
theorem
  for p1,p2,p being Point of TOP-REAL 2,a being Real st p in LSeg(p1,p2)
  & p1`1<a & p2`1<a holds p`1<a
proof
  let p1,p2,p be Point of TOP-REAL 2,a be Real;
  assume that
A1: p in LSeg(p1,p2) and
A2: p1`1<a and
A3: p2`1<a;
  consider r being Real such that
A4: p = (1-r)*p1+r*p2 and
A5: 0<=r and
A6: r<=1 by A1;
A7: p`1 = ((1-r)*p1)`1+(r*p2)`1 by A4,TOPREAL3:2
    .= ((1-r)*p1)`1+r*p2`1 by TOPREAL3:4
    .= (1-r)*p1`1+r*p2`1 by TOPREAL3:4;
  per cases;
  suppose
    0=r;
    then p=p1+(0)*p2 by A4,RLVECT_1:def 8
      .=p1+0.TOP-REAL 2 by RLVECT_1:10
      .=p1 by RLVECT_1:4;
    hence thesis by A2;
  end;
  suppose
A8: 0<>r;
A9: (1-r)*a+r*a= a;
    1-r>=0 by A6,XREAL_1:48;
    then
A10: (1-r)*p1`1<=(1-r)*a by A2,XREAL_1:64;
    r*p2`1<r*a by A3,A5,A8,XREAL_1:68;
    hence thesis by A7,A10,A9,XREAL_1:8;
  end;
end;
