
theorem  Th48: :: ACpart2:
for R being finite RelStr, A being StableSet of R,
    C being Clique-partition of R st card C = card A
  for c being set st c in C ex a being Element of A st c /\ A = {a}
proof
  let R be finite RelStr, A be StableSet of R,
      C be Clique-partition of R such that
A1: card C = card A;
    consider f being Function of A, C such that
A2: f is bijective and
A3: for x being set st x in A holds x in f.x by A1,Th47;
   let c be set such that
A4: c in C;
   rng f = C by A2,FUNCT_2:def 3;
   then consider x being object such that
A5: x in dom f and
A6: c = f.x by A4,FUNCT_1:def 3;
A7:  x in c by A5,A6,A3;
    reconsider a = x as Element of A by A5;
    take a;
    now
      let z be object;
      hereby
        assume z in c /\ A;
        then A8: z in c & z in A by XBOOLE_0:def 4;
        c is Clique of R by A4,Def11;
        hence z = a by A8,A7,Th15;
      end;
      assume z = a;
      hence z in c /\ A by A7,A5,XBOOLE_0:def 4;
    end;
    hence c /\ A = {a} by TARSKI:def 1;
end;
