
theorem Th80:
  for A be non empty closed_interval Subset of REAL,
      u be Function
  holds
   ( u is Point of
       R_Normed_Algebra_of_ContinuousFunctions(ClstoCmp(A))
    iff
   dom u = A & u is continuous PartFunc of REAL,REAL )
proof
  let A be non empty closed_interval Subset of REAL,
      u be Function;
  consider a,b be Real such that
A1: a <= b & [.a,b.] = A
  & ClstoCmp(A) = Closed-Interval-TSpace (a,b) by Def7ClstoCmp;
  hereby
    assume u is Point of
      R_Normed_Algebra_of_ContinuousFunctions(ClstoCmp(A)); then
    u in ContinuousFunctions ClstoCmp(A); then
    consider w be continuous RealMap of ClstoCmp(A) such that
A2:   u = w;
B3: dom w = the carrier of ClstoCmp(A) by FUNCT_2:def 1; then
    dom w = A by Lm1; then
    dom w c= REAL & rng w c= REAL; then
    reconsider v=w as PartFunc of REAL,REAL by RELSET_1:4;
    v is continuous PartFunc of REAL,REAL by A1,Th80b;
    hence dom u = A & u is continuous PartFunc of REAL,REAL by A2,B3,Lm1;
  end;
  assume
A4: dom u = A & u is continuous PartFunc of REAL,REAL; then
A5: dom u = the carrier of ClstoCmp(A) by Lm1;
  rng u c= REAL by A4,RELAT_1:def 19; then
  reconsider v=u as RealMap of ClstoCmp(A) by A5,FUNCT_2:2;
  v is continuous by A1,A4,Th80b; then
  v in the carrier of R_Normed_Algebra_of_ContinuousFunctions(ClstoCmp(A));
  hence u is Point of
          R_Normed_Algebra_of_ContinuousFunctions(ClstoCmp(A));
end;
