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) .|. (x - y) = x .|. x - y .|. y
proof
  (x + y) .|. (x - y) = x .|. (x - y) + y .|. (x - y) by Def2
    .= (x .|. x - x .|. y) + y .|. (x - y) by Th12
    .= (x .|. x - x .|. y) + (x .|. y - y .|. y) by Th12;
  hence thesis;
end;
