theorem Th1:
  len F = len G & (for k,v st k in dom F & v = G.k holds F.k = v *
  a) implies Sum(F) = Sum(G) * a
proof
  defpred P[Nat] means for H,I be FinSequence of V st len H = len I & len H =
$1 & (for k,v st k in dom H & v = I.k holds H.k = v * a) holds Sum(H) = Sum(I)
  * a;
  now
    let n be Nat;
    assume
A1: for H,I be FinSequence of V st len H = len I & len H = n & for k,
    v st k in dom H & v = I.k holds H.k = v * a holds Sum(H) = Sum(I) * a;
    let H,I be FinSequence of V;
    assume that
A2: len H = len I and
A3: len H = n + 1 and
A4: for k,v st k in dom H & v = I.k holds H.k = v * a;
    reconsider p = H | (Seg n),q = I | (Seg n) as FinSequence of V by
FINSEQ_1:18;
A5: n <= n + 1 by NAT_1:12;
    then
A6: len p = n by A3,FINSEQ_1:17;
A7: len q = n by A2,A3,A5,FINSEQ_1:17;
A8: now
      len p <= len H by A3,A5,FINSEQ_1:17;
      then
A9:   dom p c= dom H by FINSEQ_3:30;
      let k,v;
      assume that
A10:  k in dom p and
A11:  v = q.k;
      dom p = dom q by A6,A7,FINSEQ_3:29;
      then I.k = q.k by A10,FUNCT_1:47;
      then H.k = v * a by A4,A10,A11,A9;
      hence p.k = v * a by A10,FUNCT_1:47;
    end;
A12: n + 1 in Seg(n + 1) by FINSEQ_1:4;
    then n + 1 in dom H & n + 1 in dom I by A2,A3,FINSEQ_1:def 3;
    then reconsider v1 = H.(n + 1),v2 = I.(n + 1) as Vector of V by FINSEQ_2:11
;
    n + 1 in dom H by A3,A12,FINSEQ_1:def 3;
    then
A13: v1 = v2 * a by A4;
    thus Sum(H) = Sum(p) + v1 by A3,A6,Lm1
      .= Sum(q) * a + v2 * a by A1,A6,A7,A8,A13
      .= (Sum(q) + v2) * a by VECTSP_2:def 9
      .= Sum(I) * a by A2,A3,A7,Lm1;
  end;
  then
A14: for i be Nat st P[i] holds P[i+1];
  now
    let H,I be FinSequence of V;
    assume that
A15: len H = len I and
A16: len H = 0 and
    for k,v st k in dom H & v = I.k holds H.k = v * a;
    H = <*>(the carrier of V) by A16;
    then
A17: Sum(H) = 0.V by RLVECT_1:43;
    I = <*>(the carrier of V) by A15,A16;
    then Sum(I) = 0.V by RLVECT_1:43;
    hence Sum(H) = Sum(I) * a by A17,VECTSP_2:32;
  end;
  then
A18: P[0];
  for n being Nat holds P[n] from NAT_1:sch 2(A18,A14);
  hence thesis;
end;
