reserve S for Subset of TOP-REAL 2,
  C,C1,C2 for non empty compact Subset of TOP-REAL 2,
  p,q for Point of TOP-REAL 2;
reserve i,j,k for Nat,
  t,r1,r2,s1,s2 for Real;
reserve D1 for non vertical non empty compact Subset of TOP-REAL 2,
  D2 for non horizontal non empty compact Subset of TOP-REAL 2,
  D for non vertical non horizontal non empty compact Subset of TOP-REAL 2;

theorem Th59:
  S-bound L~SpStSeq C = S-bound C
proof
  set S1 = LSeg(NW-corner C,NE-corner C), S2 = LSeg(NE-corner C,SE-corner C),
  S3 = LSeg(SE-corner C,SW-corner C), 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: S3 \/ S4 is compact by COMPTS_1:10;
A4: (NE-corner C)`2 = N-bound C by EUCLID:52;
  then
A5: S-bound S2 = 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 S4 = S-bound C by A6,Th22,Th55;
A9: S-bound(S3 \/ S4) = min(S-bound S3,S-bound S4) by Th48
    .= min(S-bound C,S-bound C) by A1,A6,A8,Th55
    .= S-bound C;
A10: L~SpStSeq C = (S1 \/ S2) \/ (S3 \/ S4) by Th41;
A11: S1 \/ S2 is compact by COMPTS_1:10;
  S-bound(S1 \/ S2) = min(S-bound S1,S-bound S2) 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;
end;
