reserve x,y,y1,y2 for set,
  p for FinSequence,
  i,k,l,n for Nat,
  V for RealLinearSpace,
  u,v,v1,v2,v3,w for VECTOR of V,
  a,b for Real,
  F,G,H1,H2 for FinSequence of V,
  A,B for Subset of V,
  f for Function of the carrier of V, REAL;
reserve K,L,L1,L2,L3 for Linear_Combination of V;
reserve l,l1,l2 for Linear_Combination of A;
reserve e,e1,e2 for Element of LinComb(V);
reserve x,y for set,
  k,n for Nat;

theorem Th66:
  for R being add-associative right_zeroed right_complementable
  Abelian associative well-unital distributive non empty doubleLoopStr,
      a being
  Element of R for V being Abelian add-associative right_zeroed
  right_complementable vector-distributive scalar-distributive
  scalar-associative scalar-unital non empty ModuleStr over R, F,G being
FinSequence of V st len F = len G & for k for v being Element of
  V st k in dom F & v = G.k holds F.k = a * v holds Sum(F) = a * Sum(G)
proof
  let R be add-associative right_zeroed right_complementable Abelian
associative well-unital distributive non empty doubleLoopStr,
  a be Element of R;
  let V be Abelian add-associative right_zeroed right_complementable
vector-distributive scalar-distributive scalar-associative scalar-unital
non empty ModuleStr over R, F,G be FinSequence of V;
  defpred P[Nat] means
for H,I being FinSequence of
V st len H = len I & len H = $1 & (for k for v be Element of V st k in dom H &
  v = I.k holds H.k = a * v) holds Sum(H) = a * Sum(I);
A1: P[n] implies P[n+1]
  proof
    assume
A2: for H,I being FinSequence of V st len H = len I &
len H = n & for k for v being Element of V st k in dom H & v = I.k holds H.k =
    a * v holds Sum(H) = a * Sum(I);
    let H,I be FinSequence of V;
    assume that
A3: len H = len I and
A4: len H = n + 1 and
A5: for k for v being Element of V st k in dom H & v = I.k holds H.k = a * v;
    reconsider p = H | (Seg n),q = I | (Seg n) as FinSequence of V
      by FINSEQ_1:18;
A6: n <= n + 1 by NAT_1:12;
    then
A7: q = I | (dom q) by A3,A4,FINSEQ_1:17;
A8: len p = n by A4,A6,FINSEQ_1:17;
A9: len q = n by A3,A4,A6,FINSEQ_1:17;
A10: now
A11:  dom p c= dom H by A4,A6,A8,FINSEQ_3:30;
      let k;
      let v be Element of V;
      assume that
A12:  k in dom p and
A13:  v = q.k;
      dom q = dom p by A8,A9,FINSEQ_3:29;
      then I.k = q.k by A12,FUNCT_1:47;
      then H.k = a * v by A5,A12,A13,A11;
      hence p.k = a * v by A12,FUNCT_1:47;
    end;
    n + 1 in Seg(n + 1) by FINSEQ_1:4;
    then
A14: n + 1 in dom H by A4,FINSEQ_1:def 3;
    dom H = dom I by A3,FINSEQ_3:29;
    then reconsider v1 = H.(n + 1),v2 = I.(n + 1) as Element of V by A14,
FINSEQ_2:11;
A15: v1 = a * v2 by A5,A14;
    p = H | (dom p) by A4,A6,FINSEQ_1:17;
    hence Sum(H) = Sum(p) + v1 by A4,A8,RLVECT_1:38
      .= a * Sum(q) + a * v2 by A2,A8,A9,A10,A15
      .= a * (Sum(q) + v2) by VECTSP_1:def 14
      .= a * Sum(I) by A3,A4,A9,A7,RLVECT_1:38;
  end;
A16: P[0]
  proof
    let H,I be FinSequence of V;
    assume that
A17: len H = len I and
A18: len H = 0 and
    for k for v being Element of V st k in dom H & v = I.k holds H.k = a * v;
    H = <*>(the carrier of V) by A18;
    then
A19: Sum(H) = 0.V by RLVECT_1:43;
    I = <*>(the carrier of V) by A17,A18;
    then Sum(I) = 0.V by RLVECT_1:43;
    hence thesis by A19,VECTSP_1:14;
  end;
  for n holds P[n] from NAT_1:sch 2(A16,A1);
  hence thesis;
end;
