reserve k for Nat;
reserve N for with_zero set,
   S for IC-recognized
    halting IC-Ins-separated
  non empty with_non-empty_values AMI-Struct over N;
reserve
   S for IC-recognized CurIns-recognized
    halting IC-Ins-separated
  non empty with_non-empty_values AMI-Struct over N;
reserve S for relocable IC-recognized CurIns-recognized
     halting IC-Ins-separated
  non empty with_non-empty_values AMI-Struct over N;
reserve m,j for Nat;
reserve S for relocable1 relocable2
  relocable IC-recognized CurIns-recognized halting IC-Ins-separated
  non empty with_non-empty_values AMI-Struct over N;

theorem Th13:
 for q be non halt-free finite
  (the InstructionsF of S)-valued NAT-defined Function
  for p being q-halted q-autonomic non empty FinPartState of S
   st IC S in dom p
  for k being Nat holds DataPart(Result(q, p)) =
   DataPart Result(Reloc(q,k),IncIC( p,k))
proof
 let q be non halt-free finite
  (the InstructionsF of S)-valued NAT-defined Function;
  let p be q-halted q-autonomic non empty FinPartState of S
  such that
A1: IC S in dom p;
  let k be Nat;
  consider s being State of S such that
A2: p c= s by PBOOLE:141;
  consider P being
   Instruction-Sequence of S
   such that
A3: q c= P by PBOOLE:145;
A4: IncIC(p,k) is Reloc(q,k)-halted Reloc(q,k)-autonomic by A1,Th12,Th11;
A5: IncIC( p,k) is Autonomy of Reloc(q,k)
       by A4,EXTPRO_1:def 12;
  P halts_on s by A2,A3,EXTPRO_1:def 11;
  then consider j1 being Nat such that
A6: Result(P,s) = Comput(P,s,j1) and
A7: CurInstr(P, Result(P,s)) = halt S by EXTPRO_1:def 9;
  consider t being State of S such that
A8: IncIC( p,k) c= t by PBOOLE:141;
  consider Q being
   Instruction-Sequence of S
   such that
A9: Reloc(q,k) c= Q by PBOOLE:145;
  Q.(IC Comput(Q,t,j1)) = CurInstr(Q,Comput(Q,t,j1)) by PBOOLE:143
    .= IncAddr(CurInstr(P,Comput(P,s,j1)), k)
     by Def5,A3,A9,A2,A8
    .= halt S by A6,A7,COMPOS_0:4;
  then
A10: Result(Q,t) = Comput(Q,t,j1) by EXTPRO_1:7;
A11: Comput(Q,t,j1) | dom DataPart p
        = Comput(P,s,j1) | dom DataPart p
         by Def6,A9,A3,A8,A2;
A12: DataPart p = DataPart IncIC( p,k) by MEMSTR_0:51;
A13:  p is Autonomy of q by EXTPRO_1:def 12;
  thus DataPart(Result(q, p))
     = DataPart((Result(P,s)) | dom  p) by A2,A3,A13,EXTPRO_1:def 13
    .= (Result(P,s)) | (dom  p /\ Data-Locations S) by RELAT_1:71
    .= (Result(P,s)) | dom DataPart p by MEMSTR_0:14
    .= (Result(Q,t))|(dom IncIC( p,k) /\ Data-Locations S)
          by A6,A10,A11,A12,MEMSTR_0:14
    .= ((Result(Q,t)) | dom IncIC( p,k))|Data-Locations S
           by RELAT_1:71
    .= DataPart Result(Reloc(q,k),IncIC( p,k))
       by A5,A9,A8,EXTPRO_1:def 13;
end;
