reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;

theorem
  (for X,Y st X<>Y & X in A \/ B & Y in A \/ B holds X misses Y) implies
  union(A /\ B) = union A /\ union B
proof
  assume
A1: for X,Y st X<>Y & X in A \/ B & Y in A \/ B holds X misses Y;
 union A /\ union B c= union (A /\ B)
  proof
    let x;
    assume
A2: x in union A /\ union B;
    then x in union A by XBOOLE_0:def 4;
    then consider X such that
A3: x in X and
A4: X in A by TARSKI:def 4;
    x in union B by A2,XBOOLE_0:def 4;
    then consider Y such that
A5: x in Y and
A6: Y in B by TARSKI:def 4;
    now
A7:   x in X /\ Y by A3,A5,XBOOLE_0:def 4;
      assume
A8:   X<>Y;
      X in A \/ B & Y in A \/ B by A4,A6,XBOOLE_0:def 3;
      hence contradiction by A1,A8,A7,XBOOLE_0:4;
    end;
    then Y in A /\ B by A4,A6,XBOOLE_0:def 4;
    hence thesis by A5,TARSKI:def 4;
  end;
  hence thesis by Th78;
end;
