reserve a,b,c,x,y,z for object,X,Y,Z for set,
  n for Nat,
  i,j for Integer,
  r,r1,r2,r3,s for Real,
  c1,c2 for Complex,
  p for Point of TOP-REAL n;

theorem Th53:
  p <> 0.TOP-REAL n implies |. p (/) |. p .| .| = 1
  proof
A1: |.p.|^2 = Sum sqr p by TOPREAL9:5;
    assume p <> 0.TOP-REAL n;
    then |.p.| <> 0 by EUCLID_2:42;
    then Sum (sqr p (/) Sum sqr p) = 1 by A1,Th19;
    hence thesis by A1,Th18,SQUARE_1:18;
  end;
