theorem
  for f being Assign of BASSModel(R,BASSIGN) holds SIGMA(EG(f)) = gfp(S,
  TransEG(f))
proof
  let f be Assign of BASSModel(R,BASSIGN);
  set g = EG(f);
  set h = Tau(gfp(S,TransEG(f)),R,BASSIGN);
A1: SIGMA(h) = gfp(S,TransEG(f)) by Th32;
  then SIGMA(h) is_a_fixpoint_of TransEG(f) by KNASTER:5;
  then
A2: for s being Element of S holds s|= h iff s|= Fax(f,h) by Th42;
A3: SIGMA(h) c= SIGMA(g)
  proof
    let x be object;
    assume x in SIGMA(h);
    then consider s be Element of S such that
A4: x=s and
A5: s|= h;
    s|= g by A2,A5,Th41;
    hence thesis by A4;
  end;
  for s being Element of S holds s|= g iff s|= Fax(f,g) by Th39;
  then SIGMA(g) is_a_fixpoint_of TransEG(f) by Th42;
  then SIGMA(g) c= gfp(S,TransEG(f)) by KNASTER:8;
  hence thesis by A1,A3,XBOOLE_0:def 10;
end;
