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 Th30:
  for F be FinSequence of S st F is one-to-one & Carrier LS c= rng F
    holds sum LS = Sum (LS*F)
  proof
    let F be FinSequence of S such that
    A1: F is one-to-one and
    A2: Carrier LS c=rng F;
    consider G be FinSequence of S such that
    A3: G is one-to-one and
    A4: rng G=Carrier LS and
    A5: sum LS=Sum(LS*G) by Def3;
    reconsider R=rng G as Subset of rng F by A2,A4;
    reconsider FR=F-R,FR1=F-R` as FinSequence of S by FINSEQ_3:86;
    consider p be Permutation of dom F such that
    A6: F*p=(F-R`)^(F-R) by FINSEQ_3:115;
    rng F\R` = R`` .= R;
    then A7: rng FR1=R by FINSEQ_3:65;
    FR1 is one-to-one by A1,FINSEQ_3:87;
    then FR1,G are_fiberwise_equipotent by A3,A7,RFINSEQ:26;
    then consider q be Permutation of dom G such that
    A8: FR1=G*q by RFINSEQ:4;
    len(LS*G)=len G by FINSEQ_2:33;
    then A9: dom G=dom(LS*G) by FINSEQ_3:29;
    (LS*G)*q=LS*FR1 by A8,RELAT_1:36;
    then A10: sum LS=Sum(LS*FR1) by A5,A9,FINSOP_1:7;
    len(LS*F)=len F by FINSEQ_2:33;
    then A11: dom F=dom(LS*F) by FINSEQ_3:29;
    rng F\R=rng FR by FINSEQ_3:65;
    then rng FR misses Carrier LS by A4,XBOOLE_1:79;
    then A12: LS*(FR1^FR)=(LS*FR1)^(LS*FR) & Sum(LS*FR)=0 by Th29,FINSEQOP:9;
    (LS*F)*p=LS*(FR1^FR) by A6,RELAT_1:36;
    hence Sum(LS*F) = Sum(LS*(FR1^FR)) by A11,FINSOP_1:7
                       .= sum LS+0 by A10,A12,RVSUM_1:75
                       .= sum LS;
  end;
