
theorem Th10:
  for p, p1, p2 being Point of TOP-REAL 2 st p1 <> p2 &
  p in LSeg(p1,p2) holds LE p1,p,p1,p2
proof
  let p, p1, p2 be Point of TOP-REAL 2;
  assume that
A1: p1 <> p2 and
A2: p in LSeg(p1,p2);
  thus LE p1,p,p1,p2
  proof
    thus p1 in LSeg(p1,p2) & p in LSeg(p1,p2) by A2,RLTOPSP1:68;
    let r1,r2 be Real;
    assume that
    0<=r1 and r1<=1 and
A3: p1=(1-r1)*p1+r1*p2 and
A4: 0<=r2 and r2<=1
    and p=(1-r2)*p1+r2*p2;
    0.TOP-REAL 2 = (1-r1)*p1+r1*p2+-p1 by A3,RLVECT_1:5
      .= (1-r1)*p1+r1*p2-p1
      .= (1-r1)*p1+(r1*p2-p1) by RLVECT_1:def 3
      .= (1-r1)*p1+(-p1+r1*p2)
      .= (1-r1)*p1+-p1+r1*p2 by RLVECT_1:def 3
      .= ((1-r1)*p1-p1)+r1*p2;
    then -r1*p2=(((1-r1)*p1-p1)+r1*p2)+-r1*p2 by RLVECT_1:4
      .=(((1-r1)*p1-p1)+r1*p2)-r1*p2
      .=((1-r1)*p1-p1)+(r1*p2-r1*p2) by RLVECT_1:def 3
      .=((1-r1)*p1-p1)+0.TOP-REAL 2 by RLVECT_1:5
      .=(1-r1)*p1-p1 by RLVECT_1:4
      .=(1-r1)*p1-1*p1 by RLVECT_1:def 8
      .=(1-r1-1)*p1 by RLVECT_1:35
      .=(-r1)*p1
      .=-r1*p1 by RLVECT_1:79;
    then r1*p1=--r1*p2
      .= r1*p2;
    hence thesis by A1,A4,RLVECT_1:36;
  end;
end;
