%   ORIGINAL: h4/seq/SEQ__ABS__IMP
% 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/nets/NET__ABS: !x0 x g. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) ==> h4/nets/tends (\n. h4/real/abs (x n)) (h4/real/abs x0) (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g)
% Assm: h4/seq/tends__num__real: !x0 x. h4/seq/_2D_2D_3E x x0 <=> h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) h4/arithmetic/_3E_3D)
% Goal: !l f. h4/seq/_2D_2D_3E f l ==> h4/seq/_2D_2D_3E (\n. h4/real/abs (f n)) (h4/real/abs l)
%   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_netss_NETu_u_ABS]: !_0. (!x n. happ (happ _0 x) n = h4/real/abs (happ x n)) ==> (!x0 x g. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) ==> h4/nets/tends (happ _0 x) (h4/real/abs x0) (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g))
% Assm [h4s_seqs_tendsu_u_numu_u_real]: !x0 x. h4/seq/_2D_2D_3E x x0 <=> h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) h4/arithmetic/_3E_3D)
% Goal: !_0. (!f n. happ (happ _0 f) n = h4/real/abs (happ f n)) ==> (!l f. h4/seq/_2D_2D_3E f l ==> h4/seq/_2D_2D_3E (happ _0 f) (h4/real/abs l))
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_Q133633,TV_Q133629]: ![V_f, V_g]: (![V_x]: s(TV_Q133629,happ(s(t_fun(TV_Q133633,TV_Q133629),V_f),s(TV_Q133633,V_x))) = s(TV_Q133629,happ(s(t_fun(TV_Q133633,TV_Q133629),V_g),s(TV_Q133633,V_x))) => s(t_fun(TV_Q133633,TV_Q133629),V_f) = s(t_fun(TV_Q133633,TV_Q133629),V_g))).
fof(ah4s_netss_NETu_u_ABS, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_n]: s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x))),s(TV_u_27a,V_n))) = 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_x),s(TV_u_27a,V_n))))) => ![V_x0, V_x, V_g]: (p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x),s(t_h4s_realaxs_real,V_x0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(s(t_h4s_topologys_topology(t_h4s_realaxs_real),h4s_topologys_mtop(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)))))) => p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x))),s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_x0))),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(s(t_h4s_topologys_topology(t_h4s_realaxs_real),h4s_topologys_mtop(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))))))))).
fof(ah4s_seqs_tendsu_u_numu_u_real, axiom, ![V_x0, V_x]: s(t_bool,h4s_seqs_u_2du_2du_3e(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x),s(t_h4s_realaxs_real,V_x0))) = s(t_bool,h4s_netss_tends(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x),s(t_h4s_realaxs_real,V_x0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(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_topology(t_h4s_realaxs_real),h4s_topologys_mtop(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)))))).
fof(ch4s_seqs_SEQu_u_ABSu_u_IMP, conjecture, ![V_uu_0]: (![V_f, V_n]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_uu_0),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,h4s_reals_abs(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))))) => ![V_l, V_f]: (p(s(t_bool,h4s_seqs_u_2du_2du_3e(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l)))) => p(s(t_bool,h4s_seqs_u_2du_2du_3e(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_l))))))))).
