reserve X for set;
reserve n,i for Element of NAT;
reserve a,b,c,d,e,r,x0 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve f,g,h for PartFunc of REAL,REAL n;
reserve E for Element of REAL n;

theorem Th19:
  for n be non zero Element of NAT, h be PartFunc of REAL,REAL n
   st A c= dom h & h|A is bounded
   holds (|.h.|) |A is bounded
   proof
     let n be non zero Element of NAT,
     h be PartFunc of REAL,REAL n;
     assume A1: A c= dom h & h|A is bounded;
A2:  |. (h|A) .| = (|.h.|) |A by Th18,A1;
     reconsider g = h|A as Function of A,REAL n by A1,Lm9;
     g is bounded by A1;
     then |. g .| is bounded by Th14;
     hence thesis by A2,Th17;
   end;
