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

theorem
  (e_adjac(N)) * (e_adjac(N)) = e_adjac(N) &
  (e_adjac(N) \ id(e_shore(N))) * e_adjac(N) = {} &
  (e_adjac(N) \/ (e_adjac(N))~) \/ id(e_shore(N)) = e_flow(N) \/ (e_flow(N))~
proof
  set P = the entrance of N;
  set Q = the escape of N;
  set R = 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: S c= R by SYSREL:15,XBOOLE_1:36;
  (e_adjac N \/ (e_adjac N)~) = (((P \/ Q) \ R) \/ S) \/ ( (((P \/ Q) \ R
  )~) \/ (S~)) by RELAT_1:23
    .= (((P \/ Q) \ R) \/ S) \/ ( ( ((P \/ Q) \ R)~) \/ S)
    .= (((P \/ Q) \ R) \/ ( ((P \/ Q) \ R)~ ) ) \/ S by XBOOLE_1:5
    .= (((P \/ Q) \ R) \/ ( ((P \/ Q)~) \ (R~) )) \/ S by RELAT_1:24
    .= (((P \/ Q) \ R) \/ (((P \/ Q)~) \ R)) \/ S
    .= (((P \/ Q) \/ ((P \/ Q)~)) \ R) \/ S by XBOOLE_1:42;
  then
A2: (e_adjac(N) \/ (e_adjac(N))~) \/ id(e_shore(N)) = (((P \/ Q) \/ ((P
  \/ Q)~)) \ R) \/ (S \/ R) by XBOOLE_1:4
    .= (((P \/ Q) \/ ((P \/ Q)~)) \ R) \/ R by A1,XBOOLE_1:12
    .= (((P~ \/ (Q~)) \/ (P \/ Q)) \ R) \/ R by RELAT_1:23
    .= (P~ \/ ((Q \/ P) \/ (Q~)) \ R) \/ R by XBOOLE_1:4
    .= (P~ \/ (Q \/ (P \/ (Q~))) \ R) \/ R by XBOOLE_1:4
    .= ( ( (P~ \/ Q) \/ (P \/ (Q~)) ) \ R) \/ R by XBOOLE_1:4
    .= ( (P~ \/ Q) \/ (P \/ (Q~)) ) \/ R by XBOOLE_1:39
    .= e_flow N \/ ((((P \/ (Q~))~)~) \/ R) by XBOOLE_1:5
    .= e_flow N \/ (((P~ \/ ((Q~)~))~) \/ R) by RELAT_1:23
    .= e_flow N \/ (((P~ \/ Q)~) \/ (R~))
    .= e_flow N \/ (e_flow N)~ by RELAT_1:23;
  S c= R by SYSREL:15,XBOOLE_1:36;
  then
A3: S \ R = {} by XBOOLE_1:37;
  (P \ R) * (Q \ R) c= P * (Q \ R) by RELAT_1:30,XBOOLE_1:36;
  then (P \ R) * (Q \ R) c= {} by Th15;
  then
A4: (P \ R) * (Q \ R) = {} by XBOOLE_1:3;
  (P \ R) * (P \ R) c= P * (P \ R) by RELAT_1:30,XBOOLE_1:36;
  then (P \ R) * (P \ R) c= {} by Def2;
  then
A5: (P \ R) * (P \ R) = {} by XBOOLE_1:3;
  (Q \ R) * (P \ R) c= Q * (P \ R) by RELAT_1:30,XBOOLE_1:36;
  then (Q \ R) * (P \ R) c= {} by Th15;
  then
A6: (Q \ R) * (P \ R) = {} by XBOOLE_1:3;
  (Q \ R) * (Q \ R) c= Q * (Q \ R) by RELAT_1:30,XBOOLE_1:36;
  then (Q \ R) * (Q \ R) c= {} by Def2;
  then
A7: (Q \ R) * (Q \ R) = {} by XBOOLE_1:3;
A8: (e_adjac(N) \ R) * e_adjac(N) = (((((P \/ Q) \ R) \/ S
  ) \ R) * ((P \/ Q) \ R)) \/ (((((P \/ Q) \ R) \/ S) \ R) * S) by RELAT_1:32
    .= (((((P \ R) \/ (Q \ R)) \/ S) \ R) * ((P \/ Q) \ R)) \/ (((((P \/ Q)
  \ R) \/ S) \ R) * S) by XBOOLE_1:42
    .= ((( (((P \ R) \/ (Q \ R)) \ R) \/ (S \ R)) * ((P \/ Q) \ R)) \/ ((((P
  \/ Q) \ R) \/ S) \ R) * S) by XBOOLE_1:42
    .= (((( ( ((P \ R) \ R) \/ ((Q \ R) \ R)) ) \/ (S \ R)) * ((P \/ Q) \ R)
  ) \/ ((((P \/ Q) \ R) \/ S) \ R) * S) by XBOOLE_1:42
    .= (((((((P \ (R \/ R)) \/ ((Q \ R) \ R))) \/ (S \ R)) * ((P \/ Q) \ R))
  \/ ((((P \/ Q) \ R) \/ S) \ R) * S)) by XBOOLE_1:41
    .= ((((((P \ R) \/ ((Q \ (R \/ R))) \/ (S \ R)) * ((P \/ Q) \ R)) \/ (((
  (P \/ Q) \ R) \/ S) \ R) * S))) by XBOOLE_1:41
    .= (((((P \ R) \/ (Q \ R)) \/ (S \ R)) * ((P \ R) \/ (Q \ R)) \/ ((((P
  \/ Q) \ R) \/ S) \ R) * S)) by XBOOLE_1:42
    .= (((((P \ R) \/ (Q \ R) \/ (S \ R)) * ((P \ R) \/ (Q \ R))) \/ ((((P \
  R) \/ (Q \ R)) \/ S) \ R) * S)) by XBOOLE_1:42
    .= ((( ( ( ((P \ R) \/ (Q \ R)) \/ (S \ R) ) * (P \ R) ) \/ ( ( ((P \ R)
\/ (Q \ R)) \/ (S \ R) ) * (Q \ R))) \/ ((((P \ R) \/ (Q \ R)) \/ S) \ R) * S))
  by RELAT_1:32
    .= ((( ( ( ((P \ R) \/ (Q \ R)) \/ (S \ R) ) * (P \ R) ) \/ ( ( ((P \ R)
\/ (Q \ R)) * (Q \ R) ) \/ ((S \ R) * (Q \ R)) ) ) \/ ((((P \ R) \/ (Q \ R)) \/
  S) \ R) * S)) by SYSREL:6
    .= ((( ( ( ((P \ R) \/ (Q \ R)) * (P \ R)) \/ ((S \ R) ) * (P \ R)) ) \/
( ( ((P \ R) \/ (Q \ R)) * (Q \ R)) \/ ((S \ R) * (Q \ R)) ) ) \/ ((((P \ R) \/
  (Q \ R)) \/ S) \ R) * S) by SYSREL:6
    .= ((( ( ({} \/ {}) \/ ((S \ R) ) * (P \ R)) ) \/ ( ( ((P \ R) \/ (Q \ R
)) * (Q \ R)) \/ ((S \ R) * (Q \ R)) ) ) \/ ((((P \ R) \/ (Q \ R)) \/ S) \ R) *
  S) by A5,A6,SYSREL:6
    .= ((( {} * (P \ R)) \/ ({} * (Q \ R))) \/ ((((P \ R) \/ (Q \ R)) \/ S)
  \ R) * S) by A3,A4,A7,SYSREL:6
    .= ( ( ((P \ R) \/ (Q \ R)) \ R) \/ (S \ R) ) * S by XBOOLE_1:42
    .= ( ( ((P \ R) \ R) \/ ((Q \ R) \ R)) \/ (S \ R) ) * S by XBOOLE_1:42
    .= ( ((P \ (R \/ R)) \/ ((Q \ R) \ R)) \/ (S \ R) ) * S by XBOOLE_1:41
    .= ( ((P \ R) \/ (Q \ (R \/ R)) ) \/ (S \ R) ) * S by XBOOLE_1:41
    .= ((P \ R) * S) \/ ((Q \ R) * S) by A3,SYSREL:6
    .= {} \/ ((Q \ R) * S) by Th27
    .= {} \/ ((Q \ R) * T) by Th13
    .= {} by Th27;
  e_adjac(N) * e_adjac(N) = ((((P \/ Q) \ R) * (((P \/ Q) \ R) \/ S)) \/
  (S * (((P \/ Q) \ R) \/ S))) by SYSREL:6
    .= ((((P \/ Q) \ R) * ((P \/ Q) \ R)) \/ (((P \/ Q) \ R) * S)) \/ (S * (
  ((P \/ Q) \ R) \/ S)) by RELAT_1:32
    .= ((((P \/ Q) \ R) * ((P \/ Q) \ R)) \/ (((P \/ Q) \ R) * S)) \/ ((S *
  ((P \/ Q) \ R)) \/ (S * S)) by RELAT_1:32
    .= ((((P \ R) \/ (Q \ R)) * ((P \/ Q) \ R)) \/ (((P \/ Q) \ R) * S)) \/
  ((S * ((P \/ Q) \ R)) \/ (S * S)) by XBOOLE_1:42
    .= ((((P \ R) \/ (Q \ R)) * ((P \ R) \/ (Q \ R))) \/ (((P \/ Q) \ R) * S
  )) \/ ((S * ((P \/ Q) \ R)) \/ (S * S)) by XBOOLE_1:42
    .= ((((P \ R) \/ (Q \ R)) * ((P \ R) \/ (Q \ R))) \/ (((P \ R) \/ (Q \ R
  )) * S)) \/ ((S * ((P \/ Q) \ R)) \/ (S * S)) by XBOOLE_1:42
    .= ((((P \ R) \/ (Q \ R)) * ((P \ R) \/ (Q \ R))) \/ (((P \ R) \/ (Q \ R
  )) * S)) \/ ((S * ((P \ R) \/ (Q \ R))) \/ (S * S)) by XBOOLE_1:42
    .= ((((P \ R) \/ (Q \ R)) * ((P \ R)) \/ (((P \ R) \/ (Q \ R)) * (Q \ R)
)) \/ (((P \ R) \/ (Q \ R)) * S)) \/ ((S * ((P \ R) \/ (Q \ R))) \/ (S * S))
by RELAT_1:32
    .= ((({} \/ {}) \/ (((P \ R) \/ (Q \ R)) * (Q \ R))) \/ (((P \ R) \/ (Q
  \ R)) * S)) \/ ((S * ((P \ R) \/ (Q \ R))) \/ (S * S)) by A5,A6,SYSREL:6
    .= {} \/ (((P \ R) \/ (Q \ R)) * S) \/ ((S * ((P \ R) \/ (Q \ R))) \/ (S
  * S)) by A4,A7,SYSREL:6
    .= (((P \ R) \/ (Q \ R)) * S) \/ ((S * ((P \ R) \/ (Q \ R))) \/ S) by
SYSREL:12
    .= (((P \ R) * S) \/ ((Q \ R) * S)) \/ ((S * ((P \ R) \/ (Q \ R))) \/ S)
  by SYSREL:6
    .= (((P \ R) * S) \/ ((Q \ R) * S)) \/ ((S * (P \ R)) \/ (S * (Q \ R))
  \/ S) by RELAT_1:32
    .= ({} \/ ((Q \ R) * S)) \/ ((S * (P \ R)) \/ (S * (Q \ R)) \/ S) by Th27
    .= ((Q \ R) * T) \/ ((S * (P \ R)) \/ (S * (Q \ R)) \/ S) by Th13
    .= {} \/ ((S * (P \ R)) \/ (S * (Q \ R)) \/ S) by Th27
    .= ((P \ R) \/ (S * (Q \ R)) \/ S) by Th23
    .= ((P \ R) \/ (T * (Q \ R)) \/ S) by Th13
    .= ((P \ R) \/ (Q \ R)) \/ S by Th23
    .= e_adjac N by XBOOLE_1:42;
  hence thesis by A8,A2;
end;
