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
  f|X is bounded iff PreNorms f is bounded_above
proof
  now
    reconsider K = upper_bound PreNorms f as Real;
    assume
A1: PreNorms f is bounded_above;
    take K;
    now
      let t be object;
      assume t in X /\ dom f;
      then |.f.t.| in PreNorms f;
      hence |.f.t.| <= K by A1,SEQ_4:def 1;
    end;
    hence f|X is bounded by RFUNCT_1:73;
  end;
  hence thesis by Th17;
end;
