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