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 Th18:
  W = W1 \/ W2 implies len(W) <= len(W1) + len(W2)
proof
  defpred P[Nat] means for W,W1,W2 st card W1 <=$1 holds (W = W1 \/ W2 implies
  len(W) <= len(W1) + len(W2));
  set k = card W1;
A1: for k being Nat st P[k] holds P[k + 1]
  proof
    let k be Nat such that
A2: P[k];
    P[k+1]
    proof
      let W,W1,W2 such that
A3:   card W1 <=k+1;
      W = W1 \/ W2 implies len(W) <= len(W1) + len(W2)
      proof
        assume
A4:     W = W1 \/ W2;
        now
          per cases by A3,NAT_1:8;
          suppose
            card W1 <=k;
            hence thesis by A2,A4;
          end;
          suppose
            card W1 = k+1;
            then W1 <> {};
            then consider x being object such that
A5:         x in W1 by XBOOLE_0:def 1;
            x in Subformulae H by A5;
            then reconsider x as LTL-formula by MODELC_2:1;
            set X = {x};
            set Y = W1\X;
            set Z = Y \/ W2;
A6:         X c= W1 by A5,ZFMISC_1:31;
            then card (Y) = card W1 - card X by CARD_2:44
              .= card W1 - 1 by CARD_1:30;
            then card (Y) + 1 = card W1;
            then card (Y) <= k by A3,XREAL_1:6;
            then Z = Y \/ W2 implies len(Z) <= len(Y) + len(W2) by A2;
            then len(Z) <= len(W1) - len x + len(W2) by A5,Th10;
            then len(Z) <= (len(W1) + len(W2)) - len x;
            then
A7:         len(Z) + len x <= len(W1) + len(W2) by XREAL_1:19;
            Z \/ X = (Y \/ X) \/ W2 by XBOOLE_1:4
              .= W1 \/ W2 by A6,XBOOLE_1:45;
            then len(W) <= len(Z) + len x by A4,Th12;
            hence thesis by A7,XXREAL_0:2;
          end;
        end;
        hence thesis;
      end;
      hence thesis;
    end;
    hence thesis;
  end;
A8: P[0]
  proof
    let W,W1,W2;
    assume card W1 <=0;
    then
A9: W1 = {} H;
    then len(W1) = 0 by Th13;
    hence thesis by A9;
  end;
  for k being Nat holds P[k] from NAT_1:sch 2 (A8,A1);
  then P[k];
  hence thesis;
end;
