reserve x,y for set,
  k,n for Nat,
  i for Integer,
  G for Group,
  a,b,c ,d,e for Element of G,
  A,B,C,D for Subset of G,
  H,H1,H2,H3,H4 for Subgroup of G ,
  N1,N2 for normal Subgroup of G,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem
  [.a",b".] = [.a,b.] |^ (a * b)" & [.a",b".] = [.a,b.] |^ (b * a)"
proof
  thus [.a",b".] = [.b",a.] |^ a" by Th27
    .= ([.a,b.] |^ b") |^ a" by Th27
    .= [.a,b.] |^ (b" * a") by GROUP_3:24
    .= [.a,b.] |^ (a * b)" by GROUP_1:17;
  thus [.a",b".] = [.b,a".] |^ b" by Th28
    .= [.a,b.] |^ a" |^ b" by Th28
    .= [.a,b.] |^ (a" * b") by GROUP_3:24
    .= [.a,b.] |^ (b * a)" by GROUP_1:17;
end;
