theorem Th3:
  r(#)L is LinearFunc
proof
  consider g such that
A1: for p holds L.p = g*p by Def3;
A2: L is total by Def3;
  now
    let p;
     reconsider pp = p as Element of REAL by XREAL_0:def 1;
    thus (r(#)L).p = r*L.pp by A2,RFUNCT_1:57
      .= r*(g*p) by A1
      .= r*g*p;
  end;
  hence thesis by A2,Def3;
end;
