theorem Th25:
  for s0 holds ex pai being inf_path of R st pai.0 = s0
proof
  let s0;
  consider pai being sequence of S such that
A1: pai.0=s0 and
A2: for n being Nat holds [pai.n,pai.(n+1)] in R by Lm33;
  reconsider pai as inf_path of R by A2,Def39;
  take pai;
  thus thesis by A1;
end;
