
theorem TH11:
  for f,g,h be VECTOR of R_Algebra_of_Big_Oh_poly
  for f9,g9,h9 be Function of NAT,REAL
  st f9=f & g9=g & h9=h holds
  (h = f+g iff for x be Nat holds h9.x = f9.x + g9.x)
  proof
    let f,g,h be VECTOR of R_Algebra_of_Big_Oh_poly;
    reconsider f1=f,g1=g,h1=h as VECTOR of RAlgebra NAT by LM12;
    let f9,g9,h9 be Function of NAT,REAL such that
    A2: f9=f & g9=g & h9=h;
    A3: now
    assume
    A4: h = f+g;
    let x be Nat;
    LXN:x in NAT by ORDINAL1:def 12;
    h1=f1+g1 by A4,LM16;
    hence h9.x=f9.x+g9.x by A2,FUNCSDOM:1,LXN;
  end;
  now
    assume LAS:for x be Nat holds h9.x=f9.x+g9.x;
    for x be Element of NAT holds h9.x=f9.x+g9.x by LAS;
    then h1=f1+g1 by A2,FUNCSDOM:1;
    hence h =f+g by LM16;
  end;
  hence thesis by A3;
end;
