reserve a,b,r,x,y for Real,
  i,j,k,n for Nat,
  x1 for set;

theorem Th8:
  for X be Subset of REAL holds r**X = {r*x : x in X}
proof
  let X be Subset of REAL;
  thus r**X c= {r*x : x in X}
  proof
    let y be object;
    assume y in r**X;
    then consider z being Element of ExtREAL such that
A1: y = r * z & z in X by MEMBER_1:188;
    reconsider z1 = z as Element of REAL by A1;
    y = r * z1 by A1,XXREAL_3:def 5;
    hence thesis by A1;
  end;
  let y be object;
  assume y in {r*x : x in X};
  then consider z being Real such that
A2:y = r * z & z in X;
  thus thesis by A2,MEMBER_1:193;
end;
