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 Th36:
  for X being set, F,G being Subset-Family of X st COMPLEMENT F c=
  COMPLEMENT G holds F c= G
proof
  let X be set, F,G be Subset-Family of X such that
A1: COMPLEMENT F c= COMPLEMENT G;
  let x be object;
  assume
A2: x in F;
  then reconsider A = x as Subset of X;
  A in COMPLEMENT COMPLEMENT F by A2;
  then A` in COMPLEMENT F by Def7;
  then A`` in G by A1,Def7;
  hence thesis;
end;
