reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;

theorem Th24:
 for X be non empty closed_interval Subset of REAL,
     Y be RealNormSpace,
     v be Point of R_NormSpace_of_ContinuousFunctions(X,Y),
     g be PartFunc of REAL,Y st g=v
  holds for t be Real st t in X holds ||.g/.t.|| <= ||.v.||
proof
  let X be non empty closed_interval Subset of REAL,
      Y be RealNormSpace,
      v be Point of R_NormSpace_of_ContinuousFunctions(X,Y),
      g be PartFunc of REAL,Y;
  assume A1:g=v;
  consider f be continuous PartFunc of REAL,Y such that
A2: v=f & dom f = X by Def2;
  reconsider g1=g as bounded Function of X,Y by A1,Th9,A2;
  reconsider v1=v as VECTOR of R_NormSpace_of_BoundedFunctions(X,Y)
         by TARSKI:def 3;
A3: ||.v1.|| = ||.v.|| by FUNCT_1:49;
  let t be Real;
  assume t in X; then
  reconsider t1=t as Element of X;
  ||.g1.t1.|| <= ||.v1.|| by A1,RSSPACE4:16;
  hence thesis by A2,A1,PARTFUN1:def 6,A3;
end;
