
theorem Th11:
  for X be RealLinearSpace, A,B be Subset of X,
      l be Linear_Combination of (A \/ B)
  st rng l c= RAT & A misses B holds
  ex l1 be Linear_Combination of A, l2 be Linear_Combination of B
  st rng l1 c= RAT & rng l2 c= RAT & Sum l = Sum l1 + Sum l2
  proof
    let X be RealLinearSpace, A,B be Subset of X,
        l be Linear_Combination of (A \/ B);
    assume that
    A1: rng l c= RAT and
    A3: A misses B;
    consider l1 be Linear_Combination of A, l2 be Linear_Combination of B
    such that
    A4: Carrier l = Carrier l1 \/ Carrier l2 & l = l1 + l2
      & Carrier l1 = Carrier l \ B
      & Carrier l2 = Carrier l \ A by A3,Th9;
    take l1,l2;
    A5: Carrier l1 c= A & Carrier l2 c= B by RLVECT_2:def 6;
    now
      let y be object;
      assume y in rng l1; then
      consider x be object such that
      A6: x in dom l1 & y = l1.x by FUNCT_1:def 3;
      x in the carrier of X by A6; then
      x in dom l & x in dom l2 by FUNCT_2:def 1; then
      A7: l.x in rng l & l2.x in rng l2 & l.x = l1.x + l2.x
          by A4,FUNCT_1:3,RLVECT_2:def 10;
      per cases;
      suppose
        x in Carrier l1; then
        not x in Carrier l2 by A3,A5,XBOOLE_0:3; then
        l2.x = 0 by A6;
        hence y in RAT by A1,A6,A7;
      end;
      suppose
        not x in Carrier l1; then
        l1.x = 0 by A6;
        hence y in RAT by A6,RAT_1:def 2;
      end;
    end;
    hence rng l1 c= RAT;
    now
      let y be object;
      assume y in rng l2; then
      consider x be object such that
      A8: x in dom l2 & y = l2.x by FUNCT_1:def 3;
      x in the carrier of X by A8; then
      x in dom l & x in dom l1 by FUNCT_2:def 1; then
      A9: l.x in rng l & l1.x in rng l1 & l.x = l1.x + l2.x
          by A4,FUNCT_1:3,RLVECT_2:def 10;
      per cases;
      suppose
        x in Carrier l2; then
        not x in Carrier l1 by A3,A5,XBOOLE_0:3; then
        l1.x = 0 by A8;
        hence y in RAT by A1,A8,A9;
      end;
      suppose
        not x in Carrier l2; then
        l2.x = 0 by A8;
        hence y in RAT by A8,RAT_1:def 2;
      end;
    end;
    hence rng l2 c= RAT;
    thus thesis by A4,RLVECT_3:1;
  end;
