
theorem Th18:
  for X being non empty set, Y being non empty Subset of ExtREAL,
  F being Function of X,Y holds rng(- F) = - rng F
proof
  let X be non empty set, Y be non empty Subset of ExtREAL, F be Function of X
  ,Y;
  thus rng(- F) c= - rng F
  proof
    let y be object;
A1: dom F = X by FUNCT_2:def 1;
    assume
A2: y in rng(- F);
    then reconsider y as R_eal;
    dom(- F) = X by FUNCT_2:def 1;
    then consider a being object such that
A3: a in X and
A4: y = (- F).a by A2,FUNCT_1:def 3;
    reconsider a as Element of X by A3;
    y = - F.a by A4,Def4;
    then - y in rng F by A1,FUNCT_1:def 3;
    then - (- y) in - rng F;
    hence thesis;
  end;
    let y be object;
    assume
A5: y in - rng F;
    then reconsider y as R_eal;
A6: - y in - (- rng F) by A5;
    dom F = X by FUNCT_2:def 1;
    then consider a being object such that
A7: a in X and
A8: - y = F.a by A6,FUNCT_1:def 3;
    reconsider a as Element of X by A7;
    y = - F.a by A8; then
A9: y = (- F).a by Def4;
    dom (- F) = X by FUNCT_2:def 1;
    hence thesis by A9,FUNCT_1:def 3;
end;
