let C be non empty compact Subset of (TOP-REAL 2); S-bound (L~ (SpStSeq C)) = S-bound C
set S1 = LSeg ((NW-corner C),(NE-corner C));
set S2 = LSeg ((NE-corner C),(SE-corner C));
set S3 = LSeg ((SE-corner C),(SW-corner C));
set S4 = LSeg ((SW-corner C),(NW-corner C));
A1:
(SE-corner C) `2 = S-bound C
by EUCLID:52;
A2:
S-bound C <= N-bound C
by Th22;
A3:
(LSeg ((SE-corner C),(SW-corner C))) \/ (LSeg ((SW-corner C),(NW-corner C))) is compact
by COMPTS_1:10;
A4:
(NE-corner C) `2 = N-bound C
by EUCLID:52;
then A5:
S-bound (LSeg ((NE-corner C),(SE-corner C))) = S-bound C
by A1, Th22, Th55;
A6:
(SW-corner C) `2 = S-bound C
by EUCLID:52;
A7:
(NW-corner C) `2 = N-bound C
by EUCLID:52;
then A8:
S-bound (LSeg ((SW-corner C),(NW-corner C))) = S-bound C
by A6, Th22, Th55;
A9: S-bound ((LSeg ((SE-corner C),(SW-corner C))) \/ (LSeg ((SW-corner C),(NW-corner C)))) =
min ((S-bound (LSeg ((SE-corner C),(SW-corner C)))),(S-bound (LSeg ((SW-corner C),(NW-corner C)))))
by Th48
.=
min ((S-bound C),(S-bound C))
by A1, A6, A8, Th55
.=
S-bound C
;
A10:
L~ (SpStSeq C) = ((LSeg ((NW-corner C),(NE-corner C))) \/ (LSeg ((NE-corner C),(SE-corner C)))) \/ ((LSeg ((SE-corner C),(SW-corner C))) \/ (LSeg ((SW-corner C),(NW-corner C))))
by Th41;
A11:
(LSeg ((NW-corner C),(NE-corner C))) \/ (LSeg ((NE-corner C),(SE-corner C))) is compact
by COMPTS_1:10;
S-bound ((LSeg ((NW-corner C),(NE-corner C))) \/ (LSeg ((NE-corner C),(SE-corner C)))) =
min ((S-bound (LSeg ((NW-corner C),(NE-corner C)))),(S-bound (LSeg ((NE-corner C),(SE-corner C)))))
by Th48
.=
min ((N-bound C),(S-bound C))
by A7, A4, A5, Th55
.=
S-bound C
by A2, XXREAL_0:def 9
;
hence S-bound (L~ (SpStSeq C)) =
min ((S-bound C),(S-bound C))
by A10, A11, A3, A9, Th48
.=
S-bound C
;
verum