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) .|. (u - v) = x .|. u - x .|. v - y .|. u + y .|. v
proof
  (x - y) .|. (u - v) = x .|. (u - v) - y .|. (u - v) by Th11
    .= ( x .|. u - x .|. v ) - y .|. (u - v) by Th12
    .= ( x .|. u - x .|. v ) - ( y .|. u - y .|. v ) by Th12;
  hence thesis;
end;
