reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;

theorem
  |. S .|^2 <= 1 implies 0 <= delta(Sum sqr (T - S),b,Sum sqr S - 1)
  proof
    assume |.S.|^2 <= 1;
    then |.S.|^2 - 1 <= |.S.|^2 - |.S.|^2 by XREAL_1:10; then
A1: Sum sqr S - 1 <= 0 by TOPREAL9:5;
    0 <= |.T-S.|^2;
    then 0 <= Sum sqr (T - S) by TOPREAL9:5;
    hence thesis by A1,Lem08;
  end;
