 reserve x,y,z,t for object,X,Y,Z,W for set;
 reserve R,S,T for Relation;

theorem Th27:
  (x in dom CL R iff x in dom R & [x,x] in R) &
  (x in rng CL R iff x in dom R & [x,x] in R) &
  (x in rng CL R iff x in rng R & [x,x] in R) &
  (x in dom CL R iff x in rng R & [x,x] in R)
proof
A1: x in dom R & [x,x] in R implies x in dom CL R
  proof
    assume that
A2: x in dom R and
A3: [x,x] in R;
    [x,x] in id dom R by A2,RELAT_1:def 10;
    then [x,x] in (R /\ id dom R) by A3,XBOOLE_0:def 4;
    hence thesis by XTUPLE_0:def 12;
  end;
A4: x in dom CL R implies x in dom R & [x,x] in R
  proof
    assume x in dom CL R;
    then consider y being object such that
A5: [x,y] in CL R by XTUPLE_0:def 12;
    [x,y] in R & [x,y] in id dom R by A5,XBOOLE_0:def 4;
    hence thesis by RELAT_1:def 10;
  end;
  hence x in dom CL R iff x in dom R & [x,x] in R by A1;
  thus x in rng CL R iff x in dom R & [x,x] in R by A4,A1,Th26;
  thus x in rng CL R iff x in rng R & [x,x] in R
           by A4,A1,Th26,XTUPLE_0:def 12,def 13;
  thus thesis by A4,A1,XTUPLE_0:def 12,def 13;
end;
