theorem
  x,y are_orthogonal implies (x-y).|.(x-y) = x.|.x + y.|.y
proof
  assume
A1: x,y are_orthogonal;
  (x-y).|.(x-y) = x.|.x - x.|.y - y.|.x + y.|.y by Th28
    .= x.|.x + y.|.y - 0 by A1,Def12;
  hence thesis;
end;
