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 Th35:
  y = |[0,0]| & x = q & r > 0 implies AffineMap(r,q`1,r,q`2).:Ball
  (y,1) = Ball(x,r)
proof
  assume that
A1: y = |[0,0]| and
A2: x = q and
A3: r > 0;
  reconsider A = Ball(y,1), B = Ball(x,r) as Subset of TOP-REAL 2 by TOPREAL3:8
;
A4: B c= AffineMap(r,q`1,r,q`2).:A
  proof
    let u be object;
    assume
A5: u in B;
    then reconsider q1 = u as Point of TOP-REAL 2;
    reconsider x1 = q1 as Element of Euclid 2 by TOPREAL3:8;
    set q2 = AffineMap(1/r,-q`1/r,1/r,-q`2/r).q1;
    consider z1,z2 being Point of Euclid 2 such that
A6: z1 = q and
A7: z2 = r*q2+q and
A8: dist(q,r*q2+q) = dist(z1,z2) by TOPREAL6:def 1;
    consider z3,z4 being Point of Euclid 2 such that
A9: z3 = |[0,0]| and
A10: z4 = q2 and
A11: dist(|[0,0]|,q2) = dist(z3,z4) by TOPREAL6:def 1;
    z2 = AffineMap(r,q`1,r,q`2).q2 by A7,Th33
      .= (AffineMap(r,q`1,r,q`2)*AffineMap(1/r,-q`1/r,1/r,-q`2/r)).q1 by
FUNCT_2:15
      .= (id REAL 2).q1 by A3,Th34
      .= x1;
    then dist(x,x1) = dist(|[0,0]|+q,r*q2 + q) by A2,A6,A8,EUCLID:54,RLVECT_1:4
      .= dist(|[0,0]|,r*q2) by Th21
      .= |.r.|*dist(|[0,0]|,q2) by Th20
      .= r*dist(y,z4) by A1,A3,A9,A11,ABSVALUE:def 1;
    then r*dist(y,z4) < 1 *r by A5,METRIC_1:11;
    then dist(y,z4) < 1 by A3,XREAL_1:64;
    then
A12: q2 in A by A10,METRIC_1:11;
    AffineMap(r,q`1,r,q`2).q2 = (AffineMap(r,q`1,r,q`2)*AffineMap(1/r,-q
    `1/r,1/r,-q`2/r)).q1 by FUNCT_2:15
      .= (id REAL 2).q1 by A3,Th34
      .= (id TOP-REAL 2).q1 by EUCLID:22
      .= q1;
    hence thesis by A12,FUNCT_2:35;
  end;
  AffineMap(r,q`1,r,q`2).:A c= B
  proof
    let u be object;
    assume
A13: u in AffineMap(r,q`1,r,q`2).:A;
    then reconsider q1 = u as Point of TOP-REAL 2;
    consider q2 such that
A14: q2 in A and
A15: q1 = AffineMap(r,q`1,r,q`2).q2 by A13,FUNCT_2:65;
    reconsider x1 = q1, x2 = q2 as Element of Euclid 2 by TOPREAL3:8;
A16: dist(y,x2) < 1 by A14,METRIC_1:11;
A17: ex z3,z4 being Point of Euclid 2 st z3 = |[0,0]| & z4 = q2 & dist(|[0,
    0]|,q2) = dist(z3,z4) by TOPREAL6:def 1;
A18: ex z1,z2 being Point of Euclid 2 st z1 = q & z2 = r*q2+q & dist(q,r*q2
    +q) = dist(z1,z2) by TOPREAL6:def 1;
    q1 = r*q2 + q by A15,Th33;
    then dist(x,x1) = dist(|[0,0]|+q,r*q2 + q) by A2,A18,EUCLID:54,RLVECT_1:4
      .= dist(|[0,0]|,r*q2) by Th21
      .= |.r.|*dist(|[0,0]|,q2) by Th20
      .= r*dist(y,x2) by A1,A3,A17,ABSVALUE:def 1;
    then dist(x,x1) < r by A3,A16,XREAL_1:157;
    hence thesis by METRIC_1:11;
  end;
  hence thesis by A4;
end;
