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 Th23:
  (Mult_(BoundedFunctions X,RAlgebra X)).(1,F) = F
proof
  set X1 = BoundedFunctions X;
  reconsider f1 = F as Element of X1;
A1: [jj,f1] in [:REAL,X1:];
  thus (Mult_(BoundedFunctions X,RAlgebra X)).(1,F) = ((the Mult of RAlgebra X
  )| [:REAL,X1:]).(1,f1) by Def11
    .= (the Mult of RAlgebra X).(1,f1) by A1,FUNCT_1:49
    .= F by FUNCSDOM:12;
end;
