reserve x,y for Element of REAL;
reserve i,j,k for Element of NAT;
reserve a,b for Element of REAL;

theorem Th16:
  for x being Element of REAL holds *(x,x) in REAL+
proof
  let x be Element of REAL;
A1: x in REAL+ \/ [:{{}},REAL+:] by XBOOLE_0:def 5;
  per cases by A1,XBOOLE_0:def 3;
  suppose
    x in REAL+;
    then
    ex x9,y9 being Element of REAL+ st x = x9 & x = y9 & *(x,x) = x9 *' y9
    by Def2;
    hence thesis;
  end;
  suppose
    x in [:{0},REAL+:];
    then
    ex x9,y9 being Element of REAL+ st x = [0,x9] & x = [0,y9] & *(x,x) =
    y9 *' x9 by Def2;
    hence thesis;
  end;
end;
