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 Th20:
  f|X is bounded implies (BoundedFunctionsNorm X).f = upper_bound PreNorms f
proof
  reconsider f9=f as set;
  assume
A1: f|X is bounded;
  then f in BoundedFunctions X;
  then (BoundedFunctionsNorm X).f = upper_bound PreNorms(modetrans(f9,X))
   by Def17;
  hence thesis by A1,Th19;
end;
