reserve k,n for Nat,
  x,y,z,y1,y2 for object,X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for XFinSequence;
reserve D for set;
reserve i for Nat;
reserve m for Nat,
        D for non empty set;

theorem Th67:
  for F being non empty XFinSequence holds LastLoc F = card F -' 1
proof
  let F be initial non empty NAT-defined finite Function;
  consider k being Nat such that
A1: LastLoc F = k;
  reconsider k as Element of NAT by ORDINAL1:def 12;
  k < card F by A1,Lm1,VALUED_1:30;
  then
A2: k <= card F -' 1 by NAT_D:49;
  per cases by A2,XXREAL_0:1;
  suppose
    k < card F -' 1;
    then k+1 < card F -' 1 + 1 by XREAL_1:6;
    then k+1 < card F by NAT_1:14,XREAL_1:235;
    then
A3: k+1 <= k by A1,VALUED_1:32,Lm1;
    k <= k+1 by NAT_1:11;
    then k+0 = k+1 by A3,XXREAL_0:1;
    hence thesis;
  end;
  suppose
    k = card F -' 1;
    hence thesis by A1;
  end;
end;
