%   ORIGINAL: h4/pred__set/SUM__SET__THM_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/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/pred__set/SUM__IMAGE__THM_c0: !f. h4/pred__set/SUM__IMAGE f h4/pred__set/EMPTY = h4/num/0
% Assm: h4/pred__set/SUM__SET__DEF: h4/pred__set/SUM__SET = h4/pred__set/SUM__IMAGE h4/combin/I
% Goal: h4/pred__set/SUM__SET h4/pred__set/EMPTY = h4/num/0
%   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_SUMu_u_IMAGEu_u_THMu_c0]: !f. happ (h4/pred__set/SUM__IMAGE f) h4/pred__set/EMPTY = h4/num/0
% Assm [h4s_predu_u_sets_SUMu_u_SETu_u_DEF]: h4/pred__set/SUM__SET = h4/pred__set/SUM__IMAGE h4/combin/I
% Goal: happ h4/pred__set/SUM__SET h4/pred__set/EMPTY = h4/num/0
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_Q154483,TV_Q154479]: ![V_f, V_g]: (![V_x]: s(TV_Q154479,happ(s(t_fun(TV_Q154483,TV_Q154479),V_f),s(TV_Q154483,V_x))) = s(TV_Q154479,happ(s(t_fun(TV_Q154483,TV_Q154479),V_g),s(TV_Q154483,V_x))) => s(t_fun(TV_Q154483,TV_Q154479),V_f) = s(t_fun(TV_Q154483,TV_Q154479),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_SUMu_u_IMAGEu_u_THMu_c0, axiom, ![TV_u_27a]: ![V_f]: s(t_h4s_nums_num,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_predu_u_sets_SUMu_u_SETu_u_DEF, axiom, s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set) = s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_image(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_combins_i)))).
fof(ch4s_predu_u_sets_SUMu_u_SETu_u_THMu_c0, conjecture, s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_nums_num,h4s_nums_0)).
