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

theorem
  id X c= R \/ R~ implies id X c= R & id X c= R~
proof
  assume
A1: id X c= R \/ R~;
  for x being object holds x in X implies [x,x] in R & [x,x] in R~
  proof
    let x be object;
    assume x in X;
    then [x,x] in id(X) by RELAT_1:def 10; then
A2: [x,x] in R or [x,x] in R~ by A1,XBOOLE_0:def 3;
    hence [x,x] in R by RELAT_1:def 7;
    thus thesis by A2,RELAT_1:def 7;
  end; then
  (for x being object holds x in X implies [x,x] in R) &
  for x being object holds x in X implies [x,x] in R~;
  hence thesis by RELAT_1:47;
end;
