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

theorem
  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: not [x,y] in R2 by XBOOLE_0:def 5;
A3: y in Im(R1,x) by A1,Th9;
    not y in Im(R2,x) by A2,Th9;
    hence thesis by A3,XBOOLE_0:def 5;
  end;
  let y be object;
  assume
A4: y in Im(R1,x) \ Im(R2,x);
  then
A5: not y in Im(R2,x) by XBOOLE_0:def 5;
A6: [x,y] in R1 by A4,Th9;
  not [x,y] in R2 by A5,Th9;
  then [x,y] in R1\R2 by A6,XBOOLE_0:def 5;
  hence thesis by Th9;
end;
