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;
reserve N for with_zero non empty set;
reserve N for with_zero non empty set,
  S for IC-Ins-separated non empty AMI-Struct over N;
reserve m,n for Nat;

theorem
  for S being halting IC-Ins-separated
   non empty with_non-empty_values AMI-Struct over N,
   p being NAT-defined (the InstructionsF of S)-valued Function,
   s being State of S
  st p halts_on s
 holds Result(p,s) = Comput(p,s,LifeSpan(p,s))
proof
  let S be halting IC-Ins-separated
  non empty with_non-empty_values AMI-Struct over N;
  let p be NAT-defined (the InstructionsF of S)-valued Function;
  let s be State of S;
  assume
A1: p halts_on s;
  then
A2: CurInstr(p,Comput(p,s,LifeSpan(p,s))) = halt S by Def15;
  consider m such that
A3: Result(p,s) = Comput(p,s,m) and
A4: CurInstr(p, Result(p,s)) = halt S by A1,Def9;
  LifeSpan(p,s) <= m by A1,A3,A4,Def15;
  hence thesis by A2,A3,Th5;
end;
