reserve V for Universe,
  a,b,x,y,z,x9,y9 for Element of V,
  X for Subclass of V,
  o,p,q,r,s,t,u,a1,a2,a3,A,B,C,D for set,
  K,L,M for Ordinal,
  n for Element of omega,
  fs for finite Subset of omega,
  e,g,h for Function,
  E for non empty set,
  f for Function of VAR,E,
  k,k1 for Element of NAT,
  v1,v2,v3 for Element of VAR,
  H,H9 for ZF-formula;

theorem
  X is closed_wrt_A1-A7 & E in X implies Diagram(H,E) in X
proof
  defpred P[ZF-formula] means Diagram($1,E) in X;
  assume
A1: X is closed_wrt_A1-A7 & E in X;
  then
A2: for H st P[H] holds P['not' H] by Th19;
A3: for H,v1 st P[H] holds P[All(v1,H)] by A1,Th21;
A4: for H,H9 st P[H] & P[H9] holds P[H '&' H9] by A1,Th20;
A5: for v1,v2 holds P[v1 '=' v2] & P[v1 'in' v2] by A1,Th18;
  for H holds P[H] from ZF_LANG1:sch 1(A5,A2,A4,A3);
  hence thesis;
end;
