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 Th1915:
  for f be PartFunc of REAL,the carrier of Z st
    f is bounded & A c= dom f holds f|A is bounded
proof
  let f be PartFunc of REAL,the carrier of Z;
  assume that
A1: f is bounded and
A2: A c= dom f;
  consider r be Real such that
A3: for x be set st x in dom f holds ||. f/.x .|| <r by A1;
  now let x be set;
    assume x in dom(f|A); then
    ||. f/.x .|| <r & x in dom f /\ A by A3,A2,RELAT_1:61;
    hence ||. (f|A)/.x .|| <r by PARTFUN2:16;
  end;
  hence f|A is bounded;
end;
