reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;
reserve X,X1,X2 for Subset of A;
reserve Y for Subset of B;
reserve R,R1,R2 for Subset of [:A,B:];
reserve FR for Subset-Family of [:A,B:];

theorem Th48: :: (10.2)
  (R.:^X)` = R`.:X
proof
  thus (R.:^X)` c= R`.:X
  proof
    let a be object;
    assume
A1: a in (R.:^X)`;
    then not a in R.:^X by XBOOLE_0:def 5;
    then consider x being set such that
A2: x in X and
A3: not a in Im(R,x) by A1,Th25;
A4: not [x,a] in R by A3,Th9;
    [x,a] in [:A,B:] by A1,A2,ZFMISC_1:87;
    then [x,a] in [:A,B:] \ R by A4,XBOOLE_0:def 5;
    hence thesis by A2,RELAT_1:def 13;
  end;
  let a be object;
  assume a in R`.:X;
  then consider x being object such that
A5: [x,a] in R` and
A6: x in X by RELAT_1:def 13;
A7: not [x,a] in R by A5,XBOOLE_0:def 5;
  assume not thesis;
  then
A8: not a in B or a in R.:^X by XBOOLE_0:def 5;
  a in R.:^X implies for x being set st x in X holds [x,a] in R
  proof
    assume
A9: a in R.:^X;
    let x be set;
    assume x in X;
    then a in Im(R,x) by A9,Th24;
    hence thesis by Th9;
  end;
  hence contradiction by A5,A6,A7,A8,ZFMISC_1:87;
end;
