reserve X,Y,Z,x,y,z for set;
reserve T,R for Tolerance of X;

theorem Th20:
  for x,y being object st [x,y] in T
   ex Z being TolClass of T st x in Z & y in Z
proof
  let x,y be object;
  assume
A1: [x,y] in T;
  then
A2: x in X & y in X by ZFMISC_1:87;
  for a,b being set st a in {x,y} & b in {x,y} holds [a,b] in T
  proof
    let a,b be set;
    assume that
A3: a in {x,y} and
A4: b in {x,y};
A5: b = x or b = y by A4,TARSKI:def 2;
    a = x or a = y by A3,TARSKI:def 2;
    hence thesis by A1,A2,A5,Th7,EQREL_1:6;
  end;
  then reconsider PS = {x,y} as TolSet of T by Def1;
  consider Z being TolClass of T such that
A6: PS c= Z by Th19;
  take Z;
  x in {x,y} by TARSKI:def 2;
  hence x in Z by A6;
  y in {x,y} by TARSKI:def 2;
  hence thesis by A6;
end;
