theorem Th55:
  G1 is Subgroup of G2 & G2 is Subgroup of G1 implies the
  addMagma of G1 = the addMagma of G2
proof
  assume that
A1: G1 is Subgroup of G2 and
A2: G2 is Subgroup of G1;
  set g = the addF of G2;
  set f = the addF of G1;
  set B = the carrier of G2;
  set A = the carrier of G1;
A3: A c= B & B c= A by A1,A2,DefA5;
A4: A = B by A1,A2,DefA5;
  f = g||A by A1,DefA5
    .= (f||B)||A by A2,DefA5
    .= f||B by A4,RELAT_1:72
    .= g by A2,DefA5;
  hence thesis by A3,XBOOLE_0:def 10;
end;
