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 Th16;
  hence thesis by A1;
end;
