%   ORIGINAL: h4/set__relation/tc__domain__range
% 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/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__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/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/set__relation/domain__def: !r. h4/set__relation/domain r = h4/pred__set/GSPEC (\x. h4/pair/_2C x (?y. h4/bool/IN (h4/pair/_2C x y) r))
% Assm: h4/set__relation/range__def: !r. h4/set__relation/range r = h4/pred__set/GSPEC (\y. h4/pair/_2C y (?x. h4/bool/IN (h4/pair/_2C x y) r))
% Assm: h4/set__relation/tc__ind: !tc_27 r. (!x y. h4/bool/IN (h4/pair/_2C x y) r ==> tc_27 x y) /\ (!x y. (?z. tc_27 x z /\ tc_27 z y) ==> tc_27 x y) ==> (!x y. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> tc_27 x y)
% Goal: !y x r. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> h4/bool/IN x (h4/set__relation/domain r) /\ h4/bool/IN y (h4/set__relation/range r)
%   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_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_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_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_predu_u_sets_GSPECIFICATION]: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = happ f x)
% Assm [h4s_setu_u_relations_domainu_u_def]: !_0. (!r x. ?v. (v <=> (?y. h4/bool/IN (h4/pair/_2C x y) r)) /\ happ (happ _0 r) x = h4/pair/_2C x v) ==> (!r. h4/set__relation/domain r = h4/pred__set/GSPEC (happ _0 r))
% Assm [h4s_setu_u_relations_rangeu_u_def]: !_0. (!r y. ?v. (v <=> (?x. h4/bool/IN (h4/pair/_2C x y) r)) /\ happ (happ _0 r) y = h4/pair/_2C y v) ==> (!r. h4/set__relation/range r = h4/pred__set/GSPEC (happ _0 r))
% Assm [h4s_setu_u_relations_tcu_u_ind]: !tc_27 r. (!x y. h4/bool/IN (h4/pair/_2C x y) r ==> happ (happ tc_27 x) y) /\ (!x y. (?z. happ (happ tc_27 x) z /\ happ (happ tc_27 z) y) ==> happ (happ tc_27 x) y) ==> (!x y. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> happ (happ tc_27 x) y)
% Goal: !y x r. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> h4/bool/IN x (h4/set__relation/domain r) /\ h4/bool/IN y (h4/set__relation/range r)
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_Q137476,TV_Q137472]: ![V_f, V_g]: (![V_x]: s(TV_Q137472,happ(s(t_fun(TV_Q137476,TV_Q137472),V_f),s(TV_Q137476,V_x))) = s(TV_Q137472,happ(s(t_fun(TV_Q137476,TV_Q137472),V_g),s(TV_Q137476,V_x))) => s(t_fun(TV_Q137476,TV_Q137472),V_f) = s(t_fun(TV_Q137476,TV_Q137472),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_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = 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_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_pairs_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (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(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(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_predu_u_sets_GSPECIFICATION, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f)))))) <=> ?[V_x]: 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))))).
fof(ah4s_setu_u_relations_domainu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_r, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_y]: p(s(t_bool,h4s_bools_in(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(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_r]: s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_domain(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))))))).
fof(ah4s_setu_u_relations_rangeu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_r, V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_x]: p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),V_r))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),V_r))),s(TV_u_27a,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_y),s(t_bool,V_v)))) => ![V_r]: s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_range(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),V_r))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),V_r))))))).
fof(ah4s_setu_u_relations_tcu_u_ind, axiom, ![TV_u_27a]: ![V_tcu_27, V_r]: ((![V_x, V_y]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) & ![V_x, V_y]: (?[V_z]: (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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))) & 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_tcu_27),s(TV_u_27a,V_z))),s(TV_u_27a,V_y))))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))))) => ![V_x, V_y]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))))).
fof(ch4s_setu_u_relations_tcu_u_domainu_u_range, conjecture, ![TV_u_27a]: ![V_y, V_x, V_r]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) => (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_domain(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_range(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))))).
