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

theorem Th10:
  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 [x,y] in R1 \/ R2 by Th9;
    then [x,y] in R1 or [x,y] in R2 by XBOOLE_0:def 3;
    then y in Im(R1,x) or y in Im(R2,x) by Th9;
    hence thesis by XBOOLE_0:def 3;
  end;
  let y be object;
  assume
A1: y in Im(R1,x) \/ Im(R2,x);
  per cases by A1,XBOOLE_0:def 3;
  suppose y in Im(R1,x);
    then [x,y] in R1 by Th9;
    then [x,y] in R1\/R2 by XBOOLE_0:def 3;
    hence thesis by Th9;
  end;
  suppose y in Im(R2,x);
    then [x,y] in R2 by Th9;
    then [x,y] in R1\/R2 by XBOOLE_0:def 3;
    hence thesis by Th9;
  end;
end;
