%   ORIGINAL: h4/measure/SPACE__SIGMA
% 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/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/measure/space__def: !y x. h4/measure/space (h4/pair/_2C x y) = x
% Assm: h4/measure/sigma__def: !st sp. h4/measure/sigma sp st = h4/pair/_2C sp (h4/pred__set/BIGINTER (h4/pred__set/GSPEC (\s. h4/pair/_2C s (h4/pred__set/SUBSET st s /\ h4/measure/sigma__algebra (h4/pair/_2C sp s)))))
% Goal: !sp a. h4/measure/space (h4/measure/sigma sp a) = sp
%   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_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_measures_spaceu_u_def]: !y x. h4/measure/space (h4/pair/_2C x y) = x
% Assm [h4s_measures_sigmau_u_def]: !_0. (!st sp s. ?v. (v <=> h4/pred__set/SUBSET st s /\ h4/measure/sigma__algebra (h4/pair/_2C sp s)) /\ happ (happ (happ _0 st) sp) s = h4/pair/_2C s v) ==> (!st sp. h4/measure/sigma sp st = h4/pair/_2C sp (h4/pred__set/BIGINTER (h4/pred__set/GSPEC (happ (happ _0 st) sp))))
% Goal: !sp a. h4/measure/space (h4/measure/sigma sp a) = sp
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_Q112405,TV_Q112401]: ![V_f, V_g]: (![V_x]: s(TV_Q112401,happ(s(t_fun(TV_Q112405,TV_Q112401),V_f),s(TV_Q112405,V_x))) = s(TV_Q112401,happ(s(t_fun(TV_Q112405,TV_Q112401),V_g),s(TV_Q112405,V_x))) => s(t_fun(TV_Q112405,TV_Q112401),V_f) = s(t_fun(TV_Q112405,TV_Q112401),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_measures_spaceu_u_def, axiom, ![TV_u_27a]: ![V_y, V_x]: s(t_fun(TV_u_27a,t_bool),h4s_measures_space(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),V_x),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_y))))) = s(t_fun(TV_u_27a,t_bool),V_x)).
fof(ah4s_measures_sigmau_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_st, V_sp, V_s]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_st),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s)))) & p(s(t_bool,h4s_measures_sigmau_u_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),V_sp),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s)))))))) & s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool)))),V_uu_0),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_st))),s(t_fun(TV_u_27a,t_bool),V_sp))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s))) = s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s),s(t_bool,V_v)))) => ![V_st, V_sp]: s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_sigma(s(t_fun(TV_u_27a,t_bool),V_sp),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_st))) = 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),V_sp),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_biginter(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool)))),V_uu_0),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_st))),s(t_fun(TV_u_27a,t_bool),V_sp))))))))))).
fof(ch4s_measures_SPACEu_u_SIGMA, conjecture, ![TV_u_27a]: ![V_sp, V_a]: s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_sigma(s(t_fun(TV_u_27a,t_bool),V_sp),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_a))))) = s(t_fun(TV_u_27a,t_bool),V_sp)).
