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

theorem Th33:
  for R being Relation, a,b being object st a is_a_normal_form_wrt R
  & R reduces a,b holds a = b
proof
  let R be Relation;
  let a,b be object;
  assume
A1: not ex b being object st [a,b] in R;
  assume R reduces a,b;
  then consider p being FinSequence such that
A2: len p > 0 and
A3: p.1 = a and
A4: p.len p = b and
A5: for i being Nat st i in dom p & i+1 in dom p holds [p.i,
  p.(i+1)] in R by Th11;
A6: now
    assume len p > 1;
    then 1 in dom p & 1+1 in dom p by Lm3,Lm4;
    then [a, p.(1+1)] in R by A3,A5;
    hence contradiction by A1;
  end;
  len p >= 0+1 by A2,NAT_1:13;
  hence thesis by A3,A4,A6,XXREAL_0:1;
end;
