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;

theorem Th23:
  LSeg(SE-corner C, NE-corner C) = { p : p`1 = E-bound C & p`2 <=
  N-bound C & p`2 >= S-bound C }
proof
  set L = { p : p`1 = E-bound C & p`2 <= N-bound C & p`2 >= S-bound C};
  set q1 = SE-corner C, q2 = NE-corner C;
A1: q1`1 = E-bound C by EUCLID:52;
A2: q1`2 = S-bound C by EUCLID:52;
A3: q2`1 = E-bound C by EUCLID:52;
A4: q2`2 = N-bound C by EUCLID:52;
A5: S-bound C <= N-bound C by Th22;
  thus LSeg(q1,q2) c= L
  proof
    let a be object;
    assume
A6: a in LSeg(q1,q2);
    then reconsider p = a as Point of TOP-REAL 2;
A7: p`1 = E-bound C by A1,A3,A6,GOBOARD7:5;
A8: p`2 >= S-bound C by A2,A4,A5,A6,TOPREAL1:4;
    p`2 <= N-bound C by A2,A4,A5,A6,TOPREAL1:4;
    hence thesis by A7,A8;
  end;
  let a be object;
  assume a in L;
  then ex p st p = a & p`1 = E-bound C & p`2 <= N-bound C & p`2 >= S-bound C;
  hence thesis by A1,A2,A3,A4,GOBOARD7:7;
end;
