reserve X,Y,z,s for set, L,L1,L2,A,B for List of X, x for Element of X,
  O,O1,O2,O3 for Operation of X, a,b,y for Element of X, n,m for Nat;

theorem Th1:
  for R1,R2 be Relation holds R1 c= R2 iff
   for z being object holds Im(R1,z) c= Im(R2,z)
  proof
    let R1,R2 be Relation;
    hereby
      assume
A1:   R1 c= R2;
      let z be object;
      thus Im(R1,z) c= Im(R2,z)
      proof
        let s be object; assume s in Im(R1,z); then
        consider v being object such that
A2:     [v,s] in R1 & v in {z} by RELAT_1:def 13;
        thus thesis by A1,A2,RELAT_1:def 13;
      end;
    end;
    assume
A3: for z being object holds Im(R1,z) c= Im(R2,z);
    let a,b be object;
    assume [a,b] in R1; then
    b in Im(R1,a) & Im(R1,a) c= Im(R2,a) by A3,RELAT_1:169;
    hence thesis by RELAT_1:169;
  end;
