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 Th168:
  H is negative iff H/(x,y) is negative
proof
  thus H is negative implies H/(x,y) is negative
  proof
    given H1 such that
A1: H = 'not' H1;
    H/(x,y) = 'not' (H1/(x,y)) by A1,Th156;
    hence thesis;
  end;
  assume H/(x,y) is negative;
  then
A2: H/(x,y).1 = 2 by ZF_LANG:20;
  3 <= len H by ZF_LANG:13;
  then 1 <= len H by XXREAL_0:2;
  then
A3: 1 in dom H by FINSEQ_3:25;
  y <> 2 by Th135;
  then H.1 <> x by A2,A3,Def3;
  then 2 = H.1 by A2,A3,Def3;
  hence thesis by ZF_LANG:26;
end;
