reserve m, n for Nat;

theorem
  for X, Y being finite Subset of NAT st X misses Y holds support
  SMoebius X \/ support SMoebius Y = support (SMoebius X + SMoebius Y)
proof
  let X,Y be finite Subset of NAT;
  assume
A1: X misses Y;
  thus support SMoebius X \/ support SMoebius Y c= support (SMoebius X +
  SMoebius Y)
  proof
    let x be object;
    support SMoebius X = X /\ SCNAT & support SMoebius Y = Y /\ SCNAT by Def5;
    then
A2: support SMoebius X misses support SMoebius Y by A1,XBOOLE_1:76;
    assume
A3: x in support SMoebius X \/ support SMoebius Y;
    per cases by A3,XBOOLE_0:def 3;
    suppose
A4:   x in support SMoebius X;
      then not x in support SMoebius Y by A2,XBOOLE_0:3;
      then (SMoebius Y).x = 0 by PRE_POLY:def 7;
      then (SMoebius X).x + (SMoebius Y).x <> 0 by A4,PRE_POLY:def 7;
      then (SMoebius X + SMoebius Y).x <> 0 by PRE_POLY:def 5;
      hence thesis by PRE_POLY:def 7;
    end;
    suppose
A5:   x in support SMoebius Y;
      then not x in support SMoebius X by A2,XBOOLE_0:3;
      then (SMoebius X).x = 0 by PRE_POLY:def 7;
      then (SMoebius X).x + (SMoebius Y).x <> 0 by A5,PRE_POLY:def 7;
      then (SMoebius X + SMoebius Y).x <> 0 by PRE_POLY:def 5;
      hence thesis by PRE_POLY:def 7;
    end;
  end;
  thus thesis by PRE_POLY:75;
end;
