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-directed & c is_a_cluster_point_of N holds
c = sup 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-directed & c is_a_cluster_point_of N holds
c = sup N
let N be net of ; :: thesis: ( ( for x being Element of holds x "/\" is continuous ) & N is eventually-directed & c is_a_cluster_point_of N implies c = sup N )
assume A1: ( ( for x being Element of holds x "/\" is continuous ) & N is eventually-directed & c is_a_cluster_point_of N ) ; :: thesis: c = sup 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, Lm5, Lm6;
hence c = sup (rng the mapping of N) by YELLOW_0:30
.= sup N by YELLOW_2:def 5 ;
:: thesis: verum