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

theorem Th11:
  for R being Relation, a,b being object holds R reduces a,b iff
    ex p being FinSequence st len p > 0 & p.1 = a & p.len p = b &
 for i being Nat st i in dom p & i+1 in dom p holds [p.i, p.(i+1)] in R
proof
  let R be Relation, a,b be object;
  thus R reduces a,b implies ex p being FinSequence st len p > 0 & p.1 = a & p
.len p = b & for i being Nat st i in dom p & i+1 in dom p holds [p.i
  , p.(i+1)] in R
  proof
    given p being RedSequence of R such that
A1: p.1 = a & p.len p = b;
    take p;
    thus thesis by A1,Def2;
  end;
  given p being FinSequence such that
A2: len p > 0 and
A3: p.1 = a & p.len p = b and
A4: for i being Nat st i in dom p & i+1 in dom p holds [p.i,
  p.(i+1)] in R;
  reconsider p as RedSequence of R by A2,A4,Def2;
  take p;
  thus thesis by A3;
end;
