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 Th14:
  proj1 A = {x where x is Element of NAT:
    ex y being Element of NAT st [x,y] in A}
  proof
    set A1 = {x where x is Element of NAT:
                ex y being Element of NAT st [x,y] in A};
A1: proj1 A c= A1
    proof
      let x be object;
      assume x in proj1 A;
      then consider y be object such that
A2:   [x,y] in A by XTUPLE_0:def 12;
      ex x1,y1 be object st x1 in NAT & y1 in NAT & [x,y] = [x1,y1]
        by A2,ZFMISC_1:def 2;
      then reconsider x,y as Element of NAT by XTUPLE_0:1;
      [x,y] in A & y is Element of NAT by A2;
      hence thesis;
    end;
    A1 c= proj1 A
    proof
      let x be object;
      assume x in A1;
      then ex x1 be Element of NAT st x = x1 & ex y be Element of NAT st
        [x1,y] in A;
      hence thesis by XTUPLE_0:def 12;
    end;
    hence thesis by A1;
  end;
