reserve X,Y,Z for set,
  a,b,c,d,x,y,z,u for object,
  R for Relation,
  A,B,C for Ordinal;

theorem Th1:
  Y c= X implies (RelIncl X) |_2 Y = RelIncl Y
proof
  assume
A1: Y c= X;
  let a,b be object;
  thus [a,b] in (RelIncl X) |_2 Y implies [a,b] in RelIncl Y
  proof
    assume
A2: [a,b] in (RelIncl X) |_2 Y;
    then [a,b] in [:Y,Y:] by XBOOLE_0:def 4;
    then
A3: a in Y & b in Y by ZFMISC_1:87;
    reconsider a,b as set by TARSKI:1;
    [a,b] in RelIncl X by A2,XBOOLE_0:def 4;
    then a c= b by A1,A3,Def1;
    hence thesis by A3,Def1;
  end;
  assume
A4: [a,b] in RelIncl Y;
  then
A5: a in field RelIncl Y & b in field RelIncl Y by RELAT_1:15;
    reconsider a,b as set by TARSKI:1;
A6: field RelIncl Y = Y by Def1;
  then a c= b by A4,A5,Def1;
  then
A7: [a,b] in RelIncl X by A1,A5,A6,Def1;
  [a,b] in [:Y,Y:] by A5,A6,ZFMISC_1:87;
  hence thesis by A7,XBOOLE_0:def 4;
end;
