let X be non empty compact Subset of (TOP-REAL 2); :: thesis: ( (SW-corner X) `2 <= (W-min X) `2 & (SW-corner X) `2 <= (W-max X) `2 & (SW-corner X) `2 <= (NW-corner X) `2 & (W-min X) `2 <= (W-max X) `2 & (W-min X) `2 <= (NW-corner X) `2 & (W-max X) `2 <= (NW-corner X) `2 )
A1: (SW-corner X) `2 = inf (proj2 | X) by EUCLID:56;
set LX = W-most X;
A2: (W-min X) `2 = inf (proj2 | (W-most X)) by EUCLID:56;
A3: (W-max X) `2 = sup (proj2 | (W-most X)) by EUCLID:56;
A4: (NW-corner X) `2 = sup (proj2 | X) by EUCLID:56;
A5: (SW-corner X) `2 <= (W-min X) `2 by A1, A2, Th62, XBOOLE_1:17;
A6: (W-min X) `2 <= (W-max X) `2 by A2, A3, Th53;
A7: (W-max X) `2 <= (NW-corner X) `2 by A3, A4, Th63, XBOOLE_1:17;
thus (SW-corner X) `2 <= (W-min X) `2 by A1, A2, Th62, XBOOLE_1:17; :: thesis: ( (SW-corner X) `2 <= (W-max X) `2 & (SW-corner X) `2 <= (NW-corner X) `2 & (W-min X) `2 <= (W-max X) `2 & (W-min X) `2 <= (NW-corner X) `2 & (W-max X) `2 <= (NW-corner X) `2 )
thus (SW-corner X) `2 <= (W-max X) `2 by A5, A6, XXREAL_0:2; :: thesis: ( (SW-corner X) `2 <= (NW-corner X) `2 & (W-min X) `2 <= (W-max X) `2 & (W-min X) `2 <= (NW-corner X) `2 & (W-max X) `2 <= (NW-corner X) `2 )
hence (SW-corner X) `2 <= (NW-corner X) `2 by A7, XXREAL_0:2; :: thesis: ( (W-min X) `2 <= (W-max X) `2 & (W-min X) `2 <= (NW-corner X) `2 & (W-max X) `2 <= (NW-corner X) `2 )
thus (W-min X) `2 <= (W-max X) `2 by A2, A3, Th53; :: thesis: ( (W-min X) `2 <= (NW-corner X) `2 & (W-max X) `2 <= (NW-corner X) `2 )
thus (W-min X) `2 <= (NW-corner X) `2 by A6, A7, XXREAL_0:2; :: thesis: (W-max X) `2 <= (NW-corner X) `2
thus (W-max X) `2 <= (NW-corner X) `2 by A3, A4, Th63, XBOOLE_1:17; :: thesis: verum