
theorem
for X be non empty set, F be FinSequence of Funcs(X,ExtREAL) st
 F is without_+infty-valued or F is without_-infty-valued holds
  for n,m be Nat st n in dom F & m in dom F holds dom(F/.n + F/.m) = X
proof
   let X be non empty set, F be FinSequence of Funcs(X,ExtREAL);
   assume A1: F is without_+infty-valued or F is without_-infty-valued;
   per cases by A1;
   suppose A2: F is without_+infty-valued;
     let n,m be Nat;
     assume n in dom F & m in dom F; then
     (F/.n)"{+infty} = {} & (F/.m)"{+infty} = {} by A2,Th53; then
A4:  (dom(F/.n) /\ dom(F/.m)) \ ((( (F/.n)"{-infty} /\ (F/.m)"{+infty} ) \/
           ( (F/.n)"{+infty} /\ (F/.m)"{-infty} ))) = dom(F/.n) /\ dom(F/.m);
     dom (F/.n) = X & dom (F/.m) = X by FUNCT_2:def 1;
     hence dom(F/.n + F/.m) = X by A4,MESFUNC1:def 3;
   end;
   suppose A5: F is without_-infty-valued;
    let n,m be Nat;
     assume n in dom F & m in dom F; then
     (F/.n)"{-infty} = {} & (F/.m)"{-infty} = {} by A5,Th54; then
A7:  (dom(F/.n) /\ dom(F/.m)) \ ((( (F/.n)"{-infty} /\ (F/.m)"{+infty} ) \/
           ( (F/.n)"{+infty} /\ (F/.m)"{-infty} ))) = dom(F/.n) /\ dom(F/.m);
     dom (F/.n) = X & dom (F/.m) = X by FUNCT_2:def 1;
     hence dom(F/.n + F/.m) = X by A7,MESFUNC1:def 3;
   end;
end;
