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 :: (9)
  (R1.:^X) \/ (R2.:^X) c= (R1 \/ R2).:^X
proof
  let y be object;
  assume
A1: y in (R1.:^X) \/ (R2.:^X);
  per cases by A1,XBOOLE_0:def 3;
  suppose
A2: y in R1.:^X;
    then reconsider B as non empty set;
    reconsider y as Element of B by A2;
    for x being set st x in X holds y in Im(R1 \/ R2,x)
    proof
      let x be set;
      assume x in X;
      then y in Im(R1,x) by A2,Th25;
      then y in Im(R1,x) \/ Im(R2,x) by XBOOLE_0:def 3;
      hence thesis by Th10;
    end;
    hence thesis by Th25;
  end;
  suppose
A3: y in R2.:^X;
    then reconsider B as non empty set;
    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 x in X;
      then y in Im(R2,x) by A3,Th25;
      then y in Im(R1,x) \/ Im(R2,x) by XBOOLE_0:def 3;
      hence thesis by Th10;
    end;
    hence thesis by Th25;
  end;
end;
