reserve A,B,C,D,p,q,r for Element of LTLB_WFF,
        F,G,X for Subset of LTLB_WFF,
        M for LTLModel,
        i,j,n for Element of NAT,
        f,f1,f2,g for FinSequence of LTLB_WFF;

theorem th20:
  F |- ('G' (A => B)) => (('G' (A => 'X' A)) => ('G' (A => 'G' B)))
  proof
    A => B in {A =>B} by TARSKI:def 1;then
    A => B in F \/ {A =>B} by XBOOLE_0:def 3;then
    A => B in F \/ {A =>B} \/ {A => 'X' A} by XBOOLE_0:def 3;then
A1: F \/ {A =>B} \/ {A => 'X' A} |- A => B by LTLAXIO1:42;
    A => 'X' A in {A => 'X' A} by TARSKI:def 1;then
    A => 'X' A in F \/ {A =>B} \/ {A => 'X' A} by XBOOLE_0:def 3;then
    F \/ {A =>B} \/ {A => 'X' A} |- A => 'X' A by LTLAXIO1:42;then
    F \/ {A =>B} \/ {A => 'X' A} |- 'G' (A => 'G' B) by LTLAXIO1:45,54,A1;then
    F \/ {A =>B}  |- ('G' (A => 'X' A)) => 'G' (A => 'G' B) by LTLAXIO1:57;
    hence thesis by LTLAXIO1:57;
  end;
