reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;

theorem Th11:
  Im(R1 /\ R2,x) = Im(R1,x) /\ Im(R2,x)
proof
  thus Im(R1 /\ R2,x) c= Im(R1,x) /\ Im(R2,x)
  proof
    let y be object;
    assume y in Im(R1 /\ R2,x);
    then
A1: [x,y] in R1/\R2 by Th9;
    then
A2: [x,y] in R1 by XBOOLE_0:def 4;
A3: [x,y] in R2 by A1,XBOOLE_0:def 4;
A4: y in Im(R1,x) by A2,Th9;
    y in Im(R2,x) by A3,Th9;
    hence thesis by A4,XBOOLE_0:def 4;
  end;
  let y be object;
  assume
A5: y in Im(R1,x) /\ Im(R2,x);
  then
A6: y in Im(R1,x) by XBOOLE_0:def 4;
A7: y in Im(R2,x) by A5,XBOOLE_0:def 4;
A8: [x,y] in R1 by A6,Th9;
  [x,y] in R2 by A7,Th9;
  then [x,y] in R1 /\ R2 by A8,XBOOLE_0:def 4;
  hence thesis by Th9;
end;
