reserve p,q,r for FinSequence,
  x,y for object;

theorem Th25:
  for R being Relation, a,b being object st R reduces a,b holds
  a,b are_convertible_wrt R & b,a are_convertible_wrt R
proof
  let R be Relation, a,b be object;
  given p being RedSequence of R such that
A1: p.1 = a & p.len p = b;
  p is RedSequence of R \/ R~
  proof
    thus len p > 0;
    let i be Nat;
    assume i in dom p & i+1 in dom p;
    then [p.i, p.(i+1)] in R by Def2;
    hence thesis by XBOOLE_0:def 3;
  end;
  then R \/ R~ reduces a,b by A1;
  hence a,b are_convertible_wrt R;
  hence thesis by Lm5;
end;
