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;
