%   ORIGINAL: h4/set__relation/tc__strongind0
% 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/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% 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/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/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. r (h4/pair/_2C x y) ==> tc_27 (h4/pair/_2C x y)) /\ (!x y. (?z. h4/set__relation/tc r (h4/pair/_2C x z) /\ tc_27 (h4/pair/_2C x z) /\ h4/set__relation/tc r (h4/pair/_2C z y) /\ tc_27 (h4/pair/_2C z y)) ==> tc_27 (h4/pair/_2C x y)) ==> (!a0. h4/set__relation/tc r a0 ==> tc_27 a0)
%   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_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% 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_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_setu_u_relations_tcu_u_def]: !x x. 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. happ r (h4/pair/_2C x y) ==> happ tc_27 (h4/pair/_2C x y)) /\ (!x y. (?z. h4/set__relation/tc r (h4/pair/_2C x z) /\ happ tc_27 (h4/pair/_2C x z) /\ h4/set__relation/tc r (h4/pair/_2C z y) /\ happ tc_27 (h4/pair/_2C z y)) ==> happ tc_27 (h4/pair/_2C x y)) ==> (!a0. h4/set__relation/tc r a0 ==> happ tc_27 a0)
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_Q137049,TV_Q137045]: ![V_f, V_g]: (![V_x]: s(TV_Q137045,happ(s(t_fun(TV_Q137049,TV_Q137045),V_f),s(TV_Q137049,V_x))) = s(TV_Q137045,happ(s(t_fun(TV_Q137049,TV_Q137045),V_g),s(TV_Q137049,V_x))) => s(t_fun(TV_Q137049,TV_Q137045),V_f) = s(t_fun(TV_Q137049,TV_Q137045),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_CONJu_u_ASSOC, 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_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_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_setu_u_relations_tcu_u_def, axiom, ![TV_u_27a]: ![V_x, V_x0]: (p(s(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_strongind0, conjecture, ![TV_u_27a]: ![V_tcu_27, V_r]: ((![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r),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)))))) => 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_x),s(TV_u_27a,V_y))))))) & ![V_x, V_y]: (?[V_z]: (p(s(t_bool,h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r),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_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_x),s(TV_u_27a,V_z)))))) & (p(s(t_bool,h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r),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),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),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y)))))))) => ![V_a0]: (p(s(t_bool,h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_a0)))) => 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_a0))))))).
