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 Th13:
  x in Seg n implies |.(0*n)+*(x,r).| = |.r.|
  proof
    set f = (0*n)+*(x,r);
A1: n in NAT by ORDINAL1:def 12;
    assume
A2: x in Seg n;
    f^2 = (0*n)+*(x,r^2) by Th12;
    then Sum (f^2) = r^2 by A2,A1,JORDAN2B:10;
    hence thesis by COMPLEX1:72;
  end;
