%   ORIGINAL: h4/quotient__pred__set/IN__RSP
% 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/quotient__pred__set/SET__REL__MP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!s t x y. h4/quotient/_3D_3D_3D_3E R $equals s t /\ R x y ==> (h4/bool/IN x s <=> h4/bool/IN y t))
% Goal: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x1 x2 s1 s2. R x1 x2 /\ h4/quotient/_3D_3D_3D_3E R $equals s1 s2 ==> (h4/bool/IN x1 s1 <=> h4/bool/IN x2 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_quotientu_u_predu_u_sets_SETu_u_RELu_u_MP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!s t x y. h4/quotient/_3D_3D_3D_3E R $equals s t /\ happ (happ R x) y ==> (h4/bool/IN x s <=> h4/bool/IN y t))
% Goal: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x1 x2 s1 s2. happ (happ R x1) x2 /\ h4/quotient/_3D_3D_3D_3E R $equals s1 s2 ==> (h4/bool/IN x1 s1 <=> h4/bool/IN x2 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_Q85626,TV_Q85622]: ![V_f, V_g]: (![V_x]: s(TV_Q85622,happ(s(t_fun(TV_Q85626,TV_Q85622),V_f),s(TV_Q85626,V_x))) = s(TV_Q85622,happ(s(t_fun(TV_Q85626,TV_Q85622),V_g),s(TV_Q85626,V_x))) => s(t_fun(TV_Q85626,TV_Q85622),V_f) = s(t_fun(TV_Q85626,TV_Q85622),V_g))).
fof(ah4s_quotientu_u_predu_u_sets_SETu_u_RELu_u_MP, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_s, V_t, V_x, V_y]: ((p(s(t_bool,h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) => s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_t)))))).
fof(ch4s_quotientu_u_predu_u_sets_INu_u_RSP, conjecture, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x1, V_x2, V_s1, V_s2]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x1))),s(TV_u_27a,V_x2)))) & p(s(t_bool,h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals),s(t_fun(TV_u_27a,t_bool),V_s1),s(t_fun(TV_u_27a,t_bool),V_s2))))) => s(t_bool,h4s_bools_in(s(TV_u_27a,V_x1),s(t_fun(TV_u_27a,t_bool),V_s1))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x2),s(t_fun(TV_u_27a,t_bool),V_s2)))))).
