
theorem Th9:
  for X be non empty addLoopStr, A,B be Subset of X,
      l be Linear_Combination of (A \/ B)
  st A misses B holds
  ex l1 be Linear_Combination of A, l2 be Linear_Combination of B
  st Carrier l = Carrier l1 \/ Carrier l2 & l = l1 + l2
   & Carrier l1 = Carrier l \ B & Carrier l2 = Carrier l \ A
  proof
    let X be non empty addLoopStr, A,B be Subset of X,
        l be Linear_Combination of (A \/ B);
    assume
A2: A misses B;
    consider l1 be Linear_Combination of A such that
    A3: Carrier l1 = Carrier l \ B
      & for x be Element of X st x in Carrier l1 holds l1.x = l.x by Th8;
    consider l2 be Linear_Combination of B such that
    A4: Carrier l2 = Carrier l \ A
      & for x be Element of X st x in Carrier l2 holds l2.x = l.x by Th8;
    take l1,l2;
    A5: Carrier l1 \/ Carrier l2 = Carrier l \ (A /\ B) by A3,A4,XBOOLE_1:54;
    hence Carrier l1 \/ Carrier l2 = Carrier l by A2;
    A6: Carrier l1 c= A & Carrier l2 c= B by RLVECT_2:def 6;
    now
      let x be Element of X;
      per cases by A2,A5,XBOOLE_0:def 3;
      suppose
        A7: x in Carrier l1; then
        not x in Carrier l2 by A2,A6,XBOOLE_0:3; then
        l1.x = l.x & l2.x = 0 by A3,A7;
        hence l.x = l1.x + l2.x;
      end;
      suppose
        A8: x in Carrier l2; then
        not x in Carrier l1 by A2,A6,XBOOLE_0:3; then
        l1.x = 0 & l2.x = l.x by A4,A8;
        hence l.x = l1.x + l2.x;
      end;
      suppose
        A9: not x in Carrier l; then
        not x in Carrier l1 & not x in Carrier l2 by A2,A5,XBOOLE_0:def 3; then
        l.x = 0 & l1.x = 0 & l2.x = 0 by A9;
        hence l.x = l1.x + l2.x;
      end;
    end;
    hence l = l1+l2 by RLVECT_2:def 10;
    thus thesis by A3,A4;
  end;
