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 Th21:
  for p1,p2 being Point of TOP-REAL N, u being Point of Euclid N
  st p1 in Ball(u,r) & p2 in Ball(u,r) holds LSeg(p1,p2) c= Ball(u,r)
proof
  let p1,p2 be Point of TOP-REAL N, u be Point of Euclid N;
  assume that
A1: p1 in Ball(u,r) and
A2: p2 in Ball(u,r);
  reconsider p9=p1 as Point of Euclid N by A1;
A3: dist(p9,u)<r by A1,METRIC_1:11;
  thus LSeg(p1,p2) c= Ball(u,r)
  proof
    p2 in {u4 where u4 is Point of Euclid N : dist(u,u4)<r} by A2,METRIC_1:17;
    then
A4: ex u4 be Point of Euclid N st u4=p2 & dist(u,u4)<r;
    reconsider q29=u as Element of REAL N;
    let a be object;
    reconsider q2=q29 as Point of TOP-REAL N by EUCLID:22;
    assume a in LSeg(p1,p2);
    then consider lambda such that
A5: (1-lambda)*p1 + lambda*p2=a and
A6: 0 <= lambda and
A7: lambda <= 1;
A8: lambda=|.lambda.| by A6,ABSVALUE:def 1;
    set p = (1-lambda)*p1 + lambda*p2;
    reconsider p9=p,p19=p1,p29=p2 as Element of REAL N by EUCLID:22;
    reconsider u5=(1-lambda)*p1 + lambda*p2 as Point of Euclid N by EUCLID:22;
A9: (Pitag_dist N).(q29,p9)=|.q29-p9.| & (Pitag_dist N).(u,u5)=dist(u,u5)
    by EUCLID:def 6,METRIC_1:def 1;
    (Pitag_dist N).(q29,p29)=|.q29-p29.| by EUCLID:def 6;
    then
A10: |.q29+-p29.| < r by A4,METRIC_1:def 1;
A11: 0 <= 1-lambda by A7,XREAL_1:48;
    then
A12: 1-lambda= |.1-lambda.| by ABSVALUE:def 1;
    consider u3 being Point of Euclid N such that
A13: u3=p1 and
A14: dist(u,u3)<r by A3;
A15: (Pitag_dist N).(q29,p19)=|.q29-p19.| by EUCLID:def 6;
    then
A16: dist(u,u3) = |.q29-p19.| by A13,METRIC_1:def 1;
    |.1-lambda.|*|.q29+-p19.| < (1-lambda)*r & |.lambda.|*|.q29 + -p29
.| <= lambda*r or |.1-lambda.|*|.q29+-p19.| <= (1-lambda)*r & |.lambda.|*|.
    q29 + -p29.| < lambda*r
    proof
      per cases by A6;
      suppose
        lambda=0;
        hence thesis by A13,A14,A15,A12,A8,METRIC_1:def 1;
      end;
      suppose
        lambda>0;
        hence thesis by A14,A16,A10,A11,A12,A8,XREAL_1:64,68;
      end;
    end;
    then
A17: |.1-lambda.|*|.q29+-p19.| + |.lambda.|*|.q29 + -p29.| < (1-lambda)*
    r + lambda*r by XREAL_1:8;
    q29-p19 = q2-p1 by EUCLID:69;
    then
A18: q29-p9 = q2-p & (1-lambda)*(q29-p19) = (1-lambda)*(q2-p1) by EUCLID:65,69;
    q29-p29 = q2-p2 by EUCLID:69;
    then
A19: lambda*(q29 -p29) = lambda*(q2 -p2) by EUCLID:65;
    q2 - p = (1-lambda+lambda)*q2-((1-lambda)*p1 + lambda*p2) by RLVECT_1:def 8
      .= (1-lambda)*q2+lambda*q2-((1-lambda)*p1 + lambda*p2) by RLVECT_1:def 6
      .= (1-lambda)*q2+lambda*q2+-((1-lambda)*p1 + lambda*p2)
      .= (1-lambda)*q2+lambda*q2+(-(1-lambda)*p1 - lambda*p2) by RLVECT_1:30
      .= (1-lambda)*q2+lambda*q2+-(1-lambda)*p1 + -lambda*p2 by RLVECT_1:def 3
      .= (1-lambda)*q2+-(1-lambda)*p1+lambda*q2 + -lambda*p2 by RLVECT_1:def 3
      .= (1-lambda)*q2+(1-lambda)*(-p1)+lambda*q2 + -lambda*p2 by RLVECT_1:25

      .= (1-lambda)*(q2+-p1)+lambda*q2 + -lambda*p2 by RLVECT_1:def 5
      .= (1-lambda)*(q2+-p1)+(lambda*q2 + -lambda*p2) by RLVECT_1:def 3
      .= (1-lambda)*(q2+-p1)+(lambda*q2 + lambda*(-p2)) by RLVECT_1:25
      .= (1-lambda)*(q2-p1)+(lambda*(q2 -p2)) by RLVECT_1:def 5;
    then q29-p9 = (1-lambda)*(q29-p19) + lambda*(q29 -p29) by A18,A19,EUCLID:64
;
    then
A20: |. q29-p9 .| <= |.(1-lambda)*(q29-p19).| + |.lambda*(q29 -p29).| by
EUCLID:12;
    |.(1-lambda)*(q29+-p19).| + |.lambda*(q29 + -p29).| = |.1-lambda.|*
    |.q29+-p19.| + |.lambda*(q29 + -p29).| by EUCLID:11
      .= |.1-lambda.|*|.q29+-p19.| + |.lambda.|*|.q29 + -p29.| by EUCLID:11;
    then |.q29-p9.| < r by A20,A17,XXREAL_0:2;
    then a in {u6 where u6 is Element of Euclid N:dist(u,u6)<r} by A5,A9;
    hence thesis by METRIC_1:17;
  end;
end;
