reserve X for RealUnitarySpace;
reserve x for Point of X;
reserve i, n for Nat;

theorem Th7:
  for X st the addF of X is commutative associative & the addF of X
  is having_a_unity for Y be Subset of X st Y is weakly_summable_set for L be
  linear-Functional of X st L is Lipschitzian holds Y is_summable_set_by L
proof
  let X such that
A1: the addF of X is commutative associative & the addF of X is having_a_unity;
  let Y be Subset of X;
  assume Y is weakly_summable_set;
  then consider x such that
A2: for L be linear-Functional of X st L is Lipschitzian
   for e be Real st e >
  0 ex Y0 be finite Subset of X st Y0 is non empty & Y0 c= Y & for Y1 be finite
Subset of X st Y0 c= Y1 & Y1 c= Y holds |.(L.x)-(setopfunc(Y1,the carrier of
  X,REAL, L, addreal)).| < e by A1,Th6;
  let L be linear-Functional of X;
  assume L is Lipschitzian;
  then
  for e be Real st e > 0
   ex Y0 be finite Subset of X st Y0 is non empty &
Y0 c= Y & for Y1 be finite Subset of X st Y0 c= Y1 & Y1 c= Y holds |.(L.x)-(
  setopfunc(Y1,the carrier of X,REAL, L, addreal)).| < e by A2;
  hence thesis;
end;
