reserve m, n for Nat;

theorem
  Sum SMoebius NatDivisors 1 = 1
proof
  reconsider J = 1 as Element of NAT;
  reconsider M = ({1},1)-bag as Rbag of NAT;
A1: 1 in {1} by TARSKI:def 1;
A2: 1 in SCNAT by Def2,Th22;
  J in {1} by TARSKI:def 1;
  then J in {1} /\ SCNAT by A2,XBOOLE_0:def 4;
  then J in support SMoebius {1} by Def5;
  then
A3: (SMoebius {1}).1 = 1 by Def5,Th30;
  {1} c= SCNAT by A2,ZFMISC_1:31;
  then
A4: {1} /\ SCNAT = {1} by XBOOLE_1:28;
  for x being object st x in NAT holds (SMoebius {1}).x = M.x
  proof
    let x be object;
    per cases;
    suppose
A5:   x in {1};
      then x = 1 by TARSKI:def 1;
      hence thesis by A3,A5,UPROOTS:7;
    end;
    suppose
A6:   not x in {1};
      then
A7:   not x in support SMoebius {1} by A4,Def5;
      M.x = 0 by A6,UPROOTS:6
        .= (SMoebius {1}).x by A7,PRE_POLY:def 7;
      hence thesis;
    end;
  end;
  then support M = {1} & SMoebius {1} = M by PBOOLE:3,UPROOTS:8;
  then Sum SMoebius NatDivisors 1 = M.1 by Th12,Th41
    .= 1 by A1,UPROOTS:7;
  hence thesis;
end;
