theorem Th3:
  for A,B be Subset of RLS st A c= B holds conv A c= conv B
  proof
    let A,B be Subset of RLS such that
    A1: A c=B;
    A2: Convex-Family B c=Convex-Family A
    proof
      let x be object;
      assume A3: x in Convex-Family B;
      then reconsider X=x as Subset of RLS;
      B c=X by A3,CONVEX1:def 4;
      then A4: A c=X by A1;
      X is convex by A3,CONVEX1:def 4;
      hence thesis by A4,CONVEX1:def 4;
    end;
    [#]RLS is convex;
    then [#]RLS in Convex-Family B by CONVEX1:def 4;
    then A5: meet(Convex-Family A)c=meet(Convex-Family B) by A2,SETFAM_1:6;
    let y be object;
    assume y in conv A;
    hence thesis by A5;
  end;
