
theorem
  for p, q, p1 being Point of TOP-REAL 2 st p`2 <> q`2 & p1 in LSeg (p, q)
  holds ( p1`2 = p`2 implies p1 = p )
proof
  let p, q, p1 be Point of TOP-REAL 2;
  assume that
A1: p`2 <> q`2 and
A2: p1 in LSeg (p, q);
  assume
A3: p1`2 = p`2;
  assume
A4: p1 <> p;
  consider l1 be Real such that
A5: p1 = (1-l1)*p + l1*q and 0 <= l1
  and l1 <= 1 by A2;
A6: (1-l1)*p + l1*q = |[ ((1-l1)*p)`1 + (l1*q)`1 , ((1-l1)*p)`2 + (l1*q)`2 ]|
  by EUCLID:55;
A7: (1-l1)*p = |[ (1-l1)*p`1, (1-l1)*p`2 ]| by EUCLID:57;
A8: l1*q = |[ l1*q`1, l1*q`2 ]| by EUCLID:57;
  p`2 = ( (1-l1)*p )`2 + ( l1*q )`2 by A3,A5,A6,EUCLID:52
    .= (1-l1)*p`2 + (l1*q)`2 by A7,EUCLID:52
    .= (1-l1)*p`2 + l1*q`2 by A8,EUCLID:52;
  then (1 - (1-l1))*p`2 = l1*q`2;
  then l1 = 0 by A1,XCMPLX_1:5;
  then p1 = 1*p + 0.TOP-REAL 2 by A5,RLVECT_1:10
    .= p + 0.TOP-REAL 2 by RLVECT_1:def 8
    .= p by RLVECT_1:4;
  hence thesis by A4;
end;
