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 * [.a,b.]
proof
  a * [.a,b.] = a * ((a" * b") * (a * b)) by GROUP_1:def 3
             .= a * (a" * (b" * (a * b))) by GROUP_1:def 3
             .= (a * a") * (b" * (a * b)) by GROUP_1:def 3
             .= 1_G * (b" * (a * b)) by GROUP_1:def 5
             .= b" * (a * b) by GROUP_1:def 4
             .= b" * a * b by GROUP_1:def 3;
  hence thesis;
end;
