
theorem Th41:
  for X be non empty set, F be Functional_Sequence of X,ExtREAL
    st F is additive holds -F is additive
proof
    let X be non empty set, F be Functional_Sequence of X,ExtREAL;
    assume A1: F is additive;
    now let n,m be Nat;
     assume n <> m;
     let x be set;
     assume x in dom((-F).n) /\ dom((-F).m); then
     x in dom(-(F.n)) /\ dom((-F).m) by Th37; then
A3:  x in dom(-(F.n)) /\ dom(-(F.m)) by Th37; then
     x in dom(-(F.n)) & x in dom(-(F.m)) by XBOOLE_0:def 4; then
     (-(F.n)).x = -((F.n).x) & (-(F.m)).x = -((F.m).x)
       by MESFUNC1:def 7; then
A4:  ((-F).n).x = -((F.n).x) & ((-F).m).x = -((F.m).x) by Th37;
     x in dom(F.n) /\ dom(-(F.m)) by A3,MESFUNC1:def 7; then
     x in dom(F.n) /\ dom(F.m) by MESFUNC1:def 7;
     hence ((-F).n).x <> +infty or ((-F).m).x <> -infty
       by A4,XXREAL_3:6,A1,MESFUNC9:def 5;
    end;
    hence -F is additive by MESFUNC9:def 5;
end;
