reserve X for set,
  Y for non empty set;
reserve n for Nat;
reserve r for Real,
  M for non empty MetrSpace;
reserve n for Nat,
  p,q,q1,q2 for Point of TOP-REAL 2,
  r,s1,s2,t1,t2 for Real,
  x,y for Point of Euclid 2;

theorem
  Ball(x,r)` is connected Subset of TOP-REAL 2
proof
  set C = Ball(x,r)`;
  per cases;
  suppose
    r <= 0;
    then Ball(x,r) = {} TOP-REAL 2 by TBSP_1:12;
    then
A1:   C = [#] TOP-REAL 2 by TOPREAL3:8;
    thus thesis by A1;
  end;
  suppose
A2: r > 0;
    reconsider q = x as Point of TOP-REAL 2 by TOPREAL3:8;
    reconsider y = |[0,0]| as Point of Euclid 2 by TOPREAL3:8;
    reconsider Q = Ball(x,r), J = Ball(y,1) as Subset of TOP-REAL 2 by
TOPREAL3:8;
A3: Q = AffineMap(r,q`1,r,q`2).:J by A2,Th35;
    reconsider P = Q`, K = J` as Subset of TOP-REAL 2;
A4: K = (REAL 2)\ Ball(y,1) by TOPREAL3:8
      .= (REAL 2)\ {q1 : |.q1.| < 1 } by Th31;
    AffineMap(r,q`1,r,q`2) is one-to-one & AffineMap(r,q`1,r,q`2) is onto
    by A2,Th36,JGRAPH_2:44;
    then
    the carrier of TOP-REAL 2 = the carrier of Euclid 2 & P = AffineMap(r,
    q`1,r, q`2).:K by A3,Th5,TOPREAL3:8;
    hence thesis by A4,JORDAN2C:53,TOPS_2:61;
  end;
end;
