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_U_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 st o in Out_U_Inp I holds o in Output I \/ Input
  I
  proof
    let o be Object of A such that
A1: o in Out_U_Inp I;
    o in Input I or o in Output I
    proof
      assume
A2:   not o in Input I;
      per cases by A2,XBOOLE_0:def 5;
      suppose
        not o in Out_U_Inp I;
        hence thesis by A1;
      end;
      suppose
A3:     o in Out_\_Inp I;
        Out_\_Inp I c= Output I by Th3;
        hence thesis by A3;
      end;
    end;
    hence thesis by XBOOLE_0:def 3;
  end;
  hence Out_U_Inp I c= Output I \/ Input I by SUBSET_1:2;
  Output I c= Out_U_Inp I & Input I c= Out_U_Inp I by Th4,XBOOLE_1:36;
  hence thesis by XBOOLE_1:8;
end;
