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;
reserve F,G,H for Point of R_Normed_Algebra_of_BoundedFunctions X;

theorem Th26:
  f=F & f|X is bounded implies |.f.x.| <= ||.F.||
proof
  assume that
A1: f=F and
A2: f|X is bounded;
A3: |.f.x.| in PreNorms f;
  PreNorms f is non empty bounded_above by A2,Th17;
  then |.f.x.| <= upper_bound PreNorms f by A3,SEQ_4:def 1;
  hence thesis by A1,A2,Th20;
end;
