for L being non empty lower-bounded Poset
for R being extra-order Relation of L
for C being non empty strict_chain of R st C is sup-closed & ( for c being Element of L st c in C holds
ex_sup_of SetBelow (R,C,c),L ) & R satisfies_SIC_on C holds
for c being Element of L st c in C holds
c = sup (SetBelow (R,C,c))