let C be non empty compact Subset of (TOP-REAL 2); :: thesis: E-bound (L~ (SpStSeq C)) = E-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: (NW-corner C) `1 = W-bound C by EUCLID:56;
A2: W-bound C <= E-bound C by Th23;
A3: (LSeg (SE-corner C),(SW-corner C)) \/ (LSeg (SW-corner C),(NW-corner C)) is compact by COMPTS_1:19;
A4: (SW-corner C) `1 = W-bound C by EUCLID:56;
then A5: E-bound (LSeg (SW-corner C),(NW-corner C)) = W-bound C by A1, Th65;
A6: (SE-corner C) `1 = E-bound C by EUCLID:56;
A7: (NE-corner C) `1 = E-bound C by EUCLID:56;
then A8: E-bound (LSeg (NE-corner C),(SE-corner C)) = E-bound C by A6, Th65;
A9: E-bound ((LSeg (NW-corner C),(NE-corner C)) \/ (LSeg (NE-corner C),(SE-corner C))) = max (E-bound (LSeg (NW-corner C),(NE-corner C))),(E-bound (LSeg (NE-corner C),(SE-corner C))) by Th57
.= max (E-bound C),(E-bound C) by A1, A7, A8, Th23, Th65
.= E-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;
E-bound ((LSeg (SE-corner C),(SW-corner C)) \/ (LSeg (SW-corner C),(NW-corner C))) = max (E-bound (LSeg (SE-corner C),(SW-corner C))),(E-bound (LSeg (SW-corner C),(NW-corner C))) by Th57
.= max (W-bound C),(E-bound C) by A6, A4, A5, Th23, Th65
.= E-bound C by A2, XXREAL_0:def 10 ;
hence E-bound (L~ (SpStSeq C)) = max (E-bound C),(E-bound C) by A10, A11, A9, A3, Th57
.= E-bound C ;
:: thesis: verum