let x be set ; :: thesis: for N being Pnet
for y being Element of Elements N st y in the carrier' of N holds
( x in Pre (N,y) iff pre N,y,x )
let N be Pnet; :: thesis: for y being Element of Elements N st y in the carrier' of N holds
( x in Pre (N,y) iff pre N,y,x )
let y be Element of Elements N; :: thesis: ( y in the carrier' of N implies ( x in Pre (N,y) iff pre N,y,x ) )
assume A1: y in the carrier' of N ; :: thesis: ( x in Pre (N,y) iff pre N,y,x )
A2: ( pre N,y,x implies x in Pre (N,y) )

proof

assume pre N,y,x ; :: thesis: x in Pre (N,y)
then A3: [x,y] in Flow N by Def6;
then x in the carrier of N by A1, Th13;
then x in Elements N by XBOOLE_0:def 3;
hence x in Pre (N,y) by A3, Def8; :: thesis: verum

end;

( x in Pre (N,y) implies pre N,y,x )

proof

assume x in Pre (N,y) ; :: thesis: pre N,y,x
then [x,y] in Flow N by Def8;
hence pre N,y,x by A1, Def6; :: thesis: verum

end;

hence ( x in Pre (N,y) iff pre N,y,x ) by A2; :: thesis: verum