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

theorem
  for Y ex T being Tolerance of union Y st for Z st Z in Y holds Z is
  TolSet of T
proof
  let Y;
  defpred X[set,set] means ex Z st Z in Y & $1 in Z & $2 in Z;
A1: for x st x in union Y holds X[x,x]
  proof
    let x;
    assume x in union Y;
    then ex Z st x in Z & Z in Y by TARSKI:def 4;
    hence thesis;
  end;
A2: for x,y st x in union Y & y in union Y & X[x,y] holds X[y,x];
  consider T being Tolerance of union Y such that
A3: for x,y st x in union Y & y in union Y holds [x,y] in T iff X[x,y]
  from ToleranceEx(A1,A2);
  take T;
  let Z such that
A4: Z in Y;
  for x,y st x in Z & y in Z holds [x,y] in T
  proof
    let x,y;
    assume
A5: x in Z & y in Z;
    then x in union Y & y in union Y by A4,TARSKI:def 4;
    hence thesis by A3,A4,A5;
  end;
  hence thesis by Def1;
end;
