
theorem Th34: :: PbeSmax:
for R being transitive RelStr, A being StableSet of R,
    C being Clique of subrelstr Lower A, a, b being Element of R
 st a in A & a in C & b in C holds a = b or b <= a
proof
let R be transitive RelStr, A be StableSet of R,
    C be Clique of subrelstr Lower A, a, b be Element of R such that
A1: a in A and
A2: a in C and
A3: b in C;
   set ap = subrelstr Lower A;
   reconsider a9 = a as Element of ap by A2;
A4: b in the carrier of ap by A3;
   reconsider b9 = b as Element of ap by A3;
A5: b9 in Lower A by A4,YELLOW_0:def 15;
A6: C is Clique of R by Th28;
   per cases by A5,XBOOLE_0:def 3;
   suppose b9 in A;
     hence thesis by A1,A2,A3,A6,Th15;
   end;
   suppose b9 in downarrow A;
    then consider c being Element of R such that
  A7: b <= c and
  A8: c in A by WAYBEL_0:def 15;
 per cases;
  suppose A9: a <> b;
    per cases by A9,A2,A3,Th6;
    suppose a9 <= b9;
      then  a <= b by YELLOW_0:59;
      then a <= c by A7,YELLOW_0:def 2;
      hence thesis by A8,A7,A1,Def2;
    end;
    suppose b9 <= a9;
      hence thesis by YELLOW_0:59;
    end;
  end;
  suppose a = b;
    hence thesis;
  end;
 end;
end;
