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 Th6:
  [.a,b".]|^ b = [.b,a.]
proof
  thus [.a,b".]|^ b = b" * ((a" * b"" * a * b") * b) by GROUP_1:def 3
              .= b" * (((a" * b"" * a) * (b" * b))) by GROUP_1:def 3
              .= b" * ((a" * b"" * a) * 1_G) by GROUP_1:def 5
              .= b" * (a" * b"" * a)  by GROUP_1:def 4
              .= b" * (a" * (b * a)) by GROUP_1:def 3
              .= (b" * a") * (b * a) by GROUP_1:def 3
              .= b" * a" * b * a by GROUP_1:def 3
              .= [.b,a.];
end;
