%   ORIGINAL: h4/set__relation/tc__ind
% 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/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% 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/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pair/LAMBDA__PROD: !P. (\p. P p) = h4/pair/UNCURRY (\p1 p2. P (h4/pair/_2C p1 p2))
% Assm: h4/pair/PFORALL__THM: !P. (!x y. P x y) <=> $forall (h4/pair/UNCURRY (\x y. P x y))
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/set__relation/tc__def: h4/set__relation/tc = (\r a0. !tc_27. (!a00. (?x y. a00 = h4/pair/_2C x y /\ r (h4/pair/_2C x y)) \/ (?x y. a00 = h4/pair/_2C x y /\ (?z. tc_27 (h4/pair/_2C x z) /\ tc_27 (h4/pair/_2C z y))) ==> tc_27 a00) ==> tc_27 a0)
% Goal: !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)
%   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_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% 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_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_pairs_LAMBDAu_u_PROD]: !_1. (!P p1 p2. happ (happ (happ _1 P) p1) p2 = happ P (h4/pair/_2C p1 p2)) ==> (!_0. (!P p1. happ (happ _0 P) p1 = happ (happ _1 P) p1) ==> (!P x. happ P x = happ (h4/pair/UNCURRY (happ _0 P)) x))
% Assm [h4s_pairs_PFORALLu_u_THM]: !_1. (!P x y. happ (happ (happ _1 P) x) y <=> happ (happ P x) y) ==> (!_0. (!P x. happ (happ _0 P) x = happ (happ _1 P) x) ==> (!P. (!x y. happ (happ P x) y) <=> $forall (h4/pair/UNCURRY (happ _0 P))))
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_setu_u_relations_tcu_u_def]: !x x. happ (h4/set__relation/tc x) x <=> (!tc_27. (!a00. (?x y. a00 = h4/pair/_2C x y /\ happ x (h4/pair/_2C x y)) \/ (?x y. a00 = h4/pair/_2C x y /\ (?z. happ tc_27 (h4/pair/_2C x z) /\ happ tc_27 (h4/pair/_2C z y))) ==> happ tc_27 a00) ==> happ tc_27 x)
% Goal: !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)
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_Q137174,TV_Q137170]: ![V_f, V_g]: (![V_x]: s(TV_Q137170,happ(s(t_fun(TV_Q137174,TV_Q137170),V_f),s(TV_Q137174,V_x))) = s(TV_Q137170,happ(s(t_fun(TV_Q137174,TV_Q137170),V_g),s(TV_Q137174,V_x))) => s(t_fun(TV_Q137174,TV_Q137170),V_f) = s(t_fun(TV_Q137174,TV_Q137170),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_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,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_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_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),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_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_pairs_LAMBDAu_u_PROD, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P, V_p1, V_p2]: s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_1),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P))),s(TV_u_27a,V_p1))),s(TV_u_27b,V_p2))) = s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_p1),s(TV_u_27b,V_p2))))) => ![V_uu_0]: (![V_P, V_p1]: s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P))),s(TV_u_27a,V_p1))) = s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_1),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P))),s(TV_u_27a,V_p1))) => ![V_P, V_x]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))) = s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P))))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)))))).
fof(ah4s_pairs_PFORALLu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P, V_x, V_y]: 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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = 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_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) => ![V_uu_0]: (![V_P, V_x]: s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P))),s(TV_u_27a,V_x))) => ![V_P]: (![V_x, V_y]: p(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_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> p(s(t_bool,d_forall(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P)))))))))))).
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_setu_u_relations_tcu_u_def, axiom, ![TV_u_27a]: ![V_x, V_x0]: (p(s(t_bool,happ(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_x))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_x0)))) <=> ![V_tcu_27]: (![V_a00]: ((?[V_x1, V_y]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_a00) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x1),s(TV_u_27a,V_y))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_x),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x1),s(TV_u_27a,V_y))))))) | ?[V_x1, V_y]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_a00) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x1),s(TV_u_27a,V_y))) & ?[V_z]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x1),s(TV_u_27a,V_z)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(TV_u_27a,V_y))))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_x0))))))).
fof(ch4s_setu_u_relations_tcu_u_ind, conjecture, ![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))))))).
