let X1, X2, X3, X4 be set ; :: thesis: ( ( X1 c= [:X1,X2,X3,X4:] or X1 c= [:X2,X3,X4,X1:] or X1 c= [:X3,X4,X1,X2:] or X1 c= [:X4,X1,X2,X3:] ) implies X1 = {} )
assume that
A1: ( X1 c= [:X1,X2,X3,X4:] or X1 c= [:X2,X3,X4,X1:] or X1 c= [:X3,X4,X1,X2:] or X1 c= [:X4,X1,X2,X3:] ) and
A2: X1 <> {} ; :: thesis: contradiction
A3: ( [:X1,X2,X3,X4:] <> {} or [:X2,X3,X4,X1:] <> {} or [:X3,X4,X1,X2:] <> {} or [:X4,X1,X2,X3:] <> {} ) by A1, A2, XBOOLE_1:3;
then A4: ( X2 <> {} & X3 <> {} ) by Th55;
A5: X4 <> {} by A3, Th55;

per cases ( X1 c= [:X1,X2,X3,X4:] or X1 c= [:X2,X3,X4,X1:] or X1 c= [:X3,X4,X1,X2:] or X1 c= [:X4,X1,X2,X3:] ) by A1;

suppose A6: X1 c= [:X1,X2,X3,X4:] ; :: thesis: contradiction

consider v being set such that
A7: v in X1 and
A8: for x1, x2, x3, x4 being set st ( x1 in X1 or x2 in X1 ) holds
v <> [x1,x2,x3,x4] by A2, Th34;
reconsider v = v as Element of [:X1,X2,X3,X4:] by A6, A7;
v = [(v `1),(v `2),(v `3),(v `4)] by A2, A4, A5, Th60;
hence contradiction by A2, A8; :: thesis: verum

end;

suppose X1 c= [:X2,X3,X4,X1:] ; :: thesis: contradiction

then X1 c= [:[:X2,X3:],X4,X1:] by Th54;
hence contradiction by A2, Th49; :: thesis: verum

end;

suppose X1 c= [:X3,X4,X1,X2:] ; :: thesis: contradiction

then X1 c= [:[:X3,X4:],X1,X2:] by Th54;
hence contradiction by A2, Th49; :: thesis: verum

end;

suppose A9: X1 c= [:X4,X1,X2,X3:] ; :: thesis: contradiction

consider v being set such that
A10: v in X1 and
A11: for x1, x2, x3, x4 being set st ( x1 in X1 or x2 in X1 ) holds
v <> [x1,x2,x3,x4] by A2, Th34;
reconsider v = v as Element of [:X4,X1,X2,X3:] by A9, A10;
v = [(v `1),(v `2),(v `3),(v `4)] by A2, A4, A5, Th60;
hence contradiction by A2, A11; :: thesis: verum

end;

end;