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
  for F being data-only PartFunc of FinPartSt S, FinPartSt S,
      l being Nat
 for q be non halt-free finite
  (the InstructionsF of S)-valued NAT-defined Function,
      p being q-autonomic q-halted non empty FinPartState of S
        st IC S in dom p
  for k being Nat holds q,  p computes F
    iff Reloc(q,k), IncIC( p,k) computes F
proof
  let F be data-only PartFunc of FinPartSt S ,FinPartSt S,
      l be Nat;
  let q be non halt-free finite
    (the InstructionsF of S)-valued NAT-defined Function,
      p be q-autonomic q-halted non empty FinPartState of S
  such that
A1: IC S in dom p;
  let k be Nat;
  hereby
    assume
A2: q,  p computes F;
    thus Reloc(q,k), IncIC( p,k) computes F
    proof
      let x be set;
      assume
A3:   x in dom F;
      then consider d1 being FinPartState of S such that
A4:   x = d1 and
A5:    p +* d1 is Autonomy of q and
A6:   F.d1 c= Result(q, p +* d1) by A2;
      dom F c= FinPartSt S by RELAT_1:def 18;
      then reconsider d = x as FinPartState of S by A3,MEMSTR_0:76;
      reconsider d as data-only FinPartState of S by A3,MEMSTR_0:def 17;
      dom(p +* d) = dom p \/ dom d by FUNCT_4:def 1;
      then
A7:   IC S in dom(p +* d) by A1,XBOOLE_0:def 3;
A8:  p+*d is q-autonomic by A4,A5,EXTPRO_1:def 12;
    then
A9:   IncIC((p +* d) ,k) is Reloc(q,k)-autonomic by A7,Th11;
A10:    p+*d is q-halted by A4,A5,EXTPRO_1:def 12;
  reconsider pd = p +* d
   as q-halted q-autonomic non empty FinPartState of S
         by A4,A5,EXTPRO_1:def 12;
A11:   DataPart(Result(q, pd))
     = DataPart Result(Reloc(q,k),IncIC((p+*d),k)) by A7,Th13
    .= DataPart Result(Reloc(q,k),IncIC(p,k) +* d)
            by MEMSTR_0:54;
      reconsider Fs1 = F.d1 as FinPartState of S by A6;
      take d;
      thus x=d;
   IncIC(p,k) +* d = IncIC(p+*d ,k) by MEMSTR_0:54;
      hence IncIC( p,k) +* d is Autonomy of Reloc(q,k)
            by A8,A10,A9,EXTPRO_1:def 12;
A12:  Fs1 is data-only by A3,A4,MEMSTR_0:def 17;
      F.d1 c= DataPart Result(Reloc(q,k),IncIC(p,k) +* d)
         by A6,A12,A4,A11,MEMSTR_0:5;
      hence F.d c= Result(Reloc(q,k),IncIC( p,k) +* d)
      by A4,A12,MEMSTR_0:5;
    end;
  end;
  assume
A13: Reloc(q,k), IncIC( p,k) computes F;
  let x be set;
  assume
A14: x in dom F;
  then consider d1 being FinPartState of S such that
A15: x = d1 and
A16: IncIC(p,k) +* d1 is Autonomy of Reloc(q,k) and
A17: F.d1 c= Result(Reloc(q,k),IncIC(p,k) +* d1)
          by A13;
  dom F c= FinPartSt S by RELAT_1:def 18;
  then reconsider d = x as FinPartState of S by A14,MEMSTR_0:76;
  reconsider d as data-only FinPartState of S by A14,MEMSTR_0:def 17;
A18: dom(p +* d) = dom p \/ dom d by FUNCT_4:def 1;
  then
A19: IC S in dom(p +* d) by A1,XBOOLE_0:def 3;
A20: IncIC(p,k) +* d = IncIC((p +* d),k) by MEMSTR_0:54;
 IncIC(p+*d,k) is Reloc(q,k)-autonomic
  by A15,A16,A20,EXTPRO_1:def 12;
  then
A21: p +* d is q-autonomic by A19,Th11;
A22:  IncIC(p+*d,k) is Reloc(q,k)-halted
 by A15,A16,A20,EXTPRO_1:def 12;
A23: p +* d is q-halted by A19,Th12,A21,A22;
  reconsider pd = p +* d
   as q-halted q-autonomic non empty FinPartState of S
    by A19,Th12,A21,A22;
A24: IC S in dom pd by A18,A1,XBOOLE_0:def 3;
A25: DataPart Result(Reloc(q,k),IncIC(p,k) +* d1)
   = DataPart Result(Reloc(q,k),IncIC(p +* d,k)) by A15,MEMSTR_0:54
  .= DataPart(Result(q, p+*d)) by Th13,A24;
  take d;
  thus x=d;
  thus  p +* d is Autonomy of q by A21,A23,EXTPRO_1:def 12;
  reconsider Fs1 = F.d1 as FinPartState of S by A17;
A26: Fs1 is data-only by A14,A15,MEMSTR_0:def 17;
  then F.d1 c= DataPart Result(q, p +* d)
   by A25,A17,MEMSTR_0:5;
  hence thesis by A15,A26,MEMSTR_0:5;
end;
