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
  H is being_equality iff H/(x,y) is being_equality
proof
  thus H is being_equality implies H/(x,y) is being_equality
  by Th153;
  assume H/(x,y) is being_equality;
  then
A1: H/(x,y).1 = 0 by ZF_LANG:18;
  3 <= len H by ZF_LANG:13;
  then 1 <= len H by XXREAL_0:2;
  then
A2: 1 in dom H by FINSEQ_3:25;
  y <> 0 by Th135;
  then H.1 <> x by A1,A2,Def3;
  then 0 = H.1 by A1,A2,Def3;
  hence thesis by ZF_LANG:24;
end;
