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;

theorem
  not (G is_proper_subformula_of H & H is_proper_subformula_of G)
proof
  assume G is_proper_subformula_of H & H is_proper_subformula_of G;
  then G is_proper_subformula_of G by ZF_LANG:64;
  hence contradiction;
end;
