 reserve R for finite Approximation_Space;
 reserve X,Y,Z for Subset of R;
 reserve kap for RIF of R;

theorem Similar2:
  for R being finite set
  for A, B being Subset of R holds
    (JaccardDist R).(A,B) = card (A \+\ B) / card (A \/ B)
  proof
    let R be finite set;
    let A, B be Subset of R;
    per cases;
    suppose
A1:   A \/ B <> {};
      (JaccardDist R).(A,B) = 1 - JaccardIndex (A,B) by JacDef2
         .= 1 - card (A /\ B) / card (A \/ B) by JacInd,A1
         .= card (A \+\ B) / card (A \/ B) by A1,Lemacik;
      hence thesis;
    end;
    suppose
A1:   A \/ B = {};
      (JaccardDist R).(A,B) = 1 - JaccardIndex (A,B) by JacDef2
         .= 1 - 1 by JacInd,A1
         .= card (A \+\ B) / card (A \/ B) by A1;
      hence thesis;
    end;
  end;
