reserve
  x for object, X for set,
  i, n, m for Nat,
  r, s for Real,
  c, c1, c2, d for Complex,
  f, g for complex-valued Function,
  g1 for n-element complex-valued FinSequence,
  f1 for n-element real-valued FinSequence,
  T for non empty TopSpace,
  p for Element of TOP-REAL n;

theorem Th49:
  for f being Function of T,TOP-REAL n, g being Function of T,R^1 holds
  f<+>g = f<++>incl(g,n)
  proof
    let f be Function of T,TOP-REAL n;
    let g be Function of T,R^1;
    set I = incl(g,n);
    reconsider h = f<+>g as Function of T,TOP-REAL n by Th43;
    reconsider G = f<++>I as Function of T,TOP-REAL n by Th39;
    h = G
    proof
      let t be Point of T;
A1:   dom h = the carrier of T by FUNCT_2:def 1;
A2:   f.t + I.t = f.t + g.t
      proof
A3:     dom (f.t) = Seg n & dom (I.t) = Seg n by FINSEQ_1:89;
A4:     dom (f.t + I.t) = dom (f.t) /\ dom (I.t) by VALUED_1:def 1
        .= Seg n by A3;
A5:    dom (f.t + g.t) = dom (f.t) by VALUED_1:def 2;
        thus dom (f.t + I.t) = dom (f.t + g.t) by A4,FINSEQ_1:89;
        let x be object;
        assume
A6:     x in dom (f.t + I.t);
        hence (f.t + I.t).x = f.t.x + I.t.x by VALUED_1:def 1
        .= f.t.x + g.t by A4,A6,Th47
        .= (f.t + g.t).x by A3,A4,A6,A5,VALUED_1:def 2;
      end;
      dom G = the carrier of T by FUNCT_2:def 1;
      hence G.t = f.t + I.t by VALUED_2:def 45
      .= h.t by A1,A2,VALUED_2:def 41;
    end;
    hence thesis;
  end;
