%   ORIGINAL: h4/bag/BIG__BAG__UNION__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/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !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/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g 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/pred__set/SUM__IMAGE__UNION: !t s f. h4/pred__set/FINITE s /\ h4/pred__set/FINITE t ==> h4/pred__set/SUM__IMAGE f (h4/pred__set/UNION s t) = h4/arithmetic/_2D (h4/arithmetic/_2B (h4/pred__set/SUM__IMAGE f s) (h4/pred__set/SUM__IMAGE f t)) (h4/pred__set/SUM__IMAGE f (h4/pred__set/INTER s t))
% Assm: h4/bag/BAG__UNION0: !c b. h4/bag/BAG__UNION b c = (\x. h4/arithmetic/_2B (b x) (c x))
% Assm: h4/bag/BAG__DIFF0: !b2 b1. h4/bag/BAG__DIFF b1 b2 = (\x. h4/arithmetic/_2D (b1 x) (b2 x))
% Assm: h4/bag/BIG__BAG__UNION__def: !sob. h4/bag/BIG__BAG__UNION sob = (\x. h4/pred__set/SUM__IMAGE (\b. b x) sob)
% Goal: !s2 s1. h4/pred__set/FINITE s1 /\ h4/pred__set/FINITE s2 ==> h4/bag/BIG__BAG__UNION (h4/pred__set/UNION s1 s2) = h4/bag/BAG__DIFF (h4/bag/BAG__UNION (h4/bag/BIG__BAG__UNION s1) (h4/bag/BIG__BAG__UNION s2)) (h4/bag/BIG__BAG__UNION (h4/pred__set/INTER s1 s2))
%   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_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !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_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g 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_predu_u_sets_SUMu_u_IMAGEu_u_UNION]: !t s f. h4/pred__set/FINITE s /\ h4/pred__set/FINITE t ==> h4/pred__set/SUM__IMAGE f (h4/pred__set/UNION s t) = h4/arithmetic/_2D (h4/arithmetic/_2B (h4/pred__set/SUM__IMAGE f s) (h4/pred__set/SUM__IMAGE f t)) (h4/pred__set/SUM__IMAGE f (h4/pred__set/INTER s t))
% Assm [h4s_bags_BAGu_u_UNION0]: !c b x. happ (h4/bag/BAG__UNION b c) x = h4/arithmetic/_2B (happ b x) (happ c x)
% Assm [h4s_bags_BAGu_u_DIFF0]: !b2 b1 x. happ (h4/bag/BAG__DIFF b1 b2) x = h4/arithmetic/_2D (happ b1 x) (happ b2 x)
% Assm [h4s_bags_BIGu_u_BAGu_u_UNIONu_u_def]: !_0. (!x b. happ (happ _0 x) b = happ b x) ==> (!sob x. happ (h4/bag/BIG__BAG__UNION sob) x = h4/pred__set/SUM__IMAGE (happ _0 x) sob)
% Goal: !s2 s1. h4/pred__set/FINITE s1 /\ h4/pred__set/FINITE s2 ==> h4/bag/BIG__BAG__UNION (h4/pred__set/UNION s1 s2) = h4/bag/BAG__DIFF (h4/bag/BAG__UNION (h4/bag/BIG__BAG__UNION s1) (h4/bag/BIG__BAG__UNION s2)) (h4/bag/BIG__BAG__UNION (h4/pred__set/INTER s1 s2))
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_Q232826,TV_Q232822]: ![V_f, V_g]: (![V_x]: s(TV_Q232822,happ(s(t_fun(TV_Q232826,TV_Q232822),V_f),s(TV_Q232826,V_x))) = s(TV_Q232822,happ(s(t_fun(TV_Q232826,TV_Q232822),V_g),s(TV_Q232826,V_x))) => s(t_fun(TV_Q232826,TV_Q232822),V_f) = s(t_fun(TV_Q232826,TV_Q232822),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_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> 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_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_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),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_predu_u_sets_SUMu_u_IMAGEu_u_UNION, axiom, ![TV_u_27a]: ![V_t, V_s, V_f]: ((p(s(t_bool,h4s_predu_u_sets_finite(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_t))))) => s(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_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(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),V_s))),s(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),V_t))))),s(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_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))))))).
fof(ah4s_bags_BAGu_u_UNION0, axiom, ![TV_u_27a]: ![V_c, V_b, V_x]: s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_union(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b),s(t_fun(TV_u_27a,t_h4s_nums_num),V_c))),s(TV_u_27a,V_x))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b),s(TV_u_27a,V_x))),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_c),s(TV_u_27a,V_x)))))).
fof(ah4s_bags_BAGu_u_DIFF0, axiom, ![TV_u_27a]: ![V_b2, V_b1, V_x]: s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_diff(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2))),s(TV_u_27a,V_x))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(TV_u_27a,V_x))),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2),s(TV_u_27a,V_x)))))).
fof(ah4s_bags_BIGu_u_BAGu_u_UNIONu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_b]: s(t_h4s_nums_num,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_h4s_nums_num),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_h4s_nums_num)),V_uu_0),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))) = s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b),s(TV_u_27a,V_x))) => ![V_sob, V_x]: s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bigu_u_bagu_u_union(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_sob))),s(TV_u_27a,V_x))) = s(t_h4s_nums_num,h4s_predu_u_sets_sumu_u_image(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_h4s_nums_num),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_h4s_nums_num)),V_uu_0),s(TV_u_27a,V_x))),s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_sob))))).
fof(ch4s_bags_BIGu_u_BAGu_u_UNIONu_u_UNION, conjecture, ![TV_u_27a]: ![V_s2, V_s1]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_s1)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_s2))))) => s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bigu_u_bagu_u_union(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),h4s_predu_u_sets_union(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_s1),s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_s2))))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_diff(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_union(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bigu_u_bagu_u_union(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_s1))),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bigu_u_bagu_u_union(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_s2))))),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bigu_u_bagu_u_union(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),h4s_predu_u_sets_inter(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_s1),s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_s2))))))))).
