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

theorem Th9:
  rng ((Flow M)|(the carrier' of M)) c= (the carrier of M) &
  rng ((Flow M)~|(the carrier' of M)) c= (the carrier of M) &
  rng ((Flow M)|(the carrier of M)) c= (the carrier' of M) &
  rng ((Flow M)~|(the carrier of M)) c= (the carrier' of M) &
  rng id(the carrier' of M) c= (the carrier' of M) &
  dom id(the carrier' of M) c= (the carrier' of M) &
  rng id(the carrier of M) c= (the carrier of M) &
  dom id(the carrier of M) c= (the carrier of M)
proof
A1: for x being object holds x in rng ((Flow M)|(the carrier' of M)) implies
  x in (the carrier of M)
  proof
    let x be object;
    assume x in rng ((Flow M)|(the carrier' of M));
    then consider y being object such that
A2: [y,x] in (Flow M)|(the carrier' of M) by XTUPLE_0:def 13;
A3: y in (the carrier' of M) by A2,RELAT_1:def 11;
    [y,x] in (Flow M) by A2,RELAT_1:def 11;
    hence thesis by A3,Th7;
  end;
A4: for x being object holds x in rng ((Flow M)~|(the carrier' of M)) implies
  x in (the carrier of M)
  proof
    let x be object;
    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,x] in (Flow M)~ by A5,RELAT_1:def 11;
A7: y in (the carrier' of M) by A5,RELAT_1:def 11;
    [x,y] in (Flow M) by A6,RELAT_1:def 7;
    hence thesis by A7,Th7;
  end;
A8: for x being object holds x in rng ((Flow M)|(the carrier of M)) implies
  x in (the carrier' of M)
  proof
    let x be object;
    assume x in rng ((Flow M)|(the carrier of M));
    then consider y being object such that
A9: [y,x] in (Flow M)|(the carrier of M) by XTUPLE_0:def 13;
A10: y in (the carrier of M) by A9,RELAT_1:def 11;
    [y,x] in (Flow M) by A9,RELAT_1:def 11;
    hence thesis by A10,Th7;
  end;
  for x being object holds x in rng ((Flow M)~|(the carrier of M)) implies
  x in (the carrier' of M)
  proof
    let x be object;
    assume x in rng ((Flow M)~|(the carrier of M));
    then consider y being object such that
A11: [y,x] in (Flow M)~|(the carrier of M) by XTUPLE_0:def 13;
A12: [y,x] in (Flow M)~ by A11,RELAT_1:def 11;
A13: y in (the carrier of M) by A11,RELAT_1:def 11;
    [x,y] in (Flow M) by A12,RELAT_1:def 7;
    hence thesis by A13,Th7;
  end;
  hence thesis by A1,A4,A8,TARSKI:def 3;
end;
