reserve a,b,c for boolean object;
reserve p,q,r,s,A,B,C for Element of LTLB_WFF,
        F,G,X,Y for Subset of LTLB_WFF,
        i,j,k,n for Element of NAT,
        f,f1,f2,g for FinSequence of LTLB_WFF;
reserve M for LTLModel;

theorem Th37:
  A is axltl1 or A is axltl1a or A is axltl2 or A is axltl3 or
  A is axltl4 or A is axltl5 or A is axltl6 implies F|=A
 proof
  assume A1: A is axltl1 or A is axltl1a or A is axltl2 or A is axltl3 or A is
axltl4 or A is axltl5 or A is axltl6;
  let M;
  assume M|=F;
  let n be Element of NAT;
  per cases by A1;
  suppose A is axltl1;
   then consider B be Element of LTLB_WFF such that
    A2: A=('not'('X' B))=>('X'('not' B));
   thus thesis by A2,Th15;
  end;
  suppose A is axltl1a;
   then consider B be Element of LTLB_WFF such that
    A3: A=('X'('not' B))=>('not'('X' B));
   thus thesis by A3,Th16;
  end;
  suppose A is axltl2;
   then consider B,C be Element of LTLB_WFF such that
    A4: A=('X'(B=>C))=>(('X' B)=>('X' C));
   thus thesis by A4,Th17;
  end;
  suppose A is axltl3;
   then consider B be Element of LTLB_WFF such that
    A5: A=('G' B)=>(B '&&'('X'('G' B)));
   thus thesis by A5,Th18;
  end;
  suppose A is axltl4;
   then consider B,C such that
    A6: A=(B 'U' C)=>(('X' C)'or'('X'(B '&&'(B 'U' C))));
   thus thesis by A6,Th19;
  end;
  suppose A is axltl5;
   then consider B,C such that
    A7: A=(('X' C)'or'('X'(B '&&'(B 'U' C))))=>(B 'U' C);
   thus thesis by A7,Th20;
  end;
  suppose A is axltl6;
   then consider B,C such that
    A8: A=(B 'U' C)=>('X'('F' C));
   thus thesis by A8,Th21;
  end;
 end;
