theorem Th56:
  G1 is Subgroup of G2 & G2 is Subgroup of G3 implies G1 is Subgroup of G3
proof
  assume that
A1: G1 is Subgroup of G2 and
A2: G2 is Subgroup of G3;
  set h = the addF of G3;
  set C = the carrier of G3;
  set B = the carrier of G2;
  set A = the carrier of G1;
A3: A c= B by A1,DefA5;
  then
A4: [:A,A:] c= [:B,B:] by ZFMISC_1:96;
  B c= C by A2,DefA5;
  then
A5: A c= C by A3;
  the addF of G1 = (the addF of G2)||A by A1,DefA5
    .= (h||B)||A by A2,DefA5
    .= h||A by A4,FUNCT_1:51;
  hence thesis by A5,DefA5;
end;
