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 th19a:
  {A} |= 'G' 'X' A
  proof
    assume not {A} |= 'G' 'X' A;then
    consider M such that
A1: M |= {A} & not M |= 'G' 'X' A;
    consider i such that
A2: not (SAT M).[i,'G' 'X' A] = 1 by A1;
    consider j such that
A3: not (SAT M).[i+j,'X' A] = 1 by A2,LTLAXIO1:10;
A4: not (SAT M).[i+j+1,A] = 1 by A3,LTLAXIO1:9;
    A in {A} by TARSKI:def 1;then
    M |= A by A1;
    hence contradiction by A4;
end;
