reserve            x for object,
               X,Y,Z for set,
         i,j,k,l,m,n for Nat,
                 r,s for Real,
                  no for Element of OrderedNAT,
                   A for Subset of [:NAT,NAT:];

theorem Th13:
  n = no implies uparrow no = NAT \ Segm n
  proof
    assume
A1: n = no;
A2: uparrow no c= NAT \ Segm n
    proof
      let x be object;
      assume
A3:   x in uparrow no;
      then x in {x where x is Element of NAT:ex p0 be Element of NAT st
        no = p0 & p0 <= x} by CARDFIL2:50;
      then ex y be Element of NAT st y = x & ex p0 be Element of NAT st
           no = p0 & p0 <= y;
      then reconsider y = x as Element of NAT;
      not y in Segm n by A1,A3,Th12,NAT_1:44;
      hence thesis by XBOOLE_0:def 5;
    end;
    NAT \ Segm n c= uparrow no
    proof
      let x be object;
      assume
A4:   x in NAT \ Segm n; then
A5:   x in NAT & not x in Segm n by XBOOLE_0:def 5;
      reconsider y = x as Element of NAT by A4;
      reconsider n1 = n as Element of NAT by ORDINAL1:def 12;
      n1 <= y & n1 = no by A1,A5,NAT_1:44;
      then y in {x where x is Element of NAT:ex p0 be Element of NAT st
        no = p0 & p0 <= x};
      hence thesis by CARDFIL2:50;
    end;
    hence thesis by A2;
  end;
