%   ORIGINAL: h4/tc/SUBSET__FDOM__LEM
% 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/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/tc/RDOM__SUBSET__FDOM: !f. h4/pred__set/SUBSET (h4/relation/RDOM (h4/tc/FMAP__TO__RELN f)) (h4/finite__map/FDOM f)
% Assm: h4/tc/RDOM__subTC: !s R. h4/relation/RDOM (h4/tc/subTC R s) = h4/relation/RDOM R
% Goal: !s f R. h4/tc/subTC R s = h4/tc/FMAP__TO__RELN f ==> h4/pred__set/SUBSET (h4/relation/RDOM R) (h4/finite__map/FDOM f)
%   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_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_tcs_RDOMu_u_SUBSETu_u_FDOM]: !f. h4/pred__set/SUBSET (h4/relation/RDOM (h4/tc/FMAP__TO__RELN f)) (h4/finite__map/FDOM f)
% Assm [h4s_tcs_RDOMu_u_subTC]: !s R. h4/relation/RDOM (h4/tc/subTC R s) = h4/relation/RDOM R
% Goal: !s f R. h4/tc/subTC R s = h4/tc/FMAP__TO__RELN f ==> h4/pred__set/SUBSET (h4/relation/RDOM R) (h4/finite__map/FDOM f)
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_Q189873,TV_Q189869]: ![V_f, V_g]: (![V_x]: s(TV_Q189869,happ(s(t_fun(TV_Q189873,TV_Q189869),V_f),s(TV_Q189873,V_x))) = s(TV_Q189869,happ(s(t_fun(TV_Q189873,TV_Q189869),V_g),s(TV_Q189873,V_x))) => s(t_fun(TV_Q189873,TV_Q189869),V_f) = s(t_fun(TV_Q189873,TV_Q189869),V_g))).
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_tcs_RDOMu_u_SUBSETu_u_FDOM, axiom, ![TV_u_27a]: ![V_f]: p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),h4s_relations_rdom(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_tcs_fmapu_u_tou_u_reln(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_fun(TV_u_27a,t_bool)),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,t_fun(TV_u_27a,t_bool)),V_f))))))).
fof(ah4s_tcs_RDOMu_u_subTC, axiom, ![TV_u_27a]: ![V_s, V_R]: s(t_fun(TV_u_27a,t_bool),h4s_relations_rdom(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_tcs_subtc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_fun(TV_u_27a,t_bool),h4s_relations_rdom(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ch4s_tcs_SUBSETu_u_FDOMu_u_LEM, conjecture, ![TV_u_27a]: ![V_s, V_f, V_R]: (s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_tcs_subtc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_tcs_fmapu_u_tou_u_reln(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_f))) => p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),h4s_relations_rdom(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_f)))))))).
