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;
reserve A for set,
  x for Element of [:Fin A, Fin A:],
  a,b,c,d,s,t for Element of DISJOINT_PAIRS A,
  B,C,D for Element of Fin DISJOINT_PAIRS A;
reserve K,L,M for Element of Normal_forms_on A;

theorem Th35:
  b in B & c in C & b \/ c in DISJOINT_PAIRS A implies b \/ c in B ^C
proof
  assume b in B & c in C;
  then
A1: b \/ c in { s \/ t: s in B & t in C };
  assume b \/ c in DISJOINT_PAIRS A;
  hence thesis by A1,XBOOLE_0:def 4;
end;
