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 Th68:
  for N1,N2 being strict normal Subgroup of G holds [.N1,N2.] is
  normal Subgroup of G
proof
  let N1,N2 be strict normal Subgroup of G;
  now
    let a;
    now
      let b;
      assume b in [.N1,N2.] |^ a;
      then consider c such that
A1:   b = c |^ a and
A2:   c in [.N1,N2.] by GROUP_3:58;
      consider F,I such that
A3:   len F = len I and
A4:   rng F c= commutators(carr N1,carr N2) and
A5:   c = Product(F |^ I) by A2,GROUP_4:28;
A6:   len(F |^ a) = len F by Def1;
A7:   rng(F |^ a) c= commutators(carr N1,carr N2)
      proof
        let x be object;
        assume x in rng(F |^ a);
        then consider y being object such that
A8:     y in dom(F |^ a) and
A9:     (F |^ a).y = x by FUNCT_1:def 3;
        reconsider y as Element of NAT by A8;
A10:    y in dom F by A6,A8,FINSEQ_3:29;
        then
A11:    F.y = (F/.y) by PARTFUN1:def 6;
        y in dom F by A6,A8,FINSEQ_3:29;
        then F.y in rng F by FUNCT_1:def 3;
        then consider d,e such that
A12:    F.y = [.d,e.] and
A13:    d in carr N1 and
A14:    e in carr N2 by A4,Th47;
        d in N1 by A13,STRUCT_0:def 5;
        then d |^ a in N1 |^ a by GROUP_3:58;
        then d |^ a in N1 by GROUP_3:def 13;
        then
A15:    d |^ a in carr N1 by STRUCT_0:def 5;
        e in N2 by A14,STRUCT_0:def 5;
        then e |^ a in N2 |^ a by GROUP_3:58;
        then e |^ a in N2 by GROUP_3:def 13;
        then
A16:    e |^ a in carr N2 by STRUCT_0:def 5;
        x = (F/.y) |^ a by A9,A10,Def1;
        then x = [.d |^ a,e |^ a.] by A12,A11,Th23;
        hence thesis by A15,A16;
      end;
      b = Product(F |^ I |^ a) by A1,A5,Th14
        .= Product((F |^ a) |^ I) by Th15;
      hence b in [.N1,N2.] by A3,A6,A7,GROUP_4:28;
    end;
    hence [.N1,N2.] |^ a is Subgroup of [.N1,N2.] by GROUP_2:58;
  end;
  hence thesis by GROUP_3:122;
end;
