%   ORIGINAL: h4/quotient__pred__set/IN__SET__MAP
% 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/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/quotient/FUN__MAP__THM: !x h g f. h4/quotient/_2D_2D_3E f g h x = g (h (f x))
% Goal: !x s f. h4/bool/IN x (h4/quotient/_2D_2D_3E f h4/combin/I s) <=> h4/bool/IN (f x) 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_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_quotients_FUNu_u_MAPu_u_THM]: !x h g f. happ (h4/quotient/_2D_2D_3E f g h) x = happ g (happ h (happ f x))
% Goal: !x s f. h4/bool/IN x (h4/quotient/_2D_2D_3E f h4/combin/I s) <=> h4/bool/IN (happ f x) 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_Q85526,TV_Q85522]: ![V_f, V_g]: (![V_x]: s(TV_Q85522,happ(s(t_fun(TV_Q85526,TV_Q85522),V_f),s(TV_Q85526,V_x))) = s(TV_Q85522,happ(s(t_fun(TV_Q85526,TV_Q85522),V_g),s(TV_Q85526,V_x))) => s(t_fun(TV_Q85526,TV_Q85522),V_f) = s(t_fun(TV_Q85526,TV_Q85522),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_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
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_quotients_FUNu_u_MAPu_u_THM, axiom, ![TV_u_27d,TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_h, V_g, V_f]: s(TV_u_27d,happ(s(t_fun(TV_u_27a,TV_u_27d),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_fun(TV_u_27c,TV_u_27b),V_h))),s(TV_u_27a,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_h),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x)))))))).
fof(ch4s_quotientu_u_predu_u_sets_INu_u_SETu_u_MAP, conjecture, ![TV_u_27a,TV_u_27b]: ![V_x, V_s, V_f]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_bool,t_bool),h4s_combins_i),s(t_fun(TV_u_27b,t_bool),V_s))))) = s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),V_s)))).
