%   ORIGINAL: h4/nets/MR1__BOUNDED
% 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/real/REAL__ADD__SYM: !y x. h4/realax/real__add x y = h4/realax/real__add y x
% Assm: h4/real/REAL__LT__RADD: !z y x. h4/realax/real__lt (h4/realax/real__add x z) (h4/realax/real__add y z) <=> h4/realax/real__lt x y
% Assm: h4/real/REAL__LET__TRANS: !z y x. h4/real/real__lte x y /\ h4/realax/real__lt y z ==> h4/realax/real__lt x z
% Assm: h4/real/REAL__SUB__ADD: !y x. h4/realax/real__add (h4/real/real__sub x y) y = x
% Assm: h4/real/REAL__SUB__LZERO: !x. h4/real/real__sub (h4/real/real__of__num h4/num/0) x = h4/realax/real__neg x
% Assm: h4/real/ABS__NEG: !x. h4/real/abs (h4/realax/real__neg x) = h4/real/abs x
% Assm: h4/real/ABS__TRIANGLE: !y x. h4/real/real__lte (h4/real/abs (h4/realax/real__add x y)) (h4/realax/real__add (h4/real/abs x) (h4/real/abs y))
% Assm: h4/real/ABS__SUB: !y x. h4/real/abs (h4/real/real__sub x y) = h4/real/abs (h4/real/real__sub y x)
% Assm: h4/topology/MR1__DEF: !y x. h4/topology/dist h4/topology/mr1 (h4/pair/_2C x y) = h4/real/abs (h4/real/real__sub y x)
% Assm: h4/nets/bounded0: !m g f. h4/nets/bounded (h4/pair/_2C m g) f <=> (?k x N. g N N /\ (!n. g n N ==> h4/realax/real__lt (h4/topology/dist m (h4/pair/_2C (f n) x)) k))
% Goal: !g f. h4/nets/bounded (h4/pair/_2C h4/topology/mr1 g) f <=> (?k N. g N N /\ (!n. g n N ==> h4/realax/real__lt (h4/real/abs (f n)) k))
%   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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_reals_REALu_u_ADDu_u_SYM]: !y x. h4/realax/real__add x y = h4/realax/real__add y x
% Assm [h4s_reals_REALu_u_LTu_u_RADD]: !z y x. h4/realax/real__lt (h4/realax/real__add x z) (h4/realax/real__add y z) <=> h4/realax/real__lt x y
% Assm [h4s_reals_REALu_u_LETu_u_TRANS]: !z y x. h4/real/real__lte x y /\ h4/realax/real__lt y z ==> h4/realax/real__lt x z
% Assm [h4s_reals_REALu_u_SUBu_u_ADD]: !y x. h4/realax/real__add (h4/real/real__sub x y) y = x
% Assm [h4s_reals_REALu_u_SUBu_u_LZERO]: !x. h4/real/real__sub (h4/real/real__of__num h4/num/0) x = h4/realax/real__neg x
% Assm [h4s_reals_ABSu_u_NEG]: !x. h4/real/abs (h4/realax/real__neg x) = h4/real/abs x
% Assm [h4s_reals_ABSu_u_TRIANGLE]: !y x. h4/real/real__lte (h4/real/abs (h4/realax/real__add x y)) (h4/realax/real__add (h4/real/abs x) (h4/real/abs y))
% Assm [h4s_reals_ABSu_u_SUB]: !y x. h4/real/abs (h4/real/real__sub x y) = h4/real/abs (h4/real/real__sub y x)
% Assm [h4s_topologys_MR1u_u_DEF]: !y x. h4/topology/dist h4/topology/mr1 (h4/pair/_2C x y) = h4/real/abs (h4/real/real__sub y x)
% Assm [h4s_netss_bounded0]: !m g f. h4/nets/bounded (h4/pair/_2C m g) f <=> (?k x N. happ (happ g N) N /\ (!n. happ (happ g n) N ==> h4/realax/real__lt (h4/topology/dist m (h4/pair/_2C (happ f n) x)) k))
% Goal: !g f. h4/nets/bounded (h4/pair/_2C h4/topology/mr1 g) f <=> (?k N. happ (happ g N) N /\ (!n. happ (happ g n) N ==> h4/realax/real__lt (h4/real/abs (happ f n)) k))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q198797,TV_Q198793]: ![V_f, V_g]: (![V_x]: s(TV_Q198793,happ(s(t_fun(TV_Q198797,TV_Q198793),V_f),s(TV_Q198797,V_x))) = s(TV_Q198793,happ(s(t_fun(TV_Q198797,TV_Q198793),V_g),s(TV_Q198797,V_x))) => s(t_fun(TV_Q198797,TV_Q198793),V_f) = s(t_fun(TV_Q198797,TV_Q198793),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_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(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_reals_REALu_u_ADDu_u_SYM, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_reals_REALu_u_LTu_u_RADD, axiom, ![V_z, V_y, V_x]: s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) = s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))).
fof(ah4s_reals_REALu_u_LETu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) & p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z)))))).
fof(ah4s_reals_REALu_u_SUBu_u_ADD, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_realaxs_real,V_x)).
fof(ah4s_reals_REALu_u_SUBu_u_LZERO, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_reals_ABSu_u_NEG, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x))))) = s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_reals_ABSu_u_TRIANGLE, axiom, ![V_y, V_x]: p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_y))))))))).
fof(ah4s_reals_ABSu_u_SUB, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))))) = s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_topologys_MR1u_u_DEF, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(t_h4s_realaxs_real),h4s_topologys_mr1),s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))))) = s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_netss_bounded0, axiom, ![TV_u_27b,TV_u_27a]: ![V_m, V_g, V_f]: (p(s(t_bool,h4s_netss_bounded(s(t_h4s_pairs_prod(t_h4s_topologys_metric(TV_u_27a),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))),h4s_pairs_u_2c(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_g))),s(t_fun(TV_u_27b,TV_u_27a),V_f)))) <=> ?[V_k, V_x, V_N]: (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_g),s(TV_u_27b,V_N))),s(TV_u_27b,V_N)))) & ![V_n]: (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_g),s(TV_u_27b,V_n))),s(TV_u_27b,V_N)))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_f),s(TV_u_27b,V_n))),s(TV_u_27a,V_x))))),s(t_h4s_realaxs_real,V_k)))))))).
fof(ch4s_netss_MR1u_u_BOUNDED, conjecture, ![TV_u_27a]: ![V_g, V_f]: (p(s(t_bool,h4s_netss_bounded(s(t_h4s_pairs_prod(t_h4s_topologys_metric(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(s(t_h4s_topologys_metric(t_h4s_realaxs_real),h4s_topologys_mr1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g))),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f)))) <=> ?[V_k, V_N]: (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_g),s(TV_u_27a,V_N))),s(TV_u_27a,V_N)))) & ![V_n]: (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_g),s(TV_u_27a,V_n))),s(TV_u_27a,V_N)))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(TV_u_27a,V_n))))),s(t_h4s_realaxs_real,V_k)))))))).
