reserve A, B for non empty preBoolean set,
  x, y for Element of [:A,B:];
reserve X for set,
  a,b,c for Element of [:A,B:];
reserve a for Element of [:Fin X, Fin X:];
reserve A for set;
reserve x,y for Element of [:Fin X, Fin X:],
  a,b for Element of DISJOINT_PAIRS X;

theorem
  for V,W being set st V c= a`1 & W c= a`2 holds [V,W] is Element of
  DISJOINT_PAIRS X
proof
  let V,W be set;
  assume
A1: V c= a`1 & W c= a`2;
  then V is Element of Fin X & W is Element of Fin X by SETWISEO:11;
  then reconsider x = [V,W] as Element of [:Fin X, Fin X:] by ZFMISC_1:def 2;
  a`1 misses a`2 & [V,W]`1 = V by Th23;
  then x`1 misses x`2 by A1,XBOOLE_1:64;
  hence thesis by Th23;
end;
