%   ORIGINAL: h4/pred__set/countable__Uprod
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/pred__set/UNIV__NOT__EMPTY: ~(h4/pred__set/UNIV = h4/pred__set/EMPTY)
% Assm: h4/pred__set/CROSS__UNIV: h4/pred__set/UNIV = h4/pred__set/CROSS h4/pred__set/UNIV h4/pred__set/UNIV
% Assm: h4/pred__set/cross__countable__IFF: !t s. h4/pred__set/countable (h4/pred__set/CROSS s t) <=> s = h4/pred__set/EMPTY \/ t = h4/pred__set/EMPTY \/ h4/pred__set/countable s /\ h4/pred__set/countable t
% Goal: h4/pred__set/countable h4/pred__set/UNIV <=> h4/pred__set/countable h4/pred__set/UNIV /\ h4/pred__set/countable h4/pred__set/UNIV
%   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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_predu_u_sets_UNIVu_u_NOTu_u_EMPTY]: ~(h4/pred__set/UNIV = h4/pred__set/EMPTY)
% Assm [h4s_predu_u_sets_CROSSu_u_UNIV]: h4/pred__set/UNIV = h4/pred__set/CROSS h4/pred__set/UNIV h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_crossu_u_countableu_u_IFF]: !t s. h4/pred__set/countable (h4/pred__set/CROSS s t) <=> s = h4/pred__set/EMPTY \/ t = h4/pred__set/EMPTY \/ h4/pred__set/countable s /\ h4/pred__set/countable t
% Goal: h4/pred__set/countable h4/pred__set/UNIV <=> h4/pred__set/countable h4/pred__set/UNIV /\ h4/pred__set/countable h4/pred__set/UNIV
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_Q156879,TV_Q156875]: ![V_f, V_g]: (![V_x]: s(TV_Q156875,happ(s(t_fun(TV_Q156879,TV_Q156875),V_f),s(TV_Q156879,V_x))) = s(TV_Q156875,happ(s(t_fun(TV_Q156879,TV_Q156875),V_g),s(TV_Q156879,V_x))) => s(t_fun(TV_Q156879,TV_Q156875),V_f) = s(t_fun(TV_Q156879,TV_Q156875),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_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_UNIVu_u_NOTu_u_EMPTY, axiom, ![TV_u_27a]: ~ (s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))).
fof(ah4s_predu_u_sets_CROSSu_u_UNIV, axiom, ![TV_u_27a,TV_u_27b]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_univ) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_univ)))).
fof(ah4s_predu_u_sets_crossu_u_countableu_u_IFF, axiom, ![TV_u_27a,TV_u_27b]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_countable(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))))) <=> (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty) | (s(t_fun(TV_u_27b,t_bool),V_t) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_empty) | (p(s(t_bool,h4s_predu_u_sets_countable(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_predu_u_sets_countable(s(t_fun(TV_u_27b,t_bool),V_t))))))))).
fof(ch4s_predu_u_sets_countableu_u_Uprod, conjecture, ![TV_u_27a,TV_u_27b]: (p(s(t_bool,h4s_predu_u_sets_countable(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_univ)))) <=> (p(s(t_bool,h4s_predu_u_sets_countable(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ)))) & p(s(t_bool,h4s_predu_u_sets_countable(s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_univ))))))).
