%   ORIGINAL: h4/quotient/LAMBDA__PRS1
% 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/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/quotient/QUOTIENT__ABS__REP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. abs (rep a) = a)
% Assm: h4/quotient/FUN__MAP__THM: !x h g f. h4/quotient/_2D_2D_3E f g h x = g (h (f x))
% Goal: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f. (\x. f x) = h4/quotient/_2D_2D_3E rep1 abs2 (\x. h4/quotient/_2D_2D_3E abs1 rep2 f x)))
%   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_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_quotients_QUOTIENTu_u_ABSu_u_REP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. happ abs (happ rep a) = a)
% Assm [h4s_quotients_FUNu_u_MAPu_u_THM]: !x h g f. h4/quotient/_2D_2D_3E f g h x = happ g (happ h (happ f x))
% Goal: !_0. (!abs1 rep2 f x. happ (happ (happ (happ _0 abs1) rep2) f) x = h4/quotient/_2D_2D_3E abs1 rep2 f x) ==> (!rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. happ f x = h4/quotient/_2D_2D_3E rep1 abs2 (happ (happ (happ _0 abs1) rep2) f) x)))
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_Q225233,TV_Q225229]: ![V_f, V_g]: (![V_x]: s(TV_Q225229,happ(s(t_fun(TV_Q225233,TV_Q225229),V_f),s(TV_Q225233,V_x))) = s(TV_Q225229,happ(s(t_fun(TV_Q225233,TV_Q225229),V_g),s(TV_Q225233,V_x))) => s(t_fun(TV_Q225233,TV_Q225229),V_f) = s(t_fun(TV_Q225233,TV_Q225229),V_g))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
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_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_quotients_QUOTIENTu_u_ABSu_u_REP, axiom, ![TV_u_27a,TV_u_27b]: ![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_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a))).
fof(ah4s_quotients_FUNu_u_MAPu_u_THM, axiom, ![TV_u_27d,TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_h, V_g, V_f]: s(TV_u_27d,h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_fun(TV_u_27c,TV_u_27b),V_h),s(TV_u_27a,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_h),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x)))))))).
fof(ch4s_quotients_LAMBDAu_u_PRS1, conjecture, ![TV_u_27a,TV_u_27b,TV_u_27d,TV_u_27c]: ![V_uu_0]: (![V_abs1, V_rep2, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_abs1))),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(TV_u_27a,V_x))) = s(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),s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27a,V_x))) => ![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_x]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,V_x))) = s(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),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_abs1))),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(TV_u_27c,V_x))))))).
