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.]|^a
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
          .= 1_G * (b" * (a" * b * a)) by GROUP_1:def 4
          .= a" * a * (b" * (a" * b * a)) by GROUP_1:def 5
          .= a" * (a * (b" * (a" * b * a))) by GROUP_1:def 3
          .= a" * ((a * b") * ((a" * b) * a)) by GROUP_1:def 3
          .= a" * (((a * b") * (a" * b)) * a) by GROUP_1:def 3
          .= a" * ((a * b") * (a" * b)) * a by GROUP_1:def 3
          .= [.a",b.]|^a by GROUP_5:16;
end;
