reserve X for set, R,R1,R2 for Relation;
reserve x,y,z for set;
reserve n,m,k for Nat;

theorem Th19:
  (for n holds iter(R,n).:X <> {}) & X is finite implies ex x st x in X &
  for n holds Im(iter(R,n),x) <> {}
  proof
    assume that
A1: for n holds iter(R,n).:X <> {} and
A2: X is finite and
A3: for x st x in X ex n st Im(iter(R,n),x) = {};
    defpred P[object,object] means
      ex n st $2 = n & Im(iter(R,n),$1) = {};
A4: for x be object st x in X ex y be object st y in NAT & P[x,y]
    proof let x be object; assume x in X; then
      consider n such that
A5:   Im(iter(R,n),x) = {} by A3;
      take y = n; thus thesis by A5,ORDINAL1:def 12;
    end;
    consider f being Function such that
A6: dom f = X & rng f c= NAT &
    for x be object st x in X holds P[x,f.x] from FUNCT_1:sch 6(A4);
    reconsider f as Function of X,NAT by A6,FUNCT_2:2;
    consider n such that
A7: rng f c= Segm n by A2,AFINSQ_2:2;
    {{x} where x is Element of X: x in X} c= bool X
    proof let z be object;
      assume z in {{x} where x is Element of X: x in X}; then
      consider x being Element of X such that
A8:   z = {x} & x in X;
      z is Subset of X by A8,ZFMISC_1:31;
      hence thesis;
    end; then
    reconsider Y = {{x} where x is Element of X: x in X} as Subset-Family of X;
    X = union Y
    proof
      thus X c= union Y
      proof
        let x be object; assume x in X; then
        x in {x} & {x} in Y by TARSKI:def 1;
        hence thesis by TARSKI:def 4;
      end;
      let x be object; assume x in union Y; then
      consider z such that
A9:   x in z & z in Y by TARSKI:def 4;
      thus thesis by A9;
    end; then
A10: iter(R,n).:X = union {iter(R,n).:y where y is Subset of X: y in Y}
    by RELSET_2:14;
    {iter(R,n).:y where y is Subset of X: y in Y} c= {{}}
    proof
      let z be object;
      assume z in {iter(R,n).:y where y is Subset of X: y in Y}; then
      consider y being Subset of X such that
A11:   z = iter(R,n).:y & y in Y;
      consider x being Element of X such that
A12:   y = {x} & x in X by A11;
      consider m such that
A13:   f.x = m & Im(iter(R,m),x) = {} by A6,A12;
      m in rng f by A6,A12,A13,FUNCT_1:def 3; then
      m < n by A7,NAT_1:44; then
      z = {} by A11,A12,A13,Th18;
      hence thesis by TARSKI:def 1;
    end; then
    iter(R,n).:X c= union {{}} by A10,ZFMISC_1:77; then
    iter(R,n).:X c= {} by ZFMISC_1:25; then
    iter(R,n).:X = {};
    hence contradiction by A1;
  end;
