theorem Th6: :: BHSP_6:6

for X being RealUnitarySpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds

for Y being Subset of X st Y is weakly_summable_set holds

ex x being Point of X st

for L being linear-Functional of X st L is Lipschitzian holds

for e being Real st e > 0 holds

ex Y0 being finite Subset of X st

( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds

|.((L . x) - (setopfunc (Y1, the carrier of X,REAL,L,addreal))).| < e ) )

for Y being Subset of X st Y is weakly_summable_set holds

ex x being Point of X st

for L being linear-Functional of X st L is Lipschitzian holds

for e being Real st e > 0 holds

ex Y0 being finite Subset of X st

( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds

|.((L . x) - (setopfunc (Y1, the carrier of X,REAL,L,addreal))).| < e ) )