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 Th14:
  (0.REAL n)+*(x,0) = 0.REAL n
  proof
    set p = (0.REAL n)+*(x,0);
A1: dom p = Seg n by FINSEQ_1:89;
A2: dom ((0.REAL n)+*(x,0)) = dom (0.REAL n) by FUNCT_7:30;
A3: dom (0.REAL n) = Seg n;
    now
      let z be object;
      assume
A4:   z in dom p;
      per cases;
      suppose z = x;
        hence p.z = 0 by A1,A3,A4,FUNCT_7:31
        .= (0.REAL n).z;
      end;
      suppose z <> x;
        hence p.z = (0.REAL n).z by FUNCT_7:32;
      end;
    end;
    hence thesis by A2;
  end;
