
theorem P4a:
for R being commutative Ring,
    a,b being Element of R holds (a + b) * (a - b) = a^2 - b^2
proof
let R be commutative Ring; let a,b be Element of R;
thus (a + b) * (a - b) = 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 * a + ((a * (-b) + b * a) + b * (-b)) by RLVECT_1:def 3
               .= a * a + ((-(a * b) + b * a) + b * (-b)) by VECTSP_1:8
               .= a * a + (0.R + b * (-b)) by RLVECT_1:5
               .= a^2 - b^2 by VECTSP_1:8;
end;
