%   ORIGINAL: h4/extreal/CROSS__COUNTABLE__LEMMA1
% 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__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/pred__set/FINITE__INDUCT: !P. P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ P s ==> (!e. ~h4/bool/IN e s ==> P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> P s)
% Assm: h4/pred__set/CROSS__EMPTY_c1: !P. h4/pred__set/CROSS h4/pred__set/EMPTY P = h4/pred__set/EMPTY
% Assm: h4/pred__set/CROSS__EQNS_c1: !s2 s1 a. h4/pred__set/CROSS (h4/pred__set/INSERT a s1) s2 = h4/pred__set/UNION (h4/pred__set/IMAGE (\y. h4/pair/_2C a y) s2) (h4/pred__set/CROSS s1 s2)
% Assm: h4/util__prob/COUNTABLE__EMPTY: h4/util__prob/countable h4/pred__set/EMPTY
% Assm: h4/util__prob/COUNTABLE__IMAGE: !s f. h4/util__prob/countable s ==> h4/util__prob/countable (h4/pred__set/IMAGE f s)
% Assm: h4/util__prob/COUNTABLE__UNION: !t s. h4/util__prob/countable s /\ h4/util__prob/countable t ==> h4/util__prob/countable (h4/pred__set/UNION s t)
% Goal: !t s. h4/util__prob/countable s /\ h4/pred__set/FINITE s /\ h4/util__prob/countable t ==> h4/util__prob/countable (h4/pred__set/CROSS s t)
%   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_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_predu_u_sets_FINITEu_u_INDUCT]: !P. happ P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ happ P s ==> (!e. ~h4/bool/IN e s ==> happ P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> happ P s)
% Assm [h4s_predu_u_sets_CROSSu_u_EMPTYu_c1]: !P. h4/pred__set/CROSS h4/pred__set/EMPTY P = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_CROSSu_u_EQNSu_c1]: !_0. (!a y. happ (happ _0 a) y = h4/pair/_2C a y) ==> (!s2 s1 a. h4/pred__set/CROSS (h4/pred__set/INSERT a s1) s2 = h4/pred__set/UNION (h4/pred__set/IMAGE (happ _0 a) s2) (h4/pred__set/CROSS s1 s2))
% Assm [h4s_utilu_u_probs_COUNTABLEu_u_EMPTY]: h4/util__prob/countable h4/pred__set/EMPTY
% Assm [h4s_utilu_u_probs_COUNTABLEu_u_IMAGE]: !s f. h4/util__prob/countable s ==> h4/util__prob/countable (h4/pred__set/IMAGE f s)
% Assm [h4s_utilu_u_probs_COUNTABLEu_u_UNION]: !t s. h4/util__prob/countable s /\ h4/util__prob/countable t ==> h4/util__prob/countable (h4/pred__set/UNION s t)
% Goal: !t s. h4/util__prob/countable s /\ h4/pred__set/FINITE s /\ h4/util__prob/countable t ==> h4/util__prob/countable (h4/pred__set/CROSS s t)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q183030,TV_Q183026]: ![V_f, V_g]: (![V_x]: s(TV_Q183026,happ(s(t_fun(TV_Q183030,TV_Q183026),V_f),s(TV_Q183030,V_x))) = s(TV_Q183026,happ(s(t_fun(TV_Q183030,TV_Q183026),V_g),s(TV_Q183030,V_x))) => s(t_fun(TV_Q183030,TV_Q183026),V_f) = s(t_fun(TV_Q183030,TV_Q183026),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (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,t0) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_predu_u_sets_FINITEu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) & ![V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))) => ![V_e]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))))))) => ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_CROSSu_u_EMPTYu_c1, axiom, ![TV_u_27c,TV_u_27a]: ![V_P]: s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27c,t_bool),h4s_predu_u_sets_empty),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_predu_u_sets_CROSSu_u_EQNSu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_a, V_y]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_uu_0),s(TV_u_27a,V_a))),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_y))) => ![V_s2, V_s1, V_a]: 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_insert(s(TV_u_27a,V_a),s(t_fun(TV_u_27a,t_bool),V_s1))),s(t_fun(TV_u_27b,t_bool),V_s2))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_uu_0),s(TV_u_27a,V_a))),s(t_fun(TV_u_27b,t_bool),V_s2))),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_s1),s(t_fun(TV_u_27b,t_bool),V_s2))))))).
fof(ah4s_utilu_u_probs_COUNTABLEu_u_EMPTY, axiom, ![TV_u_27a]: p(s(t_bool,h4s_utilu_u_probs_countable(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))))).
fof(ah4s_utilu_u_probs_COUNTABLEu_u_IMAGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_s, V_f]: (p(s(t_bool,h4s_utilu_u_probs_countable(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_utilu_u_probs_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)))))))).
fof(ah4s_utilu_u_probs_COUNTABLEu_u_UNION, axiom, ![TV_u_27a]: ![V_t, V_s]: ((p(s(t_bool,h4s_utilu_u_probs_countable(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_utilu_u_probs_countable(s(t_fun(TV_u_27a,t_bool),V_t))))) => p(s(t_bool,h4s_utilu_u_probs_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)))))))).
fof(ch4s_extreals_CROSSu_u_COUNTABLEu_u_LEMMA1, conjecture, ![TV_u_27a,TV_u_27b]: ![V_t, V_s]: ((p(s(t_bool,h4s_utilu_u_probs_countable(s(t_fun(TV_u_27a,t_bool),V_s)))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_utilu_u_probs_countable(s(t_fun(TV_u_27b,t_bool),V_t)))))) => p(s(t_bool,h4s_utilu_u_probs_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)))))))).
