reserve T, T1 for non empty TopSpace;
reserve F,G,H for Subset-Family of T,
  A,B,C,D for Subset of T,
  O,U for open Subset of T,
  p,q for Point of T,
  x,y,X for set;
reserve Un for FamilySequence of T,
  r,r1,r2 for Real,
  n for Element of NAT;
reserve m for Function of [:the carrier of T,the carrier of T:],REAL;

theorem
  for r,m st r>0 & m is_metric_of (the carrier of T) holds min(r,m)
  is_metric_of (the carrier of T)
proof
  let r,m such that
A1: r>0 and
A2: m is_metric_of (the carrier of T);
  let a,b,c be Element of T;
  for a,b,c be Element of T holds m.(a,a)=0& m.(a,b)=m.(b,a)& m.(a,c)<=m.(
  a,b)+m.(b,c) by A2;
  then m is_a_pseudometric_of (the carrier of T) by Lm8;
  then
A3: min(r,m) is_a_pseudometric_of (the carrier of T) by A1,Th30;
  min(r,m).(a,b)=0 implies a=b
  proof
    assume min(r,m).(a,b)=0;
    then min(r,m.(a,b))=0 by Lm9;
    then m.(a,b)=0 by A1,XXREAL_0:def 9;
    hence thesis by A2;
  end;
  hence min(r,m).(a,b)=0 iff a=b by A3,Lm8;
  thus thesis by A3,Lm8;
end;
