reserve k for Nat;

theorem Th5:
  for N be with_zero set
  for S be relocable halting IC-Ins-separated
  non empty with_non-empty_values AMI-Struct over N
  for INS being Instruction of S, s being State of S,
      j, k being Nat st IC s = j+k
 holds  Exec(INS, DecIC(s,k)) =  DecIC(Exec(IncAddr(INS, k), s),k)
proof
  let N be with_zero set;
  let S be relocable halting IC-Ins-separated
  non empty with_non-empty_values AMI-Struct over N;
  let INS be Instruction of S, s be State of S,
      j,k be Nat such that
A1: IC s = j+k;
  set s1 = s +* Start-At(j,S);
A2: IncIC(s1,k) = s +* Start-At(IC s,S) by A1,MEMSTR_0:58
     .= s by MEMSTR_0:31;
 thus  Exec(INS, DecIC(s,k))
    =  Exec(INS,s1) by A1,NAT_D:34
   .= (Exec(INS,s1) +* Start-At(IC Exec(INS,s1),S)) by MEMSTR_0:31
   .= (IncIC(Exec(INS,s1),k) +*
       Start-At(IC Exec(INS,s1),S)) by FUNCT_4:114
   .= DecIC( IncIC(Exec(INS,s1),k),k) by MEMSTR_0:59
   .=  DecIC(Exec(IncAddr(INS, k), s),k) by A2,Th4;
end;
