%   ORIGINAL: h4/bag/mlt__UNION__RCANCEL
% 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/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% 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/EQ__CLAUSES_c2: !t. (F <=> 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/bag/BAG__UNION__INSERT_c1: !e b2 b1. h4/bag/BAG__UNION b1 (h4/bag/BAG__INSERT e b2) = h4/bag/BAG__INSERT e (h4/bag/BAG__UNION b1 b2)
% Assm: h4/bag/BAG__UNION__EMPTY_c0: !b. h4/bag/BAG__UNION b h4/bag/EMPTY__BAG = b
% Assm: h4/bag/STRONG__FINITE__BAG__INDUCT: !P. P h4/bag/EMPTY__BAG /\ (!b. h4/bag/FINITE__BAG b /\ P b ==> (!e. P (h4/bag/BAG__INSERT e b))) ==> (!b. h4/bag/FINITE__BAG b ==> P b)
% Assm: h4/bag/FINITE__BAG__UNION: !b2 b1. h4/bag/FINITE__BAG (h4/bag/BAG__UNION b1 b2) <=> h4/bag/FINITE__BAG b1 /\ h4/bag/FINITE__BAG b2
% Assm: h4/bag/TC__mlt1__FINITE__BAG: !b2 b1 R. h4/relation/TC (h4/bag/mlt1 R) b1 b2 ==> h4/bag/FINITE__BAG b1 /\ h4/bag/FINITE__BAG b2
% Assm: h4/bag/mlt__INSERT__CANCEL: !e b a R. h4/relation/transitive R /\ h4/relation/WF R ==> (h4/relation/TC (h4/bag/mlt1 R) (h4/bag/BAG__INSERT e a) (h4/bag/BAG__INSERT e b) <=> h4/relation/TC (h4/bag/mlt1 R) a b)
% Goal: !c b a R. h4/relation/WF R /\ h4/relation/transitive R ==> (h4/relation/TC (h4/bag/mlt1 R) (h4/bag/BAG__UNION a c) (h4/bag/BAG__UNION b c) <=> h4/relation/TC (h4/bag/mlt1 R) a b /\ h4/bag/FINITE__BAG c)
%   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_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~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_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% 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_EQu_u_CLAUSESu_c2]: !t. (F <=> 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_bags_BAGu_u_UNIONu_u_INSERTu_c1]: !e b2 b1. h4/bag/BAG__UNION b1 (h4/bag/BAG__INSERT e b2) = h4/bag/BAG__INSERT e (h4/bag/BAG__UNION b1 b2)
% Assm [h4s_bags_BAGu_u_UNIONu_u_EMPTYu_c0]: !b. h4/bag/BAG__UNION b h4/bag/EMPTY__BAG = b
% Assm [h4s_bags_STRONGu_u_FINITEu_u_BAGu_u_INDUCT]: !P. happ P h4/bag/EMPTY__BAG /\ (!b. h4/bag/FINITE__BAG b /\ happ P b ==> (!e. happ P (h4/bag/BAG__INSERT e b))) ==> (!b. h4/bag/FINITE__BAG b ==> happ P b)
% Assm [h4s_bags_FINITEu_u_BAGu_u_UNION]: !b2 b1. h4/bag/FINITE__BAG (h4/bag/BAG__UNION b1 b2) <=> h4/bag/FINITE__BAG b1 /\ h4/bag/FINITE__BAG b2
% Assm [h4s_bags_TCu_u_mlt1u_u_FINITEu_u_BAG]: !b2 b1 R. h4/relation/TC (h4/bag/mlt1 R) b1 b2 ==> h4/bag/FINITE__BAG b1 /\ h4/bag/FINITE__BAG b2
% Assm [h4s_bags_mltu_u_INSERTu_u_CANCEL]: !e b a R. h4/relation/transitive R /\ h4/relation/WF R ==> (h4/relation/TC (h4/bag/mlt1 R) (h4/bag/BAG__INSERT e a) (h4/bag/BAG__INSERT e b) <=> h4/relation/TC (h4/bag/mlt1 R) a b)
% Goal: !c b a R. h4/relation/WF R /\ h4/relation/transitive R ==> (h4/relation/TC (h4/bag/mlt1 R) (h4/bag/BAG__UNION a c) (h4/bag/BAG__UNION b c) <=> h4/relation/TC (h4/bag/mlt1 R) a b /\ h4/bag/FINITE__BAG c)
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_Q233487,TV_Q233483]: ![V_f, V_g]: (![V_x]: s(TV_Q233483,happ(s(t_fun(TV_Q233487,TV_Q233483),V_f),s(TV_Q233487,V_x))) = s(TV_Q233483,happ(s(t_fun(TV_Q233487,TV_Q233483),V_g),s(TV_Q233487,V_x))) => s(t_fun(TV_Q233487,TV_Q233483),V_f) = s(t_fun(TV_Q233487,TV_Q233483),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_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_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_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_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
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_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_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_bags_BAGu_u_UNIONu_u_INSERTu_c1, axiom, ![TV_u_27a]: ![V_e, V_b2, V_b1]: 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_b1),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2))))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),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_b1),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2)))))).
fof(ah4s_bags_BAGu_u_UNIONu_u_EMPTYu_c0, axiom, ![TV_u_27a]: ![V_b]: 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),h4s_bags_emptyu_u_bag))) = s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)).
fof(ah4s_bags_STRONGu_u_FINITEu_u_BAGu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)))) & ![V_b]: ((p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))))) => ![V_e]: p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))))))) => ![V_b]: (p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))))))).
fof(ah4s_bags_FINITEu_u_BAGu_u_UNION, axiom, ![TV_u_27a]: ![V_b2, V_b1]: (p(s(t_bool,h4s_bags_finiteu_u_bag(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_b1),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2)))))) <=> (p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1)))) & p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2))))))).
fof(ah4s_bags_TCu_u_mlt1u_u_FINITEu_u_BAG, axiom, ![TV_u_27a]: ![V_b2, V_b1, V_R]: (p(s(t_bool,h4s_relations_tc(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool)),h4s_bags_mlt1(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2)))) => (p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1)))) & p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2))))))).
fof(ah4s_bags_mltu_u_INSERTu_u_CANCEL, axiom, ![TV_u_27a]: ![V_e, V_b, V_a, V_R]: ((p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) => s(t_bool,h4s_relations_tc(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool)),h4s_bags_mlt1(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_a))),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))))) = s(t_bool,h4s_relations_tc(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool)),h4s_bags_mlt1(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_a),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))))).
fof(ch4s_bags_mltu_u_UNIONu_u_RCANCEL, conjecture, ![TV_u_27a]: ![V_c, V_b, V_a, V_R]: ((p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) => (p(s(t_bool,h4s_relations_tc(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool)),h4s_bags_mlt1(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),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_a),s(t_fun(TV_u_27a,t_h4s_nums_num),V_c))),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)))))) <=> (p(s(t_bool,h4s_relations_tc(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool)),h4s_bags_mlt1(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_a),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))) & p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_c)))))))).
