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 Th14:
  for E be finite non empty set, ASeq being SetSequence of E st
  ASeq is non-ascending
  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 m be Nat;
     reconsider mm=m as Element of NAT by ORDINAL1:def 12;
    seq.mm = card(ASeq.mm) by A2;
    hence -1 < seq.m;
  end;
  then
A3: seq is bounded_below by SEQ_2:def 4;
  assume
A4: ASeq is non-ascending;
A5: now
    let n,m be Nat;
    assume n <= m;
    then
A6: ASeq.m c= ASeq.n by A4,PROB_1:def 4;
     reconsider mm=m, nn=n as Element of NAT by ORDINAL1:def 12;
    seq.mm = card(ASeq.mm) & seq.nn = card(ASeq.nn) by A2;
    hence seq.m <=seq.n by A6,NAT_1:43;
  end;
  then seq is non-increasing by SEQM_3:8;
  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;
   reconsider NN=N as Element of NAT by ORDINAL1:def 12;
  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 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.m <= seq.N & ASeq.m c= ASeq.N by A4,A5,PROB_1:def 4;
    |.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.N qua Real)-seq.m qua Complex.| < 1 by COMPLEX1:60;
    then (seq.N qua Real)-seq.m < 1 by ABSVALUE:def 1;
    then (seq.N qua Real)-seq.m + seq.m < 1 + seq.m by XREAL_1:8;
    then (seq.NN qua Real) <=seq.m by A10,NAT_1:8;
    hence ASeq.m=ASeq.N by A10,A12,CARD_2:102,XXREAL_0:1;
  end;
  hence thesis;
end;
