%   ORIGINAL: h4/quotient/FUN__REL__EQ__REL
% 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/AND__CLAUSES_c0: !t. T /\ t <=> t
% 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/combin/W__THM: !x f. h4/combin/W f x = f x x
% Assm: h4/quotient/QUOTIENT__REL: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. R r s <=> R r r /\ R s s /\ abs r = abs s)
% Assm: h4/quotient/FUN__QUOTIENT: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient/_3D_3D_3D_3E R1 R2) (h4/quotient/_2D_2D_3E rep1 abs2) (h4/quotient/_2D_2D_3E abs1 rep2))
% Assm: h4/quotient/respects__def: h4/quotient/respects = h4/combin/W
% Goal: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g. h4/quotient/_3D_3D_3D_3E R1 R2 f g <=> h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2) f /\ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2) g /\ h4/quotient/_2D_2D_3E rep1 abs2 f = h4/quotient/_2D_2D_3E rep1 abs2 g))
%   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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% 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_combins_Wu_u_THM]: !x f. happ (happ h4/combin/W f) x = happ (happ f x) x
% Assm [h4s_quotients_QUOTIENTu_u_REL]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. happ (happ R r) s <=> happ (happ R r) r /\ happ (happ R s) s /\ happ abs r = happ abs s)
% Assm [h4s_quotients_FUNu_u_QUOTIENT]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient/_3D_3D_3D_3E R1 R2) (h4/quotient/_2D_2D_3E rep1 abs2) (h4/quotient/_2D_2D_3E abs1 rep2))
% Assm [h4s_quotients_respectsu_u_def]: h4/quotient/respects = h4/combin/W
% Goal: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g <=> happ (happ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2)) f /\ happ (happ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2)) g /\ happ (h4/quotient/_2D_2D_3E rep1 abs2) f = happ (h4/quotient/_2D_2D_3E rep1 abs2) g))
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_Q225033,TV_Q225029]: ![V_f, V_g]: (![V_x]: s(TV_Q225029,happ(s(t_fun(TV_Q225033,TV_Q225029),V_f),s(TV_Q225033,V_x))) = s(TV_Q225029,happ(s(t_fun(TV_Q225033,TV_Q225029),V_g),s(TV_Q225033,V_x))) => s(t_fun(TV_Q225033,TV_Q225029),V_f) = s(t_fun(TV_Q225033,TV_Q225029),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_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_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_combins_Wu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_w),s(t_fun(TV_u_27a,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),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_x)))).
fof(ah4s_quotients_QUOTIENTu_u_REL, 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_r, V_s]: (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_r))),s(TV_u_27a,V_s)))) <=> (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_r))),s(TV_u_27a,V_r)))) & (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_s))),s(TV_u_27a,V_s)))) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_s)))))))).
fof(ah4s_quotients_FUNu_u_QUOTIENT, axiom, ![TV_u_27a,TV_u_27c,TV_u_27d,TV_u_27b]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))))))))).
fof(ah4s_quotients_respectsu_u_def, axiom, ![TV_u_27a,TV_u_27b]: s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_respects) = s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_w)).
fof(ch4s_quotients_FUNu_u_RELu_u_EQu_u_REL, conjecture, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) <=> (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_respects),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,TV_u_27b),V_f)))) & (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_respects),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) & s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),V_g))))))))).
