
theorem Th12:
  for X be RealLinearSpace, A,B be finite Subset of X st A misses B
  holds RAT_Sums(A) + RAT_Sums(B) = RAT_Sums(A \/ B)
  proof
    let X be RealLinearSpace, A,B be finite Subset of X;
    assume
    A3: A misses B;
    set D1 = RAT_Sums(A);
    set D2 = RAT_Sums(B);
    set S1 = RAT_Sums(A) + RAT_Sums(B);
    set S2 = RAT_Sums(A \/ B);
    A4: S1 = {a+b where a,b is Point of X : a in D1 & b in D2}
        by IDEAL_1:def 19;
    now
      let p be object;
      assume p in S1; then
      consider a,b be Point of X such that
      A5: p = a+b & a in D1 & b in D2 by A4;
      consider l1 be Linear_Combination of A such that
      A6: a = Sum l1 & rng l1 c= RAT by A5;
      consider l2 be Linear_Combination of B such that
      A7: b = Sum l2 & rng l2 c= RAT by A5;
      ex l be Linear_Combination of (A \/ B)
      st Carrier l = Carrier l1 \/ Carrier l2
       & rng l c= RAT & Sum l = Sum l1 + Sum l2 by A3,A6,A7,Th10;
      hence p in S2 by A5,A6,A7;
    end; then
    A8: S1 c= S2;
    now
      let p be object;
      assume p in S2; then
      consider l be Linear_Combination of (A \/ B) such that
      A9: p = Sum l & rng l c= RAT;
      consider l1 be Linear_Combination of A, l2 be Linear_Combination of B
      such that
      A10: rng l1 c= RAT & rng l2 c= RAT & Sum l = Sum l1 + Sum l2
      by A3,A9,Th11;
      Sum l1 in D1 & Sum l2 in D2 by A10;
      hence p in S1 by A4,A9,A10;
    end; then
    S2 c= S1;
    hence thesis by A8;
  end;
