reserve x,A for set, i,j,k,m,n, l, l1, l2 for Nat;
reserve D for non empty set, z for Nat;
reserve S for COM-Struct;
reserve ins for Element of the InstructionsF of S;
reserve k, m for Nat,
  x, x1, x2, x3, y, y1, y2, y3, X,Y,Z for set;

theorem
  for S being COM-Struct, F, G being Program of S,
  f being Nat st f < card F - 1
  holds IncAddr(F,card F -' 1).f = IncAddr(F ';' G, card F -' 1).f
proof
  let S be COM-Struct, F, G be Program of S, f be Nat;
  set k = card F -' 1, P = F ';' G;
  assume
A1: f < card F - 1;
   then f < card CutLastLoc F by VALUED_1:38;
  then
A2: f in dom CutLastLoc F by AFINSQ_1:66;
A3: dom CutLastLoc F c= dom F by GRFUNC_1:2;
  CutLastLoc F c= CutLastLoc P by Th13;
  then CutLastLoc F c= P by XBOOLE_1:1;
  then
A4: dom CutLastLoc F c= dom P by GRFUNC_1:2;
A5: F.f = F/.f by A2,A3,PARTFUN1:def 6;
A6: P.f = F.f by Lm6,A1;
  thus IncAddr(F,k).f = IncAddr(F/.f,k) by A2,A3,Def9
    .= IncAddr(P/.f,k) by A2,A4,A5,A6,PARTFUN1:def 6
    .= IncAddr(P,k).f by A2,A4,Def9;
end;
