
theorem Th65:
for G being SimpleGraph holds SmallestPartition Vertices G is StableSet-wise
proof
 let G be SimpleGraph;
  set C = SmallestPartition Vertices G;
    let c be set such that
  A1: c in C;
     consider a being object such that
     a in Vertices G and
  A2: c = Class(id Vertices G, a) by A1,EQREL_1:def 3;
     reconsider cc = c as Subset of Vertices G by A1;
  A3: cc is stable proof
       let x, y be set such that
     A4: x <> y and
     A5: x in cc and
     A6: y in cc;
     A7: [a,x] in id Vertices G by A5,A2,RELAT_1:169;
     A8: [a,y] in id Vertices G by A6,A2,RELAT_1:169;
     A9: a = y by A8,RELAT_1:def 10;
       thus {x,y} nin G by A7,A9,A4,RELAT_1:def 10;
     end;
    thus c is StableSet of G by A3;
end;
