%   ORIGINAL: h4/quotient__pair/SND__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/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/quotient__pair/PAIR__REL__THM: !d c b a R2 R1. h4/quotient__pair/_23_23_23 R1 R2 (h4/pair/_2C a b) (h4/pair/_2C c d) <=> R1 a c /\ R2 b d
% Goal: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!p1 p2. h4/quotient__pair/_23_23_23 R1 R2 p1 p2 ==> R2 (h4/pair/SND p1) (h4/pair/SND p2)))
%   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_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_quotientu_u_pairs_PAIRu_u_RELu_u_THM]: !d c b a R2 R1. h4/quotient__pair/_23_23_23 R1 R2 (h4/pair/_2C a b) (h4/pair/_2C c d) <=> happ (happ R1 a) c /\ happ (happ R2 b) d
% Goal: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!p1 p2. h4/quotient__pair/_23_23_23 R1 R2 p1 p2 ==> happ (happ R2 (h4/pair/SND p1)) (h4/pair/SND p2)))
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_Q237170,TV_Q237166]: ![V_f, V_g]: (![V_x]: s(TV_Q237166,happ(s(t_fun(TV_Q237170,TV_Q237166),V_f),s(TV_Q237170,V_x))) = s(TV_Q237166,happ(s(t_fun(TV_Q237170,TV_Q237166),V_g),s(TV_Q237170,V_x))) => s(t_fun(TV_Q237170,TV_Q237166),V_f) = s(t_fun(TV_Q237170,TV_Q237166),V_g))).
fof(ah4s_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_quotientu_u_pairs_PAIRu_u_RELu_u_THM, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b,TV_u_27d]: ![V_d, V_c, V_b, V_a, V_R2, V_R1]: (p(s(t_bool,h4s_quotientu_u_pairs_u_23u_23u_23(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))),s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),h4s_pairs_u_2c(s(TV_u_27c,V_c),s(TV_u_27d,V_d)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1),s(TV_u_27a,V_a))),s(TV_u_27c,V_c)))) & p(s(t_bool,happ(s(t_fun(TV_u_27d,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2),s(TV_u_27b,V_b))),s(TV_u_27d,V_d))))))).
fof(ch4s_quotientu_u_pairs_SNDu_u_RSP, 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_p1, V_p2]: (p(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_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p1),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p2)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p1))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p2)))))))))).
