reserve i,j,k,m,n for Nat,
  D for non empty set,
  p for Element of D,
  f for FinSequence of D;
reserve D for non empty set,
  p for Element of D,
  f for FinSequence of D;
reserve f for circular FinSequence of D;

theorem Th19:
  for n being non zero Nat holds 0.REAL n <> 1.REAL n
proof
  let n be non zero Nat;
  1 <= n by NAT_1:14;
  then 1 in Seg n by FINSEQ_1:1;
  then
A1: (n |-> 0).1 = 0 & (n |-> 1).1 = 1 by FINSEQ_2:57;
  1.REAL n = 1*n by EUCLID:def 12
    .= n |-> 1 by EUCLID:def 11;
  hence thesis by A1,EUCLID:def 4;
end;
