
theorem :: ColClique:
for R being finite RelStr, C being Coloring of R holds card C >= clique# R
proof
 let R be finite RelStr, C be Coloring of R;
 assume A1: card(C) < clique# R;
 consider A being Clique of R such that
A2: card A = clique# R by Def4;
   card Segm card C = card C & card Segm card A = card A;
   then A3: card C in card A by A1,A2,NAT_1:41;
   set cR = the carrier of R;
  per cases;
  suppose  R is empty;
    hence contradiction by A1;
  end;
  suppose A4: R is non empty;
  defpred P[object,object] means
    ex D2 being set st D2 = $2 & $1 in A & $2 in C & $1 in D2;
A5: for x being object st x in A ex y being object st y in C & P[x,y] proof
     let x be object such that
   A6: x in A;
       reconsider x9 = x as Element of R by A6;
       cR is non empty by A4;
       then x9 in cR;
       then x in union C by EQREL_1:def 4;
       then consider y being set such that
   A7: x in y and
   A8: y in C by TARSKI:def 4;
       take y;
       thus thesis by A6,A7,A8;
   end;
   consider f being Function of A, C such that
A9: for x being object st x in A holds P[x,f.x] from FUNCT_2:sch 1(A5);
    consider x,y being object such that
  A10: x in A and
  A11: y in A and
  A12: x <> y and
  A13: f.x = f.y by A4,A3,FINSEQ_4:65;
    f.x in C by A10,FUNCT_2:5;
    then
  A14: f.x is StableSet of R by Def12;
      P[x,f.x] & P[y,f.y] by A10,A11,A9;
     then x in f.x & y in f.x by A13;
    hence contradiction by A14,A10,A11,A12,Th15;
  end;
end;
