
theorem Th8:
  for X be non empty set for Y be RealNormSpace for f,g,h be
  VECTOR of R_VectorSpace_of_BoundedFunctions(X,Y) for f9,g9,h9 be bounded
  Function of X,the carrier of Y st f9=f & g9=g & h9=h holds (h = f+g iff for x
  be Element of X holds h9.x = f9.x + g9.x )
proof
  let X be non empty set;
  let Y be RealNormSpace;
  let f,g,h be VECTOR of R_VectorSpace_of_BoundedFunctions(X,Y);
A1: R_VectorSpace_of_BoundedFunctions(X,Y) is Subspace of RealVectSpace(X,Y)
  by Th6,RSSPACE:11;
  then reconsider f1=f as VECTOR of RealVectSpace(X,Y) by RLSUB_1:10;
  reconsider h1=h as VECTOR of RealVectSpace(X,Y) by A1,RLSUB_1:10;
  reconsider g1=g as VECTOR of RealVectSpace(X,Y) by A1,RLSUB_1:10;
  let f9,g9,h9 be bounded Function of X,the carrier of Y such that
A2: f9=f & g9=g & h9=h;
A3: now
    assume
A4: h = f+g;
    let x be Element of X;
    h1=f1+g1 by A1,A4,RLSUB_1:13;
    hence h9.x=f9.x+g9.x by A2,LOPBAN_1:11;
  end;
  now
    assume for x be Element of X holds h9.x=f9.x+g9.x;
    then h1=f1+g1 by A2,LOPBAN_1:11;
    hence h =f+g by A1,RLSUB_1:13;
  end;
  hence thesis by A3;
end;
