%   ORIGINAL: h4/seq/SEQ__CBOUNDED
% 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/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/arithmetic/LESS__EQ__REFL: !m. h4/arithmetic/_3C_3D m m
% Assm: h4/arithmetic/GREATER__EQ: !n m. h4/arithmetic/_3E_3D n m <=> h4/arithmetic/_3C_3D m n
% Assm: h4/real/REAL__LT__01: h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% 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))
% Assm: h4/seq/cauchy0: !f. h4/seq/cauchy f <=> (!e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> (?N. !m n. h4/arithmetic/_3E_3D m N /\ h4/arithmetic/_3E_3D n N ==> h4/realax/real__lt (h4/real/abs (h4/real/real__sub (f m) (f n))) e))
% Goal: !f. h4/seq/cauchy f ==> h4/nets/bounded (h4/pair/_2C h4/topology/mr1 h4/arithmetic/_3E_3D) f
%   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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_arithmetics_LESSu_u_EQu_u_REFL]: !m. h4/arithmetic/_3C_3D m m
% Assm [h4s_arithmetics_GREATERu_u_EQ]: !n m. happ (happ h4/arithmetic/_3E_3D n) m <=> h4/arithmetic/_3C_3D m n
% Assm [h4s_reals_REALu_u_LTu_u_01]: h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% 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))
% Assm [h4s_seqs_cauchy0]: !f. h4/seq/cauchy f <=> (!e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> (?N. !m n. happ (happ h4/arithmetic/_3E_3D m) N /\ happ (happ h4/arithmetic/_3E_3D n) N ==> h4/realax/real__lt (h4/real/abs (h4/real/real__sub (happ f m) (happ f n))) e))
% Goal: !f. h4/seq/cauchy f ==> h4/nets/bounded (h4/pair/_2C h4/topology/mr1 h4/arithmetic/_3E_3D) f
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_Q133308,TV_Q133304]: ![V_f, V_g]: (![V_x]: s(TV_Q133304,happ(s(t_fun(TV_Q133308,TV_Q133304),V_f),s(TV_Q133308,V_x))) = s(TV_Q133304,happ(s(t_fun(TV_Q133308,TV_Q133304),V_g),s(TV_Q133308,V_x))) => s(t_fun(TV_Q133308,TV_Q133304),V_f) = s(t_fun(TV_Q133308,TV_Q133304),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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_IMPu_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_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_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_REFL, axiom, ![V_m]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_m))))).
fof(ah4s_arithmetics_GREATERu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_arithmetics_u_3eu_3d),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))).
fof(ah4s_reals_REALu_u_LTu_u_01, axiom, p(s(t_bool,h4s_realaxs_realu_u_lt(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,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))).
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(ah4s_seqs_cauchy0, axiom, ![V_f]: (p(s(t_bool,h4s_seqs_cauchy(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f)))) <=> ![V_e]: (p(s(t_bool,h4s_realaxs_realu_u_lt(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_e)))) => ?[V_N]: ![V_m, V_n]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_arithmetics_u_3eu_3d),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_N)))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_arithmetics_u_3eu_3d),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_N))))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),s(t_h4s_nums_num,V_m))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),s(t_h4s_nums_num,V_n))))))),s(t_h4s_realaxs_real,V_e)))))))).
fof(ch4s_seqs_SEQu_u_CBOUNDED, conjecture, ![V_f]: (p(s(t_bool,h4s_seqs_cauchy(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f)))) => p(s(t_bool,h4s_netss_bounded(s(t_h4s_pairs_prod(t_h4s_topologys_metric(t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool))),h4s_pairs_u_2c(s(t_h4s_topologys_metric(t_h4s_realaxs_real),h4s_topologys_mr1),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_arithmetics_u_3eu_3d))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f)))))).
