reserve Z for RealNormSpace;
reserve a,b,c,d,e,r for Real;
reserve A,B for non empty closed_interval Subset of REAL;

theorem Th1919:
  for g be PartFunc of REAL,the carrier of Z
   st A c= dom g & g|A is bounded
   holds (||.g.||) |A is bounded
proof
   let g be PartFunc of REAL,the carrier of Z;
   assume A1: A c= dom g & g|A is bounded; then
A2: ||. (g|A) .|| = (||.g.||) |A by Th1918;
   reconsider h = g|A as Function of A,the carrier of Z by A1,Lm3;
   h is bounded by A1;
   hence thesis by A2,Th1914;
end;
