
theorem Th2: :: 1.2(ii), p. 39
  for L being non empty reflexive transitive RelStr, u,x,y,z being Element of L
  st u <= x & x << y & y <= z holds u << z
proof
  let L be non empty reflexive transitive RelStr;
  let u,x,y,z be Element of L such that
A1: u <= x and
A2: for D being non empty directed Subset of L st y <= sup D
  ex d being Element of L st d in D & x <= d and
A3: y <= z;
  let D be non empty directed Subset of L;
  assume z <= sup D;
  then y <= sup D by A3,ORDERS_2:3;
  then consider d being Element of L such that
A4: d in D and
A5: x <= d by A2;
  take d;
  thus thesis by A1,A4,A5,ORDERS_2:3;
end;
