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

theorem Th10:
  for X be non empty closed_interval Subset of REAL, Y be RealNormSpace,
  f,g,h be VECTOR of R_VectorSpace_of_ContinuousFunctions(X,Y),
  f9,g9,h9 be continuous PartFunc of REAL,Y
     st f9=f & g9=g & h9=h & dom f9=X & dom g9=X & dom h9=X
         holds (h = f+g iff for x
  be Element of X holds h9/.x = f9/.x + g9/.x )
proof
  let X be non empty closed_interval Subset of REAL;
  let Y be RealNormSpace;
  let f,g,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;
  reconsider g1=g as VECTOR
    of R_VectorSpace_of_BoundedFunctions(X,Y) by A1,RLSUB_1:10;
  let f9,g9,h9 be continuous PartFunc of REAL,Y such that
A2: f9=f & g9=g & h9=h & dom f9=X & dom g9=X & dom h9=X;
  reconsider f90=f1, g90=g1, h90=h1 as bounded Function of X,Y
    by RSSPACE4:def 5;
A3: now
    assume
A4: h = f+g;
    let x be Element of X;
    A5: h1=f1+g1 by A1,A4,RLSUB_1:13;
     thus h9/.x =h90.x by A2,PARTFUN1:def 6
               .=f90/.x+g90/.x by A5,RSSPACE4:8
               .=f9/.x+g9/.x by A2;
  end;
  now
    assume A6: for x be Element of X holds h9/.x=f9/.x+g9/.x;
    now let x be Element of X;
       thus h90.x = h9/.x by A2,PARTFUN1:def 6
            .= f90/.x+g90/.x by A2,A6
            .= f90.x+g90.x;
    end;
    then h1=f1+g1 by RSSPACE4:8;
    hence h =f+g by A1,RLSUB_1:13;
  end;
  hence thesis by A3;
end;
