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 Th40:
  F is_proper_subformula_of G & G is_subformula_of H or F
  is_subformula_of G & G is_proper_subformula_of H or F is_subformula_of G & G
  is_immediate_constituent_of H or F is_immediate_constituent_of G & G
  is_subformula_of H or F is_proper_subformula_of G & G
  is_immediate_constituent_of H or F is_immediate_constituent_of G & G
  is_proper_subformula_of H implies F is_proper_subformula_of H
proof
A1: F is_immediate_constituent_of G implies F is_proper_subformula_of G by
ZF_LANG:61;
A2: now
    assume
A3: F is_proper_subformula_of G;
    then
A4: len F < len G by ZF_LANG:62;
    assume
A5: G is_subformula_of H;
    F is_subformula_of G by A3;
    then
A6: F is_subformula_of H by A5,ZF_LANG:65;
    len G <= len H by A5,Th39;
    hence F is_proper_subformula_of H by A4,A6;
  end;
  G is_immediate_constituent_of H implies G is_proper_subformula_of H by
ZF_LANG:61;
  hence thesis by A2,A1;
end;
