
theorem Th34:
for F be FinSequence of ExtREAL, k be Nat
 holds (F is without-infty implies F|k is without-infty)
     & (F is without+infty implies F|k is without+infty)
proof
   let F be FinSequence of ExtREAL, k be Nat;
   hereby assume A1: F is without-infty;
    now assume -infty in rng(F|k); then
     consider i be Element of NAT such that
A2:   i in dom(F|k) & -infty = (F|k).i by PARTFUN1:3;
     dom(F|k) c= dom F by RELAT_1:60; then
     i in dom F & (F|k).i = F.i by A2,FUNCT_1:47; then
     -infty in rng F by A2,FUNCT_1:3;
     hence contradiction by A1,MESFUNC5:def 3;
    end;
    hence F|k is without-infty by MESFUNC5:def 3;
   end;
   assume A3: F is without+infty;
   now assume +infty in rng(F|k); then
    consider i be Element of NAT such that
A4:  i in dom(F|k) & +infty = (F|k).i by PARTFUN1:3;
    dom(F|k) c= dom F by RELAT_1:60; then
    i in dom F & (F|k).i = F.i by A4,FUNCT_1:47; then
    +infty in rng F by A4,FUNCT_1:3;
    hence contradiction by A3,MESFUNC5:def 4;
   end;
   hence F|k is without+infty by MESFUNC5:def 4;
end;
