reserve p,p1,p2,p3,p11,p22,q,q1,q2 for Point of TOP-REAL 2,
  f,h for FinSequence of TOP-REAL 2,
  r,r1,r2,s,s1,s2 for Real,
  u,u1,u2,u5 for Point of Euclid 2,
  n,m,i,j,k for Nat,
  N for Nat,
  x,y,z for set;
reserve lambda for Real;

theorem
  |[s,r1]| in Ball(u,r) & |[s,s1]| in Ball(u,r) implies |[s,(r1+s1)/2]|
  in Ball(u,r)
proof
  set p = |[s,r1]|, q = |[s,s1]|, p3 = |[s,(r1+s1)/2]|;
  assume |[s,r1]| in Ball(u,r) & |[s,s1]| in Ball(u,r);
  then
A1: LSeg(|[s,r1]|,|[s,s1]|) c= Ball(u,r) by Th21;
A2: p3`2 = (1 - 1/2)*(p`2) + (1/2)*(q`2)
    .= ((1 - 1/2)*p)`2 + (1/2)*(q`2) by Th4
    .= ((1 - 1/2)*p)`2 + ((1/2)*q)`2 by Th4
    .= ((1 - 1/2)*p + (1/2)*q)`2 by Th2;
  p3`1 = (1 - 1/2)*(p`1) + (1/2)*(q`1)
    .= ((1 - 1/2)*p)`1 + (1/2)*(q`1) by Th4
    .= ((1 - 1/2)*p)`1 + ((1/2)*q)`1 by Th4
    .= ((1 - 1/2)*p + (1/2)*q)`1 by Th2;
  then p3 = (1 - 1/2)*p + (1/2)*q by A2,Th6;
  then p3 in {(1-lambda)*p + lambda*q: 0<=lambda & lambda<=1 };
  hence thesis by A1;
end;
