reserve a,b,r,x,y for Real,
  i,j,k,n for Nat,
  x1 for set;
reserve A, B for non empty closed_interval Subset of REAL;
reserve f, g for Function of A,REAL;
reserve D, D1, D2 for Division of A;

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;
