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 Th4:
  F |=0 'G' A & F |=0 'G' (A=>B) implies F |=0 'G' B
 proof
  assume that
   A1: F |=0 'G' A and
   A2: F |=0 'G' (A=>B);
   let M;
   assume
B10: M |=0 F;then
B11: M |=0 'G' A by A1;
B12: M |=0 'G' (A=>B) by B10,A2;
     now
       let i;
B13:   (SAT M).[0+i,A]=1 by B11,LTLAXIO1:10;
       (SAT M).[0+i,A=>B]=1 by B12,LTLAXIO1:10;then
       (SAT M).[i,A] => (SAT M).[i,B] = 1 by LTLAXIO1:def 11;
       hence (SAT M).[0+i,B]=1 by B13;
     end;
     hence M |=0 'G' B by LTLAXIO1:10;
 end;
