reserve x,y for object,X,Y for set;
reserve M for Pnet;

theorem
  (f_escape(M)) * (f_escape(M)) = f_escape(M) &
  (f_escape(M)) * (f_entrance(M)) = f_escape(M) &
  (f_entrance(M)) * (f_entrance(M)) =f_entrance(M) &
  (f_entrance(M)) * (f_escape(M)) = f_entrance(M)
proof
  set R = ((Flow M)|(the carrier of M));
  set S = id(the carrier' of M);
A1: S * R = {} by Th12;
A2: R * S = R by Th12;
A3: S * S = S by SYSREL:12;
A4: (f_escape(M)) * (f_escape(M)) =
  (R * (R \/ S)) \/ (S * (R \/ S)) by SYSREL:6
    .= ((R * R) \/ (R * S)) \/ (S * (R \/ S)) by RELAT_1:32
    .= ((R * R) \/ (R * S)) \/ ((S * R) \/ (S * S)) by RELAT_1:32
    .= ({} \/ R) \/ ({} \/ S) by A1,A2,A3,Th11
    .=f_escape(M);
A5: (f_escape(M)) * (f_entrance(M)) = f_escape(M)
  proof
    set T = ((Flow M)~|(the carrier of M));
A6: S * T = {} by Th12;
A7: R * S = R by Th12;
A8: S * S = S by SYSREL:12;
    (f_escape(M)) * (f_entrance(M)) =
    (R * (T \/ S)) \/ (S * (T \/ S)) by SYSREL:6
      .= ((R * T) \/ (R * S)) \/ (S * (T \/ S)) by RELAT_1:32
      .= ((R * T) \/ (R * S)) \/ ((S * T) \/ (S * S)) by RELAT_1:32
      .= ({} \/ R) \/ ({} \/ S) by A6,A7,A8,Th11
      .=f_escape(M);
    hence thesis;
  end;
A9: (f_entrance(M)) * (f_entrance(M)) = f_entrance(M)
  proof
    set R = ((Flow M)~|(the carrier of M));
A10: S * R = {} by Th12;
A11: R * S = R by Th12;
A12: S * S = S by SYSREL:12;
    (f_entrance(M)) * (f_entrance(M)) =
    (R * (R \/ S)) \/ (S * (R \/ S)) by SYSREL:6
      .= ((R * R) \/ (R * S)) \/ (S * (R \/ S)) by RELAT_1:32
      .= ((R * R) \/ (R * S)) \/ ((S * R) \/ (S * S)) by RELAT_1:32
      .= ({} \/ R) \/ ({} \/ S) by A10,A11,A12,Th11
      .=f_entrance(M);
    hence thesis;
  end;
  (f_entrance(M)) * (f_escape(M)) = f_entrance(M)
  proof
    set T = ((Flow M)~|(the carrier of M));
A13: T * S = T by Th12;
A14: S * R = {} by Th12;
A15: S * S = S by SYSREL:12;
    (f_entrance(M)) * (f_escape(M)) =
    (T * (R \/ S)) \/ (S * (R \/ S)) by SYSREL:6
      .= ((T * R) \/ (T * S)) \/ (S * (R \/ S)) by RELAT_1:32
      .= ((T * R) \/ (T * S)) \/ ((S * R) \/ (S * S)) by RELAT_1:32
      .= ({} \/ T) \/ ({} \/ S) by A13,A14,A15,Th11
      .=f_entrance(M);
    hence thesis;
  end;
  hence thesis by A4,A5,A9;
end;
