reserve X,Y,Z,Z1,Z2,D for set,x,y for object;
reserve SFX,SFY,SFZ for set;
reserve F,G for Subset-Family of D;
reserve P for Subset of D;

theorem Th32:
  F <> {} implies COMPLEMENT(F) <> {}
proof
  set X = the Element of F;
  assume
A1: F <> {};
  then reconsider X as Subset of D by TARSKI:def 3;
  X`` = X;
  hence thesis by A1,Def7;
end;
