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 Th15:
  for X,Y be ComplexLinearSpace, f,g,h be VECTOR of
  C_VectorSpace_of_LinearOperators(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 ComplexLinearSpace;
  let f,g,h be VECTOR of C_VectorSpace_of_LinearOperators(X,Y);
  reconsider f9=f,g9=g,h9=h as LinearOperator of X,Y by Def4;
A1: C_VectorSpace_of_LinearOperators(X,Y) is Subspace of ComplexVectSpace(
  the carrier of X,Y) by Th13,CSSPACE:11;
  then reconsider f1 = f as VECTOR of ComplexVectSpace(the carrier of X,Y) by
CLVECT_1:29;
  reconsider h1 = h as VECTOR of ComplexVectSpace(the carrier of X,Y) by A1,
CLVECT_1:29;
  reconsider g1 = g as VECTOR of ComplexVectSpace(the carrier of X,Y) by A1,
CLVECT_1:29;
A2: now
    assume
A3: h = f+g;
    let x be Element of X;
    h1 = f1 + g1 by A1,A3,CLVECT_1:32;
    hence h9.x=f9.x+g9.x by LOPBAN_1:1;
  end;
  now
    assume for x be Element of X holds h9.x=f9.x+g9.x;
    then h1 = f1 + g1 by LOPBAN_1:1;
    hence h =f+g by A1,CLVECT_1:32;
  end;
  hence thesis by A2;
end;
