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;

theorem Th11:
  H is conjunctive or H is disjunctive or H is Until or H is
  Release implies len(H) =1+ len(the_left_argument_of H)+len(
  the_right_argument_of H) & len(the_left_argument_of H) < len(H) & len(
  the_right_argument_of H) < len(H)
proof
  set iL=len(the_left_argument_of H);
  set iR=len(the_right_argument_of H);
  set iR1=iR+1;
  assume
A1: H is conjunctive or H is disjunctive or H is Until or H is Release;
  per cases by A1;
  suppose
    H is conjunctive;
    then H = (the_left_argument_of H) '&' (the_right_argument_of H) by Th6;
    then
A2: len(H) = 1+iL+iR by Lm2;
    1<=iR1 by NAT_1:11;
    then
A3: iL < iL +iR1 by NAT_1:19;
    1<=1+iL by NAT_1:11;
    hence thesis by A2,A3,NAT_1:19;
  end;
  suppose
    H is disjunctive;
    then H = (the_left_argument_of H) 'or' (the_right_argument_of H) by Th7;
    then
A4: len(H) = 1+iL+iR by Lm2;
    1<=iR1 by NAT_1:11;
    then
A5: iL < iL +iR1 by NAT_1:19;
    1<=1+iL by NAT_1:11;
    hence thesis by A4,A5,NAT_1:19;
  end;
  suppose
    H is Until;
    then H = the_left_argument_of H 'U' the_right_argument_of H by Th8;
    then
A6: len(H) = 1+iL+iR by Lm2;
    1<=iR1 by NAT_1:11;
    then
A7: iL < iL +iR1 by NAT_1:19;
    1<=1+iL by NAT_1:11;
    hence thesis by A6,A7,NAT_1:19;
  end;
  suppose
    H is Release;
    then H = the_left_argument_of H 'R' the_right_argument_of H by Th9;
    then
A8: len(H) = 1+iL+iR by Lm2;
    1<=iR1 by NAT_1:11;
    then
A9: iL < iL +iR1 by NAT_1:19;
    1<=1+iL by NAT_1:11;
    hence thesis by A8,A9,NAT_1:19;
  end;
end;
