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 tautsat:
  A is tautology iff {}PL-WFF |= A
  proof
    hereby
      assume A1: A is tautology;
      assume not {}PL-WFF |= A; then
      consider M such that
A3:   M |= {}PL-WFF & not M |= A;
      thus contradiction by A3,A1;
    end;
    assume
A4: {}PL-WFF |= A;
    assume not A is tautology;then
    consider M such that
A5: not (SAT M).A=1;
    M |= {}PL-WFF;then
    M |= A by A4;
    hence contradiction by A5;
  end;
