reserve k,n,m for Nat,
  a,x,X,Y for set,
  D,D1,D2,S for non empty set,
  p,q for FinSequence of NAT;
reserve F,F1,G,G1,H,H1,H2 for LTL-formula;
reserve sq,sq9 for FinSequence;
reserve L,L9 for FinSequence;
reserve j for Nat;
reserve j1 for Element of NAT;

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;
