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