
theorem Th1:
  for a,c,d being Real,p being Point of TOP-REAL 2
  st c <=d & p in LSeg(|[a,c]|,|[a,d]|) holds p`1=a & c <=p`2 & p`2<=d
proof
  let a,c,d be Real,p be Point of TOP-REAL 2;
  assume that
A1: c <=d and
A2: p in LSeg(|[a,c]|,|[a,d]|);
  thus p`1=a by A2,TOPREAL3:11;
A3: (|[a,c]|)`2=c by EUCLID:52;
  (|[a,d]|)`2=d by EUCLID:52;
  hence thesis by A1,A2,A3,TOPREAL1:4;
end;
