
theorem Th27:
  for M being non empty MetrSpace, P being non empty Subset of
  TopSpaceMetr M holds (dist_min P) .: P = { 0 }
proof
  let M be non empty MetrSpace, P be non empty Subset of TopSpaceMetr M;
  consider x being object such that
A1: x in P by XBOOLE_0:def 1;
  thus (dist_min P) .: P c= { 0 }
  proof
    let y be object;
    assume y in (dist_min P) .: P;
    then
    ex x being object st x in dom dist_min P & x in P & y = ( dist_min P).x
by
FUNCT_1:def 6;
    then y = 0 by Th5;
    hence thesis by TARSKI:def 1;
  end;
  let y be object;
A2: dom dist_min P = the carrier of TopSpaceMetr M by FUNCT_2:def 1;
  assume y in { 0 };
  then y = 0 by TARSKI:def 1;
  then y = (dist_min P).x by A1,Th5;
  hence thesis by A1,A2,FUNCT_1:def 6;
end;
