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 Th1914:
  for f be Function of A,the carrier of Z holds
    f is bounded iff ||.f.|| is bounded
proof
   let f be Function of A,the carrier of Z;
   hereby assume A1: f is bounded;
    consider r be Real such that
A2:  for x be set st x in dom f holds ||. f/.x .|| <r by A1;
    now let x be set;
     assume A3: x in dom ||. f .||; then
     x in dom f by NORMSP_0:def 2; then
A4:  ||. f/.x .|| <r by A2;
     ||. f/.x .|| = ||. f .||.x by A3,NORMSP_0:def 2;
     hence |. ||. f .||.x .| <r by A4,ABSVALUE:def 1;
    end;
    hence ||.f.|| is bounded by COMSEQ_2:def 3;
   end;
   assume ||.f.|| is bounded; then
   consider r be Real such that
B2: for x be set st x in dom ||. f .||
     holds |. ||. f .||.x .| <r by COMSEQ_2:def 3;
   now let x be set;
    assume x in dom f; then
B3: x in dom ||. f .|| by NORMSP_0:def 2; then
B4: |. ||. f .||.x .| <r by B2;
    ||. f .||.x = ||. f/.x .|| by B3,NORMSP_0:def 2;
    hence ||. f/.x .|| <r by B4,ABSVALUE:def 1;
   end;
   hence f is bounded;
end;
