let C be non empty compact Subset of (TOP-REAL 2); :: thesis: 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 ;
:: thesis: verum