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 Th7:
  (SAT M).[n,A '&&' B]=1 iff (SAT M).[n,A]=1 & (SAT M).[n,B]=1
 proof
  hereby assume(SAT M).[n,A '&&' B]=1;
   then (SAT M).[n,A=>(B=>TFALSUM)]=>(SAT M).[n,TFALSUM]=1 by Def11;
   then (SAT M).[n,A=>(B=>TFALSUM)]=>FALSE=1 by Def11;
   then ((SAT M).[n,A]=>(SAT M).[n,B=>TFALSUM])=0 by Def11;
   then (SAT M).[n,A]=>((SAT M).[n,B]=>(SAT M).[n,TFALSUM])=0 by Def11;
   then A1: (SAT M).[n,A]=>((SAT M).[n,B]=>FALSE)=0 by Def11;
   (SAT M).[n,A]=0 or(SAT M).[n,A]=1 by XBOOLEAN:def 3;
   hence (SAT M).[n,A]=1 & (SAT M).[n,B]=1 by A1;
  end;
  assume that
   A2: (SAT M).[n,A]=1 and
   A3: (SAT M).[n,B]=1;
  (SAT M).[n,B]=>(SAT M).[n,TFALSUM]=0 by A3,Def11;
  then (SAT M).[n,A]=>(SAT M).[n,B=>TFALSUM]=0 by A2,Def11;
  then (SAT M).[n,A=>(B=>TFALSUM)]=0 by Def11;
  then (SAT M).[n,A=>(B=>TFALSUM)]=>(SAT M).[n,TFALSUM]=1;
  hence (SAT M).[n,A '&&' B]=1 by Def11;
 end;
