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  IC-Ins-separated
   non empty with_non-empty_values AMI-Struct over N
  for P being Instruction-Sequence of S
  for s being State of S
 holds Comput(P,s,k+1) = Exec(P.IC Comput(P,s,k),Comput(P,s,k))
proof
  let S be IC-Ins-separated non empty with_non-empty_values AMI-Struct over N;
  let P be Instruction-Sequence of S;
  let s be State of S;
A1: dom P = NAT by PARTFUN1:def 2;
  thus Comput(P,s,k+1) = Following(P,Comput(P,s,k)) by Th3
   .= Exec(P.(IC Comput(P,s,k)),Comput(P,s,k)) by A1,PARTFUN1:def 6;
end;
