
theorem Th31: :: SPAChain:
for R being RelStr, S being Subset of R, A being StableSet of R
 holds A /\ S is StableSet of subrelstr S
proof
 let R be RelStr, S be Subset of R, A be StableSet of R;
   set sS = subrelstr S, AS = A /\ S;
 per cases;
 suppose R is empty;
    then A /\ S = {}sS;
   hence A /\ S is StableSet of sS;
 end;
 suppose A1: R is non empty;
   per cases;
   suppose S is empty;
     then A /\ S = {}sS;
    hence A /\ S is StableSet of sS;
   end;
   suppose A2: S is non empty;
      S = the carrier of sS by YELLOW_0:def 15;
     then reconsider AS as Subset of sS by XBOOLE_1:17;
     AS is stable proof
      let x, y be Element of sS such that
     A3: x in AS and
     A4: y in AS and
     A5: x <> y;
       reconsider x9 = x, y9 = y as Element of R by A1,A2,YELLOW_0:58;
     A6: x9 in A by A3,XBOOLE_0:def 4;
      y9 in A by A4,XBOOLE_0:def 4;
       then not x9 <= y9 & not y9 <= x9 by A6,A5,Def2;
      hence not x <= y & not y <= x by YELLOW_0:59;
      end;
     hence A /\ S is StableSet of sS;
   end;
  end;
end;
