reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem
  for E,F,G be RealLinearSpace,
          L be Function of [:E,F:],G holds
    L is Bilinear
  iff
    ( ( for x1,x2 be Point of E, y be Point of F
        holds L.(x1+x2,y) = L.(x1,y) + L.(x2,y) )
      &
      ( for x be Point of E,
            y be Point of F,
            a be Real
        holds L.(a*x,y) = a * L.(x,y) )
      &
      ( for x be Point of E, y1,y2 be Point of F
        holds L.(x,y1+y2) = L.(x,y1) + L.(x,y2) )
      &
      ( for x be Point of E,
            y be Point of F,
            a be Real
        holds L. (x, a*y) = a * L.(x,y) ) ) by LM6;
