reserve X for non empty set;
reserve x for Element of X;
reserve d1,d2 for Element of X;
reserve A for BinOp of X;
reserve M for Function of [:X,X:],X;
reserve V for Ring;
reserve V1 for Subset of V;
reserve V for Algebra;
reserve V1 for Subset of V;
reserve MR for Function of [:REAL,X:],X;
reserve a for Real;
reserve F,G,H for VECTOR of R_Algebra_of_BoundedFunctions X;
reserve f,g,h for Function of X,REAL;

theorem Th17:
  f|X is bounded implies PreNorms f is bounded_above
proof
  assume f|X is bounded;
  then consider K be Real such that
A1: for x be object st x in X /\ dom f holds |.f.x.| <= K by RFUNCT_1:73;
A2: X /\ dom f = X /\ X by FUNCT_2:def 1;
  take K;
    let r be ExtReal;
    assume r in PreNorms f;
    then ex t be Element of X st r=|.f.t.|;
    hence r <=K by A1,A2;
end;
