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 Th33:
  not y in union X & not y in union Y implies
    (X misses Y iff Ext(X,x,y) misses Ext(Y,x,y))
proof
  assume
A1: not y in union X & not y in union Y;
  thus X misses Y implies Ext(X,x,y) misses Ext(Y,x,y)
  proof
    assume that
A2:   X misses Y and
A3:   Ext(X,x,y) meets Ext(Y,x,y);
    consider a be object such that
A4:   a in Ext(X,x,y) & a in Ext(Y,x,y) by XBOOLE_0:3,A3;
    per cases by A4,XBOOLE_0:def 3;
    suppose a in {A\/{y} where A is Element of X: x in A};
      then consider A be Element of X such that
A5:     a= A\/{y} & x in A;
      per cases by A4,XBOOLE_0:def 3;
      suppose a in {A\/{y} where A is Element of Y: x in A};
        then consider B be Element of Y such that
A6:       a= B\/{y} & x in B;
A7:       X <>{} & Y <>{} by A5,A6,SUBSET_1:def 1;
        then not y in A & not y in B by A1,TARSKI:def 4;
        then {y} misses A & {y} misses B by ZFMISC_1:50;
        then A=B by A6,A5,XBOOLE_1:71;
        hence thesis by A2,A7,XBOOLE_0:3;
      end;
      suppose a in {A where A is Element of Y: not x in A & A in Y};
        then ex B be Element of Y st a= B & not x in B & B in Y;
        hence thesis by A5,ZFMISC_1:136;
      end;
    end;
    suppose a in {A where A is Element of X: not x in A & A in X};
      then consider A be Element of X such that
A8:     a= A & not x in A & A in X;
      per cases by A4,XBOOLE_0:def 3;
      suppose a in {A\/{y} where A is Element of Y: x in A};
        then ex B be Element of Y st a= B\/{y} & x in B;
        hence thesis by A8,ZFMISC_1:136;
      end;
      suppose a in {A where A is Element of Y: not x in A & A in Y};
        then ex B be Element of Y st a= B & not x in B & B in Y;
        hence thesis by A2,A8,XBOOLE_0:3;
      end;
    end;
  end;
  assume that
A9:  Ext(X,x,y) misses Ext(Y,x,y) and
A10: X meets Y;
  consider a be object such that
A11: a in X & a in Y by A10,XBOOLE_0:3;
  reconsider a as set by TARSKI:1;
  per cases;
  suppose x in a;
    then a\/{y} in {A\/{y} where A is Element of X: x in A} &
    a\/{y} in {A\/{y} where A is Element of Y: x in A} by A11;
    then a\/{y} in Ext(X,x,y) & a\/{y} in Ext(Y,x,y) by XBOOLE_0:def 3;
    hence thesis by A9,XBOOLE_0:3;
  end;
  suppose not x in a;
    then a in {A where A is Element of X:  not x in A & A in X} &
    a in {A where A is Element of Y:  not x in A & A in Y} by A11;
    then a in Ext(X,x,y) & a in Ext(Y,x,y) by XBOOLE_0:def 3;
    hence thesis by A9,XBOOLE_0:3;
  end;
end;
