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;
