%   ORIGINAL: h4/measure/BOREL__MEASURABLE__SETS_c0
% 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/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/measure/BOREL__MEASURABLE__SETS1: !c. h4/bool/IN (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__lt x c))) (h4/measure/subsets h4/measure/Borel)
% Goal: !c. h4/bool/IN (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__lt x c))) (h4/measure/subsets h4/measure/Borel)
%   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_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_measures_BORELu_u_MEASURABLEu_u_SETS1]: !_0. (!c x. happ (happ _0 c) x = h4/pair/_2C x (h4/extreal/extreal__lt x c)) ==> (!c. h4/bool/IN (h4/pred__set/GSPEC (happ _0 c)) (h4/measure/subsets h4/measure/Borel))
% Goal: !_0. (!c x. happ (happ _0 c) x = h4/pair/_2C x (h4/extreal/extreal__lt x c)) ==> (!c. h4/bool/IN (h4/pred__set/GSPEC (happ _0 c)) (h4/measure/subsets h4/measure/Borel))
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_Q114569,TV_Q114565]: ![V_f, V_g]: (![V_x]: s(TV_Q114565,happ(s(t_fun(TV_Q114569,TV_Q114565),V_f),s(TV_Q114569,V_x))) = s(TV_Q114565,happ(s(t_fun(TV_Q114569,TV_Q114565),V_g),s(TV_Q114569,V_x))) => s(t_fun(TV_Q114569,TV_Q114565),V_f) = s(t_fun(TV_Q114569,TV_Q114565),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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_measures_BORELu_u_MEASURABLEu_u_SETS1, axiom, ![V_uu_0]: (![V_c, V_x]: 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)),happ(s(t_fun(t_h4s_extreals_extreal,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_c))),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,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_c))))) => ![V_c]: p(s(t_bool,h4s_bools_in(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)),happ(s(t_fun(t_h4s_extreals_extreal,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_c))))),s(t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))))).
fof(ch4s_measures_BORELu_u_MEASURABLEu_u_SETSu_c0, conjecture, ![V_uu_0]: (![V_c, V_x]: 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)),happ(s(t_fun(t_h4s_extreals_extreal,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_c))),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,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_c))))) => ![V_c]: p(s(t_bool,h4s_bools_in(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)),happ(s(t_fun(t_h4s_extreals_extreal,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_c))))),s(t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))))).
