 reserve R for 1-sorted;
 reserve X,Y for Subset of R;
 reserve R for finite 1-sorted;
 reserve X,Y for Subset of R;
 reserve R for finite Approximation_Space;
 reserve X,Y,Z,W for Subset of R;

theorem :: Example 1, kappa is not symmetric
  for X,Y being Subset of ExampleRIFSpace
  st X = {1,2} & Y = {2,3,4} holds
    kappa (X,Y) <> kappa (Y,X)
  proof
    let X,Y be Subset of ExampleRIFSpace;
    assume
A1: X = {1,2} & Y = {2,3,4}; then
A3: card Y = 3 by CARD_2:58;
Z1: kappa (Y,X) = card (Y /\ X) / card Y by KappaDef,A1;
    set U = the carrier of ExampleRIFSpace;
    2 in Y & not 1 in Y by A1,ENUMSET1:def 1; then
    X /\ Y = {2} by ZFMISC_1:54,A1; then
A4: card (X /\ Y) = 1 by CARD_1:30;
    kappa (X,Y) = card (X /\ Y) / card X by KappaDef,A1
       .= 1 / 2 by A4,A1,CARD_2:57;
    hence thesis by A3,Z1,A4;
  end;
