%   ORIGINAL: h4/topology/TOPOLOGY__UNION
% 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/topology/TOPOLOGY_c3: !P L. h4/pred__set/SUBSET P (h4/topology/open L) ==> h4/topology/open L (h4/pred__set/BIGUNION P)
% Goal: !P L. h4/pred__set/SUBSET P (h4/topology/open L) ==> h4/topology/open L (h4/pred__set/BIGUNION P)
%   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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_topologys_TOPOLOGYu_c3]: !P L. h4/pred__set/SUBSET P (h4/topology/open L) ==> happ (h4/topology/open L) (h4/pred__set/BIGUNION P)
% Goal: !P L. h4/pred__set/SUBSET P (h4/topology/open L) ==> happ (h4/topology/open L) (h4/pred__set/BIGUNION P)
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_Q352076,TV_Q352072]: ![V_f, V_g]: (![V_x]: s(TV_Q352072,happ(s(t_fun(TV_Q352076,TV_Q352072),V_f),s(TV_Q352076,V_x))) = s(TV_Q352072,happ(s(t_fun(TV_Q352076,TV_Q352072),V_g),s(TV_Q352076,V_x))) => s(t_fun(TV_Q352076,TV_Q352072),V_f) = s(t_fun(TV_Q352076,TV_Q352072),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
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_topologys_TOPOLOGYu_c3, axiom, ![TV_u_27a]: ![V_P, V_L]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_topologys_open(s(t_h4s_topologys_topology(TV_u_27a),V_L)))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_topologys_open(s(t_h4s_topologys_topology(TV_u_27a),V_L))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))))))).
fof(ch4s_topologys_TOPOLOGYu_u_UNION, conjecture, ![TV_u_27a]: ![V_P, V_L]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_topologys_open(s(t_h4s_topologys_topology(TV_u_27a),V_L)))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_topologys_open(s(t_h4s_topologys_topology(TV_u_27a),V_L))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))))))).
