let X be ComplexBanachSpace; :: thesis: for seq being sequence of X holds
( seq is summable iff for p being Real st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.(((Partial_Sums seq) . m) - ((Partial_Sums seq) . n)).|| < p )
let seq be sequence of X; :: thesis: ( seq is summable iff for p being Real st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.(((Partial_Sums seq) . m) - ((Partial_Sums seq) . n)).|| < p )

A1: now

assume for p being Real st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.(((Partial_Sums seq) . m) - ((Partial_Sums seq) . n)).|| < p ; :: thesis: seq is summable
then Partial_Sums seq is CCauchy by Th5;
then Partial_Sums seq is convergent by CLOPBAN1:def 13;
hence seq is summable by Def2; :: thesis: verum

end;

now

assume seq is summable ; :: thesis: for p being Real st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.(((Partial_Sums seq) . m) - ((Partial_Sums seq) . n)).|| < p
then Partial_Sums seq is convergent by Def2;
hence for p being Real st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.(((Partial_Sums seq) . m) - ((Partial_Sums seq) . n)).|| < p by Th4; :: thesis: verum

end;

hence ( seq is summable iff for p being Real st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.(((Partial_Sums seq) . m) - ((Partial_Sums seq) . n)).|| < p ) by A1; :: thesis: verum