
theorem Th30: :: SPAChain1:
for R being RelStr, S being Subset of R, A being StableSet of subrelstr S
 holds A is StableSet of R
proof
 let R be RelStr, S be Subset of R, A be StableSet of subrelstr S;
   set sS = subrelstr S;
 per cases;
 suppose R is empty;
    then the carrier of sS = {} by YELLOW_0:def 15;
    then A = {}R;
   hence A is StableSet of R;
 end;
 suppose A1: R is non empty;
   per cases;
   suppose S is empty;
     then the carrier of sS = {} by YELLOW_0:def 15;
     then A = {}R;
    hence A is StableSet of R;
   end;
   suppose A2: S is non empty;
      S = the carrier of sS by YELLOW_0:def 15;
      then reconsider A as Subset of R by XBOOLE_1:1;
     A is stable proof
      let x, y be Element of R such that
     A3: x in A and
     A4: y in A and
     A5: x <> y;
       reconsider x9 = x, y9 = y as Element of sS by A3,A4;
       not x9 <= y9 & not y9 <= x9 by A3,A4,A5,Def2;
        hence not x <= y & not y <= x by A1,A2,YELLOW_0:60;
     end;
     hence thesis;
   end;
  end;
end;
