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 Th39:
  Sum (v + L) = (sum L)*v + Sum L
  proof
    defpred P[Nat] means
    for L be Linear_Combination of V st card Carrier L<=$1 holds
      Sum(v+L)=(sum L)*v+Sum L;
    A1: for n be Nat st P[n] holds P[n+1]
    proof
      let n be Nat such that
      A2: P[n];
      let L be Linear_Combination of V such that
      A3: card Carrier L<=n+1;
      per cases by A3,NAT_1:8;
      suppose card Carrier L<=n;
        hence thesis by A2;
      end;
      suppose A4: card Carrier L=n+1;
        then Carrier L is non empty;
        then consider w be object such that
        A5: w in Carrier L;
        reconsider w as Element of V by A5;
        A6: card(Carrier L\{w})=n by A4,A5,STIRL2_1:55;
        consider Lw be Linear_Combination of{w} such that
        A7: Lw.w=L.w by RLVECT_4:37;
        set LLw=L-Lw;
        LLw.w = L.w-Lw.w by RLVECT_2:54
             .= 0 by A7;
        then A8: not w in Carrier LLw by RLVECT_2:19;
        A9: Carrier Lw c={w} by RLVECT_2:def 6;
        then A10: Carrier LLw c=Carrier L\/Carrier Lw &
          Carrier L\/Carrier Lw c= Carrier L\/{w} by RLVECT_2:55,XBOOLE_1:9;
        Carrier L\/{w}=Carrier L by A5,ZFMISC_1:40;
        then Carrier LLw c=Carrier L by A10;
        then card Carrier LLw<=n by A8,A6,NAT_1:43,ZFMISC_1:34;
        then A11: Sum(v+LLw)=(sum LLw)*v+Sum LLw by A2;
        A12: LLw+Lw = L+(Lw-Lw) by RLVECT_2:40
                   .= L+ZeroLC(V) by RLVECT_2:57
                   .=L by RLVECT_2:41;
        then A13: Sum L = Sum LLw+Sum Lw by RLVECT_3:1
                       .= Sum LLw+Lw.w*w by RLVECT_2:32;
        v+Carrier Lw c= v+{w} by A9,RLTOPSP1:8;
        then v+Carrier Lw c={v+w} by Lm3;
        then Carrier(v+Lw)c={v+w} by Th16;
        then v+Lw is Linear_Combination of{v+w} by RLVECT_2:def 6;
        then A14: Sum(v+Lw) = (v+Lw).(v+w)*(v+w) by RLVECT_2:32
                           .= Lw.(v+w-v)*(v+w) by Def1
                           .= Lw.w*(v+w) by RLVECT_4:1;
        A15: sum L = sum LLw+sum Lw by A12,Th34
                  .= sum LLw+Lw.w by A9,Th32;
        v+L=(v+LLw)+(v+Lw) by A12,Th17;
        hence Sum(v+L) = Sum(v+LLw)+Sum(v+Lw) by RLVECT_3:1
                      .= (sum LLw)*v+Sum LLw+(Lw.w*v+Lw.w*w)
                           by A11,A14,RLVECT_1:def 5
                      .= (sum LLw)*v+Sum LLw+Lw.w*v+Lw.w*w by RLVECT_1:def 3
                      .= (sum LLw)*v+Lw.w*v+Sum LLw+Lw.w*w by RLVECT_1:def 3
                      .= sum L*v+Sum LLw+Lw.w*w by A15,RLVECT_1:def 6
                      .= sum L*v+Sum L by A13,RLVECT_1:def 3;
      end;
    end;
    A16: P[0 qua Nat]
    proof
      let L be Linear_Combination of V;
      assume card Carrier L<=0;
      then A17: Carrier L={}V;
      then A18: L=ZeroLC(V) & Sum L=0.V by RLVECT_2:34,def 5;
      v+Carrier L={} by A17,Th8;
      then Carrier(v+L)={} by Th16;
      hence Sum(v+L) = 0.V by RLVECT_2:34
                    .= 0.V+0.V
                    .= (sum L)*v+Sum L by A18,Th31,RLVECT_1:10;
    end;
    for n be Nat holds P[n] from NAT_1:sch 2(A16,A1);
    then P[card Carrier L];
    hence thesis;
  end;
