let X be non empty compact Subset of (TOP-REAL 2); :: thesis: ( (LSeg ((SW-corner X),(W-min X))) /\ X = {(W-min X)} & (LSeg ((W-max X),(NW-corner X))) /\ X = {(W-max X)} )
now :: thesis: for x being object holds
( ( x in (LSeg ((SW-corner X),(W-min X))) /\ X implies x = W-min X ) & ( x = W-min X implies x in (LSeg ((SW-corner X),(W-min X))) /\ X ) )
let x be object ; :: thesis: ( ( x in (LSeg ((SW-corner X),(W-min X))) /\ X implies x = W-min X ) & ( x = W-min X implies x in (LSeg ((SW-corner X),(W-min X))) /\ X ) )
A1: W-min X in LSeg ((SW-corner X),(W-min X)) by RLTOPSP1:68;
hereby :: thesis: ( x = W-min X implies x in (LSeg ((SW-corner X),(W-min X))) /\ X )
W-min X in W-most X by Th34;
then ( SW-corner X in LSeg ((SW-corner X),(NW-corner X)) & W-min X in LSeg ((SW-corner X),(NW-corner X)) ) by RLTOPSP1:68, XBOOLE_0:def 4;
then A2: LSeg ((SW-corner X),(W-min X)) c= LSeg ((SW-corner X),(NW-corner X)) by TOPREAL1:6;
assume A3: x in (LSeg ((SW-corner X),(W-min X))) /\ X ; :: thesis: x = W-min X
then A4: x in LSeg ((SW-corner X),(W-min X)) by XBOOLE_0:def 4;
reconsider p = x as Point of (TOP-REAL 2) by A3;
x in X by A3, XBOOLE_0:def 4;
then p in W-most X by A4, A2, XBOOLE_0:def 4;
then A5: (W-min X) `2 <= p `2 by Th31;
(SW-corner X) `2 <= (W-min X) `2 by Th30;
then p `2 <= (W-min X) `2 by A4, TOPREAL1:4;
then A6: p `2 = (W-min X) `2 by A5, XXREAL_0:1;
(SW-corner X) `1 = (W-min X) `1 by Th29;
then A7: p `1 = (W-min X) `1 by A4, GOBOARD7:5;
p = |[(p `1),(p `2)]| by EUCLID:53;
hence x = W-min X by A7, A6, EUCLID:53; :: thesis: verum
end;
W-min X in W-most X by Th34;
then A8: W-min X in X by XBOOLE_0:def 4;
assume x = W-min X ; :: thesis: x in (LSeg ((SW-corner X),(W-min X))) /\ X
hence x in (LSeg ((SW-corner X),(W-min X))) /\ X by A8, A1, XBOOLE_0:def 4; :: thesis: verum
end;
hence (LSeg ((SW-corner X),(W-min X))) /\ X = {(W-min X)} by TARSKI:def 1; :: thesis: (LSeg ((W-max X),(NW-corner X))) /\ X = {(W-max X)}
now :: thesis: for x being object holds
( ( x in (LSeg ((W-max X),(NW-corner X))) /\ X implies x = W-max X ) & ( x = W-max X implies x in (LSeg ((W-max X),(NW-corner X))) /\ X ) )
let x be object ; :: thesis: ( ( x in (LSeg ((W-max X),(NW-corner X))) /\ X implies x = W-max X ) & ( x = W-max X implies x in (LSeg ((W-max X),(NW-corner X))) /\ X ) )
A9: W-max X in LSeg ((W-max X),(NW-corner X)) by RLTOPSP1:68;
hereby :: thesis: ( x = W-max X implies x in (LSeg ((W-max X),(NW-corner X))) /\ X )
W-max X in W-most X by Th34;
then ( NW-corner X in LSeg ((SW-corner X),(NW-corner X)) & W-max X in LSeg ((SW-corner X),(NW-corner X)) ) by RLTOPSP1:68, XBOOLE_0:def 4;
then A10: LSeg ((W-max X),(NW-corner X)) c= LSeg ((SW-corner X),(NW-corner X)) by TOPREAL1:6;
assume A11: x in (LSeg ((W-max X),(NW-corner X))) /\ X ; :: thesis: x = W-max X
then A12: x in LSeg ((W-max X),(NW-corner X)) by XBOOLE_0:def 4;
reconsider p = x as Point of (TOP-REAL 2) by A11;
x in X by A11, XBOOLE_0:def 4;
then p in W-most X by A12, A10, XBOOLE_0:def 4;
then A13: p `2 <= (W-max X) `2 by Th31;
(W-max X) `2 <= (NW-corner X) `2 by Th30;
then (W-max X) `2 <= p `2 by A12, TOPREAL1:4;
then A14: p `2 = (W-max X) `2 by A13, XXREAL_0:1;
(NW-corner X) `1 = (W-max X) `1 by Th29;
then A15: p `1 = (W-max X) `1 by A12, GOBOARD7:5;
p = |[(p `1),(p `2)]| by EUCLID:53;
hence x = W-max X by A15, A14, EUCLID:53; :: thesis: verum
end;
W-max X in W-most X by Th34;
then A16: W-max X in X by XBOOLE_0:def 4;
assume x = W-max X ; :: thesis: x in (LSeg ((W-max X),(NW-corner X))) /\ X
hence x in (LSeg ((W-max X),(NW-corner X))) /\ X by A16, A9, XBOOLE_0:def 4; :: thesis: verum
end;
hence (LSeg ((W-max X),(NW-corner X))) /\ X = {(W-max X)} by TARSKI:def 1; :: thesis: verum