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 Th44:
  X|-p implies X|-'X' p
 proof
  assume X|-p;
  then consider f such that
   A1: f.len f=p and
   A2: 1<=len f and
   A3: for i be Nat st 1<=i & i<=len f holds prc f,X,i;
  set g=f^<*('X' p)*>;
  A4: len g=len f+len<*('X' p)*> 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
   .=p by A1,A2,FINSEQ_1:64;
  1<=len g by A2,A4,NAT_1:16;
  then g/.len g=g.len g by Lm1
   .='X'(g/.len f) by A4,A6,FINSEQ_1:42;
  then g/.len f NEX_rule g/.len g;
  then prc g,X,len g by A2,A5;
  hence X|-'X' p by A2,A3,Th40;
 end;
