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 Th8:
  for S being COM-Struct, F being preProgram of S
  holds IncAddr(IncAddr(F,k),m) = IncAddr(F,k+m)
proof
  let S be COM-Struct, F be preProgram of S;
A1: dom IncAddr(IncAddr(F,k),m) = dom IncAddr(F,k) by Def9
    .= dom F by Def9;
A2: dom IncAddr(F,k+m) = dom F by Def9;
  for x being object st x in dom F holds
  IncAddr(IncAddr(F,k),m).x = IncAddr(F,k+m).x
  proof
    let x be object such that
A3: x in dom F;
    reconsider x as Element of NAT by A3,ORDINAL1:def 12;
A4: x in dom IncAddr(F,k) by A3,Def9;
A5: IncAddr(F/.x,k) = IncAddr(F,k).x
    by A3,Def9
      .= IncAddr(F,k)/.x by A4,PARTFUN1:def 6;
    IncAddr(IncAddr(F,k),m).x
    = IncAddr(IncAddr(F,k)/.x,m) by A4,Def9
      .= IncAddr(F/.x,k+m) by A5,COMPOS_0:7
      .= IncAddr(F,k+m).x by A3,Def9;
    hence thesis;
  end;
  hence thesis by A1,A2,FUNCT_1:2;
end;
