let X be non empty compact Subset of (TOP-REAL 2); :: thesis: ( (SW-corner X) `1 <= (S-min X) `1 & (SW-corner X) `1 <= (S-max X) `1 & (SW-corner X) `1 <= (SE-corner X) `1 & (S-min X) `1 <= (S-max X) `1 & (S-min X) `1 <= (SE-corner X) `1 & (S-max X) `1 <= (SE-corner X) `1 )
A1: (SW-corner X) `1 = inf (proj1 | X) by EUCLID:56;
set LX = S-most X;
A2: (S-min X) `1 = inf (proj1 | (S-most X)) by EUCLID:56;
A3: (S-max X) `1 = sup (proj1 | (S-most X)) by EUCLID:56;
A4: (SE-corner X) `1 = sup (proj1 | X) by EUCLID:56;
A5: (SW-corner X) `1 <= (S-min X) `1 by A1, A2, Th62, XBOOLE_1:17;
A6: (S-min X) `1 <= (S-max X) `1 by A2, A3, Th53;
A7: (S-max X) `1 <= (SE-corner X) `1 by A3, A4, Th63, XBOOLE_1:17;
thus (SW-corner X) `1 <= (S-min X) `1 by A1, A2, Th62, XBOOLE_1:17; :: thesis: ( (SW-corner X) `1 <= (S-max X) `1 & (SW-corner X) `1 <= (SE-corner X) `1 & (S-min X) `1 <= (S-max X) `1 & (S-min X) `1 <= (SE-corner X) `1 & (S-max X) `1 <= (SE-corner X) `1 )
thus (SW-corner X) `1 <= (S-max X) `1 by A5, A6, XXREAL_0:2; :: thesis: ( (SW-corner X) `1 <= (SE-corner X) `1 & (S-min X) `1 <= (S-max X) `1 & (S-min X) `1 <= (SE-corner X) `1 & (S-max X) `1 <= (SE-corner X) `1 )
hence (SW-corner X) `1 <= (SE-corner X) `1 by A7, XXREAL_0:2; :: thesis: ( (S-min X) `1 <= (S-max X) `1 & (S-min X) `1 <= (SE-corner X) `1 & (S-max X) `1 <= (SE-corner X) `1 )
thus (S-min X) `1 <= (S-max X) `1 by A2, A3, Th53; :: thesis: ( (S-min X) `1 <= (SE-corner X) `1 & (S-max X) `1 <= (SE-corner X) `1 )
thus (S-min X) `1 <= (SE-corner X) `1 by A6, A7, XXREAL_0:2; :: thesis: (S-max X) `1 <= (SE-corner X) `1
thus (S-max X) `1 <= (SE-corner X) `1 by A3, A4, Th63, XBOOLE_1:17; :: thesis: verum