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;
reserve i, j, k for Nat,
  n for Nat,
  l,il for Nat;

theorem
  for g being NAT-defined (the InstructionsF of S)-valued finite Function
  for k being Nat holds
  for I being Instruction of S holds
  il in dom g & I = g.il implies
  IncAddr(I, k) = Reloc(g, k).(il + k)
proof
  let g be NAT-defined (the InstructionsF of S)-valued finite Function;
  let k be Nat;
  let I be Instruction of S;
  assume that
A1: il in dom g and
A2: I = g.il;
   reconsider ii = il as Element of NAT by ORDINAL1:def 12;
A3: il in dom IncAddr(g,k) by A1,Def9;
  thus (Reloc(g, k)).(il + k)
     = (IncAddr(g,k)).ii by A3,VALUED_1:def 12
    .= IncAddr((g)/.ii,k) by A1,Def9
    .= IncAddr(I,k) by A1,A2,PARTFUN1:def 6;
end;
