reserve X for set, R,R1,R2 for Relation;
reserve x,y,z for set;
reserve n,m,k for Nat;

theorem Th6:
  for R being total reflexive Relation of X
  for S being Subset of X holds
  R |_2 S is total reflexive Relation of S
  proof
    let R be total reflexive Relation of X;
    let S be Subset of X;
    set Q = R |_2 S;
    dom Q = S
    proof
      thus dom Q c= S;
      let x be object; assume
A1:   x in S; then x in X; then
      x in field R & R is_reflexive_in field R
      by RELAT_2:def 9,ORDERS_1:12; then
      [x,x] in R; then
      [x,x] in Q by A1,Th4;
      hence thesis by XTUPLE_0:def 12;
    end;
    hence R |_2 S is total reflexive Relation of S
    by PARTFUN1:def 2,WELLORD1:15;
  end;
