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

theorem
  (for x st x in X holds neighbourhood(x,T) is TolSet of T) implies T is
  transitive
proof
  assume
A1: for x st x in X holds neighbourhood(x,T) is TolSet of T;
A2: field T = X by ORDERS_1:12;
  for x,y,z being object
st x in field T & y in field T & z in field T & [x,y] in T & [
  y,z] in T holds [x,z] in T
  proof
    let x,y,z be object;
    assume that
    x in field T and
A3: y in field T and
    z in field T and
A4: [x,y] in T and
A5: [y,z] in T;
    reconsider N = neighbourhood(y,T) as TolSet of T by A2,A1,A3;
    [y,x] in T by A4,EQREL_1:6;
    then
A6: x in N by Th27;
    z in N by A5,Th27;
    hence thesis by A6,Def1;
  end;
  then T is_transitive_in field T;
  hence thesis;
end;
