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;
reserve S for IC-recognized CurIns-recognized halting IC-Ins-separated
  non empty with_non-empty_values AMI-Struct over N;

theorem
 for q be non halt-free finite
  (the InstructionsF of S)-valued NAT-defined Function
  for p being q-autonomic
   FinPartState of S st IC S in dom p holds IC p in dom q
proof
 let q be non halt-free finite
  (the InstructionsF of S)-valued NAT-defined Function;
 let p be q-autonomic FinPartState of S;
 assume
A1: IC S in dom p;
  then
A2: p is non empty;
  consider s being State of S such that
A3: p c= s by PBOOLE:141;
  set P = (the Instruction-Sequence of S) +* q;
A4: q c= P by FUNCT_4:25;
   IC Comput(P,s,0) in dom q by A4,Def4,A2,A3;
 hence  IC p in dom q by A3,A1,GRFUNC_1:2;
end;
