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 y for set;

theorem
  for S being homogeneous J/A-independent standard-ins non empty set,
      I being Element of S
  holds IncAddr(IncAddr(I,k),m) = IncAddr(I,k+m)
proof
  let S be homogeneous J/A-independent standard-ins non empty set,
      I be Element of S;
A1: InsCode IncAddr(IncAddr(I,k),m) = InsCode IncAddr(I,k) by Def8
    .= InsCode I by Def8
    .= InsCode IncAddr(I,k+m) by Def8;
A2: AddressPart IncAddr(IncAddr(I,k),m) = AddressPart IncAddr(I,k) by Def8
    .= AddressPart I by Def8
    .= AddressPart IncAddr(I,k+m) by Def8;
A3: JumpPart IncAddr(IncAddr(I,k),m) = m + JumpPart IncAddr(I,k) by Def8;
A4: JumpPart IncAddr(I,k) = k + JumpPart I by Def8;
A5: JumpPart IncAddr(I,k+m) = k+m + JumpPart I by Def8;
    then
A6: dom JumpPart IncAddr(I,k+m) = dom JumpPart I by VALUED_1:def 2
    .= dom JumpPart IncAddr(I,k) by A4,VALUED_1:def 2
    .= dom JumpPart IncAddr(IncAddr(I,k),m) by A3,VALUED_1:def 2;
  for n being object st n in dom JumpPart IncAddr(IncAddr(I,k),m) holds
  (JumpPart IncAddr(IncAddr(I,k),m)).n = (JumpPart IncAddr(I,k+m)).n
  proof
    let n be object;
    assume
A7:   n in dom JumpPart IncAddr(IncAddr(I,k),m);
    then
A8: n in dom JumpPart IncAddr(I,k) by A3,VALUED_1:def 2;
    then
A9: n in dom JumpPart I by A4,VALUED_1:def 2;
A10:   (JumpPart IncAddr(I,k)).n = k + (JumpPart I).n
            by A4,A8,VALUED_1:def 2;
A11:   (JumpPart IncAddr(IncAddr(I,k),m)).n = m + (JumpPart IncAddr(I,k)).n
      by A7,A3,VALUED_1:def 2;
      n in dom JumpPart IncAddr(I,k+m) by A5,A9,VALUED_1:def 2;
      then (JumpPart IncAddr(I,k+m)).n = k + m + (JumpPart I).n
       by A5,VALUED_1:def 2;
      hence thesis by A11,A10;
  end;
   then JumpPart IncAddr(IncAddr(I,k),m) = JumpPart IncAddr(I,k+m)
                   by A6;
  hence thesis by A1,A2,Th1;
end;
