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;
