
theorem Th7:
  for L being add-associative right_zeroed right_complementable
Abelian non empty addLoopStr, F1,F2 being FinSequence of L st len F1 = len F2
& for i being Element of NAT st i in dom F1 holds F1/.i = -(F2/.i) holds Sum F1
  = - Sum F2
proof
  let L be add-associative right_zeroed right_complementable Abelian non
  empty addLoopStr, F1,F2 being FinSequence of L;
  assume that
A1: len F1 = len F2 and
A2: for i being Element of NAT st i in dom F1 holds F1/.i = -(F2/.i);
  defpred P[Nat] means for F1,F2 being FinSequence of L st len F1 = $1 & len
F1 = len F2 & for i being Element of NAT st i in dom F1 holds F1/.i = -(F2/.i)
  holds Sum F1 = - Sum F2;
A3: now
    let k be Nat;
    assume
A4: P[k];
    now
      let f,g be FinSequence of L;
      assume that
A5:   len f = k+1 and
A6:   len f = len g and
A7:   for i being Element of NAT st i in dom f holds f/.i = -(g/.i);
      set f1 = f|(Seg k), g1 = g|(Seg k);
      reconsider f1, g1 as FinSequence by FINSEQ_1:15;
      reconsider f1,g1 as FinSequence of L by A5,A6,Lm1;
A8:   len f1 = k by A5,Lm1;
A9:   len g1 = k by A5,A6,Lm1;
      then
A10:  dom f1 = Seg len g1 by A8,FINSEQ_1:def 3
        .= dom g1 by FINSEQ_1:def 3;
A11:  f = f1^<*f/.(k+1)*> by A5,Lm1;
A12:  g = g1^<*g/.(k+1)*> by A5,A6,Lm1;
A13:  now
A14:    dom f1 c= dom f by A5,Lm1;
        let i be Element of NAT;
        assume
A15:    i in dom f1;
        dom g1 c= dom g by A5,A6,Lm1;
        then
A16:    g/.i = g.i by A10,A15,PARTFUN1:def 6
          .= g1.i by A12,A10,A15,FINSEQ_1:def 7
          .= g1/.i by A10,A15,PARTFUN1:def 6;
        thus f1/.i = f1.i by A15,PARTFUN1:def 6
          .= f.i by A11,A15,FINSEQ_1:def 7
          .= f/.i by A15,A14,PARTFUN1:def 6
          .= -(g1/.i) by A7,A15,A14,A16;
      end;
      1 <= k + 1 by NAT_1:11;
      then
A17:  k+1 in dom f by A5,FINSEQ_3:25;
      thus Sum f = Sum f1 + Sum <*f/.(k+1)*> by A11,RLVECT_1:41
        .= -(Sum g1) + Sum <*f/.(k+1)*> by A4,A8,A9,A13
        .= -(Sum g1) + f/.(k+1) by RLVECT_1:44
        .= -(Sum g1) + -(g/.(k+1)) by A7,A17
        .= -(Sum g1 + g/.(k+1)) by RLVECT_1:31
        .= -(Sum g1 + Sum<*g/.(k+1)*>) by RLVECT_1:44
        .= -Sum g by A12,RLVECT_1:41;
    end;
    hence P[k+1];
  end;
  now
    let f,g be FinSequence of L;
    assume that
A18: len f = 0 and
A19: len f = len g and
    for i being Element of NAT st i in dom f holds f/.i = -(g/.i);
A20: g = <*>(the carrier of L) by A18,A19;
    f = <*>(the carrier of L) by A18;
    hence Sum f = 0.L by RLVECT_1:43
      .= - 0.L by RLVECT_1:12
      .= -(Sum g) by A20,RLVECT_1:43;
  end;
  then
A21: P[0];
  for k being Nat holds P[k] from NAT_1:sch 2(A21,A3);
  hence thesis by A1,A2;
end;
