 reserve f,g for Function;
 reserve R for non empty reflexive RelStr;
 reserve R for non empty RelStr;

theorem ImCoim:
  for u,w being Element of R holds
    u in Im (the InternalRel of R,w) iff
      w in Coim (the InternalRel of R,u)
  proof
    let u,w be Element of R;
    thus u in Im (the InternalRel of R,w) implies
      w in Coim (the InternalRel of R,u)
    proof
      assume u in Im (the InternalRel of R,w); then
Z1:   [w,u] in the InternalRel of R by RELAT_1:169;
      u in {u} by TARSKI:def 1;
      hence thesis by Z1,RELAT_1:def 14;
    end;
    assume w in Coim (the InternalRel of R,u); then
    consider t being object such that
W1: [w,t] in the InternalRel of R & t in {u} by RELAT_1:def 14;
    t = u by W1,TARSKI:def 1;
    hence thesis by RELAT_1:169,W1;
  end;
