
theorem Th15: :: CliChr:
          :: more general than DILWORTH:50 for finite RelStr
for R being with_finite_clique# with_finite_chromatic# RelStr
 holds clique# R <= chromatic# R
proof
 let P be with_finite_clique# with_finite_chromatic# RelStr;
 assume A1: clique# P > chromatic# P;
 consider A being Clique of P such that
A2: card A = clique# P by DILWORTH:def 4;
 consider C being finite Coloring of P such that
A3: card C = chromatic# P by Def3;
   card Segm card C = card C & card Segm card A = card A;
   then A4: card C in card A by A3,A1,A2,NAT_1:41;
   set cP = the carrier of P;
  per cases;
  suppose P is empty;
    hence contradiction by A1;
  end;
  suppose A5: P is non empty;
  defpred P[object,object] means
    ex D2 being set st D2 = $2 & $1 in A & $2 in C & $1 in D2;
A6: for x being object st x in A ex y being object st y in C & P[x,y]
proof
     let x be object; assume
   A7: x in A; then
       reconsider xp1 = x as Element of P;
       cP is non empty by A5;
       then xp1 in cP;
       then x in union C by EQREL_1:def 4;
       then consider y being set such that
   A8: x in y and
   A9: y in C by TARSKI:def 4;
       take y;
       thus thesis by A7,A8,A9;
   end;
   consider f being Function of A, C such that
A10: for x being object st x in A holds P[x,f.x] from FUNCT_2:sch 1(A6);
    consider x,y being object such that
  A11: x in A and
  A12: y in A and
  A13: x <> y and
  A14: f.x = f.y by A5,A4,FINSEQ_4:65;
      f.x in C by A11,FUNCT_2:5; then
  A15: f.x is StableSet of P by DILWORTH:def 12;
     P[x,f.x] & P[y,f.y] by A11,A12,A10;
     then x in f.x & y in f.x by A14;
    hence contradiction by A15,A11,A12,A13,DILWORTH:15;
  end;
end;
