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 th21:
  F |=0 'G' A implies F |= A
proof
  assume
Z1: F |=0 'G' A;
  assume not F |= A;then
  consider M such that
A1: M |= F & not M |= A;
A3: M |=0 F
    proof
      let A;
      assume A in F;then
      M |= A by A1;
      hence M |=0 A;
    end;
  consider i such that
A2: not (SAT M).[i,A] = 1 by A1;
  M |=0 'G' A by A3,Z1;then
  (SAT M).[0+i,A] = 1 by LTLAXIO1:10;
  hence contradiction by A2;
end;
