reserve x,y,z for object,X,Y for set;
reserve N for e_net;

theorem
  e_entrance(N) * e_entrance(N) = e_entrance(N) &
  e_entrance(N) * e_escape(N) = e_entrance(N) &
  e_escape(N) * e_entrance(N) = e_escape(N) &
  e_escape(N) * e_escape(N) = e_escape(N)
proof
  set P = ((the escape of N) \ id the carrier of N);
  set Q = ((the entrance of N) \ id the carrier of N);
  set S = id((the carrier of N) \ rng the entrance of N);
  set T = id((the carrier of N) \ rng the escape of N);
A1: e_entrance(N) * e_entrance(N) = ((P~) * ((P~) \/ T)) \/ (T * ((P~) \/ T)
  ) by SYSREL:6
    .= (((P~) * (P~)) \/ ((P~) * T)) \/ (T * ((P~) \/ T)) by RELAT_1:32
    .= (((P~) * (P~)) \/ ((P~) * T)) \/ ((T * (P~)) \/ (T * T)) by RELAT_1:32
    .= (((P * P)~) \/ ((P~) * T)) \/ ((T * (P~)) \/ (T * T)) by RELAT_1:35
    .= (((P * P)~) \/ ((P~) * (T~))) \/ ((T * (P~)) \/ (T * T))
    .= (((P * P)~) \/ ((P~) * (T~))) \/ (((T~) * (P~)) \/ (T * T))
    .= (((P * P)~) \/ ((P~) * (T~))) \/ (((T~) * (P~)) \/ T) by SYSREL:12
    .= (((P * P)~) \/ ((T * P)~)) \/ (((T~) * (P~)) \/ T) by RELAT_1:35
    .= (((P * P)~) \/ ((T * P)~)) \/ (((P * T)~) \/ T) by RELAT_1:35
    .= (({}~) \/ ((T * P)~)) \/ (((P * T)~) \/ T) by Th24
    .= (({}~) \/ (P~)) \/ (((P * T)~) \/ T) by Th23
    .= ({} \/ (P~)) \/ ({} \/ T) by Th27
    .= e_entrance(N);
A2: e_escape(N) * e_escape(N) = ((Q~) * ((Q~) \/ S)) \/ (S * ((Q~) \/ S)) by
SYSREL:6
    .= (((Q~) * (Q~)) \/ ((Q~) * S)) \/ (S * ((Q~) \/ S)) by RELAT_1:32
    .= (((Q~) * (Q~)) \/ ((Q~) * S)) \/ ((S * (Q~)) \/ (S * S)) by RELAT_1:32
    .= (((Q * Q)~) \/ ((Q~) * S)) \/ ((S * (Q~)) \/ (S * S)) by RELAT_1:35
    .= (((Q * Q)~) \/ ((Q~) * (S~))) \/ ((S * (Q~)) \/ (S * S))
    .= (((Q * Q)~) \/ ((Q~) * (S~))) \/ (((S~) * (Q~)) \/ (S * S))
    .= (((Q * Q)~) \/ ((Q~) * (S~))) \/ (((S~) * (Q~)) \/ S) by SYSREL:12
    .= (((Q * Q)~) \/ ((S * Q)~)) \/ (((S~) * (Q~)) \/ S) by RELAT_1:35
    .= (((Q * Q)~) \/ ((S * Q)~)) \/ (((Q * S)~) \/ S) by RELAT_1:35
    .= (({}~) \/ ((S * Q)~)) \/ (((Q * S)~) \/ S) by Th24
    .= (({}~) \/ (Q~)) \/ (((Q * S)~) \/ S) by Th23
    .= ({} \/ (Q~)) \/ ({} \/ S) by Th27
    .= e_escape(N);
A3: e_escape(N) * e_entrance(N) = ((Q~) * ((P~) \/ T)) \/ (S * ((P~) \/ T))
  by SYSREL:6
    .= (((Q~) * (P~)) \/ ((Q~) * T)) \/ (S * ((P~) \/ T)) by RELAT_1:32
    .= (((Q~) * (P~)) \/ ((Q~) * T)) \/ ((S * (P~)) \/ (S * T)) by RELAT_1:32
    .= (((P * Q)~) \/ ((Q~) * T)) \/ ((S * (P~)) \/ (S * T)) by RELAT_1:35
    .= (((P * Q)~) \/ ((Q~) * (T~))) \/ ((S * (P~)) \/ (S * T))
    .= (((P * Q)~) \/ ((Q~) * (T~))) \/ (((S~) * (P~)) \/ (S * T))
    .= (((P * Q)~) \/ ((Q~) * (T~))) \/ (((S~) * (P~)) \/ (S * S)) by Th13
    .= (((P * Q)~) \/ ((Q~) * (T~))) \/ (((S~) * (P~)) \/ S) by SYSREL:12
    .= (((P * Q)~) \/ ((T * Q)~)) \/ (((S~) * (P~)) \/ S) by RELAT_1:35
    .= (((P * Q)~) \/ ((T * Q)~)) \/ (((P * S)~) \/ S) by RELAT_1:35
    .= (((P * Q)~) \/ ((S * Q)~)) \/ (((P * S)~) \/ S) by Th13
    .= (((P * Q)~) \/ ((S * Q)~)) \/ (((P * T)~) \/ S) by Th13
    .= (({}~) \/ ((S * Q)~)) \/ (((P * T)~) \/ S) by Th24
    .= (({}~) \/ (Q~)) \/ (((P * T)~) \/ S) by Th23
    .= ({} \/ (Q~)) \/ ({} \/ S) by Th27
    .= e_escape(N);
  e_entrance(N) * e_escape(N) = ((P~) * ((Q~) \/ S)) \/ (T * ((Q~) \/ S))
  by SYSREL:6
    .= (((P~) * (Q~)) \/ ((P~) * S)) \/ (T * ((Q~) \/ S)) by RELAT_1:32
    .= (((P~) * (Q~)) \/ ((P~) * S)) \/ ((T * (Q~)) \/ (T * S)) by RELAT_1:32
    .= (((Q * P)~) \/ ((P~) * S)) \/ ((T * (Q~)) \/ (T * S)) by RELAT_1:35
    .= (((Q * P)~) \/ ((P~) * (S~))) \/ ((T * (Q~)) \/ (T * S))
    .= (((Q * P)~) \/ ((P~) * (S~))) \/ (((T~) * (Q~)) \/ (T * S))
    .= (((Q * P)~) \/ ((P~) * (S~))) \/ (((T~) * (Q~)) \/ (T * T)) by Th13
    .= (((Q * P)~) \/ ((P~) * (S~))) \/ (((T~) * (Q~)) \/ T) by SYSREL:12
    .= (((Q * P)~) \/ ((S * P)~)) \/ (((T~) * (Q~)) \/ T) by RELAT_1:35
    .= (((Q * P)~) \/ ((S * P)~)) \/ (((Q * T)~) \/ T) by RELAT_1:35
    .= (((Q * P)~) \/ ((T * P)~)) \/ (((Q * T)~) \/ T) by Th13
    .= (((Q * P)~) \/ ((T * P)~)) \/ (((Q * S)~) \/ T) by Th13
    .= (({}~) \/ ((T * P)~)) \/ (((Q * S)~) \/ T) by Th24
    .= (({}~) \/ (P~)) \/ (((Q * S)~) \/ T) by Th23
    .= ({} \/ (P~)) \/ ({} \/ T) by Th27
    .= e_entrance(N);
  hence thesis by A1,A3,A2;
end;
