
theorem Th31:
  for K be non empty multMagma for V,W be non empty ModuleStr over
K for v be Vector of V, w be Vector of W, a be Element of K for f be Form of V,
  W st f is homogeneousSAF holds f.(a*v,w) = a*f.(v,w)
proof
  let K be non empty multMagma;
  let V,W be non empty ModuleStr over K;
  let v be Vector of V, y be Vector of W, r be Element of K, f be Form of V,W;
  set F=FunctionalSAF(f,y);
  assume f is homogeneousSAF;
  then
A1: F is homogeneous;
  thus f.(r*v,y) = F.(r*v) by Th9
    .= r*F.v by A1
    .= r*f.(v,y) by Th9;
end;
