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 Th45:
  O is filtering iff O = id dom O
  proof
    thus O is filtering implies O = id dom O
    proof
      assume
A1:   O c= id X;
      let z,s be object;
      thus [z,s] in O implies [z,s] in id dom O
      proof
        assume [z,s] in O; then
        z in dom O & z = s by A1,RELAT_1:def 10,XTUPLE_0:def 12;
        hence thesis by RELAT_1:def 10;
      end;
      assume [z,s] in id dom O; then
A2:   z in dom O & z = s by RELAT_1:def 10; then
      consider y being object such that
A3:   [z,y] in O by XTUPLE_0:def 12;
      thus thesis by A1,A2,A3,RELAT_1:def 10;
    end;
    assume O = id dom O;
    hence O c= id X by SYSREL:15;
  end;
