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 :: (6)
  (R1 /\ R2).:^X = (R1.:^X) /\ (R2.:^X)
proof
  thus (R1 /\ R2).:^X c= (R1.:^X) /\ (R2.:^X)
  proof
    let y be object;
    assume
A1: y in (R1 /\ R2).:^X;
    then reconsider B as non empty set;
    reconsider y as Element of B by A1;
    for x being set st x in X holds y in Im(R1,x)
    proof
      let x be set;
      assume x in X;
      then y in Im(R1 /\ R2,x) by A1,Th24;
      then y in Im(R1,x) /\ Im(R2,x) by Th11;
      hence thesis by XBOOLE_0:def 4;
    end;
    then
A2: y in R1.:^X by Th25;
    for x being set st x in X holds y in Im(R2,x)
    proof
      let x be set;
      assume x in X;
      then y in Im(R1 /\ R2,x) by A1,Th24;
      then y in Im(R1,x) /\ Im(R2,x) by Th11;
      hence thesis by XBOOLE_0:def 4;
    end;
    then y in R2.:^X by Th25;
    hence thesis by A2,XBOOLE_0:def 4;
  end;
  let y be object;
  assume
A3: y in (R1.:^X) /\ (R2.:^X);
  then
A4: y in (R1.:^X) by XBOOLE_0:def 4;
A5: y in (R2.:^X) by A3,XBOOLE_0:def 4;
  reconsider B as non empty set by A3;
  reconsider y as Element of B by A3;
  for x being set st x in X holds y in Im(R1/\R2,x)
  proof
    let x be set;
    assume
A6: x in X;
    then
A7: y in Im(R1,x) by A4,Th25;
    y in Im(R2,x) by A5,A6,Th25;
    then y in Im(R1,x) /\ Im(R2,x) by A7,XBOOLE_0:def 4;
    hence thesis by Th11;
  end;
  hence thesis by Th25;
end;
