%   ORIGINAL: h4/pred__set/countable__Usum
% 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/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/EXISTS__REFL: !a. ?x. x = a
% Assm: h4/sum/INR__INL__11_c0: !y x. h4/sum/INL x = h4/sum/INL y <=> x = y
% Assm: h4/sum/INR__INL__11_c1: !y x. h4/sum/INR x = h4/sum/INR y <=> x = y
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/pred__set/IN__IMAGE: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = f x /\ h4/bool/IN x s)
% Assm: h4/pred__set/SUM__UNIV: h4/pred__set/UNIV = h4/pred__set/UNION (h4/pred__set/IMAGE h4/sum/INL h4/pred__set/UNIV) (h4/pred__set/IMAGE h4/sum/INR h4/pred__set/UNIV)
% Assm: h4/pred__set/INJ__INL: !t s. (!x. h4/bool/IN x s ==> h4/bool/IN (h4/sum/INL x) t) ==> h4/pred__set/INJ h4/sum/INL s t
% Assm: h4/pred__set/INJ__INR: !t s. (!x. h4/bool/IN x s ==> h4/bool/IN (h4/sum/INR x) t) ==> h4/pred__set/INJ h4/sum/INR s t
% Assm: h4/pred__set/union__countable__IFF: !t s. h4/pred__set/countable (h4/pred__set/UNION s t) <=> h4/pred__set/countable s /\ h4/pred__set/countable t
% Assm: h4/pred__set/inj__image__countable__IFF: !s f. h4/pred__set/INJ f s (h4/pred__set/IMAGE f s) ==> (h4/pred__set/countable (h4/pred__set/IMAGE f s) <=> h4/pred__set/countable s)
% 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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_EXISTSu_u_REFL]: !a. ?x. x = a
% Assm [h4s_sums_INRu_u_INLu_u_11u_c0]: !y x. happ h4/sum/INL x = happ h4/sum/INL y <=> x = y
% Assm [h4s_sums_INRu_u_INLu_u_11u_c1]: !y x. happ h4/sum/INR x = happ h4/sum/INR y <=> x = y
% Assm [h4s_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_INu_u_IMAGE]: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = happ f x /\ h4/bool/IN x s)
% Assm [h4s_predu_u_sets_SUMu_u_UNIV]: h4/pred__set/UNIV = h4/pred__set/UNION (h4/pred__set/IMAGE h4/sum/INL h4/pred__set/UNIV) (h4/pred__set/IMAGE h4/sum/INR h4/pred__set/UNIV)
% Assm [h4s_predu_u_sets_INJu_u_INL]: !t s. (!x. h4/bool/IN x s ==> h4/bool/IN (happ h4/sum/INL x) t) ==> h4/pred__set/INJ h4/sum/INL s t
% Assm [h4s_predu_u_sets_INJu_u_INR]: !t s. (!x. h4/bool/IN x s ==> h4/bool/IN (happ h4/sum/INR x) t) ==> h4/pred__set/INJ h4/sum/INR s t
% Assm [h4s_predu_u_sets_unionu_u_countableu_u_IFF]: !t s. h4/pred__set/countable (h4/pred__set/UNION s t) <=> h4/pred__set/countable s /\ h4/pred__set/countable t
% Assm [h4s_predu_u_sets_inju_u_imageu_u_countableu_u_IFF]: !s f. h4/pred__set/INJ f s (h4/pred__set/IMAGE f s) ==> (h4/pred__set/countable (h4/pred__set/IMAGE f s) <=> h4/pred__set/countable s)
% 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_Q156779,TV_Q156775]: ![V_f, V_g]: (![V_x]: s(TV_Q156775,happ(s(t_fun(TV_Q156779,TV_Q156775),V_f),s(TV_Q156779,V_x))) = s(TV_Q156775,happ(s(t_fun(TV_Q156779,TV_Q156775),V_g),s(TV_Q156779,V_x))) => s(t_fun(TV_Q156779,TV_Q156775),V_f) = s(t_fun(TV_Q156779,TV_Q156775),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_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => 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_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
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_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_EXISTSu_u_REFL, axiom, ![TV_u_27a]: ![V_a]: ?[V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_a)).
fof(ah4s_sums_INRu_u_INLu_u_11u_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_x))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_sums_INRu_u_INLu_u_11u_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_x))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))) <=> s(TV_u_27b,V_x) = s(TV_u_27b,V_y))).
fof(ah4s_predu_u_sets_INu_u_UNIV, axiom, ![TV_u_27a]: ![V_x]: p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))))).
fof(ah4s_predu_u_sets_INu_u_IMAGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_s, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> ?[V_x]: (s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_SUMu_u_UNIV, axiom, ![TV_u_27a,TV_u_27b]: s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_univ) = s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))),s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_univ)))))).
fof(ah4s_predu_u_sets_INJu_u_INL, axiom, ![TV_u_27a,TV_u_27b]: ![V_t, V_s]: (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_x))),s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_t))))) => p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_t)))))).
fof(ah4s_predu_u_sets_INJu_u_INR, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_s]: (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(t_h4s_sums_sum(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,TV_u_27a)),h4s_sums_inr),s(TV_u_27a,V_x))),s(t_fun(t_h4s_sums_sum(TV_u_27b,TV_u_27a),t_bool),V_t))))) => p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,TV_u_27a)),h4s_sums_inr),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_sums_sum(TV_u_27b,TV_u_27a),t_bool),V_t)))))).
fof(ah4s_predu_u_sets_unionu_u_countableu_u_IFF, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_countable(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (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_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_inju_u_imageu_u_countableu_u_IFF, axiom, ![TV_u_27b,TV_u_27a]: ![V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))) => s(t_bool,h4s_predu_u_sets_countable(s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_bool,h4s_predu_u_sets_countable(s(t_fun(TV_u_27a,t_bool),V_s))))).
fof(ch4s_predu_u_sets_countableu_u_Usum, conjecture, ![TV_u_27a,TV_u_27b]: (p(s(t_bool,h4s_predu_u_sets_countable(s(t_fun(t_h4s_sums_sum(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))))))).
