%   ORIGINAL: h4/quotient__pred__set/IN__GSPECR
% 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/quotient__pred__set/GSPECR__def: !v f R2 R1. h4/quotient__pred__set/GSPECR R1 R2 f v <=> h4/bool/RES__EXISTS (h4/quotient/respects R1) (\x. h4/quotient__pair/_23_23_23 R2 $equals (h4/pair/_2C v T) (f x))
% Goal: !v f R2 R1. h4/bool/IN v (h4/quotient__pred__set/GSPECR R1 R2 f) <=> h4/bool/RES__EXISTS (h4/quotient/respects R1) (\x. h4/quotient__pair/_23_23_23 R2 $equals (h4/pair/_2C v T) (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_TRUTH]: T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_quotientu_u_predu_u_sets_GSPECRu_u_def]: !_0. (!R2 v f x. happ (happ (happ (happ _0 R2) v) f) x <=> h4/quotient__pair/_23_23_23 R2 $equals (h4/pair/_2C v T) (happ f x)) ==> (!v f R2 R1. happ (h4/quotient__pred__set/GSPECR R1 R2 f) v <=> h4/bool/RES__EXISTS (h4/quotient/respects R1) (happ (happ (happ _0 R2) v) f))
% Goal: !_0. (!R2 v f x. happ (happ (happ (happ _0 R2) v) f) x <=> h4/quotient__pair/_23_23_23 R2 $equals (h4/pair/_2C v T) (happ f x)) ==> (!v f R2 R1. h4/bool/IN v (h4/quotient__pred__set/GSPECR R1 R2 f) <=> h4/bool/RES__EXISTS (h4/quotient/respects R1) (happ (happ (happ _0 R2) v) 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_Q85651,TV_Q85647]: ![V_f, V_g]: (![V_x]: s(TV_Q85647,happ(s(t_fun(TV_Q85651,TV_Q85647),V_f),s(TV_Q85651,V_x))) = s(TV_Q85647,happ(s(t_fun(TV_Q85651,TV_Q85647),V_g),s(TV_Q85651,V_x))) => s(t_fun(TV_Q85651,TV_Q85647),V_f) = s(t_fun(TV_Q85651,TV_Q85647),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_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_predu_u_sets_SPECIFICATION, axiom, ![TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
fof(ah4s_quotientu_u_predu_u_sets_GSPECRu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_R2, V_v, V_f, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_v))),s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)),V_f))),s(TV_u_27a,V_x))) = s(t_bool,h4s_quotientu_u_pairs_u_23u_23u_23(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals),s(t_h4s_pairs_prod(TV_u_27b,t_bool),h4s_pairs_u_2c(s(TV_u_27b,V_v),s(t_bool,t))),s(t_h4s_pairs_prod(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)),V_f),s(TV_u_27a,V_x))))) => ![V_v, V_f, V_R2, V_R1]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),h4s_quotientu_u_predu_u_sets_gspecr(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,t_h4s_pairs_prod(TV_u_27b,t_bool)),V_f))),s(TV_u_27b,V_v))) = s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_v))),s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)),V_f))))))).
fof(ch4s_quotientu_u_predu_u_sets_INu_u_GSPECR, conjecture, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_R2, V_v, V_f, V_x]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),t_fun(TV_u_27b,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R2))),s(TV_u_27a,V_v))),s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f))),s(TV_u_27b,V_x))) = s(t_bool,h4s_quotientu_u_pairs_u_23u_23u_23(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R2),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals),s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_v),s(t_bool,t))),s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f),s(TV_u_27b,V_x))))) => ![V_v, V_f, V_R2, V_R1]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_quotientu_u_predu_u_sets_gspecr(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R2),s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f))))) = s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27b,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),t_fun(TV_u_27b,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R2))),s(TV_u_27a,V_v))),s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f))))))).
