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

theorem
  for Y st for Z being set holds Z in Y iff x in Z & Z is TolClass of T
  holds neighbourhood(x,T) = union Y
proof
  let Y such that
A1: for Z being set holds Z in Y iff x in Z & Z is TolClass of T;
  for y being object holds y in neighbourhood(x,T) iff y in union Y
  proof
    let y be object;
    thus y in neighbourhood(x,T) implies y in union Y
    proof
      assume y in neighbourhood(x,T);
      then [x,y] in T by Th27;
      then consider Z being TolClass of T such that
A2:   x in Z and
A3:   y in Z by Th20;
      Z in Y by A1,A2;
      hence thesis by A3,TARSKI:def 4;
    end;
    assume y in union Y;
    then consider Z such that
A4: y in Z and
A5: Z in Y by TARSKI:def 4;
    reconsider Z as TolClass of T by A1,A5;
    x in Z by A1,A5;
    then [x,y] in T by A4,Def1;
    hence thesis by Th27;
  end;
  hence thesis by TARSKI:2;
end;
