reserve i,j,e,u for object;
reserve I for set; 
reserve x,X,Y,Z,V for ManySortedSet of I;

theorem
  X overlaps Y or X overlaps Z implies X overlaps Y (\/) Z
proof
  assume
A1: X overlaps Y or X overlaps Z;
  per cases by A1;
  suppose
    X overlaps Y;
    then consider x such that
A2: x in X and
A3: x in Y by Th11;
    x in Y (\/) Z by A3,Th7;
    hence thesis by A2,Th10;
  end;
  suppose
    X overlaps Z;
    then consider x such that
A4: x in X and
A5: x in Z by Th11;
    x in Y (\/) Z by A5,Th7;
    hence thesis by A4,Th10;
  end;
end;
