reserve X for non empty set;
reserve Y for ComplexLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Complex;
reserve u,v,w for VECTOR of CLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem Th23:
  for X,Y be ComplexNormSpace, f,g,h be VECTOR of
C_VectorSpace_of_BoundedLinearOperators(X,Y) holds h = f+g iff for x be VECTOR
  of X holds h.x = f.x + g.x
proof
  let X,Y be ComplexNormSpace;
  let f,g,h be VECTOR of C_VectorSpace_of_BoundedLinearOperators(X,Y);
A1: C_VectorSpace_of_BoundedLinearOperators(X,Y) is Subspace of
  C_VectorSpace_of_LinearOperators(X,Y) by Th21,CSSPACE:11;
  then reconsider f1=f as VECTOR of C_VectorSpace_of_LinearOperators(X,Y) by
CLVECT_1:29;
  reconsider h1=h as VECTOR of C_VectorSpace_of_LinearOperators(X,Y) by A1,
CLVECT_1:29;
  reconsider g1=g as VECTOR of C_VectorSpace_of_LinearOperators(X,Y) by A1,
CLVECT_1:29;
  hereby
    assume
A2: h = f+g;
    let x be Element of X;
    h1=f1+g1 by A1,A2,CLVECT_1:32;
    hence h.x=f.x+g.x by Th15;
  end;
  assume for x be Element of X holds h.x=f.x+g.x;
  then h1=f1+g1 by Th15;
  hence thesis by A1,CLVECT_1:32;
end;
