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 non empty ext-real-membered set holds 0 ** A = {0}
proof
  let A be non empty ext-real-membered set;
  for e being object holds e in 0**A iff e = 0
  proof
    let e be object;
    consider r being ExtReal such that
A1: r in A by MEMBERED:8;
    hereby
      assume e in 0**A;
      then ex z being Element of ExtREAL st e = 0 * z & z in A
        by MEMBER_1:188;
      hence e = 0;
    end;
    assume e = 0;
    then e = 0*r;
    hence thesis by A1,MEMBER_1:186;
  end;
  hence thesis by TARSKI:def 1;
end;
