R ) )
=> ( ! [A5: sum_sum_a_b,B3: sum_sum_a_b] :
( ( member947389014um_a_b @ ( produc176426981um_a_b @ A5 @ B3 ) @ R )
=> ( P @ A5 ) )
=> ( P @ X2 ) ) ) ).
% Domain.inducts
thf(fact_103_Domain_ODomainI,axiom,
! [A: standard_Constant_a,B: product_prod_nat_nat,R: set_Pr1647387645at_nat] :
( ( member1696759390at_nat @ ( produc407553657at_nat @ A @ B ) @ R )
=> ( member1632892294tant_a @ A @ ( domain1060562500at_nat @ R ) ) ) ).
% Domain.DomainI
thf(fact_104_Domain_ODomainI,axiom,
! [A: nat,B: nat,R: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( member_nat @ A @ ( domain_nat_nat @ R ) ) ) ).
% Domain.DomainI
thf(fact_105_Domain_ODomainI,axiom,
! [A: nat,B: sum_sum_a_b,R: set_Pr1174980151um_a_b] :
( ( member1294585472um_a_b @ ( produc1808556047um_a_b @ A @ B ) @ R )
=> ( member_nat @ A @ ( domain1368163076um_a_b @ R ) ) ) ).
% Domain.DomainI
thf(fact_106_Domain_ODomainI,axiom,
! [A: standard_Constant_a,B: produc1548871597um_a_b,R: set_Pr409224873um_a_b] :
( ( member1998628618um_a_b @ ( produc1697725733um_a_b @ A @ B ) @ R )
=> ( member1632892294tant_a @ A @ ( domain1362581744um_a_b @ R ) ) ) ).
% Domain.DomainI
thf(fact_107_Domain_ODomainI,axiom,
! [A: sum_sum_a_b,B: sum_sum_a_b,R: set_Pr1916610317um_a_b] :
( ( member947389014um_a_b @ ( produc176426981um_a_b @ A @ B ) @ R )
=> ( member_Sum_sum_a_b @ A @ ( domain2069673178um_a_b @ R ) ) ) ).
% Domain.DomainI
thf(fact_108_Domain_Osimps,axiom,
! [A: standard_Constant_a,R: set_Pr1647387645at_nat] :
( ( member1632892294tant_a @ A @ ( domain1060562500at_nat @ R ) )
= ( ? [A6: standard_Constant_a,B4: product_prod_nat_nat] :
( ( A = A6 )
& ( member1696759390at_nat @ ( produc407553657at_nat @ A6 @ B4 ) @ R ) ) ) ) ).
% Domain.simps
thf(fact_109_Domain_Osimps,axiom,
! [A: nat,R: set_Pr1986765409at_nat] :
( ( member_nat @ A @ ( domain_nat_nat @ R ) )
= ( ? [A6: nat,B4: nat] :
( ( A = A6 )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ A6 @ B4 ) @ R ) ) ) ) ).
% Domain.simps
thf(fact_110_Domain_Osimps,axiom,
! [A: nat,R: set_Pr1174980151um_a_b] :
( ( member_nat @ A @ ( domain1368163076um_a_b @ R ) )
= ( ? [A6: nat,B4: sum_sum_a_b] :
( ( A = A6 )
& ( member1294585472um_a_b @ ( produc1808556047um_a_b @ A6 @ B4 ) @ R ) ) ) ) ).
% Domain.simps
thf(fact_111_Domain_Osimps,axiom,
! [A: standard_Constant_a,R: set_Pr409224873um_a_b] :
( ( member1632892294tant_a @ A @ ( domain1362581744um_a_b @ R ) )
= ( ? [A6: standard_Constant_a,B4: produc1548871597um_a_b] :
( ( A = A6 )
& ( member1998628618um_a_b @ ( produc1697725733um_a_b @ A6 @ B4 ) @ R ) ) ) ) ).
% Domain.simps
thf(fact_112_Domain_Osimps,axiom,
! [A: sum_sum_a_b,R: set_Pr1916610317um_a_b] :
( ( member_Sum_sum_a_b @ A @ ( domain2069673178um_a_b @ R ) )
= ( ? [A6: sum_sum_a_b,B4: sum_sum_a_b] :
( ( A = A6 )
& ( member947389014um_a_b @ ( produc176426981um_a_b @ A6 @ B4 ) @ R ) ) ) ) ).
% Domain.simps
thf(fact_113_Domain_Ocases,axiom,
! [A: standard_Constant_a,R: set_Pr1647387645at_nat] :
( ( member1632892294tant_a @ A @ ( domain1060562500at_nat @ R ) )
=> ~ ! [B3: product_prod_nat_nat] :
~ ( member1696759390at_nat @ ( produc407553657at_nat @ A @ B3 ) @ R ) ) ).
% Domain.cases
thf(fact_114_Domain_Ocases,axiom,
! [A: nat,R: set_Pr1986765409at_nat] :
( ( member_nat @ A @ ( domain_nat_nat @ R ) )
=> ~ ! [B3: nat] :
~ ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ R ) ) ).
% Domain.cases
thf(fact_115_Domain_Ocases,axiom,
! [A: nat,R: set_Pr1174980151um_a_b] :
( ( member_nat @ A @ ( domain1368163076um_a_b @ R ) )
=> ~ ! [B3: sum_sum_a_b] :
~ ( member1294585472um_a_b @ ( produc1808556047um_a_b @ A @ B3 ) @ R ) ) ).
% Domain.cases
thf(fact_116_Domain_Ocases,axiom,
! [A: standard_Constant_a,R: set_Pr409224873um_a_b] :
( ( member1632892294tant_a @ A @ ( domain1362581744um_a_b @ R ) )
=> ~ ! [B3: produc1548871597um_a_b] :
~ ( member1998628618um_a_b @ ( produc1697725733um_a_b @ A @ B3 ) @ R ) ) ).
% Domain.cases
thf(fact_117_Domain_Ocases,axiom,
! [A: sum_sum_a_b,R: set_Pr1916610317um_a_b] :
( ( member_Sum_sum_a_b @ A @ ( domain2069673178um_a_b @ R ) )
=> ~ ! [B3: sum_sum_a_b] :
~ ( member947389014um_a_b @ ( produc176426981um_a_b @ A @ B3 ) @ R ) ) ).
% Domain.cases
thf(fact_118_Domain__iff,axiom,
! [A: standard_Constant_a,R: set_Pr1647387645at_nat] :
( ( member1632892294tant_a @ A @ ( domain1060562500at_nat @ R ) )
= ( ? [Y3: product_prod_nat_nat] : ( member1696759390at_nat @ ( produc407553657at_nat @ A @ Y3 ) @ R ) ) ) ).
% Domain_iff
thf(fact_119_Domain__iff,axiom,
! [A: nat,R: set_Pr1986765409at_nat] :
( ( member_nat @ A @ ( domain_nat_nat @ R ) )
= ( ? [Y3: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ R ) ) ) ).
% Domain_iff
thf(fact_120_Domain__iff,axiom,
! [A: nat,R: set_Pr1174980151um_a_b] :
( ( member_nat @ A @ ( domain1368163076um_a_b @ R ) )
= ( ? [Y3: sum_sum_a_b] : ( member1294585472um_a_b @ ( produc1808556047um_a_b @ A @ Y3 ) @ R ) ) ) ).
% Domain_iff
thf(fact_121_Domain__iff,axiom,
! [A: standard_Constant_a,R: set_Pr409224873um_a_b] :
( ( member1632892294tant_a @ A @ ( domain1362581744um_a_b @ R ) )
= ( ? [Y3: produc1548871597um_a_b] : ( member1998628618um_a_b @ ( produc1697725733um_a_b @ A @ Y3 ) @ R ) ) ) ).
% Domain_iff
thf(fact_122_Domain__iff,axiom,
! [A: sum_sum_a_b,R: set_Pr1916610317um_a_b] :
( ( member_Sum_sum_a_b @ A @ ( domain2069673178um_a_b @ R ) )
= ( ? [Y3: sum_sum_a_b] : ( member947389014um_a_b @ ( produc176426981um_a_b @ A @ Y3 ) @ R ) ) ) ).
% Domain_iff
thf(fact_123_DomainE,axiom,
! [A: standard_Constant_a,R: set_Pr1647387645at_nat] :
( ( member1632892294tant_a @ A @ ( domain1060562500at_nat @ R ) )
=> ~ ! [B3: product_prod_nat_nat] :
~ ( member1696759390at_nat @ ( produc407553657at_nat @ A @ B3 ) @ R ) ) ).
% DomainE
thf(fact_124_DomainE,axiom,
! [A: nat,R: set_Pr1986765409at_nat] :
( ( member_nat @ A @ ( domain_nat_nat @ R ) )
=> ~ ! [B3: nat] :
~ ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ R ) ) ).
% DomainE
thf(fact_125_DomainE,axiom,
! [A: nat,R: set_Pr1174980151um_a_b] :
( ( member_nat @ A @ ( domain1368163076um_a_b @ R ) )
=> ~ ! [B3: sum_sum_a_b] :
~ ( member1294585472um_a_b @ ( produc1808556047um_a_b @ A @ B3 ) @ R ) ) ).
% DomainE
thf(fact_126_DomainE,axiom,
! [A: standard_Constant_a,R: set_Pr409224873um_a_b] :
( ( member1632892294tant_a @ A @ ( domain1362581744um_a_b @ R ) )
=> ~ ! [B3: produc1548871597um_a_b] :
~ ( member1998628618um_a_b @ ( produc1697725733um_a_b @ A @ B3 ) @ R ) ) ).
% DomainE
thf(fact_127_DomainE,axiom,
! [A: sum_sum_a_b,R: set_Pr1916610317um_a_b] :
( ( member_Sum_sum_a_b @ A @ ( domain2069673178um_a_b @ R ) )
=> ~ ! [B3: sum_sum_a_b] :
~ ( member947389014um_a_b @ ( produc176426981um_a_b @ A @ B3 ) @ R ) ) ).
% DomainE
thf(fact_128_Domain__mono,axiom,
! [R: set_Pr1174980151um_a_b,S: set_Pr1174980151um_a_b] :
( ( ord_le823954903um_a_b @ R @ S )
=> ( ord_less_eq_set_nat @ ( domain1368163076um_a_b @ R ) @ ( domain1368163076um_a_b @ S ) ) ) ).
% Domain_mono
thf(fact_129_Domain__mono,axiom,
! [R: set_Pr409224873um_a_b,S: set_Pr409224873um_a_b] :
( ( ord_le615126793um_a_b @ R @ S )
=> ( ord_le1739761029tant_a @ ( domain1362581744um_a_b @ R ) @ ( domain1362581744um_a_b @ S ) ) ) ).
% Domain_mono
thf(fact_130_Domain__mono,axiom,
! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ R @ S )
=> ( ord_less_eq_set_nat @ ( domain_nat_nat @ R ) @ ( domain_nat_nat @ S ) ) ) ).
% Domain_mono
thf(fact_131_Domain__mono,axiom,
! [R: set_Pr1647387645at_nat,S: set_Pr1647387645at_nat] :
( ( ord_le1909159005at_nat @ R @ S )
=> ( ord_le1739761029tant_a @ ( domain1060562500at_nat @ R ) @ ( domain1060562500at_nat @ S ) ) ) ).
% Domain_mono
thf(fact_132_Domain__mono,axiom,
! [R: set_Pr1916610317um_a_b,S: set_Pr1916610317um_a_b] :
( ( ord_le1059794605um_a_b @ R @ S )
=> ( ord_le192794300um_a_b @ ( domain2069673178um_a_b @ R ) @ ( domain2069673178um_a_b @ S ) ) ) ).
% Domain_mono
thf(fact_133_labeled__graph_Oexpand,axiom,
! [Labeled_graph: labele935650037_a_nat,Labeled_graph2: labele935650037_a_nat] :
( ( ( ( labele195203296_a_nat @ Labeled_graph )
= ( labele195203296_a_nat @ Labeled_graph2 ) )
& ( ( labele1810595089_a_nat @ Labeled_graph )
= ( labele1810595089_a_nat @ Labeled_graph2 ) ) )
=> ( Labeled_graph = Labeled_graph2 ) ) ).
% labeled_graph.expand
thf(fact_134_labeled__graph_Oexpand,axiom,
! [Labeled_graph: labele431970251um_a_b,Labeled_graph2: labele431970251um_a_b] :
( ( ( ( labele1939049654um_a_b @ Labeled_graph )
= ( labele1939049654um_a_b @ Labeled_graph2 ) )
& ( ( labele577278695um_a_b @ Labeled_graph )
= ( labele577278695um_a_b @ Labeled_graph2 ) ) )
=> ( Labeled_graph = Labeled_graph2 ) ) ).
% labeled_graph.expand
thf(fact_135_labeled__graph_Osel_I2_J,axiom,
! [X1: set_Pr1647387645at_nat,X22: set_nat] :
( ( labele1810595089_a_nat @ ( labele16114835_a_nat @ X1 @ X22 ) )
= X22 ) ).
% labeled_graph.sel(2)
thf(fact_136_labeled__graph_Osel_I2_J,axiom,
! [X1: set_Pr409224873um_a_b,X22: set_Sum_sum_a_b] :
( ( labele577278695um_a_b @ ( labele1729654377um_a_b @ X1 @ X22 ) )
= X22 ) ).
% labeled_graph.sel(2)
thf(fact_137_labeled__graph_Osel_I1_J,axiom,
! [X1: set_Pr1647387645at_nat,X22: set_nat] :
( ( labele195203296_a_nat @ ( labele16114835_a_nat @ X1 @ X22 ) )
= X1 ) ).
% labeled_graph.sel(1)
thf(fact_138_labeled__graph_Osel_I1_J,axiom,
! [X1: set_Pr409224873um_a_b,X22: set_Sum_sum_a_b] :
( ( labele1939049654um_a_b @ ( labele1729654377um_a_b @ X1 @ X22 ) )
= X1 ) ).
% labeled_graph.sel(1)
thf(fact_139_Domain__empty__iff,axiom,
! [R: set_Pr1647387645at_nat] :
( ( ( domain1060562500at_nat @ R )
= bot_bo1160111033tant_a )
= ( R = bot_bo810816657at_nat ) ) ).
% Domain_empty_iff
thf(fact_140_Domain__empty__iff,axiom,
! [R: set_Pr409224873um_a_b] :
( ( ( domain1362581744um_a_b @ R )
= bot_bo1160111033tant_a )
= ( R = bot_bo1262634813um_a_b ) ) ).
% Domain_empty_iff
thf(fact_141_Domain__empty__iff,axiom,
! [R: set_Pr1986765409at_nat] :
( ( ( domain_nat_nat @ R )
= bot_bot_set_nat )
= ( R = bot_bo2130386637at_nat ) ) ).
% Domain_empty_iff
thf(fact_142_Domain__empty__iff,axiom,
! [R: set_Pr1174980151um_a_b] :
( ( ( domain1368163076um_a_b @ R )
= bot_bot_set_nat )
= ( R = bot_bo575978147um_a_b ) ) ).
% Domain_empty_iff
thf(fact_143_Domain__empty__iff,axiom,
! [R: set_Pr1916610317um_a_b] :
( ( ( domain2069673178um_a_b @ R )
= bot_bo1491243248um_a_b )
= ( R = bot_bo225809273um_a_b ) ) ).
% Domain_empty_iff
thf(fact_144_edge__preserving__atomic,axiom,
! [H1: set_Pr375490359_b_nat,E1: set_Pr409224873um_a_b,E2: set_Pr1647387645at_nat,V1: sum_sum_a_b,V12: nat,V22: sum_sum_a_b,V23: nat,K: standard_Constant_a] :
( ( edge_p749155930tant_a @ H1 @ E1 @ E2 )
=> ( ( member1249152_b_nat @ ( produc522153487_b_nat @ V1 @ V12 ) @ H1 )
=> ( ( member1249152_b_nat @ ( produc522153487_b_nat @ V22 @ V23 ) @ H1 )
=> ( ( member1998628618um_a_b @ ( produc1697725733um_a_b @ K @ ( produc176426981um_a_b @ V1 @ V22 ) ) @ E1 )
=> ( member1696759390at_nat @ ( produc407553657at_nat @ K @ ( product_Pair_nat_nat @ V12 @ V23 ) ) @ E2 ) ) ) ) ) ).
% edge_preserving_atomic
thf(fact_145_edge__preserving__atomic,axiom,
! [H1: set_Pr1986765409at_nat,E1: set_Pr1647387645at_nat,E2: set_Pr1647387645at_nat,V1: nat,V12: nat,V22: nat,V23: nat,K: standard_Constant_a] :
( ( edge_p170421892tant_a @ H1 @ E1 @ E2 )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ V1 @ V12 ) @ H1 )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ V22 @ V23 ) @ H1 )
=> ( ( member1696759390at_nat @ ( produc407553657at_nat @ K @ ( product_Pair_nat_nat @ V1 @ V22 ) ) @ E1 )
=> ( member1696759390at_nat @ ( produc407553657at_nat @ K @ ( product_Pair_nat_nat @ V12 @ V23 ) ) @ E2 ) ) ) ) ) ).
% edge_preserving_atomic
thf(fact_146_edge__preserving__atomic,axiom,
! [H1: set_Pr1916610317um_a_b,E1: set_Pr409224873um_a_b,E2: set_Pr409224873um_a_b,V1: sum_sum_a_b,V12: sum_sum_a_b,V22: sum_sum_a_b,V23: sum_sum_a_b,K: standard_Constant_a] :
( ( edge_p1871801392tant_a @ H1 @ E1 @ E2 )
=> ( ( member947389014um_a_b @ ( produc176426981um_a_b @ V1 @ V12 ) @ H1 )
=> ( ( member947389014um_a_b @ ( produc176426981um_a_b @ V22 @ V23 ) @ H1 )
=> ( ( member1998628618um_a_b @ ( produc1697725733um_a_b @ K @ ( produc176426981um_a_b @ V1 @ V22 ) ) @ E1 )
=> ( member1998628618um_a_b @ ( produc1697725733um_a_b @ K @ ( produc176426981um_a_b @ V12 @ V23 ) ) @ E2 ) ) ) ) ) ).
% edge_preserving_atomic
thf(fact_147_edge__preserving__atomic,axiom,
! [H1: set_Pr1174980151um_a_b,E1: set_Pr1647387645at_nat,E2: set_Pr409224873um_a_b,V1: nat,V12: sum_sum_a_b,V22: nat,V23: sum_sum_a_b,K: standard_Constant_a] :
( ( edge_p1382426714tant_a @ H1 @ E1 @ E2 )
=> ( ( member1294585472um_a_b @ ( produc1808556047um_a_b @ V1 @ V12 ) @ H1 )
=> ( ( member1294585472um_a_b @ ( produc1808556047um_a_b @ V22 @ V23 ) @ H1 )
=> ( ( member1696759390at_nat @ ( produc407553657at_nat @ K @ ( product_Pair_nat_nat @ V1 @ V22 ) ) @ E1 )
=> ( member1998628618um_a_b @ ( produc1697725733um_a_b @ K @ ( produc176426981um_a_b @ V12 @ V23 ) ) @ E2 ) ) ) ) ) ).
% edge_preserving_atomic
thf(fact_148_labeled__graph_Oexhaust__sel,axiom,
! [Labeled_graph: labele935650037_a_nat] :
( Labeled_graph
= ( labele16114835_a_nat @ ( labele195203296_a_nat @ Labeled_graph ) @ ( labele1810595089_a_nat @ Labeled_graph ) ) ) ).
% labeled_graph.exhaust_sel
thf(fact_149_labeled__graph_Oexhaust__sel,axiom,
! [Labeled_graph: labele431970251um_a_b] :
( Labeled_graph
= ( labele1729654377um_a_b @ ( labele1939049654um_a_b @ Labeled_graph ) @ ( labele577278695um_a_b @ Labeled_graph ) ) ) ).
% labeled_graph.exhaust_sel
thf(fact_150_singleton__insert__inj__eq_H,axiom,
! [A: standard_Constant_a,A3: set_St761939237tant_a,B: standard_Constant_a] :
( ( ( insert1909710879tant_a @ A @ A3 )
= ( insert1909710879tant_a @ B @ bot_bo1160111033tant_a ) )
= ( ( A = B )
& ( ord_le1739761029tant_a @ A3 @ ( insert1909710879tant_a @ B @ bot_bo1160111033tant_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_151_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_nat_nat,A3: set_Pr1986765409at_nat,B: product_prod_nat_nat] :
( ( ( insert271595217at_nat @ A @ A3 )
= ( insert271595217at_nat @ B @ bot_bo2130386637at_nat ) )
= ( ( A = B )
& ( ord_le841296385at_nat @ A3 @ ( insert271595217at_nat @ B @ bot_bo2130386637at_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_152_singleton__insert__inj__eq_H,axiom,
! [A: produc1032616263at_nat,A3: set_Pr1647387645at_nat,B: produc1032616263at_nat] :
( ( ( insert1625259895at_nat @ A @ A3 )
= ( insert1625259895at_nat @ B @ bot_bo810816657at_nat ) )
= ( ( A = B )
& ( ord_le1909159005at_nat @ A3 @ ( insert1625259895at_nat @ B @ bot_bo810816657at_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_153_singleton__insert__inj__eq_H,axiom,
! [A: nat,A3: set_nat,B: nat] :
( ( ( insert_nat @ A @ A3 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_154_singleton__insert__inj__eq_H,axiom,
! [A: produc1124793815um_a_b,A3: set_Pr1174980151um_a_b,B: produc1124793815um_a_b] :
( ( ( insert983991207um_a_b @ A @ A3 )
= ( insert983991207um_a_b @ B @ bot_bo575978147um_a_b ) )
= ( ( A = B )
& ( ord_le823954903um_a_b @ A3 @ ( insert983991207um_a_b @ B @ bot_bo575978147um_a_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_155_singleton__insert__inj__eq_H,axiom,
! [A: produc1548871597um_a_b,A3: set_Pr1916610317um_a_b,B: produc1548871597um_a_b] :
( ( ( insert1435405693um_a_b @ A @ A3 )
= ( insert1435405693um_a_b @ B @ bot_bo225809273um_a_b ) )
= ( ( A = B )
& ( ord_le1059794605um_a_b @ A3 @ ( insert1435405693um_a_b @ B @ bot_bo225809273um_a_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_156_singleton__insert__inj__eq_H,axiom,
! [A: produc1963079155um_a_b,A3: set_Pr409224873um_a_b,B: produc1963079155um_a_b] :
( ( ( insert323157027um_a_b @ A @ A3 )
= ( insert323157027um_a_b @ B @ bot_bo1262634813um_a_b ) )
= ( ( A = B )
& ( ord_le615126793um_a_b @ A3 @ ( insert323157027um_a_b @ B @ bot_bo1262634813um_a_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_157_singleton__insert__inj__eq_H,axiom,
! [A: sum_sum_a_b,A3: set_Sum_sum_a_b,B: sum_sum_a_b] :
( ( ( insert_Sum_sum_a_b @ A @ A3 )
= ( insert_Sum_sum_a_b @ B @ bot_bo1491243248um_a_b ) )
= ( ( A = B )
& ( ord_le192794300um_a_b @ A3 @ ( insert_Sum_sum_a_b @ B @ bot_bo1491243248um_a_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_158_singleton__insert__inj__eq,axiom,
! [B: standard_Constant_a,A: standard_Constant_a,A3: set_St761939237tant_a] :
( ( ( insert1909710879tant_a @ B @ bot_bo1160111033tant_a )
= ( insert1909710879tant_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_le1739761029tant_a @ A3 @ ( insert1909710879tant_a @ B @ bot_bo1160111033tant_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_159_singleton__insert__inj__eq,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat,A3: set_Pr1986765409at_nat] :
( ( ( insert271595217at_nat @ B @ bot_bo2130386637at_nat )
= ( insert271595217at_nat @ A @ A3 ) )
= ( ( A = B )
& ( ord_le841296385at_nat @ A3 @ ( insert271595217at_nat @ B @ bot_bo2130386637at_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_160_singleton__insert__inj__eq,axiom,
! [B: produc1032616263at_nat,A: produc1032616263at_nat,A3: set_Pr1647387645at_nat] :
( ( ( insert1625259895at_nat @ B @ bot_bo810816657at_nat )
= ( insert1625259895at_nat @ A @ A3 ) )
= ( ( A = B )
& ( ord_le1909159005at_nat @ A3 @ ( insert1625259895at_nat @ B @ bot_bo810816657at_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_161_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A3: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_162_singleton__insert__inj__eq,axiom,
! [B: produc1124793815um_a_b,A: produc1124793815um_a_b,A3: set_Pr1174980151um_a_b] :
( ( ( insert983991207um_a_b @ B @ bot_bo575978147um_a_b )
= ( insert983991207um_a_b @ A @ A3 ) )
= ( ( A = B )
& ( ord_le823954903um_a_b @ A3 @ ( insert983991207um_a_b @ B @ bot_bo575978147um_a_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_163_singleton__insert__inj__eq,axiom,
! [B: produc1548871597um_a_b,A: produc1548871597um_a_b,A3: set_Pr1916610317um_a_b] :
( ( ( insert1435405693um_a_b @ B @ bot_bo225809273um_a_b )
= ( insert1435405693um_a_b @ A @ A3 ) )
= ( ( A = B )
& ( ord_le1059794605um_a_b @ A3 @ ( insert1435405693um_a_b @ B @ bot_bo225809273um_a_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_164_singleton__insert__inj__eq,axiom,
! [B: produc1963079155um_a_b,A: produc1963079155um_a_b,A3: set_Pr409224873um_a_b] :
( ( ( insert323157027um_a_b @ B @ bot_bo1262634813um_a_b )
= ( insert323157027um_a_b @ A @ A3 ) )
= ( ( A = B )
& ( ord_le615126793um_a_b @ A3 @ ( insert323157027um_a_b @ B @ bot_bo1262634813um_a_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_165_singleton__insert__inj__eq,axiom,
! [B: sum_sum_a_b,A: sum_sum_a_b,A3: set_Sum_sum_a_b] :
( ( ( insert_Sum_sum_a_b @ B @ bot_bo1491243248um_a_b )
= ( insert_Sum_sum_a_b @ A @ A3 ) )
= ( ( A = B )
& ( ord_le192794300um_a_b @ A3 @ ( insert_Sum_sum_a_b @ B @ bot_bo1491243248um_a_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_166_univalent__empty,axiom,
unival860567212at_nat @ bot_bo810816657at_nat ).
% univalent_empty
thf(fact_167_univalent__empty,axiom,
unival118648434um_a_b @ bot_bo225809273um_a_b ).
% univalent_empty
thf(fact_168_univalent__empty,axiom,
unival1487587672um_a_b @ bot_bo1262634813um_a_b ).
% univalent_empty
thf(fact_169_univalent__empty,axiom,
unival2092813468um_a_b @ bot_bo575978147um_a_b ).
% univalent_empty
thf(fact_170_mem__Collect__eq,axiom,
! [A: sum_sum_a_b,P: sum_sum_a_b > $o] :
( ( member_Sum_sum_a_b @ A @ ( collect_Sum_sum_a_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_171_mem__Collect__eq,axiom,
! [A: produc1124793815um_a_b,P: produc1124793815um_a_b > $o] :
( ( member1294585472um_a_b @ A @ ( collec492052930um_a_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_172_mem__Collect__eq,axiom,
! [A: produc1963079155um_a_b,P: produc1963079155um_a_b > $o] :
( ( member1998628618um_a_b @ A @ ( collec1254333256um_a_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_173_Collect__mem__eq,axiom,
! [A3: set_Sum_sum_a_b] :
( ( collect_Sum_sum_a_b
@ ^ [X3: sum_sum_a_b] : ( member_Sum_sum_a_b @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_174_Collect__mem__eq,axiom,
! [A3: set_Pr1174980151um_a_b] :
( ( collec492052930um_a_b
@ ^ [X3: produc1124793815um_a_b] : ( member1294585472um_a_b @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_175_Collect__mem__eq,axiom,
! [A3: set_Pr409224873um_a_b] :
( ( collec1254333256um_a_b
@ ^ [X3: produc1963079155um_a_b] : ( member1998628618um_a_b @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_176_insert__subset,axiom,
! [X2: produc1032616263at_nat,A3: set_Pr1647387645at_nat,B5: set_Pr1647387645at_nat] :
( ( ord_le1909159005at_nat @ ( insert1625259895at_nat @ X2 @ A3 ) @ B5 )
= ( ( member1696759390at_nat @ X2 @ B5 )
& ( ord_le1909159005at_nat @ A3 @ B5 ) ) ) ).
% insert_subset
thf(fact_177_insert__subset,axiom,
! [X2: nat,A3: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A3 ) @ B5 )
= ( ( member_nat @ X2 @ B5 )
& ( ord_less_eq_set_nat @ A3 @ B5 ) ) ) ).
% insert_subset
thf(fact_178_insert__subset,axiom,
! [X2: produc1548871597um_a_b,A3: set_Pr1916610317um_a_b,B5: set_Pr1916610317um_a_b] :
( ( ord_le1059794605um_a_b @ ( insert1435405693um_a_b @ X2 @ A3 ) @ B5 )
= ( ( member947389014um_a_b @ X2 @ B5 )
& ( ord_le1059794605um_a_b @ A3 @ B5 ) ) ) ).
% insert_subset
thf(fact_179_insert__subset,axiom,
! [X2: standard_Constant_a,A3: set_St761939237tant_a,B5: set_St761939237tant_a] :
( ( ord_le1739761029tant_a @ ( insert1909710879tant_a @ X2 @ A3 ) @ B5 )
= ( ( member1632892294tant_a @ X2 @ B5 )
& ( ord_le1739761029tant_a @ A3 @ B5 ) ) ) ).
% insert_subset
thf(fact_180_insert__subset,axiom,
! [X2: product_prod_nat_nat,A3: set_Pr1986765409at_nat,B5: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ ( insert271595217at_nat @ X2 @ A3 ) @ B5 )
= ( ( member701585322at_nat @ X2 @ B5 )
& ( ord_le841296385at_nat @ A3 @ B5 ) ) ) ).
% insert_subset
thf(fact_181_insert__subset,axiom,
! [X2: produc1124793815um_a_b,A3: set_Pr1174980151um_a_b,B5: set_Pr1174980151um_a_b] :
( ( ord_le823954903um_a_b @ ( insert983991207um_a_b @ X2 @ A3 ) @ B5 )
= ( ( member1294585472um_a_b @ X2 @ B5 )
& ( ord_le823954903um_a_b @ A3 @ B5 ) ) ) ).
% insert_subset
thf(fact_182_insert__subset,axiom,
! [X2: produc1963079155um_a_b,A3: set_Pr409224873um_a_b,B5: set_Pr409224873um_a_b] :
( ( ord_le615126793um_a_b @ ( insert323157027um_a_b @ X2 @ A3 ) @ B5 )
= ( ( member1998628618um_a_b @ X2 @ B5 )
& ( ord_le615126793um_a_b @ A3 @ B5 ) ) ) ).
% insert_subset
thf(fact_183_insert__subset,axiom,
! [X2: sum_sum_a_b,A3: set_Sum_sum_a_b,B5: set_Sum_sum_a_b] :
( ( ord_le192794300um_a_b @ ( insert_Sum_sum_a_b @ X2 @ A3 ) @ B5 )
= ( ( member_Sum_sum_a_b @ X2 @ B5 )
& ( ord_le192794300um_a_b @ A3 @ B5 ) ) ) ).
% insert_subset
thf(fact_184_singletonI,axiom,
! [A: standard_Constant_a] : ( member1632892294tant_a @ A @ ( insert1909710879tant_a @ A @ bot_bo1160111033tant_a ) ) ).
% singletonI
thf(fact_185_singletonI,axiom,
! [A: product_prod_nat_nat] : ( member701585322at_nat @ A @ ( insert271595217at_nat @ A @ bot_bo2130386637at_nat ) ) ).
% singletonI
thf(fact_186_singletonI,axiom,
! [A: produc1032616263at_nat] : ( member1696759390at_nat @ A @ ( insert1625259895at_nat @ A @ bot_bo810816657at_nat ) ) ).
% singletonI
thf(fact_187_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_188_singletonI,axiom,
! [A: produc1124793815um_a_b] : ( member1294585472um_a_b @ A @ ( insert983991207um_a_b @ A @ bot_bo575978147um_a_b ) ) ).
% singletonI
thf(fact_189_singletonI,axiom,
! [A: produc1548871597um_a_b] : ( member947389014um_a_b @ A @ ( insert1435405693um_a_b @ A @ bot_bo225809273um_a_b ) ) ).
% singletonI
thf(fact_190_singletonI,axiom,
! [A: produc1963079155um_a_b] : ( member1998628618um_a_b @ A @ ( insert323157027um_a_b @ A @ bot_bo1262634813um_a_b ) ) ).
% singletonI
thf(fact_191_singletonI,axiom,
! [A: sum_sum_a_b] : ( member_Sum_sum_a_b @ A @ ( insert_Sum_sum_a_b @ A @ bot_bo1491243248um_a_b ) ) ).
% singletonI
thf(fact_192_empty__subsetI,axiom,
! [A3: set_Pr1647387645at_nat] : ( ord_le1909159005at_nat @ bot_bo810816657at_nat @ A3 ) ).
% empty_subsetI
thf(fact_193_empty__subsetI,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A3 ) ).
% empty_subsetI
thf(fact_194_empty__subsetI,axiom,
! [A3: set_Pr1174980151um_a_b] : ( ord_le823954903um_a_b @ bot_bo575978147um_a_b @ A3 ) ).
% empty_subsetI
thf(fact_195_empty__subsetI,axiom,
! [A3: set_Pr1916610317um_a_b] : ( ord_le1059794605um_a_b @ bot_bo225809273um_a_b @ A3 ) ).
% empty_subsetI
thf(fact_196_empty__subsetI,axiom,
! [A3: set_Pr409224873um_a_b] : ( ord_le615126793um_a_b @ bot_bo1262634813um_a_b @ A3 ) ).
% empty_subsetI
thf(fact_197_empty__subsetI,axiom,
! [A3: set_Sum_sum_a_b] : ( ord_le192794300um_a_b @ bot_bo1491243248um_a_b @ A3 ) ).
% empty_subsetI
thf(fact_198_subset__empty,axiom,
! [A3: set_Pr1647387645at_nat] :
( ( ord_le1909159005at_nat @ A3 @ bot_bo810816657at_nat )
= ( A3 = bot_bo810816657at_nat ) ) ).
% subset_empty
thf(fact_199_subset__empty,axiom,
! [A3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
= ( A3 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_200_subset__empty,axiom,
! [A3: set_Pr1174980151um_a_b] :
( ( ord_le823954903um_a_b @ A3 @ bot_bo575978147um_a_b )
= ( A3 = bot_bo575978147um_a_b ) ) ).
% subset_empty
thf(fact_201_subset__empty,axiom,
! [A3: set_Pr1916610317um_a_b] :
( ( ord_le1059794605um_a_b @ A3 @ bot_bo225809273um_a_b )
= ( A3 = bot_bo225809273um_a_b ) ) ).
% subset_empty
thf(fact_202_subset__empty,axiom,
! [A3: set_Pr409224873um_a_b] :
( ( ord_le615126793um_a_b @ A3 @ bot_bo1262634813um_a_b )
= ( A3 = bot_bo1262634813um_a_b ) ) ).
% subset_empty
thf(fact_203_subset__empty,axiom,
! [A3: set_Sum_sum_a_b] :
( ( ord_le192794300um_a_b @ A3 @ bot_bo1491243248um_a_b )
= ( A3 = bot_bo1491243248um_a_b ) ) ).
% subset_empty
thf(fact_204_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_205_reflexivity__rule__def,axiom,
( standa63370785tant_a
= ( ^ [T: standard_Constant_a] : ( produc1676969687_a_nat @ ( labele16114835_a_nat @ bot_bo810816657at_nat @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) @ ( labele16114835_a_nat @ ( insert1625259895at_nat @ ( produc407553657at_nat @ T @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) @ bot_bo810816657at_nat ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% reflexivity_rule_def
thf(fact_206_nonempty__rule__def,axiom,
( standa1410829644tant_a
= ( produc1676969687_a_nat @ ( labele16114835_a_nat @ bot_bo810816657at_nat @ bot_bot_set_nat ) @ ( labele16114835_a_nat @ bot_bo810816657at_nat @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ).
% nonempty_rule_def
thf(fact_207_empty__iff,axiom,
! [C: produc1032616263at_nat] :
~ ( member1696759390at_nat @ C @ bot_bo810816657at_nat ) ).
% empty_iff
thf(fact_208_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_209_empty__iff,axiom,
! [C: produc1124793815um_a_b] :
~ ( member1294585472um_a_b @ C @ bot_bo575978147um_a_b ) ).
% empty_iff
thf(fact_210_empty__iff,axiom,
! [C: produc1548871597um_a_b] :
~ ( member947389014um_a_b @ C @ bot_bo225809273um_a_b ) ).
% empty_iff
thf(fact_211_empty__iff,axiom,
! [C: produc1963079155um_a_b] :
~ ( member1998628618um_a_b @ C @ bot_bo1262634813um_a_b ) ).
% empty_iff
thf(fact_212_empty__iff,axiom,
! [C: sum_sum_a_b] :
~ ( member_Sum_sum_a_b @ C @ bot_bo1491243248um_a_b ) ).
% empty_iff
thf(fact_213_all__not__in__conv,axiom,
! [A3: set_Pr1647387645at_nat] :
( ( ! [X3: produc1032616263at_nat] :
~ ( member1696759390at_nat @ X3 @ A3 ) )
= ( A3 = bot_bo810816657at_nat ) ) ).
% all_not_in_conv
thf(fact_214_all__not__in__conv,axiom,
! [A3: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_215_all__not__in__conv,axiom,
! [A3: set_Pr1174980151um_a_b] :
( ( ! [X3: produc1124793815um_a_b] :
~ ( member1294585472um_a_b @ X3 @ A3 ) )
= ( A3 = bot_bo575978147um_a_b ) ) ).
% all_not_in_conv
thf(fact_216_all__not__in__conv,axiom,
! [A3: set_Pr1916610317um_a_b] :
( ( ! [X3: produc1548871597um_a_b] :
~ ( member947389014um_a_b @ X3 @ A3 ) )
= ( A3 = bot_bo225809273um_a_b ) ) ).
% all_not_in_conv
thf(fact_217_all__not__in__conv,axiom,
! [A3: set_Pr409224873um_a_b] :
( ( ! [X3: produc1963079155um_a_b] :
~ ( member1998628618um_a_b @ X3 @ A3 ) )
= ( A3 = bot_bo1262634813um_a_b ) ) ).
% all_not_in_conv
thf(fact_218_all__not__in__conv,axiom,
! [A3: set_Sum_sum_a_b] :
( ( ! [X3: sum_sum_a_b] :
~ ( member_Sum_sum_a_b @ X3 @ A3 ) )
= ( A3 = bot_bo1491243248um_a_b ) ) ).
% all_not_in_conv
thf(fact_219_Collect__empty__eq,axiom,
! [P: produc1032616263at_nat > $o] :
( ( ( collec52779676at_nat @ P )
= bot_bo810816657at_nat )
= ( ! [X3: produc1032616263at_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_220_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_221_Collect__empty__eq,axiom,
! [P: produc1124793815um_a_b > $o] :
( ( ( collec492052930um_a_b @ P )
= bot_bo575978147um_a_b )
= ( ! [X3: produc1124793815um_a_b] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_222_Collect__empty__eq,axiom,
! [P: produc1548871597um_a_b > $o] :
( ( ( collec1769032088um_a_b @ P )
= bot_bo225809273um_a_b )
= ( ! [X3: produc1548871597um_a_b] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_223_Collect__empty__eq,axiom,
! [P: produc1963079155um_a_b > $o] :
( ( ( collec1254333256um_a_b @ P )
= bot_bo1262634813um_a_b )
= ( ! [X3: produc1963079155um_a_b] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_224_Collect__empty__eq,axiom,
! [P: sum_sum_a_b > $o] :
( ( ( collect_Sum_sum_a_b @ P )
= bot_bo1491243248um_a_b )
= ( ! [X3: sum_sum_a_b] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_225_empty__Collect__eq,axiom,
! [P: produc1032616263at_nat > $o] :
( ( bot_bo810816657at_nat
= ( collec52779676at_nat @ P ) )
= ( ! [X3: produc1032616263at_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_226_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_227_empty__Collect__eq,axiom,
! [P: produc1124793815um_a_b > $o] :
( ( bot_bo575978147um_a_b
= ( collec492052930um_a_b @ P ) )
= ( ! [X3: produc1124793815um_a_b] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_228_empty__Collect__eq,axiom,
! [P: produc1548871597um_a_b > $o] :
( ( bot_bo225809273um_a_b
= ( collec1769032088um_a_b @ P ) )
= ( ! [X3: produc1548871597um_a_b] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_229_empty__Collect__eq,axiom,
! [P: produc1963079155um_a_b > $o] :
( ( bot_bo1262634813um_a_b
= ( collec1254333256um_a_b @ P ) )
= ( ! [X3: produc1963079155um_a_b] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_230_empty__Collect__eq,axiom,
! [P: sum_sum_a_b > $o] :
( ( bot_bo1491243248um_a_b
= ( collect_Sum_sum_a_b @ P ) )
= ( ! [X3: sum_sum_a_b] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_231_subsetI,axiom,
! [A3: set_Pr1174980151um_a_b,B5: set_Pr1174980151um_a_b] :
( ! [X4: produc1124793815um_a_b] :
( ( member1294585472um_a_b @ X4 @ A3 )
=> ( member1294585472um_a_b @ X4 @ B5 ) )
=> ( ord_le823954903um_a_b @ A3 @ B5 ) ) ).
% subsetI
thf(fact_232_subsetI,axiom,
! [A3: set_Pr409224873um_a_b,B5: set_Pr409224873um_a_b] :
( ! [X4: produc1963079155um_a_b] :
( ( member1998628618um_a_b @ X4 @ A3 )
=> ( member1998628618um_a_b @ X4 @ B5 ) )
=> ( ord_le615126793um_a_b @ A3 @ B5 ) ) ).
% subsetI
thf(fact_233_subsetI,axiom,
! [A3: set_Sum_sum_a_b,B5: set_Sum_sum_a_b] :
( ! [X4: sum_sum_a_b] :
( ( member_Sum_sum_a_b @ X4 @ A3 )
=> ( member_Sum_sum_a_b @ X4 @ B5 ) )
=> ( ord_le192794300um_a_b @ A3 @ B5 ) ) ).
% subsetI
thf(fact_234_subset__antisym,axiom,
! [A3: set_Sum_sum_a_b,B5: set_Sum_sum_a_b] :
( ( ord_le192794300um_a_b @ A3 @ B5 )
=> ( ( ord_le192794300um_a_b @ B5 @ A3 )
=> ( A3 = B5 ) ) ) ).
% subset_antisym
thf(fact_235_insertCI,axiom,
! [A: produc1032616263at_nat,B5: set_Pr1647387645at_nat,B: produc1032616263at_nat] :
( ( ~ ( member1696759390at_nat @ A @ B5 )
=> ( A = B ) )
=> ( member1696759390at_nat @ A @ ( insert1625259895at_nat @ B @ B5 ) ) ) ).
% insertCI
thf(fact_236_insertCI,axiom,
! [A: nat,B5: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B5 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B5 ) ) ) ).
% insertCI
thf(fact_237_insertCI,axiom,
! [A: produc1548871597um_a_b,B5: set_Pr1916610317um_a_b,B: produc1548871597um_a_b] :
( ( ~ ( member947389014um_a_b @ A @ B5 )
=> ( A = B ) )
=> ( member947389014um_a_b @ A @ ( insert1435405693um_a_b @ B @ B5 ) ) ) ).
% insertCI
thf(fact_238_insertCI,axiom,
! [A: standard_Constant_a,B5: set_St761939237tant_a,B: standard_Constant_a] :
( ( ~ ( member1632892294tant_a @ A @ B5 )
=> ( A = B ) )
=> ( member1632892294tant_a @ A @ ( insert1909710879tant_a @ B @ B5 ) ) ) ).
% insertCI
thf(fact_239_insertCI,axiom,
! [A: product_prod_nat_nat,B5: set_Pr1986765409at_nat,B: product_prod_nat_nat] :
( ( ~ ( member701585322at_nat @ A @ B5 )
=> ( A = B ) )
=> ( member701585322at_nat @ A @ ( insert271595217at_nat @ B @ B5 ) ) ) ).
% insertCI
thf(fact_240_insertCI,axiom,
! [A: sum_sum_a_b,B5: set_Sum_sum_a_b,B: sum_sum_a_b] :
( ( ~ ( member_Sum_sum_a_b @ A @ B5 )
=> ( A = B ) )
=> ( member_Sum_sum_a_b @ A @ ( insert_Sum_sum_a_b @ B @ B5 ) ) ) ).
% insertCI
thf(fact_241_insertCI,axiom,
! [A: produc1124793815um_a_b,B5: set_Pr1174980151um_a_b,B: produc1124793815um_a_b] :
( ( ~ ( member1294585472um_a_b @ A @ B5 )
=> ( A = B ) )
=> ( member1294585472um_a_b @ A @ ( insert983991207um_a_b @ B @ B5 ) ) ) ).
% insertCI
thf(fact_242_insertCI,axiom,
! [A: produc1963079155um_a_b,B5: set_Pr409224873um_a_b,B: produc1963079155um_a_b] :
( ( ~ ( member1998628618um_a_b @ A @ B5 )
=> ( A = B ) )
=> ( member1998628618um_a_b @ A @ ( insert323157027um_a_b @ B @ B5 ) ) ) ).
% insertCI
thf(fact_243_insert__iff,axiom,
! [A: produc1032616263at_nat,B: produc1032616263at_nat,A3: set_Pr1647387645at_nat] :
( ( member1696759390at_nat @ A @ ( insert1625259895at_nat @ B @ A3 ) )
= ( ( A = B )
| ( member1696759390at_nat @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_244_insert__iff,axiom,
! [A: nat,B: nat,A3: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A3 ) )
= ( ( A = B )
| ( member_nat @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_245_insert__iff,axiom,
! [A: produc1548871597um_a_b,B: produc1548871597um_a_b,A3: set_Pr1916610317um_a_b] :
( ( member947389014um_a_b @ A @ ( insert1435405693um_a_b @ B @ A3 ) )
= ( ( A = B )
| ( member947389014um_a_b @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_246_insert__iff,axiom,
! [A: standard_Constant_a,B: standard_Constant_a,A3: set_St761939237tant_a] :
( ( member1632892294tant_a @ A @ ( insert1909710879tant_a @ B @ A3 ) )
= ( ( A = B )
| ( member1632892294tant_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_247_insert__iff,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A3: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ A @ ( insert271595217at_nat @ B @ A3 ) )
= ( ( A = B )
| ( member701585322at_nat @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_248_insert__iff,axiom,
! [A: sum_sum_a_b,B: sum_sum_a_b,A3: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b @ A @ ( insert_Sum_sum_a_b @ B @ A3 ) )
= ( ( A = B )
| ( member_Sum_sum_a_b @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_249_insert__iff,axiom,
! [A: produc1124793815um_a_b,B: produc1124793815um_a_b,A3: set_Pr1174980151um_a_b] :
( ( member1294585472um_a_b @ A @ ( insert983991207um_a_b @ B @ A3 ) )
= ( ( A = B )
| ( member1294585472um_a_b @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_250_insert__iff,axiom,
! [A: produc1963079155um_a_b,B: produc1963079155um_a_b,A3: set_Pr409224873um_a_b] :
( ( member1998628618um_a_b @ A @ ( insert323157027um_a_b @ B @ A3 ) )
= ( ( A = B )
| ( member1998628618um_a_b @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_251_insert__absorb2,axiom,
! [X2: produc1032616263at_nat,A3: set_Pr1647387645at_nat] :
( ( insert1625259895at_nat @ X2 @ ( insert1625259895at_nat @ X2 @ A3 ) )
= ( insert1625259895at_nat @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_252_insert__absorb2,axiom,
! [X2: nat,A3: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A3 ) )
= ( insert_nat @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_253_insert__absorb2,axiom,
! [X2: produc1124793815um_a_b,A3: set_Pr1174980151um_a_b] :
( ( insert983991207um_a_b @ X2 @ ( insert983991207um_a_b @ X2 @ A3 ) )
= ( insert983991207um_a_b @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_254_insert__absorb2,axiom,
! [X2: produc1963079155um_a_b,A3: set_Pr409224873um_a_b] :
( ( insert323157027um_a_b @ X2 @ ( insert323157027um_a_b @ X2 @ A3 ) )
= ( insert323157027um_a_b @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_255_insert__absorb2,axiom,
! [X2: sum_sum_a_b,A3: set_Sum_sum_a_b] :
( ( insert_Sum_sum_a_b @ X2 @ ( insert_Sum_sum_a_b @ X2 @ A3 ) )
= ( insert_Sum_sum_a_b @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_256_insert__absorb2,axiom,
! [X2: produc1548871597um_a_b,A3: set_Pr1916610317um_a_b] :
( ( insert1435405693um_a_b @ X2 @ ( insert1435405693um_a_b @ X2 @ A3 ) )
= ( insert1435405693um_a_b @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_257_insert__absorb2,axiom,
! [X2: standard_Constant_a,A3: set_St761939237tant_a] :
( ( insert1909710879tant_a @ X2 @ ( insert1909710879tant_a @ X2 @ A3 ) )
= ( insert1909710879tant_a @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_258_insert__absorb2,axiom,
! [X2: product_prod_nat_nat,A3: set_Pr1986765409at_nat] :
( ( insert271595217at_nat @ X2 @ ( insert271595217at_nat @ X2 @ A3 ) )
= ( insert271595217at_nat @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_259_graph__homomorphism__empty,axiom,
! [G: labele935650037_a_nat,F: set_Pr1986765409at_nat] :
( ( graph_2130075512at_nat @ ( labele16114835_a_nat @ bot_bo810816657at_nat @ bot_bot_set_nat ) @ G @ F )
= ( ( F = bot_bo2130386637at_nat )
& ( G
= ( restri572569417_a_nat @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_260_graph__homomorphism__empty,axiom,
! [G: labele935650037_a_nat,F: set_Pr375490359_b_nat] :
( ( graph_165730638_b_nat @ ( labele1729654377um_a_b @ bot_bo1262634813um_a_b @ bot_bo1491243248um_a_b ) @ G @ F )
= ( ( F = bot_bo1923972003_b_nat )
& ( G
= ( restri572569417_a_nat @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_261_graph__homomorphism__empty,axiom,
! [G: labele431970251um_a_b,F: set_Pr1174980151um_a_b] :
( ( graph_1452133198um_a_b @ ( labele16114835_a_nat @ bot_bo810816657at_nat @ bot_bot_set_nat ) @ G @ F )
= ( ( F = bot_bo575978147um_a_b )
& ( G
= ( restri1162247455um_a_b @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_262_graph__homomorphism__empty,axiom,
! [G: labele935650037_a_nat,F: set_Pr1902304780_b_nat] :
( ( graph_1203757137_b_nat @ ( labele2034032468um_a_b @ bot_bo1208513655um_a_b @ bot_bo575978147um_a_b ) @ G @ F )
= ( ( F = bot_bo949957280_b_nat )
& ( G
= ( restri572569417_a_nat @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_263_graph__homomorphism__empty,axiom,
! [G: labele431970251um_a_b,F: set_Pr1916610317um_a_b] :
( ( graph_611893540um_a_b @ ( labele1729654377um_a_b @ bot_bo1262634813um_a_b @ bot_bo1491243248um_a_b ) @ G @ F )
= ( ( F = bot_bo225809273um_a_b )
& ( G
= ( restri1162247455um_a_b @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_264_graph__homomorphism__empty,axiom,
! [G: labele935650037_a_nat,F: set_Pr689416536at_nat] :
( ( graph_149544303at_nat @ ( labele1813985418at_nat @ bot_bo1524397311at_nat @ bot_bo810816657at_nat ) @ G @ F )
= ( ( F = bot_bo1582968772at_nat )
& ( G
= ( restri572569417_a_nat @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_265_graph__homomorphism__empty,axiom,
! [G: labele431970251um_a_b,F: set_Pr653770722um_a_b] :
( ( graph_502887975um_a_b @ ( labele2034032468um_a_b @ bot_bo1208513655um_a_b @ bot_bo575978147um_a_b ) @ G @ F )
= ( ( F = bot_bo1892785270um_a_b )
& ( G
= ( restri1162247455um_a_b @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_266_graph__homomorphism__empty,axiom,
! [G: labele935650037_a_nat,F: set_Pr507214050_b_nat] :
( ( graph_1815313447_b_nat @ ( labele989994um_a_b @ bot_bo1262778871um_a_b @ bot_bo225809273um_a_b ) @ G @ F )
= ( ( F = bot_bo1746228598_b_nat )
& ( G
= ( restri572569417_a_nat @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_267_graph__homomorphism__empty,axiom,
! [G: labele431970251um_a_b,F: set_Pr304595758um_a_b] :
( ( graph_166346565um_a_b @ ( labele1813985418at_nat @ bot_bo1524397311at_nat @ bot_bo810816657at_nat ) @ G @ F )
= ( ( F = bot_bo627753370um_a_b )
& ( G
= ( restri1162247455um_a_b @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_268_graph__homomorphism__empty,axiom,
! [G: labele431970251um_a_b,F: set_Pr55236280um_a_b] :
( ( graph_1763806717um_a_b @ ( labele989994um_a_b @ bot_bo1262778871um_a_b @ bot_bo225809273um_a_b ) @ G @ F )
= ( ( F = bot_bo1217542988um_a_b )
& ( G
= ( restri1162247455um_a_b @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_269_edge__preserving__subset,axiom,
! [R_1: set_Pr1174980151um_a_b,R_2: set_Pr1174980151um_a_b,E_1: set_Pr1647387645at_nat,E_2: set_Pr1647387645at_nat,G: set_Pr409224873um_a_b] :
( ( ord_le823954903um_a_b @ R_1 @ R_2 )
=> ( ( ord_le1909159005at_nat @ E_1 @ E_2 )
=> ( ( edge_p1382426714tant_a @ R_2 @ E_2 @ G )
=> ( edge_p1382426714tant_a @ R_1 @ E_1 @ G ) ) ) ) ).
% edge_preserving_subset
thf(fact_270_subrelI,axiom,
! [R: set_Pr1647387645at_nat,S: set_Pr1647387645at_nat] :
( ! [X4: standard_Constant_a,Y4: product_prod_nat_nat] :
( ( member1696759390at_nat @ ( produc407553657at_nat @ X4 @ Y4 ) @ R )
=> ( member1696759390at_nat @ ( produc407553657at_nat @ X4 @ Y4 ) @ S ) )
=> ( ord_le1909159005at_nat @ R @ S ) ) ).
% subrelI
thf(fact_271_subrelI,axiom,
! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
( ! [X4: nat,Y4: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ S ) )
=> ( ord_le841296385at_nat @ R @ S ) ) ).
% subrelI
thf(fact_272_subrelI,axiom,
! [R: set_Pr1174980151um_a_b,S: set_Pr1174980151um_a_b] :
( ! [X4: nat,Y4: sum_sum_a_b] :
( ( member1294585472um_a_b @ ( produc1808556047um_a_b @ X4 @ Y4 ) @ R )
=> ( member1294585472um_a_b @ ( produc1808556047um_a_b @ X4 @ Y4 ) @ S ) )
=> ( ord_le823954903um_a_b @ R @ S ) ) ).
% subrelI
thf(fact_273_subrelI,axiom,
! [R: set_Pr409224873um_a_b,S: set_Pr409224873um_a_b] :
( ! [X4: standard_Constant_a,Y4: produc1548871597um_a_b] :
( ( member1998628618um_a_b @ ( produc1697725733um_a_b @ X4 @ Y4 ) @ R )
=> ( member1998628618um_a_b @ ( produc1697725733um_a_b @ X4 @ Y4 ) @ S ) )
=> ( ord_le615126793um_a_b @ R @ S ) ) ).
% subrelI
thf(fact_274_subrelI,axiom,
! [R: set_Pr1916610317um_a_b,S: set_Pr1916610317um_a_b] :
( ! [X4: sum_sum_a_b,Y4: sum_sum_a_b] :
( ( member947389014um_a_b @ ( produc176426981um_a_b @ X4 @ Y4 ) @ R )
=> ( member947389014um_a_b @ ( produc176426981um_a_b @ X4 @ Y4 ) @ S ) )
=> ( ord_le1059794605um_a_b @ R @ S ) ) ).
% subrelI
thf(fact_275_restrict__subsD,axiom,
! [G: labele935650037_a_nat] :
( ( ord_le1909159005at_nat @ ( labele195203296_a_nat @ G ) @ ( labele195203296_a_nat @ ( restri572569417_a_nat @ G ) ) )
=> ( G
= ( restri572569417_a_nat @ G ) ) ) ).
% restrict_subsD
thf(fact_276_restrict__subsD,axiom,
! [G: labele431970251um_a_b] :
( ( ord_le615126793um_a_b @ ( labele1939049654um_a_b @ G ) @ ( labele1939049654um_a_b @ ( restri1162247455um_a_b @ G ) ) )
=> ( G
= ( restri1162247455um_a_b @ G ) ) ) ).
% restrict_subsD
thf(fact_277_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_278_emptyE,axiom,
! [A: produc1032616263at_nat] :
~ ( member1696759390at_nat @ A @ bot_bo810816657at_nat ) ).
% emptyE
thf(fact_279_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_280_emptyE,axiom,
! [A: produc1124793815um_a_b] :
~ ( member1294585472um_a_b @ A @ bot_bo575978147um_a_b ) ).
% emptyE
thf(fact_281_emptyE,axiom,
! [A: produc1548871597um_a_b] :
~ ( member947389014um_a_b @ A @ bot_bo225809273um_a_b ) ).
% emptyE
thf(fact_282_emptyE,axiom,
! [A: produc1963079155um_a_b] :
~ ( member1998628618um_a_b @ A @ bot_bo1262634813um_a_b ) ).
% emptyE
thf(fact_283_emptyE,axiom,
! [A: sum_sum_a_b] :
~ ( member_Sum_sum_a_b @ A @ bot_bo1491243248um_a_b ) ).
% emptyE
thf(fact_284_equals0D,axiom,
! [A3: set_Pr1647387645at_nat,A: produc1032616263at_nat] :
( ( A3 = bot_bo810816657at_nat )
=> ~ ( member1696759390at_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_285_equals0D,axiom,
! [A3: set_nat,A: nat] :
( ( A3 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_286_equals0D,axiom,
! [A3: set_Pr1174980151um_a_b,A: produc1124793815um_a_b] :
( ( A3 = bot_bo575978147um_a_b )
=> ~ ( member1294585472um_a_b @ A @ A3 ) ) ).
% equals0D
thf(fact_287_equals0D,axiom,
! [A3: set_Pr1916610317um_a_b,A: produc1548871597um_a_b] :
( ( A3 = bot_bo225809273um_a_b )
=> ~ ( member947389014um_a_b @ A @ A3 ) ) ).
% equals0D
thf(fact_288_equals0D,axiom,
! [A3: set_Pr409224873um_a_b,A: produc1963079155um_a_b] :
( ( A3 = bot_bo1262634813um_a_b )
=> ~ ( member1998628618um_a_b @ A @ A3 ) ) ).
% equals0D
thf(fact_289_equals0D,axiom,
! [A3: set_Sum_sum_a_b,A: sum_sum_a_b] :
( ( A3 = bot_bo1491243248um_a_b )
=> ~ ( member_Sum_sum_a_b @ A @ A3 ) ) ).
% equals0D
thf(fact_290_equals0I,axiom,
! [A3: set_Pr1647387645at_nat] :
( ! [Y4: produc1032616263at_nat] :
~ ( member1696759390at_nat @ Y4 @ A3 )
=> ( A3 = bot_bo810816657at_nat ) ) ).
% equals0I
thf(fact_291_equals0I,axiom,
! [A3: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A3 )
=> ( A3 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_292_equals0I,axiom,
! [A3: set_Pr1174980151um_a_b] :
( ! [Y4: produc1124793815um_a_b] :
~ ( member1294585472um_a_b @ Y4 @ A3 )
=> ( A3 = bot_bo575978147um_a_b ) ) ).
% equals0I
thf(fact_293_equals0I,axiom,
! [A3: set_Pr1916610317um_a_b] :
( ! [Y4: produc1548871597um_a_b] :
~ ( member947389014um_a_b @ Y4 @ A3 )
=> ( A3 = bot_bo225809273um_a_b ) ) ).
% equals0I
thf(fact_294_equals0I,axiom,
! [A3: set_Pr409224873um_a_b] :
( ! [Y4: produc1963079155um_a_b] :
~ ( member1998628618um_a_b @ Y4 @ A3 )
=> ( A3 = bot_bo1262634813um_a_b ) ) ).
% equals0I
thf(fact_295_equals0I,axiom,
! [A3: set_Sum_sum_a_b] :
( ! [Y4: sum_sum_a_b] :
~ ( member_Sum_sum_a_b @ Y4 @ A3 )
=> ( A3 = bot_bo1491243248um_a_b ) ) ).
% equals0I
thf(fact_296_ex__in__conv,axiom,
! [A3: set_Pr1647387645at_nat] :
( ( ? [X3: produc1032616263at_nat] : ( member1696759390at_nat @ X3 @ A3 ) )
= ( A3 != bot_bo810816657at_nat ) ) ).
% ex_in_conv
thf(fact_297_ex__in__conv,axiom,
! [A3: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A3 ) )
= ( A3 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_298_ex__in__conv,axiom,
! [A3: set_Pr1174980151um_a_b] :
( ( ? [X3: produc1124793815um_a_b] : ( member1294585472um_a_b @ X3 @ A3 ) )
= ( A3 != bot_bo575978147um_a_b ) ) ).
% ex_in_conv
thf(fact_299_ex__in__conv,axiom,
! [A3: set_Pr1916610317um_a_b] :
( ( ? [X3: produc1548871597um_a_b] : ( member947389014um_a_b @ X3 @ A3 ) )
= ( A3 != bot_bo225809273um_a_b ) ) ).
% ex_in_conv
thf(fact_300_ex__in__conv,axiom,
! [A3: set_Pr409224873um_a_b] :
( ( ? [X3: produc1963079155um_a_b] : ( member1998628618um_a_b @ X3 @ A3 ) )
= ( A3 != bot_bo1262634813um_a_b ) ) ).
% ex_in_conv
thf(fact_301_ex__in__conv,axiom,
! [A3: set_Sum_sum_a_b] :
( ( ? [X3: sum_sum_a_b] : ( member_Sum_sum_a_b @ X3 @ A3 ) )
= ( A3 != bot_bo1491243248um_a_b ) ) ).
% ex_in_conv
thf(fact_302_in__mono,axiom,
! [A3: set_Pr1174980151um_a_b,B5: set_Pr1174980151um_a_b,X2: produc1124793815um_a_b] :
( ( ord_le823954903um_a_b @ A3 @ B5 )
=> ( ( member1294585472um_a_b @ X2 @ A3 )
=> ( member1294585472um_a_b @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_303_in__mono,axiom,
! [A3: set_Pr409224873um_a_b,B5: set_Pr409224873um_a_b,X2: produc1963079155um_a_b] :
( ( ord_le615126793um_a_b @ A3 @ B5 )
=> ( ( member1998628618um_a_b @ X2 @ A3 )
=> ( member1998628618um_a_b @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_304_in__mono,axiom,
! [A3: set_Sum_sum_a_b,B5: set_Sum_sum_a_b,X2: sum_sum_a_b] :
( ( ord_le192794300um_a_b @ A3 @ B5 )
=> ( ( member_Sum_sum_a_b @ X2 @ A3 )
=> ( member_Sum_sum_a_b @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_305_subsetD,axiom,
! [A3: set_Pr1174980151um_a_b,B5: set_Pr1174980151um_a_b,C: produc1124793815um_a_b] :
( ( ord_le823954903um_a_b @ A3 @ B5 )
=> ( ( member1294585472um_a_b @ C @ A3 )
=> ( member1294585472um_a_b @ C @ B5 ) ) ) ).
% subsetD
thf(fact_306_subsetD,axiom,
! [A3: set_Pr409224873um_a_b,B5: set_Pr409224873um_a_b,C: produc1963079155um_a_b] :
( ( ord_le615126793um_a_b @ A3 @ B5 )
=> ( ( member1998628618um_a_b @ C @ A3 )
=> ( member1998628618um_a_b @ C @ B5 ) ) ) ).
% subsetD
thf(fact_307_subsetD,axiom,
! [A3: set_Sum_sum_a_b,B5: set_Sum_sum_a_b,C: sum_sum_a_b] :
( ( ord_le192794300um_a_b @ A3 @ B5 )
=> ( ( member_Sum_sum_a_b @ C @ A3 )
=> ( member_Sum_sum_a_b @ C @ B5 ) ) ) ).
% subsetD
thf(fact_308_equalityE,axiom,
! [A3: set_Sum_sum_a_b,B5: set_Sum_sum_a_b] :
( ( A3 = B5 )
=> ~ ( ( ord_le192794300um_a_b @ A3 @ B5 )
=> ~ ( ord_le192794300um_a_b @ B5 @ A3 ) ) ) ).
% equalityE
thf(fact_309_subset__eq,axiom,
( ord_le823954903um_a_b
= ( ^ [A7: set_Pr1174980151um_a_b,B6: set_Pr1174980151um_a_b] :
! [X3: produc1124793815um_a_b] :
( ( member1294585472um_a_b @ X3 @ A7 )
=> ( member1294585472um_a_b @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_310_subset__eq,axiom,
( ord_le615126793um_a_b
= ( ^ [A7: set_Pr409224873um_a_b,B6: set_Pr409224873um_a_b] :
! [X3: produc1963079155um_a_b] :
( ( member1998628618um_a_b @ X3 @ A7 )
=> ( member1998628618um_a_b @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_311_subset__eq,axiom,
( ord_le192794300um_a_b
= ( ^ [A7: set_Sum_sum_a_b,B6: set_Sum_sum_a_b] :
! [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b @ X3 @ A7 )
=> ( member_Sum_sum_a_b @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_312_equalityD1,axiom,
! [A3: set_Sum_sum_a_b,B5: set_Sum_sum_a_b] :
( ( A3 = B5 )
=> ( ord_le192794300um_a_b @ A3 @ B5 ) ) ).
% equalityD1
thf(fact_313_equalityD2,axiom,
! [A3: set_Sum_sum_a_b,B5: set_Sum_sum_a_b] :
( ( A3 = B5 )
=> ( ord_le192794300um_a_b @ B5 @ A3 ) ) ).
% equalityD2
thf(fact_314_subset__iff,axiom,
( ord_le823954903um_a_b
= ( ^ [A7: set_Pr1174980151um_a_b,B6: set_Pr1174980151um_a_b] :
! [T: produc1124793815um_a_b] :
( ( member1294585472um_a_b @ T @ A7 )
=> ( member1294585472um_a_b @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_315_subset__iff,axiom,
( ord_le615126793um_a_b
= ( ^ [A7: set_Pr409224873um_a_b,B6: set_Pr409224873um_a_b] :
! [T: produc1963079155um_a_b] :
( ( member1998628618um_a_b @ T @ A7 )
=> ( member1998628618um_a_b @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_316_subset__iff,axiom,
( ord_le192794300um_a_b
= ( ^ [A7: set_Sum_sum_a_b,B6: set_Sum_sum_a_b] :
! [T: sum_sum_a_b] :
( ( member_Sum_sum_a_b @ T @ A7 )
=> ( member_Sum_sum_a_b @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_317_subset__refl,axiom,
! [A3: set_Sum_sum_a_b] : ( ord_le192794300um_a_b @ A3 @ A3 ) ).
% subset_refl
thf(fact_318_Collect__mono,axiom,
! [P: sum_sum_a_b > $o,Q: sum_sum_a_b > $o] :
( ! [X4: sum_sum_a_b] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le192794300um_a_b @ ( collect_Sum_sum_a_b @ P ) @ ( collect_Sum_sum_a_b @ Q ) ) ) ).
% Collect_mono
thf(fact_319_subset__trans,axiom,
! [A3: set_Sum_sum_a_b,B5: set_Sum_sum_a_b,C2: set_Sum_sum_a_b] :
( ( ord_le192794300um_a_b @ A3 @ B5 )
=> ( ( ord_le192794300um_a_b @ B5 @ C2 )
=> ( ord_le192794300um_a_b @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_320_set__eq__subset,axiom,
( ( ^ [Y5: set_Sum_sum_a_b,Z: set_Sum_sum_a_b] : Y5 = Z )
= ( ^ [A7: set_Sum_sum_a_b,B6: set_Sum_sum_a_b] :
( ( ord_le192794300um_a_b @ A7 @ B6 )
& ( ord_le192794300um_a_b @ B6 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_321_Collect__mono__iff,axiom,
! [P: sum_sum_a_b > $o,Q: sum_sum_a_b > $o] :
( ( ord_le192794300um_a_b @ ( collect_Sum_sum_a_b @ P ) @ ( collect_Sum_sum_a_b @ Q ) )
= ( ! [X3: sum_sum_a_b] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_322_insertE,axiom,
! [A: produc1032616263at_nat,B: produc1032616263at_nat,A3: set_Pr1647387645at_nat] :
( ( member1696759390at_nat @ A @ ( insert1625259895at_nat @ B @ A3 ) )
=> ( ( A != B )
=> ( member1696759390at_nat @ A @ A3 ) ) ) ).
% insertE
thf(fact_323_insertE,axiom,
! [A: nat,B: nat,A3: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A3 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A3 ) ) ) ).
% insertE
thf(fact_324_insertE,axiom,
! [A: produc1548871597um_a_b,B: produc1548871597um_a_b,A3: set_Pr1916610317um_a_b] :
( ( member947389014um_a_b @ A @ ( insert1435405693um_a_b @ B @ A3 ) )
=> ( ( A != B )
=> ( member947389014um_a_b @ A @ A3 ) ) ) ).
% insertE
thf(fact_325_insertE,axiom,
! [A: standard_Constant_a,B: standard_Constant_a,A3: set_St761939237tant_a] :
( ( member1632892294tant_a @ A @ ( insert1909710879tant_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member1632892294tant_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_326_insertE,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A3: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ A @ ( insert271595217at_nat @ B @ A3 ) )
=> ( ( A != B )
=> ( member701585322at_nat @ A @ A3 ) ) ) ).
% insertE
thf(fact_327_insertE,axiom,
! [A: sum_sum_a_b,B: sum_sum_a_b,A3: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b @ A @ ( insert_Sum_sum_a_b @ B @ A3 ) )
=> ( ( A != B )
=> ( member_Sum_sum_a_b @ A @ A3 ) ) ) ).
% insertE
thf(fact_328_insertE,axiom,
! [A: produc1124793815um_a_b,B: produc1124793815um_a_b,A3: set_Pr1174980151um_a_b] :
( ( member1294585472um_a_b @ A @ ( insert983991207um_a_b @ B @ A3 ) )
=> ( ( A != B )
=> ( member1294585472um_a_b @ A @ A3 ) ) ) ).
% insertE
thf(fact_329_insertE,axiom,
! [A: produc1963079155um_a_b,B: produc1963079155um_a_b,A3: set_Pr409224873um_a_b] :
( ( member1998628618um_a_b @ A @ ( insert323157027um_a_b @ B @ A3 ) )
=> ( ( A != B )
=> ( member1998628618um_a_b @ A @ A3 ) ) ) ).
% insertE
thf(fact_330_insertI1,axiom,
! [A: product_prod_nat_nat,B5: set_Pr1986765409at_nat] : ( member701585322at_nat @ A @ ( insert271595217at_nat @ A @ B5 ) ) ).
% insertI1
thf(fact_331_insertI1,axiom,
! [A: sum_sum_a_b,B5: set_Sum_sum_a_b] : ( member_Sum_sum_a_b @ A @ ( insert_Sum_sum_a_b @ A @ B5 ) ) ).
% insertI1
thf(fact_332_insertI1,axiom,
! [A: produc1124793815um_a_b,B5: set_Pr1174980151um_a_b] : ( member1294585472um_a_b @ A @ ( insert983991207um_a_b @ A @ B5 ) ) ).
% insertI1
thf(fact_333_insertI1,axiom,
! [A: produc1963079155um_a_b,B5: set_Pr409224873um_a_b] : ( member1998628618um_a_b @ A @ ( insert323157027um_a_b @ A @ B5 ) ) ).
% insertI1
thf(fact_334_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_335_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_336_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_337_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_338_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_339_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_340_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_341_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_342_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_343_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_344_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_345_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_346_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_347_gr_I1_J,axiom,
( ( getRel1465196352um_a_b @ standard_S_Idt_a @ g )
= ( id_on_Sum_sum_a_b @ ( labele577278695um_a_b @ g ) ) ) ).
% gr(1)
thf(fact_348_gr_I2_J,axiom,
( ( getRel1465196352um_a_b @ standard_S_Bot_a @ g )
= bot_bo225809273um_a_b ) ).
% gr(2)
thf(fact_349__092_060open_062maintained_A_Isymmetry__rule_AS__Idt_J_AG_092_060close_062,axiom,
mainta522127984um_a_b @ ( standa997693288tant_a @ standard_S_Idt_a ) @ g ).
% \<open>maintained (symmetry_rule S_Idt) G\<close>
thf(fact_350__092_060open_062maintained_A_Itransitive__rule_AS__Idt_J_AG_092_060close_062,axiom,
mainta522127984um_a_b @ ( standa1795879409tant_a @ standard_S_Idt_a ) @ g ).
% \<open>maintained (transitive_rule S_Idt) G\<close>
thf(fact_351__092_060open_062maintained_A_Icongruence__rule_AS__Idt_Al_J_AG_092_060close_062,axiom,
mainta522127984um_a_b @ ( standa1343274079tant_a @ standard_S_Idt_a @ l ) @ g ).
% \<open>maintained (congruence_rule S_Idt l) G\<close>
% Conjectures (1)
thf(conj_0,conjecture,
( ( ( labele1810595089_a_nat @ ( labele16114835_a_nat @ ( insert1625259895at_nat @ ( produc407553657at_nat @ standard_S_Idt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) @ bot_bo810816657at_nat ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
= ( domain1368163076um_a_b @ ( insert983991207um_a_b @ ( produc1808556047um_a_b @ zero_zero_nat @ v ) @ bot_bo575978147um_a_b ) ) )
& ( ( labele16114835_a_nat @ ( insert1625259895at_nat @ ( produc407553657at_nat @ standard_S_Idt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) @ bot_bo810816657at_nat ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) )
= ( restri572569417_a_nat @ ( labele16114835_a_nat @ ( insert1625259895at_nat @ ( produc407553657at_nat @ standard_S_Idt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) @ bot_bo810816657at_nat ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) )
& ( g
= ( restri1162247455um_a_b @ g ) )
& ( ord_le192794300um_a_b @ ( image_256773707um_a_b @ ( insert983991207um_a_b @ ( produc1808556047um_a_b @ zero_zero_nat @ v ) @ bot_bo575978147um_a_b ) @ ( labele1810595089_a_nat @ ( labele16114835_a_nat @ ( insert1625259895at_nat @ ( produc407553657at_nat @ standard_S_Idt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) @ bot_bo810816657at_nat ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) @ ( labele577278695um_a_b @ g ) )
& ( unival2092813468um_a_b @ ( insert983991207um_a_b @ ( produc1808556047um_a_b @ zero_zero_nat @ v ) @ bot_bo575978147um_a_b ) )
& ( edge_p1382426714tant_a @ ( insert983991207um_a_b @ ( produc1808556047um_a_b @ zero_zero_nat @ v ) @ bot_bo575978147um_a_b ) @ ( labele195203296_a_nat @ ( labele16114835_a_nat @ ( insert1625259895at_nat @ ( produc407553657at_nat @ standard_S_Idt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) @ bot_bo810816657at_nat ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) @ ( labele1939049654um_a_b @ g ) ) ) ).
%------------------------------------------------------------------------------