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 Th19:
  [.1_G,a.] = 1_G & [.a,1_G.] = 1_G
proof
  thus [.1_G,a.] = ((1_G)" * a") * a by GROUP_1:def 4
    .= (1_G * a") * a by GROUP_1:8
    .= a" * a by GROUP_1:def 4
    .= 1_G by GROUP_1:def 5;
  thus [.a,1_G.] = (a" * (1_G)") * a by GROUP_1:def 4
    .= (a" * 1_G) * a by GROUP_1:8
    .= a" * a by GROUP_1:def 4
    .= 1_G by GROUP_1:def 5;
end;
