%   ORIGINAL: h4/util__prob/SCHROEDER__CLOSE
% 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/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/util__prob/schroeder__close__def: !x s f. h4/util__prob/schroeder__close f s x <=> (?n. h4/bool/IN x (h4/arithmetic/FUNPOW (h4/pred__set/IMAGE f) n s))
% Goal: !x s f. h4/bool/IN x (h4/util__prob/schroeder__close f s) <=> (?n. h4/bool/IN x (h4/arithmetic/FUNPOW (h4/pred__set/IMAGE f) n s))
%   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_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_utilu_u_probs_schroederu_u_closeu_u_def]: !x s f. happ (h4/util__prob/schroeder__close f s) x <=> (?n. h4/bool/IN x (h4/arithmetic/FUNPOW (h4/pred__set/IMAGE f) n s))
% Goal: !x s f. h4/bool/IN x (h4/util__prob/schroeder__close f s) <=> (?n. h4/bool/IN x (h4/arithmetic/FUNPOW (h4/pred__set/IMAGE f) n s))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q183959,TV_Q183955]: ![V_f, V_g]: (![V_x]: s(TV_Q183955,happ(s(t_fun(TV_Q183959,TV_Q183955),V_f),s(TV_Q183959,V_x))) = s(TV_Q183955,happ(s(t_fun(TV_Q183959,TV_Q183955),V_g),s(TV_Q183959,V_x))) => s(t_fun(TV_Q183959,TV_Q183955),V_f) = s(t_fun(TV_Q183959,TV_Q183955),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_predu_u_sets_SPECIFICATION, axiom, ![TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
fof(ah4s_utilu_u_probs_schroederu_u_closeu_u_def, axiom, ![TV_u_27a]: ![V_x, V_s, V_f]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),h4s_utilu_u_probs_schroederu_u_close(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27a,V_x)))) <=> ?[V_n]: p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_arithmetics_funpow(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_nums_num,V_n),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ch4s_utilu_u_probs_SCHROEDERu_u_CLOSE, conjecture, ![TV_u_27a]: ![V_x, V_s, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_utilu_u_probs_schroederu_u_close(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> ?[V_n]: p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_arithmetics_funpow(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_nums_num,V_n),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
