
theorem P4:
for R being commutative Ring,
    a,b being Element of R holds (a - b)^2 = a^2 - 2 '*' a * b + b^2
proof
let R be commutative Ring; let a,b be Element of R;
thus (a - b)^2 = a^2 + 2 '*' a * (-b) + ((-b)^2) by P3
              .= a^2 + 2 '*' (a * (-b)) + ((-b)^2) by c1
              .= a^2 + 2 '*' (-(a * b)) + (-b)^2 by VECTSP_1:8
              .= a^2 + -(2 '*' (a * b)) + (-b)^2 by c1a
              .= a^2 - (2 '*' a * b) + (-b)^2 by c1
              .= a^2 - (2 '*' a * b) + b^2 by VECTSP_1:10;
end;
