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 Th23:
  [.a,b.] |^ c = [.a |^ c,b |^ c.]
proof
  thus [.a,b.] |^ c = c" * (a" * 1_G * b" * a * b) * c by GROUP_1:def 4
    .= c" * (a" * (c * c") * b" * a * b) * c by GROUP_1:def 5
    .= c" * (a" * (c * c") * b" * (a * b)) * c by GROUP_1:def 3
    .= c" * (a" * (c * c") * (b" * (a * b))) * c by GROUP_1:def 3
    .= c" * (a" * ((c * c") * (b" * (a * b)))) * c by GROUP_1:def 3
    .= c" * a" * ((c * c") * (b" * (a * b))) * c by GROUP_1:def 3
    .= c" * a" * (c * (c" * (b" * (a * b)))) * c by GROUP_1:def 3
    .= (a" |^ c) * (c" * (b" * (a * b))) * c by GROUP_1:def 3
    .= (a |^ c)" * (c" * (b" * (a * b))) * c by GROUP_3:26
    .= (a |^ c)" * (c" * (b" * 1_G * (a * b))) * c by GROUP_1:def 4
    .= (a |^ c)" * (c" * (b" * (c * c") * (a * b))) * c by GROUP_1:def 5
    .= (a |^ c)" * (c" * (b" * ((c * c") * (a * b)))) * c by GROUP_1:def 3
    .= (a |^ c)" * (c" * b" * ((c * c") * (a * b))) * c by GROUP_1:def 3
    .= (a |^ c)" * (c" * b" * (c * (c" * (a * b)))) * c by GROUP_1:def 3
    .= (a |^ c)" * ((b" |^ c) * (c" * (a * b))) * c by GROUP_1:def 3
    .= (a |^ c)" * ((b |^ c)" * (c" * (a * b))) * c by GROUP_3:26
    .= (a |^ c)" * ((b |^ c)" * (c" * (a * 1_G * b))) * c by GROUP_1:def 4
    .= (a |^ c)" * ((b |^ c)" * (c" * (a * (c * c") * b))) * c by GROUP_1:def 5
    .= (a |^ c)" * ((b |^ c)" * (c" * (a * ((c * c") * b)))) * c by
GROUP_1:def 3
    .= (a |^ c)" * ((b |^ c)" * (c" * a * ((c * c") * b))) * c by GROUP_1:def 3
    .= (a |^ c)" * ((b |^ c)" * (c" * a * (c * (c" * b)))) * c by GROUP_1:def 3
    .= (a |^ c)" * ((b |^ c)" * ((a |^ c) * (c" * b))) * c by GROUP_1:def 3
    .= (a |^ c)" * (((b |^ c)" * ((a |^ c) * (c" * b))) * c) by GROUP_1:def 3
    .= (a |^ c)" * (((b |^ c)" * ((a |^ c) * (c" * b) * c))) by GROUP_1:def 3
    .= (a |^ c)" * (((b |^ c)" * ((a |^ c) * (b |^ c)))) by GROUP_1:def 3
    .= [.a |^ c,b |^ c.] by Th16;
end;
