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 Th9:
  (SAT M).[n,'X' A]=(SAT M).[n+1,A]
 proof
  set f=TFALSUM,sm=SAT M;
  per cases by XBOOLEAN:def 3;
  suppose A1: (SAT M).[n,f 'U' A]=1;
   then consider i such that
    A2: 0<i & sm.[n+i,A]=1 and
    A3: for j st 1<=j & j<i holds sm.[n+j,f]=1 by Def11;
   now assume A4: 1<i;
    not sm.[n+1,f]=1 by Def11;
    hence contradiction by A3,A4;
   end;
   hence thesis by A1,A2,NAT_1:25;
  end;
  suppose A5: (SAT M).[n,f 'U' A]=0;
   now assume A6: sm.[n+1,A]=1;
    not 0<1 or not sm.[n+1,A]=1 or not for j st 1<=j & j<1 holds sm.[n+j,f]=1
by A5,Def11;
    hence contradiction by A6;
   end;
   hence thesis by A5,XBOOLEAN:def 3;
  end;
 end;
