theorem Th89:
  [:X,Y:] = {} iff X = {} or Y = {}
proof
  thus [:X,Y:] = {} implies X={} or Y={}
  proof
    assume
A1: [:X,Y:] = {};
    assume X<>{};
    then consider x such that
A2: x in X by XBOOLE_0:7;
    assume Y<>{};
    then consider y such that
A3: y in Y by XBOOLE_0:7;
    [x,y] in [:X,Y:] by A2,A3,Def2;
    hence contradiction by A1;
  end;
  assume
A4: not thesis;
  then consider z such that
A5: z in [:X,Y:] by XBOOLE_0:7;
  ex x,y st x in X & y in Y & [x,y]=z by A5,Def2;
 hence contradiction by A4;
end;
