reserve N for with_zero set;

theorem
  for A being with_non_trivial_ObjectKinds IC-Ins-separated
non empty with_non-empty_values AMI-Struct over N, I being Instruction of A
 holds
  Out_\_Inp I = Output I \ Input I
proof
  let A be with_non_trivial_ObjectKinds 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 Out_\_Inp I iff o in Output I \ Input
  I
  proof
    let o be Object of A;
    hereby
A1:   Out_\_Inp I c= Output I by Th3;
      assume
A2:   o in Out_\_Inp I;
      Out_\_Inp I misses Input I by XBOOLE_1:85;
      then not o in Input I by A2,XBOOLE_0:3;
      hence o in Output I \ Input I by A2,A1,XBOOLE_0:def 5;
    end;
    assume
A3: o in Output I \ Input I;
    then
A4: not o in Input I by XBOOLE_0:def 5;
    per cases by A4,XBOOLE_0:def 5;
    suppose
A5:   not o in Out_U_Inp I;
      Output I c= Out_U_Inp I by Th4;
      then not o in Output I by A5;
      hence thesis by A3,XBOOLE_0:def 5;
    end;
    suppose
      o in Out_\_Inp I;
      hence thesis;
    end;
  end;
  hence thesis by SUBSET_1:3;
end;
