reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;

theorem Th11:
  for X be non empty closed_interval Subset of REAL, Y be RealNormSpace,
   f,h be VECTOR of R_VectorSpace_of_ContinuousFunctions(X,Y),
   f9,h9 be continuous PartFunc of REAL,Y
      st f9=f & h9=h & dom f9=X & dom h9=X
      holds h = a*f iff for x be Element of X holds h9/.x = a*f9/.x
proof
  let X be non empty closed_interval Subset of REAL;
  let Y be RealNormSpace;
  let f,h be VECTOR
       of R_VectorSpace_of_ContinuousFunctions(X,Y);
A1: R_VectorSpace_of_ContinuousFunctions(X,Y)
      is Subspace of R_VectorSpace_of_BoundedFunctions(X,Y) by RSSPACE:11;
  then reconsider f1=f as VECTOR
    of R_VectorSpace_of_BoundedFunctions(X,Y) by RLSUB_1:10;
  reconsider h1=h as VECTOR
    of R_VectorSpace_of_BoundedFunctions(X,Y) by A1,RLSUB_1:10;
  let f9,h9 be continuous PartFunc of REAL,Y such that
A2: f9=f & h9=h & dom f9=X & dom h9=X;
  reconsider f90=f1 as bounded Function of X,Y by RSSPACE4:def 5;
  reconsider h90=h1 as bounded Function of X,Y by RSSPACE4:def 5;
A3: now
    assume
A4: h = a*f;
    let x be Element of X;
    A5: h1=a*f1 by A1,A4,RLSUB_1:14;
    thus h9/.x =h90.x by A2,PARTFUN1:def 6
              .=a*f90/.x by A5,RSSPACE4:9
              .=a*f9/.x by A2;
  end;
  now
    assume A6: for x be Element of X holds h9/.x=a*f9/.x;
    now let x be Element of X;
      thus h90.x = h9/.x by A2,PARTFUN1:def 6
            .= a*f90/.x by A2,A6
            .= a*f90.x;
    end;
    then h1=a*f1 by RSSPACE4:9;
    hence h =a*f by A1,RLSUB_1:14;
  end;
  hence thesis by A3;
end;
