reserve p,q,r,s,A,B for Element of PL-WFF,
  F,G,H for Subset of PL-WFF,
  k,n for Element of NAT,
  f,f1,f2 for FinSequence of PL-WFF;
reserve M for PLModel;

theorem finin:
  for F st not F is consistent
  ex G st G is finite & not G is consistent & G c= F
  proof
    let F;
    assume not F is consistent;then
    consider A such that
A1: F |- A & F |- 'not' A;
    consider G such that
A2: G c= F & G is finite & G |- A by exfin,A1;
    consider H such that
A2a: H c= F & H is finite & H |- 'not' A by exfin,A1;
A5: G \/ H |- A by A2,monmp,XBOOLE_1:11;
    G \/ H |- 'not' A by A2a,XBOOLE_1:11,monmp;then
    not G \/ H is consistent by A5;
    hence thesis by XBOOLE_1:8,A2,A2a;
  end;
