reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;
reserve r for Real;

theorem Th41:
  P = Ball(e,r) & p = e implies proj1.:P = ].p`1-r,p`1+r.[
proof
  assume that
A1: P = Ball(e,r) and
A2: p = e;
  hereby
    let a be object;
    assume a in proj1.:P;
    then consider x being object such that
A3: x in the carrier of TOP-REAL 2 and
A4: x in P and
A5: a = proj1.x by FUNCT_2:64;
    reconsider b = a as Real by A5;
    reconsider x as Point of TOP-REAL 2 by A3;
A6: a = x`1 by A5,PSCOMP_1:def 5;
    then
A7: b < p`1+r by A1,A2,A4,Th37;
    p`1-r < b by A1,A2,A4,A6,Th37;
    hence a in ].p`1-r,p`1+r.[ by A7,XXREAL_1:4;
  end;
  let a be object;
  assume
A8: a in ].p`1-r,p`1+r.[;
  then reconsider b = a as Real;
  reconsider f = |[b,p`2]| as Point of Euclid 2 by TOPREAL3:8;
A9: dist(f,e) = (Pitag_dist 2).(f,e) by METRIC_1:def 1
    .= sqrt ((|[b,p`2]|`1 - p`1)^2 + (|[b,p`2]|`2 - p`2)^2 ) by A2,TOPREAL3:7
    .= sqrt ((b - p`1)^2 + (|[b,p`2]|`2 - p`2)^2)
    .= sqrt ((b - p`1)^2 + (p`2 - p`2)^2)
    .= sqrt ((b - p`1)^2 + 0);
  b < p`1+r by A8,XXREAL_1:4;
  then
A10: b - p`1 < p`1+r - p`1 by XREAL_1:9;
  now
    per cases;
    case
      0 <= b - p`1;
      hence dist(f,e) < r by A10,A9,SQUARE_1:22;
    end;
    case
A11:  0 > b - p`1;
      p`1 - r < b by A8,XXREAL_1:4;
      then p`1 - r + r < b + r by XREAL_1:6;
      then
A12:  p`1 - b < r + b - b by XREAL_1:9;
      sqrt ((b - p`1)^2) = sqrt ((-(b - p`1))^2)
        .= -(b - p`1) by A11,SQUARE_1:22;
      hence dist(f,e) < r by A9,A12;
    end;
  end;
  then
A13: |[b,p`2]| in P by A1,METRIC_1:11;
  a = |[b,p`2]|`1
    .= proj1.(|[b,p`2]|) by PSCOMP_1:def 5;
  hence thesis by A13,FUNCT_2:35;
end;
