let x, y, k be R_eal; :: thesis: ( k <= 0 implies ( k * (min (x,y)) = max ((k * x),(k * y)) & k * (max (x,y)) = min ((k * x),(k * y)) ) )
assume A1: k <= 0 ; :: thesis: ( k * (min (x,y)) = max ((k * x),(k * y)) & k * (max (x,y)) = min ((k * x),(k * y)) )

hereby :: thesis: k * (max (x,y)) = min ((k * x),(k * y))

per cases ( max (x,y) = x or max (x,y) = y ) by XXREAL_0:16;

suppose max (x,y) = x ; :: thesis: k * (min (x,y)) = max ((k * x),(k * y))

then A2: y <= x by XXREAL_0:def 10;
then k * x <= k * y by A1, XXREAL_3:101;
then max ((k * x),(k * y)) = k * y by XXREAL_0:def 10;
hence k * (min (x,y)) = max ((k * x),(k * y)) by A2, XXREAL_0:def 9; :: thesis: verum

end;

suppose max (x,y) = y ; :: thesis: k * (min (x,y)) = max ((k * x),(k * y))

then A3: x <= y by XXREAL_0:def 10;
then k * y <= k * x by A1, XXREAL_3:101;
then max ((k * x),(k * y)) = k * x by XXREAL_0:def 10;
hence k * (min (x,y)) = max ((k * x),(k * y)) by A3, XXREAL_0:def 9; :: thesis: verum

end;

end;

end;

per cases ( min (x,y) = x or min (x,y) = y ) by XXREAL_0:15;

suppose min (x,y) = x ; :: thesis: k * (max (x,y)) = min ((k * x),(k * y))

then A4: x <= y by XXREAL_0:def 9;
then k * y <= k * x by A1, XXREAL_3:101;
then min ((k * x),(k * y)) = k * y by XXREAL_0:def 9;
hence k * (max (x,y)) = min ((k * x),(k * y)) by A4, XXREAL_0:def 10; :: thesis: verum

end;

suppose min (x,y) = y ; :: thesis: k * (max (x,y)) = min ((k * x),(k * y))

then A5: y <= x by XXREAL_0:def 9;
then k * x <= k * y by A1, XXREAL_3:101;
then min ((k * y),(k * x)) = k * x by XXREAL_0:def 9;
hence k * (max (x,y)) = min ((k * x),(k * y)) by A5, XXREAL_0:def 10; :: thesis: verum

end;

end;