theorem Th103:
  x in a * H iff ex g st x = a * g & g in H
proof
  thus x in a * H implies ex g st x = a * g & g in H
  proof
    assume x in a * H;
    then consider g such that
A1: x = a * g & g in carr(H) by Th27;
    take g;
    thus thesis by A1;
  end;
  given g such that
A2: x = a * g and
A3: g in H;
  g in carr(H) by A3;
  hence thesis by A2,Th27;
end;
