let M be non empty MetrSpace; :: thesis: for P being non empty Subset of (TopSpaceMetr M)
for x, y being Point of M st y in P holds
(dist_min P) . x <= dist (x,y)
let P be non empty Subset of (TopSpaceMetr M); :: thesis: for x, y being Point of M st y in P holds
(dist_min P) . x <= dist (x,y)
let x, y be Point of M; :: thesis: ( y in P implies (dist_min P) . x <= dist (x,y) )
A1: ( dom (dist x) = the carrier of (TopSpaceMetr M) & dist (x,y) = (dist x) . y ) by FUNCT_2:def 1, WEIERSTR:def 4;
consider X being non empty Subset of REAL such that
A2: X = (dist x) .: P and
A3: lower_bound ((dist x) .: P) = lower_bound X by Th10;
assume y in P ; :: thesis: (dist_min P) . x <= dist (x,y)
then A4: dist (x,y) in X by A2, A1, FUNCT_1:def 6;
( (dist_min P) . x = lower_bound X & X is bounded_below ) by A2, A3, Th12, WEIERSTR:def 6;
hence (dist_min P) . x <= dist (x,y) by A4, SEQ_4:def 2; :: thesis: verum