reserve x for object;
reserve x0,r,r1,r2,g,g1,g2,p,y0 for Real;
reserve n,m,k,l for Element of NAT;
reserve a,b,d for Real_Sequence;
reserve h,h1,h2 for non-zero 0-convergent Real_Sequence;
reserve c,c1 for constant Real_Sequence;
reserve A for open Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;
reserve L for LinearFunc;
reserve R for RestFunc;

theorem Th4:
  rng c = {x0} implies c is convergent & lim c = x0 & h + c is
  convergent & lim(h + c) = x0
proof
  assume
A1: rng c = {x0};
  thus c is convergent;
  x0 in rng c by A1,TARSKI:def 1;
  hence
A2: lim c = x0 by SEQ_4:25;
  thus h + c is convergent;
  lim h = 0;
  hence lim (h + c) = 0 + x0 by A2,SEQ_2:6
    .= x0;
end;
