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;
reserve
  i,j,k for Instruction of S,
  I,J,K for Program of S;
reserve k1,k2 for Integer;
reserve l,l1,loc for Nat;
reserve i1,i2 for Instruction of S;
reserve
  i,j,k for Instruction of S,
  I,J,K for Program of S;
reserve m for Nat;

theorem
 for S being COM-Struct
  for k being Nat holds
    for P being preProgram of S holds
    il in dom P iff il + k in dom Reloc(P,k)
proof
 let S be COM-Struct;
  let k be Nat;
  let P be preProgram of S;
  A1: dom Reloc(P,k) = { j+k where j is Nat: j in dom P } by Th21;
   reconsider il1 = il as Element of NAT by ORDINAL1:def 12;
    il1 in dom P implies il1 + k in dom Reloc(P,k) by A1;
  hence il in dom P implies il + k in dom Reloc(P,k);
  assume il + k in dom Reloc(P,k);
  then ex j being Nat st il + k = j+k & j in dom P by A1;
  hence thesis;
end;
