
theorem Th27:
  for K be non empty addLoopStr for V,W be non empty ModuleStr
  over K for v be Vector of V, u,w be Vector of W, f be Form of V,W st f is
  additiveFAF holds f.(v,u+w) = f.(v,u) + f.(v,w)
proof
  let K be non empty addLoopStr;
  let V,W be non empty ModuleStr over K;
  let v be Vector of V, y,z be Vector of W, f be Form of V,W;
  set F=FunctionalFAF(f,v);
  assume f is additiveFAF;
  then
A1: F is additive;
  thus f.(v,y+z) = F.(y+z) by Th8
    .= F.y+F.z by A1
    .= f.(v,y) + F.z by Th8
    .= f.(v,y) + f.(v,z) by Th8;
end;
