%   ORIGINAL: h4/measure/INF__MEASURE__OUTER
% 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/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% 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/measure/outer__measure__space__def: !m. h4/measure/outer__measure__space m <=> h4/measure/positive m /\ h4/measure/increasing m /\ h4/measure/countably__subadditive m
% Assm: h4/measure/INF__MEASURE__POSITIVE: !m. h4/measure/algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m ==> h4/measure/positive (h4/pair/_2C (h4/measure/m__space m) (h4/pair/_2C (h4/pred__set/POW (h4/measure/m__space m)) (h4/measure/inf__measure m)))
% Assm: h4/measure/INF__MEASURE__INCREASING: !m. h4/measure/algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m ==> h4/measure/increasing (h4/pair/_2C (h4/measure/m__space m) (h4/pair/_2C (h4/pred__set/POW (h4/measure/m__space m)) (h4/measure/inf__measure m)))
% Assm: h4/measure/INF__MEASURE__COUNTABLY__SUBADDITIVE: !m. h4/measure/algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/measure/increasing m ==> h4/measure/countably__subadditive (h4/pair/_2C (h4/measure/m__space m) (h4/pair/_2C (h4/pred__set/POW (h4/measure/m__space m)) (h4/measure/inf__measure m)))
% Goal: !m. h4/measure/algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/measure/increasing m ==> h4/measure/outer__measure__space (h4/pair/_2C (h4/measure/m__space m) (h4/pair/_2C (h4/pred__set/POW (h4/measure/m__space m)) (h4/measure/inf__measure m)))
%   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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% 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_measures_outeru_u_measureu_u_spaceu_u_def]: !m. h4/measure/outer__measure__space m <=> h4/measure/positive m /\ h4/measure/increasing m /\ h4/measure/countably__subadditive m
% Assm [h4s_measures_INFu_u_MEASUREu_u_POSITIVE]: !m. h4/measure/algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m ==> h4/measure/positive (h4/pair/_2C (h4/measure/m__space m) (h4/pair/_2C (h4/pred__set/POW (h4/measure/m__space m)) (h4/measure/inf__measure m)))
% Assm [h4s_measures_INFu_u_MEASUREu_u_INCREASING]: !m. h4/measure/algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m ==> h4/measure/increasing (h4/pair/_2C (h4/measure/m__space m) (h4/pair/_2C (h4/pred__set/POW (h4/measure/m__space m)) (h4/measure/inf__measure m)))
% Assm [h4s_measures_INFu_u_MEASUREu_u_COUNTABLYu_u_SUBADDITIVE]: !m. h4/measure/algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/measure/increasing m ==> h4/measure/countably__subadditive (h4/pair/_2C (h4/measure/m__space m) (h4/pair/_2C (h4/pred__set/POW (h4/measure/m__space m)) (h4/measure/inf__measure m)))
% Goal: !m. h4/measure/algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/measure/increasing m ==> h4/measure/outer__measure__space (h4/pair/_2C (h4/measure/m__space m) (h4/pair/_2C (h4/pred__set/POW (h4/measure/m__space m)) (h4/measure/inf__measure m)))
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_Q111552,TV_Q111548]: ![V_f, V_g]: (![V_x]: s(TV_Q111548,happ(s(t_fun(TV_Q111552,TV_Q111548),V_f),s(TV_Q111552,V_x))) = s(TV_Q111548,happ(s(t_fun(TV_Q111552,TV_Q111548),V_g),s(TV_Q111552,V_x))) => s(t_fun(TV_Q111552,TV_Q111548),V_f) = s(t_fun(TV_Q111552,TV_Q111548),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> 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_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_measures_outeru_u_measureu_u_spaceu_u_def, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_outeru_u_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) <=> (p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (p(s(t_bool,h4s_measures_increasing(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_measures_countablyu_u_subadditive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))).
fof(ah4s_measures_INFu_u_MEASUREu_u_POSITIVE, axiom, ![TV_u_27a]: ![V_m]: ((p(s(t_bool,h4s_measures_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))) => p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_infu_u_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))))))).
fof(ah4s_measures_INFu_u_MEASUREu_u_INCREASING, axiom, ![TV_u_27a]: ![V_m]: ((p(s(t_bool,h4s_measures_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))) => p(s(t_bool,h4s_measures_increasing(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_infu_u_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))))))).
fof(ah4s_measures_INFu_u_MEASUREu_u_COUNTABLYu_u_SUBADDITIVE, axiom, ![TV_u_27a]: ![V_m]: ((p(s(t_bool,h4s_measures_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & (p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_measures_increasing(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => p(s(t_bool,h4s_measures_countablyu_u_subadditive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_infu_u_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))))))).
fof(ch4s_measures_INFu_u_MEASUREu_u_OUTER, conjecture, ![TV_u_27a]: ![V_m]: ((p(s(t_bool,h4s_measures_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & (p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_measures_increasing(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => p(s(t_bool,h4s_measures_outeru_u_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_infu_u_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))))))).
