let S be Hausdorff compact TopLattice; :: thesis: for c being Point of
for N being net of st ( for x being Element of holds x "/\" is continuous ) & N is eventually-filtered & c is_a_cluster_point_of N holds
c = inf N
let c be Point of ; :: thesis: for N being net of st ( for x being Element of holds x "/\" is continuous ) & N is eventually-filtered & c is_a_cluster_point_of N holds
c = inf N
let N be net of ; :: thesis: ( ( for x being Element of holds x "/\" is continuous ) & N is eventually-filtered & c is_a_cluster_point_of N implies c = inf N )
assume A1: ( ( for x being Element of holds x "/\" is continuous ) & N is eventually-filtered & c is_a_cluster_point_of N ) ; :: thesis: c = inf N
reconsider c' = c as Element of ;
( c' is_<=_than rng the mapping of N & ( for b being Element of st rng the mapping of N is_>=_than b holds
c' >= b ) ) by A1, Lm7, Lm8;
hence c = inf (rng the mapping of N) by YELLOW_0:31
.= inf N by YELLOW_2:def 6 ;
:: thesis: verum