reserve X for non empty CUNITSTR;
reserve a, b for Complex;
reserve x, y for Point of X;
reserve X for ComplexUnitarySpace;
reserve x, y, z, u, v for Point of X;

theorem
  (x+y).|.(x-y) = x.|.x - x.|.y + y.|.x - y.|.y
proof
  (x + y) .|. (x - y) = x .|. (x - y) + y .|. (x - y) by Def11
    .= (x .|. x - x .|. y) + y .|. (x - y) by Th22
    .= (x .|. x - x .|. y) + (y .|. x - y .|. y) by Th22
    .= (x.|.x - x.|.y) + y.|.x + -y.|.y;
  hence thesis;
end;
