reserve a,b,m,x,y,i1,i2,i3,i for Integer,
  k,p,q,n for Nat,
  c,c1,c2 for Element of NAT,
  z for set;

theorem Th4:
  for X being real-membered set, a being Real holds (a = 0 &
X is non empty implies a ** X = {0}) & (a ** X = {0} implies a = 0 or X = {0})
proof
  let X be real-membered set, a be Real;
  thus a = 0 & X is non empty implies a ** X = {0}
  proof
    assume that
A1: a = 0 and
A2: X is non empty;
    thus a ** X c= {0}
    proof
      let i be object;
      assume
A3:   i in a ** X;
      then reconsider i as Real;
      ex x being Complex st
      i = a * x & x in X by A3,MEMBER_1:195;
      hence thesis by A1,TARSKI:def 1;
    end;
    then a ** X = {} or a ** X = {0} by ZFMISC_1:33;
    hence thesis by A2,INTEGRA2:7;
  end;
  assume that
A4: a ** X = {0} and
A5: a <> 0;
  X,a ** X are_equipotent by A5,Th3;
  then consider x being object such that
A6: X = {x} by A4,CARD_1:28;
A7: x in X by A6,TARSKI:def 1;
  then reconsider x as Real;
  a * x in a ** X by A7,MEMBER_1:193;
  then a * x = 0 by A4,TARSKI:def 1;
  hence thesis by A5,A6,XCMPLX_1:6;
end;
