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 * c.] = [.a,c.] * [.a,b.] * [.a,b,c.]
proof
  [.a,c.] * [.a,b.] * [.a,b,c.] = [.a,c.] * ([.a,b.] * [.[.a,b.],c.]) by
GROUP_1:def 3
    .= [.a,c.] * ((a" * b") * (a * b) * ([.a,b.]" * c" * [.a,b.] * c)) by Th16
    .= [.a,c.] * ((a" * b") * (a * b) * ([.b,a.] * c" * [.a,b.] * c)) by Th22
    .= [.a,c.] * ((a" * b") * (a * b) * ((b" * a" * (b * a)) * c" * [.a,b.]
  * c)) by Th16
    .= [.a,c.] * ((a" * b") * ((a * b) * (((b" * a") * (b * a)) * c" * [.a,b
  .] * c))) by GROUP_1:def 3
    .= [.a,c.] * ((a" * b") * ((a * b) * (((b" * a") * (b * a)) * c" * ([.a,
  b.] * c)))) by GROUP_1:def 3
    .= [.a,c.] * ((a" * b") * ((a * b) * (((b" * a") * (b * a)) * (c" * ([.a
  ,b.] * c))))) by GROUP_1:def 3
    .= [.a,c.] * ((a" * b") * ((a * b) * ((b" * a") * (b * a)) * (c" * ([.a,
  b.] * c)))) by GROUP_1:def 3
    .= [.a,c.] * ((a" * b") * ((a * b) * (b" * a") * (b * a) * (c" * ([.a,b
  .] * c)))) by GROUP_1:def 3
    .= [.a,c.] * ((a" * b") * ((a * b) * (a * b)" * (b * a) * (c" * ([.a,b.]
  * c)))) by GROUP_1:17
    .= [.a,c.] * ((a" * b") * (1_G * (b * a) * (c" * ([.a,b.] * c)))) by
GROUP_1:def 5
    .= [.a,c.] * ((a" * b") * ((b * a) * (c" * ([.a,b.] * c)))) by
GROUP_1:def 4
    .= [.a,c.] * ((a" * b") * (b * a) * (c" * ([.a,b.] * c))) by GROUP_1:def 3
    .= [.a,c.] * ((b * a)" * (b * a) * (c" * ([.a,b.] * c))) by GROUP_1:17
    .= [.a,c.] * (1_G * (c" * ([.a,b.] * c))) by GROUP_1:def 5
    .= [.a,c.] * (c" * ([.a,b.] * c)) by GROUP_1:def 4
    .= (a" * c") * (a * c) * (c" * ([.a,b.] * c)) by Th16
    .= (a" * c") * (a * c) * (c" * ((a" * (b" * a * b)) * c)) by Th16
    .= (a" * c") * (a * c) * (c" * (a" * (b" * a * b)) * c) by GROUP_1:def 3
    .= (a" * c") * (a * c) * ((c" * a") * (b" * a * b) * c) by GROUP_1:def 3
    .= (a" * c") * (a * c) * ((c" * a") * ((b" * a * b) * c)) by GROUP_1:def 3
    .= (a" * c") * (a * c) * (c" * a") * ((b" * a * b) * c) by GROUP_1:def 3
    .= (a" * c") * ((a * c) * (c" * a")) * ((b" * a * b) * c) by GROUP_1:def 3
    .= (a" * c") * ((a * c) * (a * c)") * ((b" * a * b) * c) by GROUP_1:17
    .= (a" * c") * 1_G * ((b" * a * b) * c) by GROUP_1:def 5
    .= (a" * c") * ((b" * a * b) * c) by GROUP_1:def 4
    .= (a" * c") * (b" * a * b) * c by GROUP_1:def 3
    .= (a" * c") * (b" * (a * b)) * c by GROUP_1:def 3
    .= a" * c" * b" * (a * b) * c by GROUP_1:def 3
    .= a" * (c" * b") * (a * b) * c by GROUP_1:def 3
    .= a" * (c" * b") * ((a * b) * c) by GROUP_1:def 3
    .= a" * (c" * b") * (a * (b * c)) by GROUP_1:def 3
    .= a" * (b * c)" * (a * (b * c)) by GROUP_1:17;
  hence thesis by Th16;
end;
