reserve Omega for non empty set;
reserve r for Real;
reserve Sigma for SigmaField of Omega;
reserve P for Probability of Sigma;
reserve E for finite non empty set;

theorem Th16:
  for E be finite non empty set, ASeq being SetSequence of E st
ASeq is non-descending
 ex N be Nat st for m be Nat st N <= m holds ASeq.N = ASeq.m
proof
  let E be finite non empty set,ASeq being SetSequence of E;
  defpred P[Element of NAT,set] means $2=card(ASeq.$1);
A1: for x be Element of NAT ex y be Element of REAL st P[x,y]
  proof
    let x be Element of NAT;
    card(ASeq.x) in REAL by NUMBERS:19;
    hence thesis;
  end;
  consider seq being sequence of REAL such that
A2: for n being Element of NAT holds P[n,seq.n] from FUNCT_2:sch 3(A1);
  now
    let n be Nat;
     reconsider nn=n as Element of NAT by ORDINAL1:def 12;
    card(ASeq.nn) <= card E by NAT_1:43;
    then card(ASeq.nn) < card E + 1 by NAT_1:13;
    hence seq.n < card E + 1 by A2;
  end;
  then
A3: seq is bounded_above by SEQ_2:def 3;
  assume
A4: ASeq is non-descending;
A5: now
    let n,m be Nat;
    reconsider mm=m, nn=n as Element of NAT by ORDINAL1:def 12;
    assume n <= m;
    then
A6: ASeq.n c= ASeq.m by A4,PROB_1:def 5;
    seq.mm = card(ASeq.mm) & seq.nn = card(ASeq.nn) by A2;
    hence seq.n <=seq.m by A6,NAT_1:43;
  end;
  then seq is non-decreasing by SEQM_3:6;
  then consider g be Real such that
A7: for p be Real st 0<p ex n be Nat st for m be
  Nat st n<=m holds |.seq.m-g qua Complex.| < p by A3,SEQ_2:def 6;
  consider N be Nat such that
A8: for m be Nat st N<=m holds |.seq.m-g qua Complex.| < 1/2 by A7;
  take N;
  now
    |.seq.N-g qua Complex.| < 1/2 by A8;
    then
A9: |.g-seq.N qua Complex.| < 1/2 by COMPLEX1:60;
    let m be Nat;
   reconsider NN=N, mm=m as Element of NAT by ORDINAL1:def 12;
A10: seq.NN =card(ASeq.NN) & seq.mm =card(ASeq.mm) by A2;
    assume
A11: N<=m;
    then
A12: seq.N <= seq.m & ASeq.N c= ASeq.m by A4,A5,PROB_1:def 5;
    |.seq.m-g qua Complex.| < 1/2 by A8,A11;
    then
A13: |.seq.m-g qua Complex.| + |.g-seq.N qua Complex.| <1/2 + 1/2
          by A9,XREAL_1:8;
    |.seq.m-(seq.N qua Real) qua Complex.|
     <= |.seq.m-g qua Complex.| + |.g-seq.N qua Complex.|
    by COMPLEX1:63;
    then |.seq.m-(seq.N qua Real) qua Complex.| < 1 by A13,XXREAL_0:2;
    then seq.m-(seq.N qua Real) < 1 by ABSVALUE:def 1;
    then seq.m-(seq.N qua Real) +
       (seq.N qua Real) < 1 + seq.N by XREAL_1:8;
    then seq.m <=(seq.N qua Real) by A10,NAT_1:8;
    hence ASeq.m=ASeq.N by A10,A12,CARD_2:102,XXREAL_0:1;
  end;
  hence thesis;
end;
