reserve k,m,n for Element of NAT,
  a,X,Y for set,
  D,D1,D2 for non empty set;
reserve p,q for FinSequence of NAT;
reserve x,y,z,t for Variable;
reserve F,F1,G,G1,H,H1 for ZF-formula;
reserve sq,sq9 for FinSequence;

theorem Th52:
  F is_immediate_constituent_of 'not' H iff F = H
proof
  thus F is_immediate_constituent_of 'not' H implies F = H
  proof
A1: now
      given x such that
A2:   'not' H = All(x,F);
      ('not' H).1 = 2 by FINSEQ_1:41;
      hence contradiction by A2,Th17;
    end;
A3: now
      given H1 such that
A4:   'not' H = F '&' H1 or 'not' H = H1 '&' F;
      ('not' H).1 = 2 by FINSEQ_1:41;
      hence contradiction by A4,Th16;
    end;
    assume F is_immediate_constituent_of 'not' H;
    then
    'not' H = 'not' F or ( ex H1 st 'not' H = F '&' H1 or 'not' H = H1 '&'
    F ) or ex x st 'not' H = All(x,F);
    hence thesis by A3,A1,FINSEQ_1:33;
  end;
  thus thesis;
end;
