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.] |^ c)
proof
  thus [.a,b * c.] = (a" * (b * c)") * (a * (b * c)) by Th16
    .= (a" * (c" * b")) * (a * (b * c)) by GROUP_1:17
    .= (a" * (c" * 1_G * b")) * (a * (b * c)) by GROUP_1:def 4
    .= (a" * (c" * (a * a") * b")) * (a * (b * c)) by GROUP_1:def 5
    .= (a" * (c" * (a * 1_G * a") * b")) * (a * (b * c)) by GROUP_1:def 4
    .= (a" * (c" * (a * (c * c") * a") * b")) * (a * (b * c)) by GROUP_1:def 5
    .= (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)) * (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)) * (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.] * (c" * ((a" * b") * (a * b)) * c) by Th16
    .= [.a,c.] * ([.a,b.] |^ c) by Th16;
end;
