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 Th5:
  for S being homogeneous J/A-independent standard-ins non empty set,
  I, J being Element of S st
  ex k being Nat st IncAddr(I,k) = IncAddr(J,k) holds I = J
proof
  let S be homogeneous J/A-independent standard-ins non empty set,
      I, J be Element of S;
  given k being Nat such that
A1: IncAddr(I,k) = IncAddr(J,k);
A2: InsCode I = InsCode IncAddr(I,k) by Def8
    .= InsCode J by A1,Def8;
A3: AddressPart I = AddressPart IncAddr(I,k) by Def8
       .= AddressPart J by A1,Def8;
A4: JumpPart IncAddr(I,k) = k + JumpPart I by Def8;
    then
A5: dom JumpPart I = dom  JumpPart IncAddr(I,k) by VALUED_1:def 2;
A6: JumpPart IncAddr(J,k) = k + JumpPart J by Def8;
    then
A7: dom JumpPart J = dom  JumpPart IncAddr(J,k) by VALUED_1:def 2;
A8: dom JumpPart I = dom JumpPart J by A2,Def5;
  for x being object st x in dom JumpPart I holds
  (JumpPart I).x = (JumpPart J).x
  proof
    let x be object;
    assume
A9: x in dom JumpPart I;
A10:   (JumpPart IncAddr(I,k)).x = k + (JumpPart I).x
             by A4,A5,A9,VALUED_1:def 2;
A11:   (JumpPart IncAddr(J,k)).x = k + (JumpPart J).x
             by A6,A8,A9,A7,VALUED_1:def 2;
      thus thesis by A1,A10,A11;
  end;
   then JumpPart I = JumpPart J by A8;
  hence thesis by A2,A3,Th1;
end;
