
theorem Th3:
 for N being with_zero set
 for S being halting IC-Ins-separated
   non empty with_non-empty_values AMI-Struct over N
 holds S is IC-recognized iff
  for q be non halt-free finite
   (the InstructionsF of S)-valued NAT-defined Function
  for p being q-autonomic
  FinPartState of S st DataPart p <> {}
   holds IC S in dom p
proof
 let N be with_zero set;
 let S be halting IC-Ins-separated
   non empty with_non-empty_values AMI-Struct over N;
 thus S is IC-recognized implies
  for q be non halt-free finite
   (the InstructionsF of S)-valued NAT-defined Function
  for p being q-autonomic FinPartState of S st DataPart p <> {}
   holds IC S in dom p
 proof assume
A1: S is IC-recognized;
  let q be non halt-free finite
   (the InstructionsF of S)-valued NAT-defined Function;
  let p be q-autonomic FinPartState of S;
  assume DataPart p <> {};
   then  p is non empty;
  hence thesis by A1;
 end;
 assume
A2: for q be non halt-free finite
    (the InstructionsF of S)-valued NAT-defined Function
 for p being q-autonomic FinPartState of S st DataPart p <> {}
   holds IC S in dom p;
  let q be non halt-free finite
   (the InstructionsF of S)-valued NAT-defined Function;
  let p be q-autonomic FinPartState of S;
A3: dom  p c= {IC S} \/ dom DataPart p by MEMSTR_0:32;
  assume
A4:  p is non empty;
  per cases;
  suppose
A5: IC S in dom  p;
   thus IC S in dom p by A5;
  end;
  suppose not IC S in dom  p;
   then {IC S} misses dom  p by ZFMISC_1:50;
   then dom DataPart p is non empty by A4,A3,XBOOLE_1:3,73;
   then DataPart p is non empty;
  hence IC S in dom p by A2;
  end;
end;
