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 mon:
  F c= G & F |-0 A implies G |-0 A
proof
  assume A0: F c= G & F |-0 A;then
  consider f such that
A1: f.len f = A & 1 <= len f and
A2: for i be Nat st 1<=i & i<=len f holds prc0 f,F,i;
    now
      let i be Nat;
      assume 1<=i & i<=len f;then
      per cases by Def29,A2;
      suppose f.i in F;
        hence prc0 f,G,i by A0;
      end;
      suppose f.i in LTL0_axioms or
        (ex j,k be Nat st 1<=j & j<i & 1<=k & k<i & (
        f/.j,f/.k MP_rule f/.i or f/.j,f/.k MP0_rule f/.i
        or f/.j,f/.k IND0_rule f/.i)) or
        ex j be Nat st 1<=j & j<i &
        (f/.j NEX0_rule f/.i or f/.j REFL0_rule f/.i);
        hence prc0 f,G,i;
      end;
    end;
  hence G |-0 A by A1;
end;
