
theorem Th26:
  for K be non empty addLoopStr for V,W be non empty ModuleStr
  over K for v,u be Vector of V, w be Vector of W, f be Form of V,W st f is
  additiveSAF holds f.(v+u,w) = f.(v,w) + f.(u,w)
proof
  let K be non empty addLoopStr;
  let V,W be non empty ModuleStr over K;
  let v,w be Vector of V, y be Vector of W, f be Form of V,W;
  set F=FunctionalSAF(f,y);
  assume f is additiveSAF;
  then
A1: F is additive;
  thus f.(v+w,y) = F.(v+w) by Th9
    .= F.v+F.w by A1
    .= f.(v,y) + F.w by Th9
    .= f.(v,y) + f.(w,y) by Th9;
end;
