theorem
  F is_proper_subformula_of 'X' H implies F is_subformula_of H
proof
  assume
A1: F is_proper_subformula_of 'X' H;
A2: 'X' H is next;
  then not 'X' H is negative by Lm19;
  then the_argument_of ('X' H) = H by A2,Def18;
  hence thesis by A1,A2,Th37;
end;
