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
  not f/.1 in Ball(u,r) & 1<=m & m<=len f - 1 & (for i st 1<=i & i<=len
f - 1 & LSeg(f,i) /\ Ball(u,r) <> {} holds m<=i) implies not f/.m in Ball(u,r)
proof
  assume that
A1: not f/.1 in Ball(u,r) and
A2: 1<=m and
A3: m<=len f - 1 and
A4: for i st 1<=i & i<=len f - 1 & LSeg(f,i) /\ Ball(u,r) <> {} holds m <=i;
  assume
A5: f/.m in Ball(u,r);
  per cases by A2,XXREAL_0:1;
  suppose
    1=m;
    hence contradiction by A1,A5;
  end;
  suppose
A6: 1<m;
    reconsider k=m-1 as Element of NAT by A6,INT_1:5;
    1+1<=m by A6,NAT_1:13;
    then
A7: 1<=m-1 by XREAL_1:19;
    m-1<=m by XREAL_1:43;
    then
A8: k<=len f - 1 by A3,XXREAL_0:2;
    then k+1<=len f by XREAL_1:19;
    then f/.(k+1) in LSeg(f,k) by A7,TOPREAL1:21;
    then LSeg(f,k) /\ Ball(u,r) <> {} by A5,XBOOLE_0:def 4;
    then m<=k by A4,A7,A8;
    then m+1<=m by XREAL_1:19;
    hence contradiction by NAT_1:13;
  end;
end;
