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
  for N1,N2,N3 being strict normal Subgroup of G holds [.N1,N2 "\/" N3.]
  = [.N1,N2.] "\/" [.N1,N3.]
proof
  let N1,N2,N3 be strict normal Subgroup of G;
  N2 "\/" N3 is normal Subgroup of G by GROUP_4:54;
  hence [.N1,N2 "\/" N3.] = [.N2 "\/" N3,N1.] by Th69
    .= [.N2,N1.] "\/" [.N3,N1.] by Th70
    .= [.N1,N2.] "\/" [.N3,N1.] by Th69
    .= [.N1,N2.] "\/" [.N1,N3.] by Th69;
end;
