let X be non empty set ; :: thesis: for F being Functional_Sequence of X,ExtREAL
for n, m being Nat
for z being set st z in dom ((Partial_Sums F) . n) & m <= n holds
z in dom ((Partial_Sums F) . m)
let F be Functional_Sequence of X,ExtREAL ; :: thesis: for n, m being Nat
for z being set st z in dom ((Partial_Sums F) . n) & m <= n holds
z in dom ((Partial_Sums F) . m)
let n, m be Nat; :: thesis: for z being set st z in dom ((Partial_Sums F) . n) & m <= n holds
z in dom ((Partial_Sums F) . m)
let z be set ; :: thesis: ( z in dom ((Partial_Sums F) . n) & m <= n implies z in dom ((Partial_Sums F) . m) )
set PF = Partial_Sums F;
assume A1: ( z in dom ((Partial_Sums F) . n) & m <= n ) ; :: thesis: z in dom ((Partial_Sums F) . m)
assume A2: not z in dom ((Partial_Sums F) . m) ; :: thesis: contradiction
defpred S₁[ Nat] means ( m <= $1 & $1 <= n implies not z in dom ((Partial_Sums F) . $1) );
A3: S₁[ 0 ] by A2;
A4: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A5: S₁[k] ; :: thesis: S₁[k + 1]
assume A6: ( m <= k + 1 & k + 1 <= n ) ; :: thesis: not z in dom ((Partial_Sums F) . (k + 1))

per cases ( m <= k or m = k + 1 ) by A6, NAT_1:8;

suppose m <= k ; :: thesis: not z in dom ((Partial_Sums F) . (k + 1))

then A7: not z in (dom ((Partial_Sums F) . k)) /\ (dom (F . (k + 1))) by A5, A6, NAT_1:13, XBOOLE_0:def 4;
(Partial_Sums F) . (k + 1) = ((Partial_Sums F) . k) + (F . (k + 1)) by Def0;
then dom ((Partial_Sums F) . (k + 1)) = ((dom ((Partial_Sums F) . k)) /\ (dom (F . (k + 1)))) \ (((((Partial_Sums F) . k) " {-infty }) /\ ((F . (k + 1)) " {+infty })) \/ ((((Partial_Sums F) . k) " {+infty }) /\ ((F . (k + 1)) " {-infty }))) by MESFUNC1:def 3;
hence not z in dom ((Partial_Sums F) . (k + 1)) by A7, XBOOLE_0:def 5; :: thesis: verum

end;

suppose m = k + 1 ; :: thesis: not z in dom ((Partial_Sums F) . (k + 1))

hence not z in dom ((Partial_Sums F) . (k + 1)) by A2; :: thesis: verum

end;

end;

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A3, A4);
hence contradiction by A1; :: thesis: verum