
theorem Th2:
  for a,c,d being Real,p being Point of TOP-REAL 2
  st c <d & p`1=a & c <=p`2 & p`2<=d holds p in LSeg(|[a,c]|,|[a,d]|)
proof
  let a,c,d be Real,p be Point of TOP-REAL 2;
  assume that
A1: c <d and
A2: p`1=a and
A3: c <=p`2 and
A4: p`2<=d;
A5: d-c>0 by A1,XREAL_1:50;
  reconsider w=(p`2-c)/(d-c) as Real;
A6: (1-w)*(|[a,c]|)+w*(|[a,d]|) =|[(1-w)*a,(1-w)*c]|+w*(|[a,d]|) by EUCLID:58
    .=|[(1-w)*a,(1-w)*c]|+(|[w*a,w*d]|) by EUCLID:58
    .=|[(1-w)*a+w*a,(1-w)*c+w*d]| by EUCLID:56
    .= |[a,c+w*(d-c)]|
    .= |[a,c+(p`2-c)]| by A5,XCMPLX_1:87
    .= p by A2,EUCLID:53;
A7: p`2-c>=0 by A3,XREAL_1:48;
  p`2-c <=d-c by A4,XREAL_1:9;
  then w<=(d-c)/(d-c) by A5,XREAL_1:72;
  then w<=1 by A5,XCMPLX_1:60;
  hence thesis by A5,A6,A7;
end;
