
theorem Th10:
  for X be RealLinearSpace, A,B be Subset of X,
      l1 be Linear_Combination of A, l2 be Linear_Combination of B
  st rng l1 c= RAT & rng l2 c= RAT & A misses B holds
  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
  proof
    let X be RealLinearSpace, A,B be Subset of X,
        l1 be Linear_Combination of A, l2 be Linear_Combination of B;
    assume that
    A1: rng l1 c= RAT & rng l2 c= RAT and
    A2: A misses B;
    consider l be Linear_Combination of (A \/ B) such that
    A3: Carrier l = Carrier l1 \/ Carrier l2 & l = l1 + l2 by A2,Th7;
    take l;
    thus Carrier l = Carrier l1 \/ Carrier l2 by A3;
    now
      let y be object;
      assume y in rng l; then
      consider x be object such that
      A4: x in dom l & y = l.x by FUNCT_1:def 3;
      A5: x in the carrier of X by A4;
      A6: l.x = l1.x + l2.x by A3,A4,RLVECT_2:def 10;
      x in dom l1 & x in dom l2 by A5,FUNCT_2:def 1; then
      l1.x in rng l1 & l2.x in rng l2 by FUNCT_1:3;
      hence y in RAT by A1,A4,A6,RAT_1:def 2;
    end;
    hence rng l c= RAT;
    thus Sum l = Sum l1 + Sum l2 by A3,RLVECT_3:1;
  end;
