
theorem Th3:
  for fs being FinSequence of NAT holds Sum fs = 0 iff fs = (len fs) |-> 0
proof
  let fs be FinSequence of NAT;
  hereby
    assume
A1: Sum fs = 0;
A2: Seg len fs = dom fs by FINSEQ_1:def 3;
A3: Seg len fs = dom ((len fs) |-> 0) by FUNCOP_1:13;
    now
      let k be Nat such that
A4:   k in Seg len fs;
      now
        assume
A5:     fs.k <> 0;
        for k being Nat st k in dom fs holds 0 <= fs.k;
        hence contradiction by A1,A2,A4,A5,RVSUM_1:85;
      end;
      hence fs.k = ((len fs) |-> 0).k by A4,FUNCOP_1:7;
    end;
    hence fs = (len fs) |-> 0 by A2,A3,FINSEQ_1:13;
  end;
  assume fs = (len fs) |-> 0;
  hence thesis by RVSUM_1:81;
end;
