
theorem
for X,Y be non empty set, p be set holds
   X-section({}[:X,Y:],p) = {} & Y-section({}[:X,Y:],p) = {}
& ( p in X implies X-section([#][:X,Y:],p) = Y )
& ( p in Y implies Y-section([#][:X,Y:],p) = X )
proof
   let X,Y be non empty set, p be set;
   thus X-section({}[:X,Y:],p) = {};
   thus Y-section({}[:X,Y:],p) = {};
A3: [#]X = X & [#]Y = Y by SUBSET_1:def 3; then
A4: [#][:X,Y:] = [:[#]X,[#]Y:] by SUBSET_1:def 3;
   hence p in X implies X-section([#][:X,Y:],p) = Y by A3,Th16;
   thus thesis by A3,A4,Th16;
end;
