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