reserve N for with_zero set;

theorem Th4:
  for A being IC-Ins-separated non empty
  with_non-empty_values AMI-Struct over N,
  I being Instruction of A holds Output I c= Out_U_Inp I
proof
  let A be IC-Ins-separated non empty with_non-empty_values AMI-Struct over
  N, I be Instruction of A;
  for o being Object of A holds o in Output I implies o in Out_U_Inp I
  proof
    let o be Object of A;
    assume
A1: not thesis;
    for s being State of A holds s.o = Exec(I,s).o
    proof
      let s be State of A;
      reconsider so = s.o as Element of Values o by MEMSTR_0:77;
A2:   Exec(I,s+*(o,so)) = Exec(I,s) +* (o,so) by A1,Def5;
      dom Exec(I,s) = the carrier of A by PARTFUN1:def 2;
      hence s.o = (Exec(I,s) +* (o,so)).o by FUNCT_7:31
        .= Exec(I,s).o by A2,FUNCT_7:35;
    end;
    hence contradiction by A1,Def3;
  end;
  hence thesis by SUBSET_1:2;
end;
