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

theorem Th13:
  ((Flow M)~|(the carrier' of M)) misses (id(Elements(M))) &
  ((Flow M)|(the carrier' of M)) misses (id(Elements(M))) &
  ((Flow M)~|(the carrier of M)) misses (id(Elements(M))) &
  ((Flow M)|(the carrier of M)) misses (id(Elements(M)))
proof
  set T = id(Elements(M));
  thus ((Flow M)~|(the carrier' of M)) misses (id(Elements(M)))
  proof
    set R = (Flow M)~|(the carrier' of M);
    for x,y being object holds not [x,y] in R /\ T
    proof
      let x,y be object;
      assume
A1:   [x,y] in R /\ T;
      then
A2:   [x,y] in R by XBOOLE_0:def 4;
A3:   [x,y] in T by A1,XBOOLE_0:def 4;
A4:   [x,y] in (Flow M)~ by A2,RELAT_1:def 11;
A5:   x in (the carrier' of M) by A2,RELAT_1:def 11;
      [y,x] in (Flow M) by A4,RELAT_1:def 7;
      then x <> y by A5,Th7;
      hence thesis by A3,RELAT_1:def 10;
    end;
    then R /\ T = {} by RELAT_1:37;
    hence thesis by XBOOLE_0:def 7;
  end;
  thus ((Flow M)|(the carrier' of M)) misses (id(Elements(M)))
  proof
    set R = (Flow M)|(the carrier' of M);
    for x,y being object holds not [x,y] in R /\ T
    proof
      let x,y be object;
      assume
A6:   [x,y] in R /\ T;
      then
A7:   [x,y] in R by XBOOLE_0:def 4;
A8:   [x,y] in T by A6,XBOOLE_0:def 4;
A9:   x in (the carrier' of M) by A7,RELAT_1:def 11;
      [x,y] in (Flow M) by A7,RELAT_1:def 11;
      then x <> y by A9,Th7;
      hence thesis by A8,RELAT_1:def 10;
    end;
    then R /\ T = {} by RELAT_1:37;
    hence thesis by XBOOLE_0:def 7;
  end;
  thus ((Flow M)~|(the carrier of M)) misses (id(Elements(M)))
  proof
    set R = (Flow M)~|(the carrier of M);
    for x,y being object holds not [x,y] in R /\ T
    proof
      let x,y be object;
      assume
A10:  [x,y] in R /\ T;
      then
A11:  [x,y] in R by XBOOLE_0:def 4;
A12:  [x,y] in T by A10,XBOOLE_0:def 4;
A13:  [x,y] in (Flow M)~ by A11,RELAT_1:def 11;
A14:  x in the carrier of M by A11,RELAT_1:def 11;
      [y,x] in Flow M by A13,RELAT_1:def 7;
      then x <> y by A14,Th7;
      hence thesis by A12,RELAT_1:def 10;
    end;
    then R /\ T = {} by RELAT_1:37;
    hence thesis by XBOOLE_0:def 7;
  end;
  set R = (Flow M)|(the carrier of M);
  for x,y being object holds not [x,y] in R /\ T
  proof
    let x,y be object;
    assume
A15: [x,y] in R /\ T;
    then
A16: [x,y] in R by XBOOLE_0:def 4;
A17: [x,y] in T by A15,XBOOLE_0:def 4;
A18: x in the carrier of M by A16,RELAT_1:def 11;
    [x,y] in Flow M by A16,RELAT_1:def 11;
    then x <> y by A18,Th7;
    hence thesis by A17,RELAT_1:def 10;
  end;
  then R /\ T = {} by RELAT_1:37;
  hence thesis by XBOOLE_0:def 7;
end;
