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
  (SAT M).(A <=> B) = 1 iff (SAT M).A = (SAT M).B
  proof
A2: (SAT M).B = TRUE or (SAT M).B = FALSE by XBOOLEAN:def 3;
    hereby
      assume (SAT M).(A <=> B) = 1;then
      (SAT M).A <=> (SAT M).B = 1 by semequ2;
      hence (SAT M).A = (SAT M).B by A2;
    end;
    assume A3:(SAT M).A = (SAT M).B;
    thus (SAT M).(A <=> B) = (SAT M).A <=> (SAT M).B by semequ2
    .=1 by A3,A2;
  end;
