
theorem Th31:
  for V,W be non empty ModuleStr over F_Complex, f,g be Form of V,
  W holds (f+g)*'=f*'+ g*'
proof
  let V,W be non empty ModuleStr over F_Complex, f,g be Form of V,W;
  now
    let v be Vector of V,w be Vector of W;
    thus (f+g)*'.(v,w) = ((f+g).(v,w))*' by Def8
      .= (f.(v,w) +g.(v,w))*' by BILINEAR:def 2
      .= (f.(v,w))*'+(g.(v,w))*' by COMPLFLD:51
      .= f*'.(v,w) + (g.(v,w))*' by Def8
      .= f*'.(v,w) + g*'.(v,w) by Def8
      .= (f*'+ g*').(v,w) by BILINEAR:def 2;
  end;
  hence thesis;
end;
