reserve i,s,t,m,n,k for Nat,
        c,d,e for Element of NAT,
        fn for FinSequence of NAT,
        x,y for Integer;

theorem Th3:
 for X being Subset of INT, a being Integer holds
 x in a ** X iff ex y being Integer st y in X & x = a * y
proof
  let X be Subset of INT, a be Integer;
  hereby
    assume x in a ** X;
    then consider z being Complex such that
A1: x = a * z and
A2: z in X by MEMBER_1:195;
    reconsider z as Integer by A2;
    take z;
    thus z in X & x = a * z by A2,A1;
  end;
  thus thesis by MEMBER_1:193;
end;
