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
  |(f1,(0.REAL n)+*(x,r))| = f1.x * r
  proof
A1: mlt(f1,(0.REAL n)+*(x,r)) = (0.REAL n)+*(x,f1.x*r) by Th15;
A2: dom f1 = Seg n by FINSEQ_1:89;
A3: n in NAT by ORDINAL1:def 12;
    per cases;
    suppose x in dom f1;
      hence thesis by A1,A2,A3,JORDAN2B:10;
    end;
    suppose not x in dom f1;
      then
A4:   f1.x = 0 by FUNCT_1:def 2;
      hence |(f1,(0.REAL n)+*(x,r))| = Sum 0.REAL n by A1,Th14
      .= f1.x * r by A4,A3,JORDAN2B:9;
    end;
  end;
