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.] |^ b) * ([.b,c",a.] |^ c) * ([.c,a",b.] |^ a) = 1_G
proof
  set d = a" * [.b,c".] * a;
  set e = c" * [.a,b".] * c;
  set f = b" * [.c,a".] * b;
  set x = [.a,b",c.] |^ b;
  set y = [.b,c",a.] |^ c;
  set z = [.c,a",b.] |^ a;
A1: [.c",b.] = c"" * (b" * c" * b) by Th16;
A2: [.a",c.] = a"" * (c" * a" * c) by Th16;
A3: [.b",a.] = b"" * (a" * b" * a) by Th16;
  [.a,b",c.] = [.a,b".]" * e by Th16
    .= [.b",a.] * e by Th22;
  then
A4: [.a,b",c.] |^ b = b" * (b"" * ((a" * b" * a) * e)) * b by A3,GROUP_1:def 3
    .= (a" * b" * a) * e * b by GROUP_3:1;
  [.c,a",b.] = [.c,a".]" * f by Th16
    .= [.a",c.] * f by Th22;
  then
A5: z = a" * (a"" * ((c" * a" * c) * f)) * a by A2,GROUP_1:def 3
    .= (c" * a" * c) * f * a by GROUP_3:1;
  [.b,c",a.] = [.b,c".]" * (a" * [.b,c".] * a) by Th16
    .= [.c",b.] * (a" * [.b,c".] * a) by Th22;
  then [.b,c",a.] |^ c = c" * (c"" * ((b" * c" * b) * d)) * c by A1,
GROUP_1:def 3
    .= (b" * c" * b) * d * c by GROUP_3:1
    .= (b" * c" * b) * (d * c) by GROUP_1:def 3
    .= b" * (c" * b) * (d * c) by GROUP_1:def 3
    .= b" * ((c" * b) * (d * c)) by GROUP_1:def 3;
  then x * y = (a" * b" * a) * e * (b * (b" * ((c" * b) * (d * c)))) by A4,
GROUP_1:def 3
    .= (a" * b" * a) * e * ((c" * b) * (d * c)) by GROUP_3:1
    .= (a" * b" * a) * (c" * ([.a,b".] * c)) * ((c" * b) * (d * c)) by
GROUP_1:def 3
    .= (a" * b" * a) * c" * ([.a,b".] * c) * ((c" * b) * (d * c)) by
GROUP_1:def 3
    .= (a" * b" * a) * c" * (([.a,b".] * c) * ((c" * b) * (d * c))) by
GROUP_1:def 3
    .= (a" * b" * a) * c" * (([.a,b".] * c) * (c" * b) * (d * c)) by
GROUP_1:def 3
    .= (a" * b" * a) * c" * (([.a,b".] * c) * c" * b * (d * c)) by
GROUP_1:def 3
    .= (a" * b" * a) * c" * ((a" * b"" * a) * b" * b * (d * c)) by GROUP_3:1
    .= (a" * b" * a) * c" * ((a" * b"" * a) * (d * c)) by GROUP_3:1
    .= (a" * b" * a) * c" * ((a" * b"" * a) * (a" * [.b,c".] * (a * c))) by
GROUP_1:def 3
    .= (a" * b" * a) * c" * ((a" * b"" * a) * (a" * ([.b,c".] * (a * c))))
  by GROUP_1:def 3
    .= (a" * b" * a) * c" * ((a" * b"" * a) * a" * ([.b,c".] * (a * c))) by
GROUP_1:def 3
    .= (a" * b" * a) * c" * ((a" * b"") * ([.b,c".] * (a * c))) by GROUP_3:1
    .= (a" * b" * a) * c" * ((a" * b"") * [.b,c".] * (a * c)) by GROUP_1:def 3
    .= (a" * b" * a) * c" * ((a" * b"") * (b" * (c"" * b * c")) * (a * c))
  by Th16
    .= (a" * b" * a) * c" * ((a" * b"") * ((b" * (c"" * b * c")) * (a * c)))
  by GROUP_1:def 3
    .= (a" * b" * a) * c" * ((a" * b"") * (b" * ((c"" * b * c") * (a * c))))
  by GROUP_1:def 3
    .= (a" * b" * a) * c" * ((a" * b"") * b" * ((c"" * b * c") * (a * c)))
  by GROUP_1:def 3
    .= (a" * b" * a) * c" * (a" * ((c"" * b * c") * (a * c))) by GROUP_3:1;
  then
  x * y * z = ((a" * b" * a) * c" * (a" * (c"" * b * c") * (a * c))) * ((
  c" * a" * c) * f * a) by A5,GROUP_1:def 3
    .= ((a" * b" * a) * (c" * (a" * (c"" * b * c") * (a * c)))) * ((c" * a"
  * c) * f * a) by GROUP_1:def 3
    .= ((a" * b" * a) * (c" * (a" * ((c"" * b * c") * (a * c))))) * ((c" * a
  " * c) * f * a) by GROUP_1:def 3
    .= ((a" * b" * a) * c" * (a" * ((c"" * b * c") * (a * c)))) * ((c" * a"
  * c) * f * a) by GROUP_1:def 3
    .= ((a" * b" * a) * c" * (a" * (c"" * (b * c") * (a * c)))) * ((c" * a"
  * c) * f * a) by GROUP_1:def 3
    .= ((a" * b" * a) * c" * (a" * (c"" * (b * c" * (a * c))))) * ((c" * a"
  * c) * f * a) by GROUP_1:def 3
    .= ((a" * b" * a) * c" * a" * (c"" * (b * c" * (a * c)))) * ((c" * a" *
  c) * f * a) by GROUP_1:def 3
    .= ((a" * b" * a) * c" * a" * c"" * (b * c" * (a * c))) * (c" * a" * c *
  f * a) by GROUP_1:def 3
    .= ((a" * b" * a) * c" * a" * c"" * (b * (c" * (a * c)))) * (c" * a" * c
  * f * a) by GROUP_1:def 3
    .= ((a" * b" * a) * c" * a" * c"" * b * (c" * (a * c))) * (c" * a" * c *
  f * a) by GROUP_1:def 3
    .= (((a" * b" * a) * c" * a" * c"") * b * c" * (a * c)) * (c" * a" * c *
  f * a) by GROUP_1:def 3
    .= (((a" * b") * a * c") * a" * c"" * b * c") * ((a * c) * (c" * a" * c
  * f * a)) by GROUP_1:def 3
    .= (((a" * b") * a * c") * a" * c"" * b * c") * ((a * c) * (c" * a" * c
  * (f * a))) by GROUP_1:def 3
    .= (((a" * b") * a * c") * a" * c"" * b * c") * ((a * c) * (c" * a" * (c
  * (f * a)))) by GROUP_1:def 3
    .= (((a" * b") * a * c") * a" * c"" * b * c") * ((a * c) * (c" * a") * (
  c * (f * a))) by GROUP_1:def 3
    .= (((a" * b") * a * c") * a" * c"" * b * c") * ((a * c) * (a * c)" * (c
  * (f * a))) by GROUP_1:17
    .= (((a" * b") * a * c") * a" * c"" * b * c") * (1_G * (c * (f * a))) by
GROUP_1:def 5
    .= (((a" * b") * a * c") * a" * c"" * b * c") * ((c * (f * a))) by
GROUP_1:def 4
    .= (((a" * b") * a * c") * a" * c"" * b * c") * c * (f * a) by
GROUP_1:def 3
    .= (((a" * b") * a * c") * a" * c"" * b) * (f * a) by GROUP_3:1
    .= (((a" * b") * a * c") * a" * c"" * b) * (b" * [.c,a".] * (b * a)) by
GROUP_1:def 3
    .= (((a" * b") * a * c") * a" * c"" * b) * (b" * ([.c,a".] * (b * a)))
  by GROUP_1:def 3
    .= (((a" * b") * a * c") * a" * c"" * b) * b" * ([.c,a".] * (b * a)) by
GROUP_1:def 3
    .= (((a" * b") * a * c") * a" * c"") * ([.c,a".] * (b * a)) by GROUP_3:1
    .= (((a" * b") * a * c") * a" * c"") * (c" * (a"" * c * a") * (b * a))
  by Th16
    .= (((a" * b") * a * c") * a" * c"") * (c" * ((a"" * c * a") * (b * a)))
  by GROUP_1:def 3
    .= (((a" * b") * a * c") * a" * c"") * c" * ((a"" * c * a") * (b * a))
  by GROUP_1:def 3
    .= (((a" * b") * a * c") * a") * ((a"" * c * a") * (b * a)) by GROUP_3:1
    .= (((a" * b") * a * c") * a") * (a"" * (c * a") * (b * a)) by
GROUP_1:def 3
    .= (((a" * b") * a * c") * a") * (a"" * ((c * a") * (b * a))) by
GROUP_1:def 3
    .= (((a" * b") * a * c") * a") * a"" * ((c * a") * (b * a)) by
GROUP_1:def 3
    .= ((a" * b") * a * c") * ((c * a") * (b * a)) by GROUP_3:1
    .= ((a" * b") * a * c") * (c * a") * (b * a) by GROUP_1:def 3
    .= (a" * b") * (a * c") * (c * a") * (b * a) by GROUP_1:def 3
    .= (a" * b") * (a * c") * (c"" * a") * (b * a)
    .= (a" * b") * (a * c") * (a * c")" * (b * a) by GROUP_1:17
    .= (a" * b") * ((a * c") * (a * c")") * (b * a) by GROUP_1:def 3
    .= (a" * b") * 1_G * (b * a) by GROUP_1:def 5
    .= (a" * b") * (b * a) by GROUP_1:def 4
    .= (b * a)" * (b * a) by GROUP_1:17;
  hence thesis by GROUP_1:def 5;
end;
