%   ORIGINAL: h4/extreal/OPP__IN__Q
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/pair/CLOSED__PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/pred__set/IN__UNION: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm: h4/extreal/neg__neg: !x. h4/extreal/extreal__ainv (h4/extreal/extreal__ainv x) = x
% Assm: h4/extreal/Q__set__def: h4/extreal/Q__set = h4/pred__set/UNION (h4/pred__set/GSPEC (\x. h4/pair/_2C x (?a b. x = h4/extreal/extreal__div (h4/extreal/extreal__of__num a) (h4/extreal/extreal__of__num b) /\ h4/extreal/extreal__lt (h4/extreal/extreal__of__num h4/num/0) (h4/extreal/extreal__of__num b)))) (h4/pred__set/GSPEC (\x. h4/pair/_2C x (?a b. x = h4/extreal/extreal__ainv (h4/extreal/extreal__div (h4/extreal/extreal__of__num a) (h4/extreal/extreal__of__num b)) /\ h4/extreal/extreal__lt (h4/extreal/extreal__of__num h4/num/0) (h4/extreal/extreal__of__num b))))
% Goal: !x. h4/bool/IN x h4/extreal/Q__set ==> h4/bool/IN (h4/extreal/extreal__ainv x) h4/extreal/Q__set
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_pairs_CLOSEDu_u_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_predu_u_sets_GSPECIFICATION]: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = happ f x)
% Assm [h4s_predu_u_sets_INu_u_UNION]: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm [h4s_extreals_negu_u_neg]: !x. h4/extreal/extreal__ainv (h4/extreal/extreal__ainv x) = x
% Assm [h4s_extreals_Qu_u_setu_u_def]: !_1. (!x. ?v. (v <=> (?a b. x = h4/extreal/extreal__ainv (h4/extreal/extreal__div (h4/extreal/extreal__of__num a) (h4/extreal/extreal__of__num b)) /\ h4/extreal/extreal__lt (h4/extreal/extreal__of__num h4/num/0) (h4/extreal/extreal__of__num b))) /\ happ _1 x = h4/pair/_2C x v) ==> (!_0. (!x. ?v. (v <=> (?a b. x = h4/extreal/extreal__div (h4/extreal/extreal__of__num a) (h4/extreal/extreal__of__num b) /\ h4/extreal/extreal__lt (h4/extreal/extreal__of__num h4/num/0) (h4/extreal/extreal__of__num b))) /\ happ _0 x = h4/pair/_2C x v) ==> h4/extreal/Q__set = h4/pred__set/UNION (h4/pred__set/GSPEC _0) (h4/pred__set/GSPEC _1))
% Goal: !x. h4/bool/IN x h4/extreal/Q__set ==> h4/bool/IN (h4/extreal/extreal__ainv x) h4/extreal/Q__set
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q182642,TV_Q182638]: ![V_f, V_g]: (![V_x]: s(TV_Q182638,happ(s(t_fun(TV_Q182642,TV_Q182638),V_f),s(TV_Q182642,V_x))) = s(TV_Q182638,happ(s(t_fun(TV_Q182642,TV_Q182638),V_g),s(TV_Q182642,V_x))) => s(t_fun(TV_Q182642,TV_Q182638),V_f) = s(t_fun(TV_Q182642,TV_Q182638),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))))).
fof(ah4s_pairs_CLOSEDu_u_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_predu_u_sets_GSPECIFICATION, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f)))))) <=> ?[V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_v),s(t_bool,t))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f),s(TV_u_27b,V_x))))).
fof(ah4s_predu_u_sets_INu_u_UNION, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_extreals_negu_u_neg, axiom, ![V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ainv(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ainv(s(t_h4s_extreals_extreal,V_x))))) = s(t_h4s_extreals_extreal,V_x)).
fof(ah4s_extreals_Qu_u_setu_u_def, axiom, ![V_uu_1]: (![V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_a, V_b]: (s(t_h4s_extreals_extreal,V_x) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ainv(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_div(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,V_a))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,V_b))))))) & p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,V_b)))))))) & s(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool),happ(s(t_fun(t_h4s_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool)),V_uu_1),s(t_h4s_extreals_extreal,V_x))) = s(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool),h4s_pairs_u_2c(s(t_h4s_extreals_extreal,V_x),s(t_bool,V_v)))) => ![V_uu_0]: (![V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_a, V_b]: (s(t_h4s_extreals_extreal,V_x) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_div(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,V_a))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,V_b))))) & p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,V_b)))))))) & s(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool),happ(s(t_fun(t_h4s_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool)),V_uu_0),s(t_h4s_extreals_extreal,V_x))) = s(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool),h4s_pairs_u_2c(s(t_h4s_extreals_extreal,V_x),s(t_bool,V_v)))) => s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_extreals_qu_u_set) = s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool)),V_uu_0))),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool)),V_uu_1)))))))).
fof(ch4s_extreals_OPPu_u_INu_u_Q, conjecture, ![V_x]: (p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,V_x),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_extreals_qu_u_set)))) => p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ainv(s(t_h4s_extreals_extreal,V_x))),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_extreals_qu_u_set)))))).
