theorem Th47:
  u + v = u + w implies v = w
proof
  assume
A1: u + v = u + w;
  consider u9 such that
A2: u + u9 = ID(M) by Th45;
  v = v + ID(M) by Th44
    .= (u + u9) + v by A2,Th46
    .= (u9 + u) + v by Th46
    .= u9 + (u + w) by A1,Th43
    .= (u9 + u) + w by Th43
    .= ID(M) + w by A2,Th46
    .= w + ID(M) by Th46;
  hence thesis by Th44;
end;
