reserve x,y for set;
reserve N for PT_net_Str;

theorem
  for N being Pnet for x for X being set holds X c= Elements(N) implies
(x in Output(N,X) iff ex y being Element of Elements(N) st y in X & x in exit(N
  ,y))
proof
  let N be Pnet;
  let x;
  let X be set;
A1: x in Output(N,X) implies ex y being Element of Elements(N) st y in X & x
  in exit(N,y)
  proof
    assume x in Output(N,X);
    then consider y being set such that
A2: x in y and
A3: y in Ext(N,X) by TARSKI:def 4;
    ex z being Element of Elements(N) st z in X & y = exit(N,z) by A3,Def14;
    hence thesis by A2;
  end;
  assume
A4: X c= Elements(N);
  (ex y being Element of Elements(N) st y in X & x in exit(N,y)) implies x
  in Output(N,X)
  proof
    given y being Element of Elements(N) such that
A5: y in X and
A6: x in exit(N,y);
    exit(N,y) in Ext(N,X) by A4,A5,Th22;
    hence thesis by A6,TARSKI:def 4;
  end;
  hence thesis by A1;
end;
