
theorem Th10: :: Tsr0:
for R being RelStr, C being Coloring of R, S being Subset of R
 holds C | S is Coloring of subrelstr S
proof
 let R be RelStr, C be Coloring of R, S be Subset of R;
 set sS = subrelstr S;
A1: the carrier of sS = S by YELLOW_0:def 15;
  now
    let x be set;
    assume x in C | S;
     then consider c being Element of C such that
    A2: x = c /\ S and
    A3: c meets S;
        consider z being object such that z in c and
    A4: z in S by A3,XBOOLE_0:3;
    A5: sS is non empty by A4,YELLOW_0:def 15;
    A6: R is non empty by A4;
        reconsider Rp1 = R as non empty RelStr by A4;
           reconsider xp1= x as Subset of sS by A1,A2,XBOOLE_1:17;
        xp1 is stable proof
          let a, b be Element of sS such that
        A7: a in xp1 and
        A8: b in xp1 and
        A9: a <> b;
        A10: a in c & b in c by A7,A8,A2,XBOOLE_0:def 4;
            reconsider ap1 = a, bp1 = b as Element of R by A5,A6,YELLOW_0:58;
         C is Coloring of Rp1;
            then c in C;
            then reconsider cp1 = c as Subset of R;
        A11: cp1 is stable by DILWORTH:def 12;
          assume a <= b or b <= a;
          then ap1 <= bp1 or bp1 <= ap1 by YELLOW_0:59;
         hence contradiction by A9,A10,A11;
        end;
    hence x is StableSet of sS;
  end;
 hence C | S is Coloring of sS by A1,DILWORTH:def 12;
end;
