let X, Y, Z be set ; :: thesis: ( [:(X \ Y),Z:] = [:X,Z:] \ [:Y,Z:] & [:Z,(X \ Y):] = [:Z,X:] \ [:Z,Y:] )
A1: for x, y being set holds
( [x,y] in [:(X \ Y),Z:] iff [x,y] in [:X,Z:] \ [:Y,Z:] )

proof

let x, y be set ; :: thesis: ( [x,y] in [:(X \ Y),Z:] iff [x,y] in [:X,Z:] \ [:Y,Z:] )
thus ( [x,y] in [:(X \ Y),Z:] implies [x,y] in [:X,Z:] \ [:Y,Z:] ) :: thesis: ( [x,y] in [:X,Z:] \ [:Y,Z:] implies [x,y] in [:(X \ Y),Z:] )

proof

assume A2: [x,y] in [:(X \ Y),Z:] ; :: thesis: [x,y] in [:X,Z:] \ [:Y,Z:]
then A3: x in X \ Y by Lm17;
then A4: x in X by XBOOLE_0:def 5;
not x in Y by A3, XBOOLE_0:def 5;
then A5: not [x,y] in [:Y,Z:] by Lm17;
y in Z by A2, Lm17;
then [x,y] in [:X,Z:] by A4, Lm17;
hence [x,y] in [:X,Z:] \ [:Y,Z:] by A5, XBOOLE_0:def 5; :: thesis: verum

end;

assume A6: [x,y] in [:X,Z:] \ [:Y,Z:] ; :: thesis: [x,y] in [:(X \ Y),Z:]
then A7: [x,y] in [:X,Z:] by XBOOLE_0:def 5;
then A8: y in Z by Lm17;
not [x,y] in [:Y,Z:] by A6, XBOOLE_0:def 5;
then A9: ( not x in Y or not y in Z ) by Lm17;
x in X by A7, Lm17;
then x in X \ Y by A7, A9, Lm17, XBOOLE_0:def 5;
hence [x,y] in [:(X \ Y),Z:] by A8, Lm17; :: thesis: verum

end;

[:X,Z:] \ [:Y,Z:] c= [:X,Z:] by XBOOLE_1:36;
hence A10: [:(X \ Y),Z:] = [:X,Z:] \ [:Y,Z:] by A1, Lm18; :: thesis: [:Z,(X \ Y):] = [:Z,X:] \ [:Z,Y:]
A11: for y, x being set holds
( [y,x] in [:Z,(X \ Y):] iff [y,x] in [:Z,X:] \ [:Z,Y:] )

proof

let y, x be set ; :: thesis: ( [y,x] in [:Z,(X \ Y):] iff [y,x] in [:Z,X:] \ [:Z,Y:] )
A12: ( [x,y] in [:X,Z:] & not [x,y] in [:Y,Z:] iff ( [y,x] in [:Z,X:] & not [y,x] in [:Z,Y:] ) ) by Th107;
( [y,x] in [:Z,(X \ Y):] iff [x,y] in [:(X \ Y),Z:] ) by Th107;
hence ( [y,x] in [:Z,(X \ Y):] iff [y,x] in [:Z,X:] \ [:Z,Y:] ) by A10, A12, XBOOLE_0:def 5; :: thesis: verum

end;

[:Z,X:] \ [:Z,Y:] c= [:Z,X:] by XBOOLE_1:36;
hence [:Z,(X \ Y):] = [:Z,X:] \ [:Z,Y:] by A11, Lm18; :: thesis: verum