%   ORIGINAL: h4/finite__map/MAP__KEYS__using__LINV
% 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/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__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/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/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% 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/finite__map/fmap__EQ__THM: !g f. h4/finite__map/FDOM f = h4/finite__map/FDOM g /\ (!x. h4/bool/IN x (h4/finite__map/FDOM f) ==> h4/finite__map/FAPPLY f x = h4/finite__map/FAPPLY g x) <=> f = g
% Assm: h4/finite__map/MAP__KEYS__witness: h4/bool/LET (\m. !f fm. h4/finite__map/FDOM (m f fm) = h4/pred__set/IMAGE f (h4/finite__map/FDOM fm) /\ (h4/pred__set/INJ f (h4/finite__map/FDOM fm) h4/pred__set/UNIV ==> (!x. h4/bool/IN x (h4/finite__map/FDOM fm) ==> h4/finite__map/FAPPLY (m f fm) (f x) = h4/finite__map/FAPPLY fm x))) (\f fm. h4/bool/COND (h4/pred__set/INJ f (h4/finite__map/FDOM fm) h4/pred__set/UNIV) (h4/finite__map/f__o__f fm (h4/finite__map/FUN__FMAP (h4/pred__set/LINV f (h4/finite__map/FDOM fm)) (h4/pred__set/IMAGE f (h4/finite__map/FDOM fm)))) (h4/finite__map/FUN__FMAP h4/bool/ARB (h4/pred__set/IMAGE f (h4/finite__map/FDOM fm))))
% Assm: h4/finite__map/MAP__KEYS__def_c0: !fm f. h4/finite__map/FDOM (h4/finite__map/MAP__KEYS f fm) = h4/pred__set/IMAGE f (h4/finite__map/FDOM fm)
% Assm: h4/finite__map/MAP__KEYS__def_c1: !fm f. h4/pred__set/INJ f (h4/finite__map/FDOM fm) h4/pred__set/UNIV ==> (!x. h4/bool/IN x (h4/finite__map/FDOM fm) ==> h4/finite__map/FAPPLY (h4/finite__map/MAP__KEYS f fm) (f x) = h4/finite__map/FAPPLY fm x)
% Goal: !fm f. h4/pred__set/INJ f (h4/finite__map/FDOM fm) h4/pred__set/UNIV ==> h4/finite__map/MAP__KEYS f fm = h4/finite__map/f__o__f fm (h4/finite__map/FUN__FMAP (h4/pred__set/LINV f (h4/finite__map/FDOM fm)) (h4/pred__set/IMAGE f (h4/finite__map/FDOM fm)))
%   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_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_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_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_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% 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_finiteu_u_maps_fmapu_u_EQu_u_THM]: !g f. h4/finite__map/FDOM f = h4/finite__map/FDOM g /\ (!x. h4/bool/IN x (h4/finite__map/FDOM f) ==> h4/finite__map/FAPPLY f x = h4/finite__map/FAPPLY g x) <=> f = g
% Assm [h4s_finiteu_u_maps_MAPu_u_KEYSu_u_witness]: !_2. (!f fm. happ (happ _2 f) fm = h4/bool/COND (h4/pred__set/INJ f (h4/finite__map/FDOM fm) h4/pred__set/UNIV) (h4/finite__map/f__o__f fm (h4/finite__map/FUN__FMAP (h4/pred__set/LINV f (h4/finite__map/FDOM fm)) (h4/pred__set/IMAGE f (h4/finite__map/FDOM fm)))) (h4/finite__map/FUN__FMAP h4/bool/ARB (h4/pred__set/IMAGE f (h4/finite__map/FDOM fm)))) ==> (!_1. (!f. happ _1 f = happ _2 f) ==> (!_0. (!m. happ _0 m <=> (!f fm. h4/finite__map/FDOM (happ (happ m f) fm) = h4/pred__set/IMAGE f (h4/finite__map/FDOM fm) /\ (h4/pred__set/INJ f (h4/finite__map/FDOM fm) h4/pred__set/UNIV ==> (!x. h4/bool/IN x (h4/finite__map/FDOM fm) ==> h4/finite__map/FAPPLY (happ (happ m f) fm) (happ f x) = h4/finite__map/FAPPLY fm x)))) ==> h4/bool/LET _0 _1))
% Assm [h4s_finiteu_u_maps_MAPu_u_KEYSu_u_defu_c0]: !fm f. h4/finite__map/FDOM (h4/finite__map/MAP__KEYS f fm) = h4/pred__set/IMAGE f (h4/finite__map/FDOM fm)
% Assm [h4s_finiteu_u_maps_MAPu_u_KEYSu_u_defu_c1]: !fm f. h4/pred__set/INJ f (h4/finite__map/FDOM fm) h4/pred__set/UNIV ==> (!x. h4/bool/IN x (h4/finite__map/FDOM fm) ==> h4/finite__map/FAPPLY (h4/finite__map/MAP__KEYS f fm) (happ f x) = h4/finite__map/FAPPLY fm x)
% Goal: !fm f. h4/pred__set/INJ f (h4/finite__map/FDOM fm) h4/pred__set/UNIV ==> h4/finite__map/MAP__KEYS f fm = h4/finite__map/f__o__f fm (h4/finite__map/FUN__FMAP (h4/pred__set/LINV f (h4/finite__map/FDOM fm)) (h4/pred__set/IMAGE f (h4/finite__map/FDOM fm)))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q76866,TV_Q76862]: ![V_f, V_g]: (![V_x]: s(TV_Q76862,happ(s(t_fun(TV_Q76866,TV_Q76862),V_f),s(TV_Q76866,V_x))) = s(TV_Q76862,happ(s(t_fun(TV_Q76866,TV_Q76862),V_g),s(TV_Q76866,V_x))) => s(t_fun(TV_Q76866,TV_Q76862),V_f) = s(t_fun(TV_Q76866,TV_Q76862),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(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_f),s(TV_u_27a,V_x)))).
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_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_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_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
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_finiteu_u_maps_fmapu_u_EQu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_g, V_f]: ((s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))) => s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))) <=> s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))).
fof(ah4s_finiteu_u_maps_MAPu_u_KEYSu_u_witness, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_2]: (![V_f, V_fm]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),V_uu_2),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_bools_cond(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),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_univ))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_fu_u_ou_u_f(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_funu_u_fmap(s(t_fun(TV_u_27b,TV_u_27a),h4s_predu_u_sets_linv(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))))),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),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))))))))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_funu_u_fmap(s(t_fun(TV_u_27b,TV_u_27c),h4s_bools_arb),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),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))))))))) => ![V_uu_1]: (![V_f]: s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),V_uu_1),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),V_uu_2),s(t_fun(TV_u_27a,TV_u_27b),V_f))) => ![V_uu_0]: (![V_m]: (p(s(t_bool,happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),t_bool),V_uu_0),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),V_m)))) <=> ![V_f, V_fm]: (s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),V_m),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))))) = 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),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))))) & (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),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_univ)))) => ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm)))))) => s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),V_m),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))),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_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm),s(TV_u_27a,V_x))))))) => p(s(t_bool,h4s_bools_let(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),t_bool),V_uu_0),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),V_uu_1)))))))).
fof(ah4s_finiteu_u_maps_MAPu_u_KEYSu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_fm, V_f]: s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_mapu_u_keys(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))))) = 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),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm)))))).
fof(ah4s_finiteu_u_maps_MAPu_u_KEYSu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_fm, 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),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_univ)))) => ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm)))))) => s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_mapu_u_keys(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))),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_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm),s(TV_u_27a,V_x)))))).
fof(ch4s_finiteu_u_maps_MAPu_u_KEYSu_u_usingu_u_LINV, conjecture, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_fm, 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),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_univ)))) => s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_mapu_u_keys(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_fu_u_ou_u_f(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_funu_u_fmap(s(t_fun(TV_u_27b,TV_u_27a),h4s_predu_u_sets_linv(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))))),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),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))))))))))).
