let N be Pnet; :: thesis: for X being Subset of (Elements N)
for x being Element of Elements N st Elements N <> {} holds
( x in Input (N,X) iff ( ( x in X & x in the carrier of N ) or ex y being Element of Elements N st
( y in X & y in the carrier' of N & pre N,y,x ) ) )
let X be Subset of (Elements N); :: thesis: for x being Element of Elements N st Elements N <> {} holds
( x in Input (N,X) iff ( ( x in X & x in the carrier of N ) or ex y being Element of Elements N st
( y in X & y in the carrier' of N & pre N,y,x ) ) )
let x be Element of Elements N; :: thesis: ( Elements N <> {} implies ( x in Input (N,X) iff ( ( x in X & x in the carrier of N ) or ex y being Element of Elements N st
( y in X & y in the carrier' of N & pre N,y,x ) ) ) )
A1: ( ( ( x in X & x in the carrier of N ) or ex y being Element of Elements N st
( y in X & y in the carrier' of N & pre N,y,x ) ) implies x in Input (N,X) )

proof

A2: ( ex y being Element of Elements N st
( y in X & y in the carrier' of N & pre N,y,x ) implies x in Input (N,X) )

proof

given y being Element of Elements N such that A3: y in X and
A4: y in the carrier' of N and
A5: pre N,y,x ; :: thesis: x in Input (N,X)
[x,y] in Flow N by A5;
then x in Pre (N,y) by A4, Def6;
then A6: x in enter (N,y) by A4, Def8;
enter (N,y) in Entr (N,X) by A3, Th21;
hence x in Input (N,X) by A6, TARSKI:def 4; :: thesis: verum

end;

A7: ( x in X & x in the carrier of N implies x in Input (N,X) )

proof

assume that
A8: x in X and
A9: x in the carrier of N ; :: thesis: x in Input (N,X)
( enter (N,x) = {x} & {x} c= Elements N ) by A9, Def8, ZFMISC_1:31;
then A10: {x} in Entr (N,X) by A8, Def13;
x in {x} by TARSKI:def 1;
hence x in Input (N,X) by A10, TARSKI:def 4; :: thesis: verum

end;

assume ( ( x in X & x in the carrier of N ) or ex y being Element of Elements N st
( y in X & y in the carrier' of N & pre N,y,x ) ) ; :: thesis: x in Input (N,X)
hence x in Input (N,X) by A7, A2; :: thesis: verum

end;

assume A11: Elements N <> {} ; :: thesis: ( x in Input (N,X) iff ( ( x in X & x in the carrier of N ) or ex y being Element of Elements N st
( y in X & y in the carrier' of N & pre N,y,x ) ) )
( not x in Input (N,X) or ( x in X & x in the carrier of N ) or ex y being Element of Elements N st
( y in X & y in the carrier' of N & pre N,y,x ) )

proof

assume x in Input (N,X) ; :: thesis: ( ( x in X & x in the carrier of N ) or ex y being Element of Elements N st
( y in X & y in the carrier' of N & pre N,y,x ) )
then ex y being set st
( x in y & y in Entr (N,X) ) by TARSKI:def 4;
then consider y being set such that
A12: y in Entr (N,X) and
A13: x in y ;
consider z being Element of Elements N such that
A14: z in X and
A15: y = enter (N,z) by A12, Def13;
A16: ( z in the carrier' of N implies y = Pre (N,z) ) by A15, Def8;
A17: ( z in the carrier' of N implies ex y being Element of Elements N st
( y in X & y in the carrier' of N & pre N,y,x ) )

proof

assume A18: z in the carrier' of N ; :: thesis: ex y being Element of Elements N st
( y in X & y in the carrier' of N & pre N,y,x )
take z ; :: thesis: ( z in X & z in the carrier' of N & pre N,z,x )
[x,z] in Flow N by A13, A16, A18, Def6;
hence ( z in X & z in the carrier' of N & pre N,z,x ) by A14, A18; :: thesis: verum

end;

A19: ( z in the carrier of N or z in the carrier' of N ) by A11, XBOOLE_0:def 3;
( z in the carrier of N implies y = {z} ) by A15, Def8;
hence ( ( x in X & x in the carrier of N ) or ex y being Element of Elements N st
( y in X & y in the carrier' of N & pre N,y,x ) ) by A13, A14, A19, A17, TARSKI:def 1; :: thesis: verum

end;

hence ( x in Input (N,X) iff ( ( x in X & x in the carrier of N ) or ex y being Element of Elements N st
( y in X & y in the carrier' of N & pre N,y,x ) ) ) by A1; :: thesis: verum