reserve m, n for Nat;

theorem
  for X, Y being finite Subset of NAT st X misses Y holds SMoebius (X \/
  Y) = SMoebius X + SMoebius Y
proof
  let X,Y be finite Subset of NAT;
A1: support SMoebius (X \/ Y) = (X \/ Y) /\ SCNAT by Def5
    .= (X /\ SCNAT) \/ (Y /\ SCNAT) by XBOOLE_1:23;
  assume
A2: X misses Y;
  for x being object st x in NAT
   holds (SMoebius (X \/ Y)).x = (SMoebius X + SMoebius Y).x
  proof
    let x be object;
    per cases;
    suppose
A3:   x in support SMoebius (X \/ Y);
      per cases by A1,A3,XBOOLE_0:def 3;
      suppose
A4:     x in X /\ SCNAT;
        then reconsider k = x as Element of NAT;
A5:     k in support SMoebius X by A4,Def5;
        not x in Y /\ SCNAT by A2,A4,Lm1;
        then
A6:     not k in support SMoebius Y by Def5;
        (SMoebius (X \/ Y)).x = Moebius k + (0 qua Nat) by A3,Def5
          .= Moebius k + SMoebius Y.k by A6,PRE_POLY:def 7
          .= SMoebius X.k + SMoebius Y.k by A5,Def5
          .= (SMoebius X + SMoebius Y).x by PRE_POLY:def 5;
        hence thesis;
      end;
      suppose
A7:     x in Y /\ SCNAT;
        then consider k being Element of NAT such that
A8:     k = x;
        not x in X /\ SCNAT by A2,A7,Lm1;
        then
A9:     not k in support SMoebius X by A8,Def5;
A10:    k in support SMoebius Y by A7,A8,Def5;
        (SMoebius (X \/ Y)).x = Moebius k + (0 qua Nat) by A3,A8,Def5
          .= Moebius k + SMoebius X.k by A9,PRE_POLY:def 7
          .= SMoebius Y.k + SMoebius X.k by A10,Def5
          .= (SMoebius X + SMoebius Y).x by A8,PRE_POLY:def 5;
        hence thesis;
      end;
    end;
    suppose
A11:  not x in support SMoebius (X \/ Y);
      then not x in Y /\ SCNAT by A1,XBOOLE_0:def 3;
      then
A12:  not x in support SMoebius Y by Def5;
      not x in X /\ SCNAT by A1,A11,XBOOLE_0:def 3;
      then
A13:  not x in support SMoebius X by Def5;
      (SMoebius (X \/ Y)).x = 0 by A11,PRE_POLY:def 7
        .= SMoebius Y.x + 0 by A12,PRE_POLY:def 7
        .= SMoebius Y.x + SMoebius X.x by A13,PRE_POLY:def 7
        .= (SMoebius X + SMoebius Y).x by PRE_POLY:def 5;
      hence thesis;
    end;
  end;
  hence thesis by PBOOLE:3;
end;
