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 )

proof

x in Elements N by A1;
then A3: for y being set 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;

( exit N,x = {x} or exit N,x = Post N,x ) by A1, Th25;
hence exit N,x c= Elements N by A2, Th18; :: thesis: verum