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

theorem
  for R being connected reflexive total Relation of X
  for R2 being Relation of X holds R\,R2 = R
  proof
    let R be connected reflexive total Relation of X;
    let R2 be Relation of X;
    let x,y be object;
    reconsider xx = x, yy = y as set by TARSKI:1;
    hereby assume [x,y] in R\,R2; then xx,yy in R\,R2; then
      xx,yy in R or yy,xx nin R & xx,yy in R2 by Th9; then
      [x,y] in R or [y,x] nin R & x in X & y in X & field R = X &
      R is_connected_in field R & R is_reflexive_in field R & (x = y or x <> y)
      by Th2,RELAT_2:def 9,def 14,ORDERS_1:12;
      hence [x,y] in R;
    end;
    assume [x,y] in R; then
    xx,yy in R; then
    xx,yy in R\,R2 by Th9;
    hence thesis;
  end;
