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

theorem JacRefl:
  for R being finite set
  for A,B being Subset of R holds
    JaccardIndex (A,B) = 1 iff A = B
  proof
    let R be finite set;
    let A,B be Subset of R;
    hereby
      assume
T1:   JaccardIndex (A,B) = 1;
      per cases;
      suppose A \/ B = {}; then
        A = {} & B = {};
        hence A = B;
      end;
      suppose A \/ B <> {}; then
        JaccardIndex (A,B) = card (A /\ B) / card (A \/ B) by JacInd; then
        card (A /\ B) = card (A \/ B) by XCMPLX_1:58,T1;
        hence A = B by LemmaCard;
      end;
    end;
    assume
A0: A = B;
    per cases;
    suppose A = {}; then
      A \/ B = {} by A0;
      hence thesis by JacInd;
    end;
    suppose
B1:   A <> {}; then
      JaccardIndex (A,B) = card (A /\ A) / card (A \/ A) by JacInd,A0
       .= card A / card A;
      hence thesis by B1,XCMPLX_1:60;
    end;
  end;
