theorem Th63:
  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
  assume
A1: 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;
  then F is_subformula_of G & G is_subformula_of H by Th52;
  hence F is_subformula_of H by Th57;
  thus thesis by A1,Th59,Th61;
end;
