theorem BLTh15:
  for V, W being non empty ModuleStr over INT.Ring, f being Form of V,W,
  a being Element of INT.Ring, v being Vector of V holds
  FunctionalFAF(a*f,v) = a*FunctionalFAF(f,v)
  proof
    let V, W be non empty ModuleStr over INT.Ring, f be Form of V,W,
        a be Element of INT.Ring,
    w be Vector of V;
    now
      let v be Vector of W;
      thus (FunctionalFAF(a*f,w)).v = (a*f).(w,v) by BLTh8
      .= a*f.(w,v) by BLDef3
      .= a*(FunctionalFAF(f,w)).v by BLTh8
      .= (a* FunctionalFAF(f,w)).v by HAHNBAN1:def 6;
    end;
    hence thesis by FUNCT_2:63;
  end;
