
theorem Th15:
  for K be non empty doubleLoopStr for V,W be non empty ModuleStr
  over K for f be Form of V,W, a be Element of K, v be Vector of V holds
  FunctionalFAF(a*f,v) = a*FunctionalFAF(f,v)
proof
  let K be non empty doubleLoopStr, V,W be non empty ModuleStr over K, f be
  Form of V,W, a be Element of K, w be Vector of V;
  now
    let v be Vector of W;
    thus (FunctionalFAF(a*f,w)).v = (a*f).(w,v) by Th8
      .= a*f.(w,v) by Def3
      .= a*(FunctionalFAF(f,w)).v by Th8
      .= (a* FunctionalFAF(f,w)).v by HAHNBAN1:def 6;
  end;
  hence thesis by FUNCT_2:63;
end;
