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 Th28:
  F = 0.R_Normed_Algebra_of_BoundedFunctions X implies 0 = ||.F.||
proof
  set z = X --> In(0,REAL);
  reconsider z as Function of X, REAL;
  F in BoundedFunctions X;
  then consider g be Function of X,REAL such that
A1: g=F and
A2: g|X is bounded;
A3: PreNorms g is non empty bounded_above by A2,Th17;
  consider r0 be object such that
A4: r0 in PreNorms g by XBOOLE_0:def 1;
  reconsider r0 as Real by A4;
A5: (for s be Real st s in PreNorms g holds s <= 0) implies upper_bound
  PreNorms g <= 0 by SEQ_4:45;
  assume
A6: F = 0.R_Normed_Algebra_of_BoundedFunctions X;
A7: now
    let r be Real;
    assume r in PreNorms g;
    then consider t be Element of X such that
A8: r=|.g.t.|;
    z=g by A6,A1,Th25;
    then |.g.t.| = |.0 .|;
    hence 0 <= r & r <=0 by A8,ABSVALUE:2;
  end;
  then 0 <= r0 by A4;
  then upper_bound PreNorms g = 0 by A7,A3,A4,A5,SEQ_4:def 1;
  hence thesis by A1,A2,Th20;
end;
