reserve x,y for set,
        r,s for Real,
        S for non empty addLoopStr,
        LS,LS1,LS2 for Linear_Combination of S,
        G for Abelian add-associative right_zeroed right_complementable
          non empty addLoopStr,
        LG,LG1,LG2 for Linear_Combination of G,
        g,h for Element of G,
        RLS for non empty RLSStruct,
        R for vector-distributive scalar-distributive scalar-associative
        scalar-unitalnon empty RLSStruct,
        AR for Subset of R,
        LR,LR1,LR2 for Linear_Combination of R,
        V for RealLinearSpace,
        v,v1,v2,w,p for VECTOR of V,
        A,B for Subset of V,
        F1,F2 for Subset-Family of V,
        L,L1,L2 for Linear_Combination of V;

theorem Th34:
  sum (LS1+LS2) = sum LS1 + sum LS2
  proof
    set C1=Carrier LS1;
    set C2=Carrier LS2;
    consider p112 be FinSequence such that
    A1: rng p112=C1\C2 and
    A2: p112 is one-to-one by FINSEQ_4:58;
    consider p12 be FinSequence such that
    A3: rng p12=C1/\C2 and
    A4: p12 is one-to-one by FINSEQ_4:58;
    consider p211 be FinSequence such that
    A5: rng p211=C2\C1 and
    A6: p211 is one-to-one by FINSEQ_4:58;
    reconsider p112,p12,p211 as FinSequence of S by A1,A3,A5,FINSEQ_1:def 4;
    C1\C2 misses C1/\C2 by XBOOLE_1:89;
    then A7: p112^p12 is one-to-one by A1,A2,A3,A4,FINSEQ_3:91;
    set p1=p112^p12;
    A8: rng p1 = C1\C2\/C1/\C2 by A1,A3,FINSEQ_1:31
              .= C1 by XBOOLE_1:51;
    then A9: rng(p112^p12^p211) = C1\/(C2\C1) by A5,FINSEQ_1:31
                               .= C1\/C2 by XBOOLE_1:39;
    set p2=p12^p211;
    A10: rng p2 = C1/\C2\/(C2\C1) by A3,A5,FINSEQ_1:31
               .= C2 by XBOOLE_1:51;
    set pp=p1^p211;
    pp=p112^p2 by FINSEQ_1:32;
    then A11: LS2*pp=(LS2*p112)^(LS2*p2) by FINSEQOP:9;
    C2\C1 misses C1/\C2 by XBOOLE_1:89;
    then A12: p12^p211 is one-to-one by A3,A4,A5,A6,FINSEQ_3:91;
    C2 misses C1\C2 by XBOOLE_1:79;
    then Sum(LS2*p112)=0 by A1,Th29;
    then A13: Sum(LS2*pp) = 0+Sum(LS2*p2) by A11,RVSUM_1:75
                         .= sum LS2 by A10,A12,Def3;
    len(LS1*pp)=len pp & len(LS2*pp)=len pp by FINSEQ_2:33;
    then reconsider L1p=LS1*pp,L2p=LS2*pp as Element of len pp-tuples_on REAL
      by FINSEQ_2:92;
    A14: (LS1+LS2)*pp=L1p+L2p by Th13;
    A15: C1 misses C2\C1 by XBOOLE_1:79;
    then LS1*pp=(LS1*p1)^(LS1*p211) & Sum(LS1*p211)=0 by A5,Th29,FINSEQOP:9;
    then A16: Sum(LS1*pp) = Sum(LS1*p1)+0 by RVSUM_1:75
                         .= sum LS1 by A7,A8,Def3;
    A17: Carrier(LS1+LS2)c=C1\/C2
    proof
      let x be object;
      assume x in Carrier(LS1+LS2);
      then consider u be Element of S such that
      A18: x=u and
      A19: (LS1+LS2).u<>0;
      (LS1+LS2).u=LS1.u+LS2.u by RLVECT_2:def 10;
      then LS1.u<>0 or LS2.u<>0 by A19;
      then x in C1 or x in C2 by A18;
      hence thesis by XBOOLE_0:def 3;
    end;
    p112^p12^p211 is one-to-one by A5,A6,A7,A8,A15,FINSEQ_3:91;
    hence sum(LS1+LS2) = Sum(L1p+L2p) by A9,A14,A17,Th30
                          .= sum LS1+sum LS2 by A13,A16,RVSUM_1:89;
  end;
