let X be non empty set ; :: thesis: for F being Functional_Sequence of X,REAL

for n being Nat st F is with_the_same_dom holds

dom ((Partial_Sums F) . n) = dom (F . 0)

let F be Functional_Sequence of X,REAL; :: thesis: for n being Nat st F is with_the_same_dom holds

dom ((Partial_Sums F) . n) = dom (F . 0)

let n be Nat; :: thesis: ( F is with_the_same_dom implies dom ((Partial_Sums F) . n) = dom (F . 0) )

assume F is with_the_same_dom ; :: thesis: dom ((Partial_Sums F) . n) = dom (F . 0)

then dom ((Partial_Sums (R_EAL F)) . n) = dom ((R_EAL F) . 0) by Th9, MESFUNC9:29;

then dom ((R_EAL (Partial_Sums F)) . n) = dom ((R_EAL F) . 0) by Th7;

hence dom ((Partial_Sums F) . n) = dom (F . 0) ; :: thesis: verum

for n being Nat st F is with_the_same_dom holds

dom ((Partial_Sums F) . n) = dom (F . 0)

let F be Functional_Sequence of X,REAL; :: thesis: for n being Nat st F is with_the_same_dom holds

dom ((Partial_Sums F) . n) = dom (F . 0)

let n be Nat; :: thesis: ( F is with_the_same_dom implies dom ((Partial_Sums F) . n) = dom (F . 0) )

assume F is with_the_same_dom ; :: thesis: dom ((Partial_Sums F) . n) = dom (F . 0)

then dom ((Partial_Sums (R_EAL F)) . n) = dom ((R_EAL F) . 0) by Th9, MESFUNC9:29;

then dom ((R_EAL (Partial_Sums F)) . n) = dom ((R_EAL F) . 0) by Th7;

hence dom ((Partial_Sums F) . n) = dom (F . 0) ; :: thesis: verum