theorem ThB42:
  x in g * A iff ex h st x = g * h & h in A
proof
  thus x in g * A implies ex h st x = g * h & h in A
  proof
    assume x in g * A;
    then consider a,b such that
A1: x = a * b and
A2: a in {g} and
A3: b in A;
    a = g by A2,TARSKI:def 1;
    hence thesis by A1,A3;
  end;
  given h such that
A4: x = g * h & h in A;
  g in {g} by TARSKI:def 1;
  hence thesis by A4;
end;
