theorem Th25:
  [.a * b,c.] = [.a,c.] |^ b * [.b,c.]
proof
  thus [.a * b,c.] = ((a * b)" * c") * (a * b * c) by Th16
    .= (b" * a" * c") * (a * b * c) by GROUP_1:17
    .= (b" * a" * c") * (a * 1_G * b * c) by GROUP_1:def 4
    .= (b" * a" * c") * (a * (c * c") * b * c) by GROUP_1:def 5
    .= (b" * a" * c") * (a * (c * 1_G * c") * b * c) by GROUP_1:def 4
    .= (b" * a" * c") * (a * (c * (b * b") * c") * b * c) by GROUP_1:def 5
    .= b" * (a" * c") * (a * (c * (b * b") * c") * b * c) by GROUP_1:def 3
    .= b" * (a" * c") * (a * (c * (b * b") * c") * (b * c)) by GROUP_1:def 3
    .= b" * (a" * c") * (a * (c * b * b" * c") * (b * c)) by GROUP_1:def 3
    .= b" * (a" * c") * (a * (c * b * (b" * c")) * (b * c)) by GROUP_1:def 3
    .= b" * (a" * c") * (a * (c * (b * (b" * c"))) * (b * c)) by GROUP_1:def 3
    .= b" * (a" * c") * (a * c * (b * (b" * c")) * (b * c)) by GROUP_1:def 3
    .= b" * (a" * c") * (a * c * ((b * (b" * c")) * (b * c))) by GROUP_1:def 3
    .= b" * (a" * c") * (a * c) * ((b * (b" * c")) * (b * c)) by GROUP_1:def 3
    .= b" * ((a" * c") * (a * c)) * ((b * (b" * c")) * (b * c)) by
GROUP_1:def 3
    .= b" * ((a" * c") * (a * c)) * (b * ((b" * c") * (b * c))) by
GROUP_1:def 3
    .= b" * ((a" * c") * (a * c)) * b * ((b" * c") * (b * c)) by GROUP_1:def 3
    .= [.a,c.] |^ b * ((b" * c") * (b * c)) by Th16
    .= [.a,c.] |^ b * [.b,c.] by Th16;
end;
