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

theorem
  f_adjac(M) * f_adjac(M) = f_adjac(M) &
  (f_adjac(M) \ id(Elements(M))) * f_adjac(M) = {} &
  (f_adjac(M) \/ ((f_adjac(M))~)) \/ id(Elements(M)) = f_circulation(M) \/
  (f_circulation(M))~
proof
  set R = (Flow M)|(the carrier' of M);
  set S = (Flow M)~|(the carrier' of M);
  set T = id(the carrier' of M);
  set Q = id(Elements(M));
A1: ((R \/ S) \/ T) \ Q = ((R \/ T) \/ (S \/ T)) \ Q by XBOOLE_1:5
    .= ((R \/ T) \ (id(Elements(M))) \/
  ((S \/ T) \ (id(Elements(M))))) by XBOOLE_1:42
    .= R \/
  ((S \/ T) \ (id(Elements(M)))) by Th14
    .= R \/ S by Th14;
A2: (R \/ S) * (R \/ S) = ((R \/ S) * R) \/ ((R \/ S) *S) by RELAT_1:32
    .= ((R * R) \/ (S * R)) \/ ((R \/ S) *S) by SYSREL:6
    .= ((R * R) \/ (S * R)) \/ ((R * S) \/ (S *S)) by SYSREL:6
    .= ({} \/ (S * R)) \/ ((R * S) \/ (S *S)) by Th11
    .= ({} \/ {}) \/ ((R * S) \/ (S *S)) by Th11
    .= ({} \/ {}) \/ ({} \/ (S *S)) by Th11
    .= {} by Th11;
A3: R \/ S~ = R \/ (((Flow M)|(the carrier of M))~)~ by Th15
    .= Flow M by Th16;
A4: R~ \/ S = R~ \/ ((Flow M)|(the carrier of M))~ by Th15
    .= (Flow M)~ by Th16;
A5: (R \/ S)~ \/ (R \/ S) = (R~ \/ S~) \/ (R \/ S) by RELAT_1:23
    .= (R~ \/ (S \/ R)) \/ S~ by XBOOLE_1:4
    .= ((R~ \/ S) \/ R) \/ S~ by XBOOLE_1:4
    .= (Flow M) \/ (Flow M)~ by A3,A4,XBOOLE_1:4;
A6: f_adjac(M) \/ (f_adjac(M))~ =
  ((R \/ S) \/ T) \/ ((R \/ S)~ \/ T~) by RELAT_1:23
    .= (((R \/ S) \/ T) \/ (R \/ S)~) \/ T~ by XBOOLE_1:4
    .= (((R \/ S) \/ (R \/ S)~) \/ T) \/ T~ by XBOOLE_1:4
    .= ((R \/ S) \/ (R \/ S)~) \/ (T \/ T~) by XBOOLE_1:4
    .= ((R \/ S) \/ (R \/ S)~) \/ (T \/ T)
    .= ((Flow M) \/ (Flow M)~) \/
  id(the carrier' of M) by A5;
A7: id(the carrier' of M) c= id(Elements(M)) by SYSREL:15,XBOOLE_1:7;
A8: f_adjac(M) * f_adjac(M) =
  (((R \/ S) \/ T) * (R \/ S)) \/ (((R \/ S) \/ T) * T) by RELAT_1:32
    .= ((((R \/ S) \/ T) * R ) \/ (((R \/ S) \/ T) * S)) \/
  (((R \/ S) \/ T) * T) by RELAT_1:32
    .= (((R \/ S) * R ) \/ (T * R )) \/ (((R \/ S) \/ T) * S) \/
  (((R \/ S) \/ T) * T) by SYSREL:6
    .= (((R * R) \/ (S * R)) \/ (T * R )) \/ (((R \/ S) \/ T) * S) \/
  (((R \/ S) \/ T) * T) by SYSREL:6
    .= (((R * R) \/ (S * R)) \/ (T * R )) \/ (((R \/ S) * S) \/ (T * S)) \/
  (((R \/ S) \/ T) * T) by SYSREL:6
    .= (((R * R) \/ (S * R)) \/ (T * R )) \/ (((R * S) \/ (S * S)) \/ (T * S))
  \/ (((R \/ S) \/ T) * T) by SYSREL:6
    .= (((R * R) \/ (S * R)) \/ (T * R )) \/ (((R * S) \/ (S * S)) \/ (T * S))
  \/ (((R \/ S) * T) \/ (T * T)) by SYSREL:6
    .= (((R * R) \/ (S * R)) \/ (T * R )) \/ (((R * S) \/ (S * S)) \/ (T * S))
  \/ (((R * T) \/ (S * T)) \/ (T * T)) by SYSREL:6
    .= (({} \/ (S * R)) \/ (T * R )) \/ (((R * S) \/ (S * S)) \/ (T * S)) \/
  (((R * T) \/ (S * T)) \/ (T * T)) by Th11
    .= (({} \/ {}) \/ (T * R )) \/ (((R * S) \/ (S * S)) \/ (T * S)) \/
  (((R * T) \/ (S * T)) \/ (T * T)) by Th11
    .= (({} \/ {}) \/ (T * R )) \/ (({} \/ (S * S)) \/ (T * S)) \/
  (((R * T) \/ (S * T)) \/ (T * T)) by Th11
    .= (T * R ) \/ ({} \/ (T * S)) \/ (((R * T) \/ (S * T)) \/ (T * T)) by Th11
    .= R \/ (T * S) \/ (((R * T) \/ (S * T)) \/ (T * T)) by Th12
    .= R \/ S \/ (((R * T) \/ (S * T)) \/ (T * T)) by Th12
    .= R \/ S \/ (((R * T) \/ (S * T)) \/ T) by SYSREL:12
    .= R \/ S \/ (({} \/ (S * T)) \/ T) by Th12
    .= R \/ S \/ ({} \/ T) by Th12
    .= f_adjac(M);
A9: (f_adjac(M) \ id(Elements(M))) * f_adjac(M) = {} \/ ((R \/ S) * T) by A1,A2
,RELAT_1:32
    .= (R * T) \/ (S * T) by SYSREL:6
    .= {} \/ (S * T) by Th12
    .= {} by Th12;
  (f_adjac(M) \/ (f_adjac(M))~) \/ id(Elements(M)) =
  (((Flow M) \/ (Flow M)~) \/
  (id(the carrier' of M) \/ id(Elements(M)))) by A6,XBOOLE_1:4
    .= (((Flow M) \/ (Flow M)~) \/ id(Elements(M)))
  by A7,XBOOLE_1:12
    .= ((Flow M) \/ id(Elements(M))) \/
  ((Flow M)~ \/ id(Elements(M))) by XBOOLE_1:5
    .= ((Flow M) \/ id(Elements(M))) \/
  ((Flow M)~ \/ (id(Elements(M)))~)
    .= f_circulation(M) \/ (f_circulation(M))~ by RELAT_1:23;
  hence thesis by A8,A9;
end;
