theorem Th104:
  x in H * a iff ex g st x = g * a & g in H
proof
  thus x in H * a implies ex g st x = g * a & g in H
  proof
    assume x in H * a;
    then consider g such that
A1: x = g * a & g in carr(H) by Th28;
    take g;
    thus thesis by A1;
  end;
  given g such that
A2: x = g * a and
A3: g in H;
  g in carr(H) by A3;
  hence thesis by A2,Th28;
end;
