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;
