theorem
  for A being Subset of REAL,
      a,x being Real st x in a ** A holds
   ex b being Real st b in A & x = a * b
proof
  let A be Subset of REAL,
      a,x be Real;
  assume x in a ** A; then
  x in {a*b where b is Element of ExtREAL : b in A} by MEMBER_1:187; then
  consider b being Element of ExtREAL such that
A1:  x = a*b & b in A;
  reconsider b as Real by A1;
  take b;
  x = a*b by A1,XXREAL_3:def 5;
  hence thesis by A1;
end;
