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 Th31:
  (for P holds P in F iff P in G) implies F=G
proof
  assume
A1: for P holds P in F iff P in G;
  thus F c= G
  by A1;
  let x be object;
  assume x in G;
  hence thesis by A1;
end;
