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;

theorem Th3:
  1<= len H
proof
  per cases by Th2;
  suppose
    H is atomic;
    then ex n st H = atom.n;
    hence thesis by FINSEQ_1:40;
  end;
  suppose
    H is negative;
    then consider H1 such that
A1: H = 'not' H1;
    len(H) = 1+len(H1) by A1,FINSEQ_5:8;
    hence thesis by NAT_1:11;
  end;
  suppose
    H is conjunctive;
    then consider F,G such that
A2: H = F '&' G;
A3: 1 <= 1+len(F) & 1+len(F) <= 1+len(F)+len(G) by NAT_1:11;
    len(H) = 1+len(F)+len(G) by A2,Lm2;
    hence thesis by A3,XXREAL_0:2;
  end;
  suppose
    H is disjunctive;
    then consider F,G such that
A4: H = F 'or' G;
A5: 1 <= 1+len(F) & 1+len(F) <= 1+len(F)+len(G) by NAT_1:11;
    len(H) = 1+len(F)+len(G) by A4,Lm2;
    hence thesis by A5,XXREAL_0:2;
  end;
  suppose
    H is next;
    then consider H1 such that
A6: H = 'X' H1;
    len(H) = 1+len(H1) by A6,FINSEQ_5:8;
    hence thesis by NAT_1:11;
  end;
  suppose
    H is Until;
    then consider F,G such that
A7: H = F 'U' G;
A8: 1 <= 1+len(F) & 1+len(F) <= 1+len(F)+len(G) by NAT_1:11;
    len(H) = 1+len(F)+len(G) by A7,Lm2;
    hence thesis by A8,XXREAL_0:2;
  end;
  suppose
    H is Release;
    then consider F,G such that
A9: H = F 'R' G;
A10: 1 <= 1+len(F) & 1+len(F) <= 1+len(F)+len(G) by NAT_1:11;
    len(H) = 1+len(F)+len(G) by A9,Lm2;
    hence thesis by A10,XXREAL_0:2;
  end;
end;
