 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 2, simplified: all three RIFs are different
  for X,Y being Subset of ExampleRIFSpace
  st X = {1,2} & Y = {2,3,4} holds
    kappa (X,Y), kappa_1 (X,Y), kappa_2 (X,Y) are_mutually_distinct
  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;
    set U = the carrier of ExampleRIFSpace;
    1,2,3,4,5 are_mutually_distinct by ZFMISC_1:def 7; then
a1: card U = 5 by CARD_2:63;
    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;
a5: X \/ Y = {1,2} \/ ({2} \/ {3,4}) by A1,ENUMSET1:2
      .= {1,2} \/ {2} \/ {3,4} by XBOOLE_1:4
      .= {1,2} \/ {3,4} by ZFMISC_1:9
      .= {1,2,3,4} by ENUMSET1:5;
    not 1 in {3,4,5} & not 2 in {3,4,5} by ENUMSET1:def 1; then
W2: {1,2} misses {3,4,5} by ZFMISC_1:51;
WW: 3 in {3,4,5} & 4 in {3,4,5} by ENUMSET1:def 1;
    X` = {1,2} \/ {3,4,5} \ {1,2} by A1,ENUMSET1:8
      .= {3,4,5} by XBOOLE_1:88,W2; then
    X` \/ Y = {3,4,5} \/ ({2} \/ {3,4}) by A1,ENUMSET1:2
           .= {2} \/ ({3,4} \/ {3,4,5}) by XBOOLE_1:4
           .= {2} \/ {3,4,5} by ZFMISC_1:42,WW; then
    X` \/ Y = {2,3,4,5} by ENUMSET1:4; then
W3: card (X` \/ Y) / card U = 4 / 5 by a1,CARD_2:59;
W4: kappa (X,Y) = card (X /\ Y) / card X by KappaDef,A1
       .= 1 / 2 by A4,A1,CARD_2:57;
    kappa_1 (X,Y) = card Y / card (X \/ Y) by Kappa1,A1
       .= 3 / 4 by A3,a5,CARD_2:59;
    hence thesis by ZFMISC_1:def 5,W4,W3;
  end;
