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 Th56:
  F|-q implies F\/{p}|-q
 proof
  assume F|-q;
  then consider f such that
   A1: f.len f=q & 1<=len f and
   A2: for i be Nat st 1<=i & i<=len f holds prc f,F,i;
  now let i be Nat;
   assume 1<=i & i<=len f;
   then f.i in LTL_axioms or f.i in F 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 IND_rule f/.i) or ex j be Nat st 1<=j
& j<i & f/.j NEX_rule f/.i by Def29,A2;
   then f.i in LTL_axioms or f.i in F\/{p} 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 IND_rule f/.i) or ex j be Nat st
1<=j & j<i & f/.j NEX_rule f/.i by XBOOLE_0:def 3;
   hence prc f,F\/{p},i;
  end;
  hence F\/{p}|-q by A1;
 end;
