
theorem
  for I being non empty set,
      L being AbGroup,
      x being (the carrier of L)-valued ManySortedSet of I,
      J being Element of Fin I holds
  for e being Element of Fin I st e={} holds
    Sum(x,e)=0.L &
    for e,f being Element of Fin I st e misses f holds
    Sum(x,e\/f)=Sum(x,e)+Sum(x,f)
  proof
    let I be non empty set,
    L be AbGroup,
    x be (the carrier of L)-valued ManySortedSet of I,
    J be Element of Fin I;
A1: now
      let e be Element of Fin I;
      assume
A2:   e={};
      consider p be one-to-one FinSequence of I  such that
A3:   rng p = e and
A4:   Sum(x,e)=(the addF of L) "**" #(p,x) by Def1;
      p={} by A3,A2;
      then #(p,x)={} & the addF of L is having_a_unity &
      len #(p,x) =0 by FVSUM_1:8;
      then Sum(x,e)=the_unity_wrt the addF of L by A4,FINSOP_1:def 1;
      hence Sum(x,e)=0.L by FVSUM_1:7;
    end;
    now
      let e,f be Element of Fin I;
      assume
A5:   e misses f;
      consider pe be one-to-one FinSequence of I  such that
A6:   rng pe = e and
A7:   Sum(x,e)=(the addF of L) "**" #(pe,x) by Def1;
      consider pf be one-to-one FinSequence of I  such that
A8:   rng pf = f and
A9:   Sum(x,f)=(the addF of L) "**" #(pf,x) by Def1;
      reconsider pepf=pe^pf as one-to-one FinSequence of I
      by A5,A6,A8,FINSEQ_3:91;
A10:   #(pepf,x)=#(pe,x)^#(pf,x) by Th3;
      rng pepf=e\/f by A6,A8,FINSEQ_1:31;
      then Sum(x,e\/f)=(the addF of L) "**" #(pepf,x) by Def1;
      hence Sum(x,e\/f)=Sum(x,e)+Sum(x,f)
      by A7,A9,A10,FINSOP_1:5,FVSUM_1:8;
    end;
    hence thesis by A1;
  end;
