%   ORIGINAL: h4/set__relation/reln__rel__conv__thms_c0
% 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/set__relation/in__rel__to__reln: !xy R. h4/bool/IN xy (h4/set__relation/rel__to__reln R) <=> R (h4/pair/FST xy) (h4/pair/SND xy)
% Goal: !xy R. h4/bool/IN xy (h4/set__relation/rel__to__reln R) <=> R (h4/pair/FST xy) (h4/pair/SND xy)
%   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_setu_u_relations_inu_u_relu_u_tou_u_reln]: !xy R. h4/bool/IN xy (h4/set__relation/rel__to__reln R) <=> happ (happ R (h4/pair/FST xy)) (h4/pair/SND xy)
% Goal: !xy R. h4/bool/IN xy (h4/set__relation/rel__to__reln R) <=> happ (happ R (h4/pair/FST xy)) (h4/pair/SND xy)
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_Q139776,TV_Q139772]: ![V_f, V_g]: (![V_x]: s(TV_Q139772,happ(s(t_fun(TV_Q139776,TV_Q139772),V_f),s(TV_Q139776,V_x))) = s(TV_Q139772,happ(s(t_fun(TV_Q139776,TV_Q139772),V_g),s(TV_Q139776,V_x))) => s(t_fun(TV_Q139776,TV_Q139772),V_f) = s(t_fun(TV_Q139776,TV_Q139772),V_g))).
fof(ah4s_setu_u_relations_inu_u_relu_u_tou_u_reln, axiom, ![TV_u_27a,TV_u_27b]: ![V_xy, V_R]: s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_xy),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_setu_u_relations_relu_u_tou_u_reln(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R))))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_xy))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_xy)))))).
fof(ch4s_setu_u_relations_relnu_u_relu_u_convu_u_thmsu_c0, conjecture, ![TV_u_27a,TV_u_27b]: ![V_xy, V_R]: s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_xy),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_setu_u_relations_relu_u_tou_u_reln(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R))))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_xy))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_xy)))))).
