
theorem Th15:
  for a,b,r being Real
  for P,Q,R being Element of TOP-REAL 2 st
  Q in LSeg(P,R) & P in inside_of_circle(a,b,r) & R in inside_of_circle(a,b,r)
  holds Q in inside_of_circle(a,b,r)
  proof
    let a,b,r be Real;
    let P,Q,R be Element of TOP-REAL 2;
    assume that
A1: Q in LSeg(P,R) and
A2: P in inside_of_circle(a,b,r) and
A3: R in inside_of_circle(a,b,r);
    consider s be Real such that
A4: 0 <= s & s <= 1 and
A5: Q = (1 - s)*P + s * R by A1,RLTOPSP1:76;
    s in [. 0,1 .] by A4,XXREAL_1:1;
    then s in ]. 0,1 .[ or s = 0 or s = 1 by XXREAL_1:5;
    then per cases by XXREAL_1:4;
    suppose
A6:   0 < s & s < 1;
      Q = s * R + (1 - s)* P by A5;
      hence thesis by A2,A3,A6,CONVEX1:def 2;
    end;
    suppose
      s = 0;
      then Q = P + 0 * R by A5,RVSUM_1:52
            .= P by GTARSKI2:2;
      hence thesis by A2;
    end;
    suppose s = 1;
      then Q = R + 0 * P by A5,RVSUM_1:52
            .= R by GTARSKI2:2;
      hence thesis by A3;
    end;
  end;
