reserve p,p1,p2,q,r,F,G,G1,G2,H,H1,H2 for ZF-formula,
  x,x1,x2,y,y1,y2,z,z1,z2,s,t for Variable,
  a,X for set;
reserve M for non empty set,
  m,m9 for Element of M,
  v,v9 for Function of VAR,M;
reserve i,j for Element of NAT;

theorem Th174:
  H is disjunctive iff H/(x,y) is disjunctive
proof
  set G = H/(x,y);
  thus H is disjunctive implies H/(x,y) is disjunctive
  proof
    given H1,H2 such that
A1: H = H1 'or' H2;
    H/(x,y) = (H1/(x,y)) 'or' (H2/(x,y)) by A1,Th161;
    hence thesis;
  end;
  given G1,G2 such that
A2: G = G1 'or' G2;
  G is negative by A2;
  then H is negative by Th168;
  then consider H9 being ZF-formula such that
A3: H = 'not' H9;
A4: 'not' G1 '&' 'not' G2 = H9/(x,y) by A2,A3,Th156;
  then H9/(x,y) is conjunctive;
  then H9 is conjunctive by Th169;
  then consider H1,H2 such that
A5: H9 = H1 '&' H2;
  'not' G2 = H2/(x,y) by A4,A5,Th158;
  then H2/(x,y) is negative;
  then H2 is negative by Th168;
  then consider p2 such that
A6: H2 = 'not' p2;
  'not' G1 = H1/(x,y) by A4,A5,Th158;
  then H1/(x,y) is negative;
  then H1 is negative by Th168;
  then consider p1 such that
A7: H1 = 'not' p1;
  H = p1 'or' p2 by A3,A5,A7,A6;
  hence thesis;
end;
