theorem Th58:
  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 x in carr H * a by Def6A;
    then consider b such that
A1: x = b * a & b in carr H by Th41;
    take b;
    thus thesis by A1;
  end;
  given g such that
A2: x = g * a and
A3: g in H;
  x in carr H * a by A2,A3,Th41;
  hence thesis by Def6A;
end;
