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 Th47:
  p in E-most Z implies p`1 = (E-min Z)`1 & (Z is compact implies
  (E-min Z)`2 <= p`2 & p`2 <= (E-max Z)`2)
proof
A1: (SE-corner Z)`1 = E-bound Z & (E-min Z)`1 = E-bound Z by EUCLID:52;
A2: (NE-corner Z)`1 = E-bound Z by EUCLID:52;
  assume
A3: p in E-most Z;
  then p in LSeg(SE-corner Z, NE-corner Z) by XBOOLE_0:def 4;
  hence p`1 = (E-min Z)`1 by A1,A2,GOBOARD7:5;
  assume Z is compact;
  then reconsider Z as non empty compact Subset of TOP-REAL 2;
  (E-min Z)`2 = lower_bound (proj2|E-most Z) &
  (E-max Z)`2 = upper_bound (proj2|E-most Z )
  by EUCLID:52;
  hence thesis by A3,Lm3;
end;
