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(SW-corner X, S-min X)/\X = {S-min X} & LSeg(S-max X, SE-corner X)
  /\X = {S-max X}
proof
  now
    let x be object;
A1: S-min X in LSeg(SW-corner X, S-min X) by RLTOPSP1:68;
    hereby
      S-min X in S-most X by Th58;
      then SW-corner X in LSeg(SW-corner X, SE-corner X) & S-min X in LSeg(
      SW-corner X, SE-corner X) by RLTOPSP1:68,XBOOLE_0:def 4;
      then
A2:   LSeg
(SW-corner X, S-min X) c= LSeg(SW-corner X, SE-corner X) by TOPREAL1:6;
      assume
A3:   x in LSeg(SW-corner X, S-min X)/\X;
      then
A4:   x in LSeg(SW-corner X, S-min X) by XBOOLE_0:def 4;
      reconsider p = x as Point of TOP-REAL 2 by A3;
      x in X by A3,XBOOLE_0:def 4;
      then p in S-most X by A4,A2,XBOOLE_0:def 4;
      then
A5:   (S-min X)`1 <= p`1 by Th55;
      (SW-corner X)`1 <= (S-min X)`1 by Th54;
      then p`1 <= (S-min X)`1 by A4,TOPREAL1:3;
      then
A6:   p`1 = (S-min X)`1 by A5,XXREAL_0:1;
      (SW-corner X)`2 = (S-min X)`2 by Th53;
      then
A7:   p`2 = (S-min X)`2 by A4,GOBOARD7:6;
      p = |[p`1, p`2]| by EUCLID:53;
      hence x = S-min X by A7,A6,EUCLID:53;
    end;
    S-min X in S-most X by Th58;
    then
A8: S-min X in X by XBOOLE_0:def 4;
    assume x = S-min X;
    hence x in LSeg(SW-corner X, S-min X)/\X by A8,A1,XBOOLE_0:def 4;
  end;
  hence LSeg(SW-corner X, S-min X)/\X = {S-min X} by TARSKI:def 1;
  now
    let x be object;
A9: S-max X in LSeg(S-max X, SE-corner X) by RLTOPSP1:68;
    hereby
      S-max X in S-most X by Th58;
      then SE-corner X in LSeg(SW-corner X, SE-corner X) & S-max X in LSeg(
      SW-corner X, SE-corner X) by RLTOPSP1:68,XBOOLE_0:def 4;
      then
A10:  LSeg
(S-max X, SE-corner X) c= LSeg(SW-corner X, SE-corner X) by TOPREAL1:6;
      assume
A11:  x in LSeg(S-max X, SE-corner X)/\X;
      then
A12:  x in LSeg(S-max X, SE-corner X) by XBOOLE_0:def 4;
      reconsider p = x as Point of TOP-REAL 2 by A11;
      x in X by A11,XBOOLE_0:def 4;
      then p in S-most X by A12,A10,XBOOLE_0:def 4;
      then
A13:  p`1 <= (S-max X)`1 by Th55;
      (S-max X)`1 <= (SE-corner X)`1 by Th54;
      then (S-max X)`1 <= p`1 by A12,TOPREAL1:3;
      then
A14:  p`1 = (S-max X)`1 by A13,XXREAL_0:1;
      (SE-corner X)`2 = (S-max X)`2 by Th53;
      then
A15:  p`2 = (S-max X)`2 by A12,GOBOARD7:6;
      p = |[p`1, p`2]| by EUCLID:53;
      hence x = S-max X by A15,A14,EUCLID:53;
    end;
    S-max X in S-most X by Th58;
    then
A16: S-max X in X by XBOOLE_0:def 4;
    assume x = S-max X;
    hence x in LSeg(S-max X, SE-corner X)/\X by A16,A9,XBOOLE_0:def 4;
  end;
  hence thesis by TARSKI:def 1;
end;
