 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 monotone wrt 1st coordinate
  for X,Y,U being Subset of ExampleRIFSpace
  st X = {1,2} & Y = {1,2,3} & U = {2,4,5} holds
    not kappa (X,U) <= kappa (Y,U)
  proof
    let X,Y,U be Subset of ExampleRIFSpace;
    assume
A1: X = {1,2} & Y = {1,2,3} & U = {2,4,5}; then
A3: card Y = 3 by CARD_2:58;
    2 in U & not 1 in U by A1,ENUMSET1:def 1; then
    X /\ U = {2} by A1,ZFMISC_1:54; then
R1: card (X /\ U) = 1 by CARD_1:30;
R2: card X = 2 by CARD_2:57,A1;
R0: U = {2} \/ {4,5} by A1,ENUMSET1:2;
ro: 2 in Y by A1,ENUMSET1:def 1;
w1: not 4 in Y by A1,ENUMSET1:def 1;
    not 5 in Y by A1,ENUMSET1:def 1; then
rr: Y misses {4,5} by ZFMISC_1:51,w1;
    Y /\ U = Y /\ {2} \/ (Y /\ {4,5}) by R0,XBOOLE_1:23
      .= Y /\ {2} by rr; then
    Y /\ U = {2} by XBOOLE_1:28,ro,ZFMISC_1:31; then
R3: card (Y /\ U) = 1 by CARD_1:30;
W4: kappa (X,U) = 1 / 2 by R1,R2,A1,KappaDef;
    kappa (Y,U) = 1 / 3 by A3,R3,A1,KappaDef;
    hence thesis by W4;
  end;
