reserve y for object, X for set;
reserve x,x0,x1,x2,g,g1,g2,r,r1,s,p,p1 for Real;
reserve n,m,k for Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem Th1:
  (for r holds r in Y iff r in REAL) iff Y = REAL
proof
  thus (for r holds r in Y iff r in REAL) implies Y=REAL
  proof
    assume for r holds r in Y iff r in REAL;
    then for y being object holds y in Y iff y in REAL;
    hence thesis by TARSKI:2;
  end;
  assume
A1: Y=REAL;
  let r;
  thus r in Y implies r in REAL;
  thus thesis by A1;
end;
