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

theorem Th27:
 for X be non empty closed_interval Subset of REAL,
     Y be RealNormSpace,
     K be Real,
     v1,v2 being Point of R_NormSpace_of_ContinuousFunctions(X,Y),
     g1,g2 be PartFunc of REAL,Y
  st g1=v1 & g2=v2 &
   for t be Real st t in X holds
           ||.g1/.t - g2/.t .|| <= K
  holds ||.v1-v2.|| <= K
proof
  let X be non empty closed_interval Subset of REAL,
      Y be RealNormSpace,
      K be Real,
      v1,v2 being Point of R_NormSpace_of_ContinuousFunctions(X,Y),
      g1,g2 be PartFunc of REAL,Y;
  assume A1:g1=v1 & g2=v2 &
    for t be Real st t in X holds ||.g1/.t - g2/.t .|| <= K;
  consider f1 be continuous PartFunc of REAL,Y such that
A2: v1=f1 & dom f1 = X by Def2;
  consider f2 be continuous PartFunc of REAL,Y such that
A3: v2=f2 & dom f2 = X by Def2;
  consider f12 be continuous PartFunc of REAL,Y such that
A4: v1-v2=f12 & dom f12 = X by Def2;
  for t be Real st t in X holds ||.f12/.t .|| <= K
  proof
    let t be Real;
    assume t in X;
    then reconsider t1=t as Element of X;
    f12/.t1 = g1/.t - g2/.t by A1,A2,A3,A4,Th20;
    hence ||.f12/.t .|| <= K by A1;
  end;
  hence thesis by Th25,A4;
end;
