let C be non empty compact Subset of (TOP-REAL 2); :: thesis: S-min (L~ (SpStSeq C)) = SW-corner C
set X = L~ (SpStSeq C);
set S = S-most (L~ (SpStSeq C));
A1: S-most (L~ (SpStSeq C)) = LSeg ((SW-corner C),(SE-corner C)) by Th68;
A2: W-bound C <= E-bound C by Th21;
lower_bound (proj1 | (S-most (L~ (SpStSeq C)))) = lower_bound (rng (proj1 | (S-most (L~ (SpStSeq C))))) by RELSET_1:22
.= lower_bound (proj1 .: (S-most (L~ (SpStSeq C)))) by RELAT_1:115
.= lower_bound [.(W-bound C),(E-bound C).] by A1, Th73
.= W-bound C by A2, JORDAN5A:19 ;
hence S-min (L~ (SpStSeq C)) = SW-corner C by Th59; :: thesis: verum