
theorem P3:
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 * (a + b) + b * (a + b) by VECTSP_1:def 7
              .= (a * a + a * b) + b * (a + b) by VECTSP_1:def 7
              .= (a * a + a * b) + (b * a + b * b) by VECTSP_1:def 7
              .= a * a + (a * b + (b * a + b * b)) by RLVECT_1:def 3
              .= a^2 + ((a * b + b * a) + b * b) by RLVECT_1:def 3
              .= a^2 + (2 '*' (a * b) + b^2) by RING_5:2
              .= a^2 + 2 '*' (a * b) + b^2 by RLVECT_1:def 3
              .= a^2 + 2 '*' a * b + b^2 by c1;
end;
