reserve k,n,n1,m,m1,m0,h,i,j for Nat,
  a,x,y,X,X1,X2,X3,X4,Y for set;
reserve L,L1,L2 for FinSequence;
reserve F,F1,G,G1,H for LTL-formula;
reserve W,W1,W2 for Subset of Subformulae H;
reserve v for LTL-formula;
reserve N,N1,N2,N10,N20,M for strict LTLnode over v;
reserve w for Element of Inf_seq(AtomicFamily);
reserve R1,R2 for Real_Sequence;

theorem Th12:
  W1 = W \/ {F} implies len(W1) <= len(W) + len F
proof
  assume
A1: W1 = W \/ {F};
  now
    per cases;
    suppose
      F in W;
      then {F} c= W by ZFMISC_1:31;
      then W1 = W by A1,XBOOLE_1:12;
      hence thesis by XREAL_1:31;
    end;
    suppose
      not F in W;
      hence thesis by A1,Th11;
    end;
  end;
  hence thesis;
end;
