
theorem Th55: :: CSR
for R being finite antisymmetric transitive RelStr, r, s  being Nat
 st card R = r*s+1
  holds (ex C being Clique of R st card C >= r+1) or
        (ex A being StableSet of R st card A >= s+1)
proof
  let R be finite antisymmetric transitive RelStr, r, s be Nat such that
A1: card R = r*s+1 and
A2: for C being Clique of R holds card C < r+1 and
A3: for A being StableSet of R holds card A < s+1;
  consider p being Clique-partition of R such that
A4: card p = stability# R by Th51;
  consider A being finite StableSet of R such that
A5: card A = stability# R by Def6;
   stability# R < s+1 by A3,A5; then
A6: stability# R <= s by NAT_1:13;
   set wR = stability# R;   set cR = the carrier of R;   set f = canFS p;
A7: len f = card p by FINSEQ_1:93;
A8: dom f = Seg len f by FINSEQ_1:def 3;
   set f = canFS p;
A9: rng f = p by FUNCT_2:def 3;
   then reconsider f as FinSequence of bool cR by FINSEQ_1:def 4;
   now
     let d,e be Nat such that
   A10: d in dom f and
   A11: e in dom f and
   A12: d<>e;
   A13: f.d in rng f by A10,FUNCT_1:3;
   A14: f.e in rng f by A11,FUNCT_1:3;
       then reconsider fd = f.d, fe = f.e as Subset of cR by A13,A9;
       fd <> fe by A12,A10,A11,FUNCT_1:def 4;
     hence f.d misses f.e by A13,A14,A9,EQREL_1:def 4;
   end;
   then
A15: card union rng f = Sum Card f by INT_5:48;
   reconsider n9 = r as Element of NAT by ORDINAL1:def 12;
   reconsider h = wR |-> n9 as Element of wR-tuples_on NAT;
A16: Sum h = wR*r by RVSUM_1:80;
    dom f = dom Card f by CARD_3:def 2;
    then
 len Card f = wR by A7,A4,FINSEQ_3:29;
   then
reconsider Cf = Card f as Element of wR-tuples_on NAT by FINSEQ_2:92;
   now
     let j be Nat such that
   A19: j in Seg wR;
   A20: h.j = r by A19,FINSEQ_2:57;
   A21: Cf.j = card (f.j) by A19,A7,A8,A4,CARD_3:def 2;
      f.j in p by A9,A19,A4,A7,A8,FUNCT_1:3;
      then
       reconsider fj = f.j as Clique of R by Def11;
       card fj < r+1 by A2;
     hence Cf.j <= h.j by A20,A21,NAT_1:13;
   end;
   then
A22: Sum Cf <= Sum h by RVSUM_1:82;
   wR*r <= s*r by A6,XREAL_1:64;
   then Sum Cf <= s*r by A16,A22,XXREAL_0:2;
   then r*s+1 <= r*s+0 by A15,A9,A1,EQREL_1:def 4;
  hence contradiction by XREAL_1:6;
end;
