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
  for F being non empty XFinSequence,
      G be non empty NAT-defined finite Function
   st F c= G & LastLoc F = LastLoc G
  holds F = G
proof
  let F be initial non empty NAT-defined finite Function, G be non empty NAT
  -defined finite Function such that
A1: F c= G and
A2: LastLoc F = LastLoc G;
  dom F = dom G
  proof
    thus dom F c= dom G by A1,GRFUNC_1:2;
    let x be object;
    assume
A3: x in dom G;
    dom G c= NAT by RELAT_1:def 18;
    then reconsider x as Element of NAT by A3;
A4: LastLoc F in dom F by VALUED_1:30;
    x <= LastLoc F by A2,A3,VALUED_1:32;
    then x < LastLoc F or x = LastLoc F by XXREAL_0:1;
    hence thesis by A4,Def12;
  end;
  hence thesis by A1,GRFUNC_1:3;
end;
