
theorem Th47: :: Tchro0:
for R being with_finite_chromatic# RelStr, S being Subset of R
 holds chromatic# R >= chromatic# subrelstr S
proof
 let R be with_finite_chromatic# RelStr, S be Subset of R;
   consider C be finite Coloring of R such that
A1: card C = chromatic# R by Def3;
   C | S is Coloring of subrelstr S by Th10;
   then
A2: card (C | S) >= chromatic# subrelstr S by Def3;
   card C >= card (C | S) by Th8;
 hence chromatic# R >= chromatic# subrelstr S by A1,A2,XXREAL_0:2;
end;
