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 conco:
  F is consistent iff ex A st not F |- A
  proof
    hereby assume
A0:   F is consistent;
      assume
A1:   for A holds F |- A;then
A2:   F |- Prop 0;
      F |- 'not' Prop 0 by A1;
      hence contradiction by A2,A0;
    end;
    assume
A4: ex A st not F |- A;
    assume not F is consistent;then
    consider A such that
A3: F |- A & F |- 'not' A;
    now
      let B;
      F |- 'not' A => (A => B) by naab;then
      F|- A => B by A3,th43;
      hence F |- B by A3,th43;
    end;
    hence contradiction by A4;
  end;
