let X be non empty set ; :: thesis: for S being SigmaField of X
for F being Functional_Sequence of X,ExtREAL
for n being Nat st ( for m being Nat holds F . m is_simple_func_in S ) holds
( F is additive & (Partial_Sums F) . n is_simple_func_in S )
let S be SigmaField of X; :: thesis: for F being Functional_Sequence of X,ExtREAL
for n being Nat st ( for m being Nat holds F . m is_simple_func_in S ) holds
( F is additive & (Partial_Sums F) . n is_simple_func_in S )
let F be Functional_Sequence of X,ExtREAL; :: thesis: for n being Nat st ( for m being Nat holds F . m is_simple_func_in S ) holds
( F is additive & (Partial_Sums F) . n is_simple_func_in S )
let n be Nat; :: thesis: ( ( for m being Nat holds F . m is_simple_func_in S ) implies ( F is additive & (Partial_Sums F) . n is_simple_func_in S ) )
defpred S₁[ Nat] means (Partial_Sums F) . $1 is_simple_func_in S;
assume A1: for m being Nat holds F . m is_simple_func_in S ; :: thesis: ( F is additive & (Partial_Sums F) . n is_simple_func_in S )

hereby :: according to MESFUNC9:def 5 :: thesis: (Partial_Sums F) . n is_simple_func_in S

let n, m be Nat; :: thesis: ( n <> m implies for x being set holds
( not x in (dom (F . n)) /\ (dom (F . m)) or (F . n) . x <> +infty or (F . m) . x <> -infty ) )
assume n <> m ; :: thesis: for x being set holds
( not x in (dom (F . n)) /\ (dom (F . m)) or (F . n) . x <> +infty or (F . m) . x <> -infty )
F . n is_simple_func_in S by A1;
then F . n is without+infty by MESFUNC5:14;
hence for x being set holds
( not x in (dom (F . n)) /\ (dom (F . m)) or (F . n) . x <> +infty or (F . m) . x <> -infty ) ; :: thesis: verum

end;

A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
F . (k + 1) is_simple_func_in S by A1;
then ((Partial_Sums F) . k) + (F . (k + 1)) is_simple_func_in S by A3, MESFUNC5:38;
hence S₁[k + 1] by Def4; :: thesis: verum

end;

(Partial_Sums F) . 0 = F . 0 by Def4;
then A4: S₁[ 0 ] by A1;
for k being Nat holds S₁[k] from NAT_1:sch 2(A4, A2);
hence (Partial_Sums F) . n is_simple_func_in S ; :: thesis: verum