reserve N for with_zero set;
reserve N for with_zero set;
reserve x,y,z,A,B for set,
  f,g,h for Function,
  i,j,k for Nat;
reserve S for IC-Ins-separated non empty
     with_non-empty_values AMI-Struct over N,
  s for State of S;
reserve N for non empty with_zero set,
 S for IC-Ins-separated non empty with_non-empty_values AMI-Struct over N,
  s for State of S;
reserve n for Nat;

theorem
  for S being halting IC-Ins-separated
   non empty with_non-empty_values AMI-Struct over N
  for p being non halt-free
   NAT-defined (the InstructionsF of S)-valued Function
  for d being FinPartState of S
  holds d is Autonomy of p iff p,d computes {} .--> Result(p,d)
proof
  let S be halting IC-Ins-separated
  non empty with_non-empty_values AMI-Struct over N;
  let p be non halt-free
   NAT-defined (the InstructionsF of S)-valued Function;
  let d be FinPartState of S;
  thus d is Autonomy of p implies p,d computes {} .--> Result(p,d)
  proof
    assume
A2:  d is Autonomy of p;
    let x be set;
    assume
A3:   x in dom({} .--> Result(p,d));
    then
A4: x = {} by TARSKI:def 1;
    then reconsider s = x as FinPartState of S by FUNCT_1:104,RELAT_1:171;
    take s;
    thus x = s;
    d +* {} = d;
    hence d +* s is Autonomy of p by A2,A3,TARSKI:def 1;
    thus thesis by A4,FUNCOP_1:72;
  end;
A5: {} in dom({} .--> Result(p,d)) by TARSKI:def 1;
  assume p,d computes {} .--> Result(p,d);
  then ex s being FinPartState of S
    st {} = s & d +* s is Autonomy of p &
      ({} .--> Result(p,d)).s c= Result(p,d+* s)
     by A5;
  hence thesis;
end;
