reserve x,y for set,
  G for Group,
  A,B,H,H1,H2 for Subgroup of G,
  a,b,c for Element of G,
  F,F1 for FinSequence of the carrier of G,
  I,I1 for FinSequence of INT,
  i,j for Element of NAT;

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