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

theorem Th2:
  for X st the addF of X is commutative associative & the addF of X
  is having_a_unity for Y1, Y2 be finite Subset of X st Y1 misses Y2 for Z be
  finite Subset of X st Z = Y1 \/ Y2 holds setsum(Z) = setsum(Y1) + setsum(Y2)
proof
  let X such that
A1: the addF of X is commutative associative & the addF of X is having_a_unity;
  reconsider I = id the carrier of X as Function of the carrier of X, the
  carrier of X;
  let Y1, Y2 be finite Subset of X such that
A2: Y1 misses Y2;
A3: dom I = the carrier of X by FUNCT_2:def 1;
  let Z be finite Subset of X;
  assume Z = Y1 \/ Y2;
  then
A4: setopfunc(Z,the carrier of X,the carrier of X,I,the addF of X) =
setopfunc(Y1,the carrier of X,the carrier of X,I,the addF of X) + setopfunc(Y2,
  the carrier of X,the carrier of X,I,the addF of X) by A1,A2,A3,BHSP_5:14;
A5: for x be set st x in the carrier of X holds I.x = x by FUNCT_1:18;
  then setsum(Y1) = setopfunc(Y1,the carrier of X,the carrier of X,I,the addF
of X) & setsum(Y2) = setopfunc(Y2,the carrier of X,the carrier of X,I,the addF
  of X) by A1,A3,Th1;
  hence thesis by A1,A5,A3,A4,Th1;
end;
