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(I, 0) = I
proof
  let S be homogeneous J/A-independent standard-ins non empty set,
  I be Element of S;
A1: InsCode IncAddr(I, 0) = InsCode I by Def8;
A2: AddressPart IncAddr(I, 0) = AddressPart I by Def8;
A3:  JumpPart IncAddr(I, 0) = (0 qua Nat) + JumpPart I by Def8;
   then
A4: dom JumpPart I = dom JumpPart IncAddr(I, 0) by VALUED_1:def 2;
  for k being Nat st k in dom JumpPart I holds
  (JumpPart IncAddr(I, 0)).k = (JumpPart I).k
  proof
    let k be Nat;
    assume k in dom JumpPart I;
    hence (JumpPart IncAddr(I, 0)).k
        = (0 qua Nat) + (JumpPart I).k by A4,A3,VALUED_1:def 2
       .= (JumpPart I).k;
  end;
   then JumpPart IncAddr(I, 0) = JumpPart I by A4;
  hence thesis by A1,A2,Th1;
end;
