reserve X for non empty UNITSTR;
reserve a, b for Real;
reserve x, y for Point of X;
reserve X for RealUnitarySpace;
reserve x, y, z, u, v for Point of X;

theorem
  x, y are_orthogonal implies (x - y) .|. (x - y) = x .|. x + y .|. y
proof
  assume x, y are_orthogonal;
  then
A1: x .|. y = 0;
  (x - y) .|. (x - y) = x .|. x - 2 * x .|. y + y .|. y by Th18
    .= x .|. x + y .|. y - 0 by A1;
  hence thesis;
end;
