
theorem ch1:
for R being comRing
for a being Element of R,
    i being Integer holds (i '*' a)^2 = i^2 '*' a^2
proof
let R be comRing, a be Element of R, i be Integer;
thus (i '*' a)^2 = (i '*' a) * (i '*' a) by O_RING_1:def 1
                .= i '*' (a * (i '*' a)) by REALALG2:5
                .= i '*' ((i '*' a) * a) by GROUP_1:def 12
                .= i '*' (i '*' (a * a)) by REALALG2:5
                .= (i * i) '*' (a * a) by RING_3:65
                .= (i^2) '*' (a * a) by SQUARE_1:def 1
                .= (i^2) '*' a^2 by O_RING_1:def 1;
end;
