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

theorem
  T is transitive implies for x st x in X holds neighbourhood(x,T) is
  TolClass of T
proof
  assume
A1: T is transitive;
  let x;
  assume
A2: x in X;
  set N = Class(T,x);
  field T = X by ORDERS_1:12;
  then
A3: T is_transitive_in X by A1;
  for y,z st y in N & z in N holds [y,z] in T
  proof
    let y,z such that
A4: y in N and
A5: z in N;
    [x,y] in T by A4,Th27;
    then
A6: [y,x] in T by EQREL_1:6;
    [x,z] in T by A5,Th27;
    hence thesis by A3,A2,A4,A5,A6;
  end;
  then reconsider Z = N as TolSet of T by Def1;
  for x st not x in Z & x in X ex y st y in Z & not [x,y] in T
  proof
    let y such that
A7: not y in Z and
    y in X;
A8: x in Z by A2,EQREL_1:20;
    assume for z st z in Z holds [y,z] in T;
    then [y,x] in T by A8;
    then [x,y] in T by EQREL_1:6;
    hence contradiction by A7,Th27;
  end;
  hence thesis by Def2;
end;
