
theorem Th1:
  for a,b,d being Real,p being Point of TOP-REAL 2 st a <b &
  p`2=d & a <=p`1 & p`1<=b holds p in LSeg(|[a,d]|,|[b,d]|)
proof
  let a,b,d be Real,p be Point of TOP-REAL 2;
  assume that
A1: a <b and
A2: p`2=d and
A3: a <=p`1 and
A4: p`1<=b;
  reconsider w=(p`1-a)/(b-a) as Real;
A5: b-a>0 by A1,XREAL_1:50;
  p`1-a <=b-a by A4,XREAL_1:9;
  then w<=(b-a)/(b-a) by A5,XREAL_1:72;
  then
A6: w<=1 by A5,XCMPLX_1:60;
  p`1-a>=0 by A3,XREAL_1:48;
  then
A7: 0<=w by A5,XREAL_1:136;
  (1-w)*(|[a,d]|)+w*(|[b,d]|) =|[(1-w)*a,(1-w)*d]|+w*(|[b,d]|) by EUCLID:58
    .=|[(1-w)*a,(1-w)*d]|+(|[w*b,w*d]|) by EUCLID:58
    .=|[(1-w)*a+w*b,(1-w)*d+w*d]| by EUCLID:56
    .= |[a+w*(b-a),d]|
    .= |[a+(p`1-a),d]| by A5,XCMPLX_1:87
    .= p by A2,EUCLID:53;
  hence thesis by A7,A6;
end;
