reserve a,b,c,x,y,z for object,X,Y,Z for set,
  n for Nat,
  i,j for Integer,
  r,r1,r2,r3,s for Real,
  c1,c2 for Complex,
  p for Point of TOP-REAL n;

theorem Th38:
  for p being Point of TOP-REAL 3, e being Point of Euclid 3 holds
  p = e & p`3 = 0 implies product ((1,2,3) --> (].p`1-r/sqrt 2,p`1+r/sqrt 2.[,
  ].p`2-r/sqrt 2,p`2+r/sqrt 2.[,{0})) c= Ball(e,r)
proof
  let p be Point of TOP-REAL 3, e be Point of Euclid 3;
  set A = ].p`1-r/sqrt 2,p`1+r/sqrt 2.[, B = ].p`2-r/sqrt 2,p`2+r/sqrt 2.[,
  C = {0}, f = (1,2,3) --> (A,B,C);
  assume that
A1: p = e and
A2: p`3 = 0;
  let a be object;
  assume a in product f;
  then consider g being Function such that
A3: a = g and
A4: dom g = dom f and
A5: for x being object st x in dom f holds g.x in f.x by CARD_3:def 5;
A6: A = {m where m is Real:
  p`1-r/sqrt 2 < m & m < p`1+r/sqrt 2 } by RCOMP_1:def 2;
A7: B = {n where n is Real : p`2-r/sqrt 2 < n & n < p`2+r/sqrt 2 }
  by RCOMP_1:def 2;
A8: dom f = {1,2,3} by FUNCT_4:128;
  then
A9: 1 in dom f & 2 in dom f & 3 in dom f by ENUMSET1:def 1;
A10: f.1 = A & f.2 = B & f.3 = C by FUNCT_4:135,134;
  then
A11: g.1 in A & g.2 in B & g.3 in C by A5,A9;
  then consider m being Real such that
A12: m = g.1 and p`1-r/sqrt 2 < m & m < p`1+r/sqrt 2 by A6;
  consider n being Real such that
A13: n = g.2 and p`2-r/sqrt 2 < n & n < p`2+r/sqrt 2 by A7,A11;
  g.3 in f.3 by A5,A9;
  then
A14: g.3 = 0 by A10,TARSKI:def 1;
  p`1+r/sqrt 2 > p`1-r/sqrt 2 by A11,XXREAL_1:28;
  then p`1-(p`1+r/sqrt 2) < p`1-(p`1-r/sqrt 2) by XREAL_1:10;
  then -r/sqrt 2+r/sqrt 2 < r/sqrt 2+r/sqrt 2 by XREAL_1:6;
  then
A15: 0 < r;
A16: dom <*g.1,g.2,g.3*> = Seg len <*g.1,g.2,g.3*> by FINSEQ_1:def 3
     .= {1,2,3} by FINSEQ_1:45,FINSEQ_3:1;
  now
    let k be object;
    assume k in dom g;
    then k = 1 or k = 2 or k = 3 by A4,A8,ENUMSET1:def 1;
    hence g.k = <*g.1,g.2,g.3*>.k;
  end;
  then
A17: a = |[m,n,0]| by A3,A4,A12,A13,A16,A14,FUNCT_4:128,FUNCT_1:2;
  then reconsider c = a as Point of TOP-REAL 3;
  reconsider b = c as Point of Euclid 3 by TOPREAL3:8;
  |.m-p`1.| < r/sqrt 2 & |.n-p`2.| < r/sqrt 2 by A11,A12,A13,RCOMP_1:1;
  then (|.m-p`1.|)^2 < (r/sqrt 2)^2 & (|.n-p`2.|)^2 < (r/sqrt 2)^2
  by SQUARE_1:16;
  then (|.m-p`1.|)^2 < r^2/(sqrt 2)^2 & (|.n-p`2.|)^2 < r^2/(sqrt 2)^2
  by XCMPLX_1:76;
  then (|.m-p`1.|)^2 < r^2/2 & (|.n-p`2.|)^2 < r^2/2 by SQUARE_1:def 2;
  then (m-p`1)^2 < r^2/2 & (n-p`2)^2 < r^2/2 by COMPLEX1:75;
  then (m-p`1)^2 + (n-p`2)^2 < r^2/2 + r^2/2 by XREAL_1:8;
  then sqrt((m-p`1)^2 + (n-p`2)^2) < sqrt(r^2) by SQUARE_1:27;
  then
A18: sqrt((m-p`1)^2 + (n-p`2)^2) < r by A15,SQUARE_1:22;
A19: m = c`1 & n = c`2 by A17,EUCLID_5:2;
  dist(b,e) = (Pitag_dist 3).(b,e) by METRIC_1:def 1
    .= sqrt ((c`1 - p`1)^2 + (c`2 - p`2)^2  + (c`3 - p`3)^2) by A1,Th37
    .= sqrt ((c`1 - p`1)^2 + (c`2 - p`2)^2  + (Q-Q)^2) by A2,A17,EUCLID_5:2
    .= sqrt ((c`1 - p`1)^2 + (c`2 - p`2)^2  + Q);
  hence a in Ball(e,r) by A18,A19,METRIC_1:11;
end;
