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 Th4:
  (FuncAdd(X,Y)).(f,(FuncAdd(X,Y)).(g,h)) = (FuncAdd(X,Y)).((
  FuncAdd(X,Y)).(f,g),h)
proof
  now
    let x be Element of X;
    thus ((FuncAdd(X,Y)).(f,(FuncAdd(X,Y)).(g,h))).x = f.x + ((FuncAdd(X,Y)).(
    g,h)).x by LOPBAN_1:1
      .= f.x + (g.x + h.x) by LOPBAN_1:1
      .= (f.x + g.x) + h.x by RLVECT_1:def 3
      .= ((FuncAdd(X,Y)).(f,g)).x + h.x by LOPBAN_1:1
      .= ((FuncAdd(X,Y)).((FuncAdd(X,Y)).(f,g),h)).x by LOPBAN_1:1;
  end;
  hence thesis by FUNCT_2:63;
end;
