theorem
  F is_proper_subformula_of G 'U' H implies F is_subformula_of G or F
  is_subformula_of H
proof
  assume
A1: F is_proper_subformula_of G 'U' H;
A2: G 'U' H is Until;
  then
  the_left_argument_of (G 'U' H) = G & the_right_argument_of (G 'U' H) =H
  by Def19,Def20;
  hence thesis by A1,A2,Th38;
end;
