
theorem Th2:
  for R being non empty Poset, s1,s2 being Element of R st s1 <= s2
  holds [s1,s2] in Path_Rel R
proof
  let R be non empty Poset, s1,s2 be Element of R such that
A1: s1 <= s2;
  set p = <* s1, s2 *>;
A2: len p = 2 by FINSEQ_1:44;
A3: p.1 = s1;
A4: for n being Nat st 2 <= n & n <= len p holds [p.n,p.(n-1)] in the
  InternalRel of R or [p.(n-1),p.n] in the InternalRel of R
  proof
    let n1 be Nat;
    assume 2 <= n1 & n1 <= len p;
    then
A5: n1 = 2 by A2,XXREAL_0:1;
    [s1,s2] in the InternalRel of R by A1,ORDERS_2:def 5;
    hence thesis by A5;
  end;
  p.(len p) = s2 by A2;
  hence thesis by A2,A3,A4,Def3;
end;
