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

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