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' F '&' 'not' G is_proper_subformula_of F 'or' G & 'not' F
is_proper_subformula_of F 'or' G & 'not' G is_proper_subformula_of F 'or' G & F
  is_proper_subformula_of F 'or' G & G is_proper_subformula_of F 'or' G
proof
  'not' F '&' 'not' G is_immediate_constituent_of F 'or' G;
  hence
A1: 'not' F '&' 'not' G is_proper_subformula_of F 'or' G by ZF_LANG:61;
  'not' F is_immediate_constituent_of 'not' F '&' 'not' G & 'not' G
  is_immediate_constituent_of 'not' F '&' 'not' G;
  hence
A2: 'not' F is_proper_subformula_of F 'or' G & 'not' G
  is_proper_subformula_of F 'or' G by A1,Th40;
  F is_immediate_constituent_of 'not' F & G is_immediate_constituent_of
  'not' G;
  hence thesis by A2,Th40;
end;
