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 Th5:
  rng a = {r} & rng b = {r} implies a = b
proof
  assume that
A1: rng a = {r} and
A2: rng b = {r};
  now
    let n;
    a.n in rng a by VALUED_0:28;
    then
A3: a.n = r by A1,TARSKI:def 1;
    b.n in rng b by VALUED_0:28;
    hence a.n = b.n by A2,A3,TARSKI:def 1;
  end;
  hence thesis;
end;
