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 Th30:
  for r,m st r>0 & m is_a_pseudometric_of (the carrier of T) holds
  min(r,m) is_a_pseudometric_of (the carrier of T)
proof
  let r,m such that
A1: r>0 and
A2: m is_a_pseudometric_of (the carrier of T);
  now
    let a,b,c be Element of T;
    m.(a,a)=0 by A2,Th28;
    then min(r,m.(a,a))=0 by A1,XXREAL_0:def 9;
    hence min(r,m).(a,a)=0 by Lm9;
    thus min(r,m).(a,c) <= min(r,m).(a,b)+min(r,m).(c,b)
    proof
      now
        per cases;
        suppose
A3:       min(r,m).(a,b)+min(r,m).(c,b)>=r;
          min(r,m.(a,c))<=r by XXREAL_0:17;
          then min(r,m).(a,c)<=r by Lm9;
          hence thesis by A3,XXREAL_0:2;
        end;
        suppose
A4:       min(r,m).(a,b)+min(r,m).(c,b)<r;
          m.(c,b)>=0 by A2,Th29;
          then 0<=min(r,m.(c,b)) by A1,XXREAL_0:20;
          then
A5:       0<=min(r,m).(c,b) by Lm9;
          m.(a,b)>=0 by A2,Th29;
          then 0<=min(r,m.(a,b)) by A1,XXREAL_0:20;
          then
A6:       0<=min(r,m).(a,b) by Lm9;
          then min(r,m).(a,b)<r by A4,A5,Lm1;
          then min(r,m.(a,b))<r by Lm9;
          then min(r,m.(a,b))=m.(a,b) by XXREAL_0:def 9;
          then
A7:       min(r,m).(a,b)=m.(a,b) by Lm9;
          min(r,m).(c,b)<r by A4,A6,A5,Lm1;
          then min(r,m.(c,b))<r by Lm9;
          then min(r,m.(c,b))=m.(c,b) by XXREAL_0:def 9;
          then
A8:       min(r,m).(c,b)=m.(c,b) by Lm9;
          min(r,m.(a,c))<=m.(a,c) & m.(a,c)<=m.(a,b)+m.(c,b) by A2,Th28,
XXREAL_0:17;
          then min(r,m.(a,c))<=m.(a,b)+m.(c,b) by XXREAL_0:2;
          hence thesis by A7,A8,Lm9;
        end;
      end;
      hence thesis;
    end;
  end;
  hence thesis by Th28;
end;
