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

theorem Th25:
 for X be non empty closed_interval Subset of REAL,
     Y be RealNormSpace,
     K be Real,
     v being Point of R_NormSpace_of_ContinuousFunctions(X,Y),
     g be PartFunc of REAL,Y st g=v &
     for t be Real st t in X holds ||.g/.t.|| <= K
 holds ||.v.|| <= K
proof
  let X be non empty closed_interval Subset of REAL,
      Y be RealNormSpace,
      K be Real,
      v be Point of R_NormSpace_of_ContinuousFunctions(X,Y),
      g be PartFunc of REAL,Y;
  assume A1:g=v & for t be Real st t in X holds ||.g/.t.|| <= K;
  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: for t be Element of X holds ||.g1.t.|| <= K
  proof
    let t be Element of X;
    reconsider t1=t as Real;
    ||.g/.t1.|| <= K by A1;
    hence ||.g1.t.|| <= K by A2,A1,PARTFUN1:def 6;
  end;
A4: for x be Element of REAL st x in PreNorms(g1) holds x <= K
  proof
    let x be Element of REAL;
    assume x in PreNorms(g1); then
    consider t be Element of X such that
  A5: x= ||.g1.t.||;
    thus thesis by A3,A5;
  end;
  upper_bound PreNorms(g1) <= K
    proof
      assume A6: not upper_bound PreNorms(g1) <= K;
      reconsider s = upper_bound PreNorms(g1) - K as Real;
    A7: 0 < s by A6,XREAL_1:50;
      PreNorms(g1) is non empty bounded_above by RSSPACE4:11; then
      consider r be Real such that
    A8: r in PreNorms(g1) & (upper_bound PreNorms(g1))-s<r by A7,SEQ_4:def 1;
      thus contradiction by A8,A4;
    end;
  then ||.v1.|| <= K by A1,RSSPACE4:14;
  hence thesis by FUNCT_1:49;
end;
