reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;

theorem Th36:
  {z} = X \/ Y implies X = {z} & Y = {z} or X = {} & Y = {z} or X
  = {z} & Y = {}
proof
  assume
A1: {z} = X \/ Y;
  X <> {} or Y <> {} by A1;
  hence thesis by A1,Lm3,XBOOLE_1:7;
end;
