reserve m for Nat;
reserve P,PP,P1,P2 for Instruction-Sequence of SCM+FSA;

theorem Th2:
  for I,J being Program of SCM+FSA, k being Nat
     st card I <= k & k < card I + card J
  for i being Instruction of SCM+FSA st i = J.(k -' card I)
   holds (I ";" J). k = IncAddr(i,card I)
proof
  let I,J be Program of SCM+FSA;
  let k be Nat;
  assume
A1: card I <= k;
  assume k < card I + card J;
  then
A2: k + 0 < card J + card I;
  k -' card I = k - card I by A1,XREAL_1:233;
  then k -' card I < card J - 0 by A2,XREAL_1:21;
  then
A3:  (k -' card I) in dom J by AFINSQ_1:66;
  let i be Instruction of SCM+FSA;
  assume
A4: i = J. (k -' card I);
A5: k -' card I + card I = k - card I + card I by A1,XREAL_1:233
    .= k;
  then  k in {m + card I where m is Nat: m in dom J} by A3;
  then
  k in dom Reloc(J, card I) by COMPOS_1:33;
  hence (I ";" J). k = (Reloc(J,card I)). k by SCMFSA6A:40
    .= IncAddr(i,card I) by A4,A3,A5,COMPOS_1:35;
end;
