
theorem ch0:
for R being comRing,
    a,b being Element of R holds (a * b)^2 = a^2 * b^2
proof
let R be comRing, a,b be Element of R;
thus (a * b)^2 = (a * b) * (a * b) by O_RING_1:def 1
              .= a * (b * (a * b)) by GROUP_1:def 3
              .= a * (b * (b * a)) by GROUP_1:def 12
              .= a * ((b * b) * a) by GROUP_1:def 3
              .= a * (a * (b * b)) by GROUP_1:def 12
              .= (a * a) * (b * b) by GROUP_1:def 3
              .= (a * a) * (b^2) by O_RING_1:def 1
              .= (a^2) * (b^2) by O_RING_1:def 1;
end;
