theorem Th45:
  ex v st u+v = ID(M)
proof
  set a = the Element of M;
  consider b such that
A1: u = vect(a,b) by Th35;
  u + vect(b,a) = vect(a,a) by A1,Th40
    .= ID(M) by Th32;
  hence thesis;
end;
