
theorem Th2:
  for L be add-associative right_zeroed right_complementable non
empty addLoopStr, r be FinSequence of L st len r >= 2 & for k being Element of
  NAT st 2 < k & k in dom r holds r.k = 0.L holds Sum r = r/.1 + r/.2
proof
  let L be add-associative right_zeroed right_complementable non empty
  addLoopStr, r being FinSequence of L such that
A1: len r >= 2 and
A2: for k being Element of NAT st 2 < k & k in dom r holds r.k = 0.L;
A3: 2 in dom r by A1,FINSEQ_3:25;
  1 <= len r by A1,XXREAL_0:2;
  then
A4: 1 in dom r by FINSEQ_3:25;
  r is not empty by A1;
  then consider a being Element of L, r1 being FinSequence of L such that
A5: a = r.1 and
A6: r = <*a*>^r1 by FINSEQ_3:102;
A7: len <*a*> = 1 by FINSEQ_1:40;
  then
A8: r.2 = r1.(2-1) by A1,A6,FINSEQ_1:24
    .= r1.1;
  len r = 1 + len r1 by A6,A7,FINSEQ_1:22;
  then r1 is non empty by A1;
  then consider b being Element of L, r2 being FinSequence of L such that
A9: b = r1.1 and
A10: r1 = <*b*>^r2 by FINSEQ_3:102;
A11: len <*b*> = 1 by FINSEQ_1:40;
A12: now
    let i be Element of NAT such that
A13: i in dom r2;
A14: 1+i in dom r1 by A10,A11,A13,FINSEQ_1:28;
    1 <= i by A13,FINSEQ_3:25;
    then 1 < 1+i by NAT_1:13;
    then
A15: 1+1 < 1+(1+i) by XREAL_1:8;
    thus r2.i = r1.(1+i) by A10,A11,A13,FINSEQ_1:def 7
      .= r.(1+(1+i)) by A6,A7,A14,FINSEQ_1:def 7
      .= 0.L by A2,A6,A7,A14,A15,FINSEQ_1:28;
  end;
  thus Sum r = a + Sum r1 by A6,FVSUM_1:72
    .= a + (b + Sum r2) by A10,FVSUM_1:72
    .= a + (b + 0.L) by A12,POLYNOM3:1
    .= a+b by RLVECT_1:def 4
    .= r/.1 + b by A5,A4,PARTFUN1:def 6
    .= r/.1 + r/.2 by A9,A3,A8,PARTFUN1:def 6;
end;
