
theorem Th18:
  for V be non empty ModuleStr over F_Complex, f,g be Functional
  of V holds (f+g)*'=f*'+ g*'
proof
  let V be non empty ModuleStr over F_Complex, f,g be Functional of V;
  now
    let v be Vector of V;
    thus (f+g)*'.v = ((f+g).v)*' by Def2
      .= (f.v +g.v)*' by HAHNBAN1:def 3
      .= (f.v)*'+(g.v)*' by COMPLFLD:51
      .= f*'.v + (g.v)*' by Def2
      .= f*'.v + g*'.v by Def2
      .= (f*'+ g*').v by HAHNBAN1:def 3;
  end;
  hence thesis by FUNCT_2:63;
end;
