reserve x,y for object,X,Y for set;
reserve M for Pnet;

theorem
  dom(f_escape(M)) c= Elements(M) & rng(f_escape(M)) c= Elements(M) &
  dom(f_entrance(M)) c= Elements(M) & rng(f_entrance(M)) c= Elements(M)
proof
A1: for x being object holds x in dom(f_escape(M)) implies x in Elements(M)
  proof
    let x be object;
    assume x in dom(f_escape(M));
    then x in dom((Flow M)|(the carrier of M)) \/
    dom(id(the carrier' of M)) by XTUPLE_0:23;
    then x in dom((Flow M)|(the carrier of M)) or
    x in dom(id(the carrier' of M)) by XBOOLE_0:def 3;
    then x in (the carrier of M) or x in the carrier' of M by RELAT_1:57;
    hence thesis by XBOOLE_0:def 3;
  end;
A2: for x being object holds x in rng(f_escape(M)) implies x in Elements(M)
  proof
    let x be object;
    assume x in rng(f_escape(M));
    then
A3: x in rng((Flow M)|(the carrier of M)) \/
    rng(id(the carrier' of M)) by RELAT_1:12;
A4: x in rng((Flow M)|(the carrier of M)) implies thesis
    proof
      assume x in rng((Flow M)|(the carrier of M));
      then consider y being object such that
A5:   [y,x] in (Flow M)|(the carrier of M) by XTUPLE_0:def 13;
A6:   y in (the carrier of M) by A5,RELAT_1:def 11;
      [y,x] in (Flow M) by A5,RELAT_1:def 11;
      then x in (the carrier of M) or x in the carrier' of M by A6,Th7;
      hence thesis by XBOOLE_0:def 3;
    end;
    x in rng(id(the carrier' of M)) implies thesis by XBOOLE_0:def 3;
    hence thesis by A3,A4,XBOOLE_0:def 3;
  end;
A7: for x being object holds x in dom(f_entrance(M)) implies x in Elements(M)
  proof
    let x be object;
    assume x in dom(f_entrance(M));
    then x in dom((Flow M)~|(the carrier of M)) \/
    dom(id(the carrier' of M)) by XTUPLE_0:23;
    then x in dom((Flow M)~|(the carrier of M)) or
    x in dom(id(the carrier' of M)) by XBOOLE_0:def 3;
    then x in (the carrier of M) or x in the carrier' of M by RELAT_1:57;
    hence thesis by XBOOLE_0:def 3;
  end;
  for x being object holds x in rng(f_entrance(M)) implies x in Elements(M)
  proof
    let x be object;
    assume x in rng(f_entrance(M));
    then
A8: x in rng((Flow M)~|(the carrier of M)) \/
    rng(id(the carrier' of M)) by RELAT_1:12;
A9: x in rng((Flow M)~|(the carrier of M)) implies thesis
    proof
      assume x in rng((Flow M)~|(the carrier of M));
      then consider y being object such that
A10:  [y,x] in (Flow M)~|(the carrier of M) by XTUPLE_0:def 13;
A11:  [y,x] in (Flow M)~ by A10,RELAT_1:def 11;
A12:  y in (the carrier of M) by A10,RELAT_1:def 11;
      [x,y] in (Flow M) by A11,RELAT_1:def 7;
      then x in (the carrier of M) or x in the carrier' of M by A12,Th7;
      hence thesis by XBOOLE_0:def 3;
    end;
    x in rng(id(the carrier' of M)) implies thesis by XBOOLE_0:def 3;
    hence thesis by A8,A9,XBOOLE_0:def 3;
  end;
  hence thesis by A1,A2,A7,TARSKI:def 3;
end;
