let N be Pnet; :: thesis: for x being Element of Elements N st Elements N <> {} holds

exit (N,x) c= Elements N

let x be Element of Elements N; :: thesis: ( Elements N <> {} implies exit (N,x) c= Elements N )

assume A1: Elements N <> {} ; :: thesis: exit (N,x) c= Elements N

A2: ( exit (N,x) = {x} implies exit (N,x) c= Elements N )

hence exit (N,x) c= Elements N by A2, Th11; :: thesis: verum

exit (N,x) c= Elements N

let x be Element of Elements N; :: thesis: ( Elements N <> {} implies exit (N,x) c= Elements N )

assume A1: Elements N <> {} ; :: thesis: exit (N,x) c= Elements N

A2: ( exit (N,x) = {x} implies exit (N,x) c= Elements N )

proof

( exit (N,x) = {x} or exit (N,x) = Post (N,x) )
by A1, Th18;
x in Elements N
by A1;

then A3: for y being object st y in {x} holds

y in Elements N by TARSKI:def 1;

assume exit (N,x) = {x} ; :: thesis: exit (N,x) c= Elements N

hence exit (N,x) c= Elements N by A3, TARSKI:def 3; :: thesis: verum

end;then A3: for y being object st y in {x} holds

y in Elements N by TARSKI:def 1;

assume exit (N,x) = {x} ; :: thesis: exit (N,x) c= Elements N

hence exit (N,x) c= Elements N by A3, TARSKI:def 3; :: thesis: verum

hence exit (N,x) c= Elements N by A2, Th11; :: thesis: verum