
theorem Th82:
for G being finitely_colorable SimpleGraph
 holds chromatic# G = cliquecover# Complement G
proof
 let G be finitely_colorable SimpleGraph;
 set CG = Complement G; set k = cliquecover# CG;
   consider C being finite Clique-partition of CG such that
 A1: card C = k by Def18;
 A2: C is Coloring of G by Th78;
   now
    let C be finite Coloring of G;
    assume A3: k > card C;
     C is Clique-partition of CG by Th80;
    hence contradiction by A3,Def18;
   end;
 hence chromatic# G = cliquecover# Complement G by A2,A1,Def22;
end;
