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;
  thus thesis by A2,A3,Th27;
end;
