reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;

theorem
  s in LSeg(p,q) & s <> p & s <> q & p`2 < q`2 implies p`2 < s`2 & s`2 < q`2
proof
  assume that
A1: s in LSeg(p,q) and
A2: s <> p and
A3: s <> q and
A4: p`2 < q`2;
A5: p`2-q`2 < 0 by A4,XREAL_1:49;
  consider r such that
A6: s = (1-r)*p + r*q and
A7: 0 <= r and
A8: r <= 1 by A1;
  (1-r)*p = |[(1-r)*p`1,(1-r)*p`2]| by Th24;
  then
A9: ((1-r)*p)`2 = (1-r)*p`2;
  r*q = |[r*q`1,r*q`2]| by Th24;
  then
A10: (r*q)`2 = r*q`2;
  s = |[((1-r)*p)`1 + (r*q)`1,((1-r)*p)`2 + (r*q)`2]| by A6,EUCLID:55;
  then
A11: s`2 = (1-r)*p`2 + r*q`2 by A9,A10;
  then
A12: p`2 - s`2 = r*(p`2-q`2);
  r < 1 by A3,A6,A8,Th26;
  then
A13: 1-r > 0 by XREAL_1:50;
A14: q`2-p`2 > 0 by A4,XREAL_1:50;
  r > 0 by A2,A6,A7,Th25;
  then
A15: p`2 - s`2 < 0 by A12,A5,XREAL_1:132;
  q`2 - s`2 = (1-r)*(q`2-p`2) by A11;
  then q`2 - s`2 > 0 by A14,A13,XREAL_1:129;
  hence thesis by A15,XREAL_1:47,48;
end;
