reserve r, s, t, g for Real,

          r3, r1, r2, q3, p3 for Real;
reserve T for TopStruct,
  f for RealMap of T;
reserve p for Point of TOP-REAL 2,
  P for Subset of TOP-REAL 2,
  Z for non empty Subset of TOP-REAL 2,
  X for non empty compact Subset of TOP-REAL 2;

theorem
  LSeg(S-min X, S-max X) c= LSeg(SW-corner X, SE-corner X)
proof
A1: (SW-corner X)`2 = S-bound X & (SE-corner X)`2 = S-bound X by EUCLID:52;
A2: (S-max X)`1 <= (SE-corner X)`1 by Th54;
  (S-max X)`2 = S-bound X & (SW-corner X)`1 <= (S-max X)`1 by Th54,EUCLID:52;
  then
A3: S-max X in LSeg(SW-corner X, SE-corner X) by A1,A2,GOBOARD7:8;
A4: (S-min X)`1 <= (SE-corner X)`1 by Th54;
  (S-min X)`2 = S-bound X & (SW-corner X)`1 <= (S-min X)`1 by Th54,EUCLID:52;
  then S-min X in LSeg(SW-corner X, SE-corner X) by A1,A4,GOBOARD7:8;
  hence thesis by A3,TOPREAL1:6;
end;
