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:56;
A2: S-bound C <= N-bound C by Th24;
A3: (LSeg (SE-corner C),(SW-corner C)) \/ (LSeg (SW-corner C),(NW-corner C)) is compact by COMPTS_1:19;
A4: (NE-corner C) `2 = N-bound C by EUCLID:56;
then A5: S-bound (LSeg (NE-corner C),(SE-corner C)) = S-bound C by A1, Th24, Th63;
A6: (SW-corner C) `2 = S-bound C by EUCLID:56;
A7: (NW-corner C) `2 = N-bound C by EUCLID:56;
then A8: S-bound (LSeg (SW-corner C),(NW-corner C)) = S-bound C by A6, Th24, Th63;
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 Th55
.= min (S-bound C),(S-bound C) by A1, A6, A8, Th63
.= 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 Th43;
A11: (LSeg (NW-corner C),(NE-corner C)) \/ (LSeg (NE-corner C),(SE-corner C)) is compact by COMPTS_1:19;
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 Th55
.= min (N-bound C),(S-bound C) by A7, A4, A5, Th63
.= 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, Th55
.= S-bound C ;
:: thesis: verum