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 Th1935b:
  for X be set, f be PartFunc of REAL,the carrier of Z
    st f|X is bounded holds (-f)|X is bounded
proof
   let X be set, f be PartFunc of REAL,the carrier of Z;
   assume f|X is bounded; then
   consider s be Real such that
A2: for x be set st x in dom(f|X) holds ||. (f|X)/.x .|| <s;
   now let x be set;
    assume x in dom ((-f)|X); then
A4: x in dom (-(f|X)) by VFUNCT_1:29; then
    x in dom (f|X) by VFUNCT_1:def 5; then
A5: ||. (f|X)/.x .|| <s by A2;
    ((-f)|X)/.x = (-(f|X))/.x by VFUNCT_1:29
               .= -((f|X)/.x) by A4,VFUNCT_1:def 5;
    hence ||. (-f)|X/.x .|| <s by A5,NORMSP_1:2;
   end;
   hence (-f)|X is bounded;
end;
