let X be non empty set ; :: thesis: for n, m being Nat
for z being set
for F being Functional_Sequence of X,COMPLEX st z in dom ((Partial_Sums F) . n) & m <= n holds
( z in dom ((Partial_Sums F) . m) & z in dom (F . m) )
let n, m be Nat; :: thesis: for z being set
for F being Functional_Sequence of X,COMPLEX st z in dom ((Partial_Sums F) . n) & m <= n holds
( z in dom ((Partial_Sums F) . m) & z in dom (F . m) )
let z be set ; :: thesis: for F being Functional_Sequence of X,COMPLEX st z in dom ((Partial_Sums F) . n) & m <= n holds
( z in dom ((Partial_Sums F) . m) & z in dom (F . m) )
let F be Functional_Sequence of X,COMPLEX; :: thesis: ( z in dom ((Partial_Sums F) . n) & m <= n implies ( z in dom ((Partial_Sums F) . m) & z in dom (F . m) ) )
assume A1: ( z in dom ((Partial_Sums F) . n) & m <= n ) ; :: thesis: ( z in dom ((Partial_Sums F) . m) & z in dom (F . m) )
A2: dom ((Partial_Sums F) . n) = dom ((Re (Partial_Sums F)) . n) by MESFUN7C:def 11
.= dom ((Partial_Sums (Re F)) . n) by Th29 ;
dom ((Partial_Sums (Re F)) . m) = dom ((Re (Partial_Sums F)) . m) by Th29
.= dom ((Partial_Sums F) . m) by MESFUN7C:def 11 ;
hence z in dom ((Partial_Sums F) . m) by A1, A2, Th8; :: thesis: z in dom (F . m)
z in dom ((Re F) . m) by A1, A2, Th8;
hence z in dom (F . m) by MESFUN7C:def 11; :: thesis: verum