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 N being with_zero non empty set,
      S being IC-Ins-separated
           non empty with_non-empty_values AMI-Struct over N,
      P being Instruction-Sequence of S,
      s being State of S, i being Instruction of S
   holds Exec(P.IC s,s).IC S = IC Following(P,s)
proof
  let N;
  let S be IC-Ins-separated
           non empty with_non-empty_values AMI-Struct over N,
      P be Instruction-Sequence of S,
      s be State of S, i being Instruction of S;
 NAT = dom P by PARTFUN1:def 2;
 hence Exec(P.IC s,s).IC S = IC Following(P,s) by PARTFUN1:def 6;
end;
