reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;

theorem Th12:
  swap({},x,y)={}
proof
  assume swap({},x,y)<>{};
  then consider a be object such that
A1: a in swap({},x,y) by XBOOLE_0:def 1;
  per cases by A1,XBOOLE_0:def 3;
  suppose a in {(A\{x})\/{y} where A is Element of {}: x in A};
    then ex A be Element of {} st a =(A\{x})\/{y} & x in A;
    hence thesis by SUBSET_1:def 1;
  end;
  suppose a in {A\/{x} where A is Element of {}: not x in A & A in {}};
    then ex A be Element of {} st a =A\/{x} & not x in A & A in {};
    hence thesis;
  end;
end;
