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 Th55:
  p in S-most Z implies p`2 = (S-min Z)`2 & (Z is compact implies
  (S-min Z)`1 <= p`1 & p`1 <= (S-max Z)`1)
proof
A1: (SW-corner Z)`2 = S-bound Z & (S-min Z)`2 = S-bound Z by EUCLID:52;
A2: (SE-corner Z)`2 = S-bound Z by EUCLID:52;
  assume
A3: p in S-most Z;
  then p in LSeg(SW-corner Z, SE-corner Z) by XBOOLE_0:def 4;
  hence p`2 = (S-min Z)`2 by A1,A2,GOBOARD7:6;
  assume Z is compact;
  then reconsider Z as non empty compact Subset of TOP-REAL 2;
  (S-min Z)`1 = lower_bound (proj1|S-most Z) &
  (S-max Z)`1 = upper_bound (proj1|S-most Z )
  by EUCLID:52;
  hence thesis by A3,Lm3;
end;
