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 6;
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 Th12;
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 12;
( (dist_min P) . x = lower_bound X & X is bounded_below ) by A2, A3, Th14, WEIERSTR:def 8;
hence (dist_min P) . x <= dist x,y by A4, SEQ_4:def 5; :: thesis: verum