
theorem Th29: :: SPClique1:
for R being RelStr, S being Subset of R, C being Clique of R
 holds C /\ S is Clique of subrelstr S
proof
 let R be RelStr, S be Subset of R, C be Clique of R;
 set sS = subrelstr S, CS = C /\ S;
A1: CS c= S by XBOOLE_1:17;
A2: S = the carrier of sS by YELLOW_0:def 15;
  now
   let a, b be Element of sS;
   assume A3: a in CS;
   assume A4: b in CS;
   assume A5: a <> b;
   A6: a in S & b in S by A3,A4,XBOOLE_0:def 4;
     A7: S is non empty by A3;
      R is non empty by A3;
     then reconsider a9 = a, b9 = b as Element of R by A7,YELLOW_0:58;
     a9 in C & b9 in C by A3,A4,XBOOLE_0:def 4;
     then a9 <= b9 or b9 <= a9 by A5,Th6;
   hence a <= b or b <= a by A6,A2,YELLOW_0:60;
  end;
 hence C /\ S is Clique of subrelstr S by Th6,A1,YELLOW_0:def 15;
end;
