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 th12:
  F |-0 'G' A implies F |-0 'G' 'X' A
  proof
    assume F |-0 'G' A;
    then consider f such that
A1: f.len f= 'G' A and
A2: 1<=len f and
A3: for i be Nat st 1<=i & i<=len f holds prc0 f,F,i;
    set g=f^<*('G' 'X' A)*>;
A4: len g=len f+len<*('G' 'X' A)*> by FINSEQ_1:22
   .=len f+1 by FINSEQ_1:39;
   then A5: len f<len g by NAT_1:16;
   then A6: g/.len f=g.len f by A2,Lm1
   .= 'G' A by A1,A2,FINSEQ_1:64;
   1<=len g by A2,A4,NAT_1:16;then
   g/.len g=g.len g by Lm1
   .= 'G' 'X' A by A4,FINSEQ_1:42;
   then g/.len f NEX0_rule g/.len g by A6;
   then prc0 g,F,len g by A2,A5;
   hence F |-0 'G' 'X' A by A2,A3,Th40;
 end;
