let F, H be LTL-formula; :: thesis:
let W, W1 be Subset of (Subformulae H); :: thesis:
assume A1: W1 = W \/ {F} ; :: thesis:

now :: thesis:

per cases ;

suppose F in W ; :: thesis:

then {F} c= W by ZFMISC_1:31;
then W1 = W by A1, XBOOLE_1:12;
hence len W1 <= (len W) + (len F) by XREAL_1:31; :: thesis:

end;

suppose not F in W ; :: thesis:

hence len W1 <= (len W) + (len F) by A1, Th11; :: thesis:

end;

end;

end;

hence len W1 <= (len W) + (len F) ; :: thesis: