  reserve n,m,i for Nat,
          p,q for Point of TOP-REAL n,
          r,s for Real,
          R for real-valued FinSequence;

theorem Th2:
  i in Seg n implies PROJ(n,i).:OpenHypercube(p,r) = ]. p.i - r, p.i + r .[
proof
  assume
A1: i in Seg n;
  consider e be Point of Euclid n such that
A2: p=e
  and
A3: OpenHypercube(p,r) = OpenHypercube(e,r) by Def1;
  set OH=OpenHypercube(e,r),I=Intervals(e,r);
A4:OH = product I by EUCLID_9:def 4;
  len e = n by CARD_1:def 7;
  then
A5:dom e = Seg n by FINSEQ_1:def 3;
  then
A6:I.i = ]. e.i-r,e.i+r.[ by A1,EUCLID_9:def 3;
  hereby
    let y be object;
    assume y in PROJ(n,i).:OpenHypercube(p,r);
    then consider x be object such that
A7:     x in dom PROJ(n,i)
      and
A8:     x in OH
      and
A9:     PROJ(n,i).x = y by A3,FUNCT_1:def 6;
    reconsider x as Point of TOP-REAL n by A7;
A10:dom I = dom x by A8,A4,CARD_3:9;
    then
A11:dom x = Seg n by EUCLID_9:def 3,A5;
    then
A12: x/.i=x.i by A1,PARTFUN1:def 6;
    x.i in I.i by A11,A10,CARD_3:9,A8,A4,A1;
    hence y in ]. p.i - r, p.i + r .[ by A2,TOPREALC:def 6,A12,A6, A9;
  end;
  let y be object;
  assume
A13:y in ]. p.i - r, p.i + r .[;
  then reconsider s=y as Real;
A14: s < e.i+r by A2,A13, XXREAL_1:4;
  set ps = p+*(i,s);
  e.i -r < s by A2,A13, XXREAL_1:4;
  then e.i +- r < e.i +r by A14,XXREAL_0:2;
  then r>0 by XREAL_1:6;
  then
A15: ps in OH by A2,A3,EUCLID_9:11,Th1,A13;
  len ps= n by CARD_1:def 7;
  then
A16: dom ps = Seg n by FINSEQ_1:def 3;
  len p = n by CARD_1:def 7;
  then
A17: dom p = Seg n by FINSEQ_1:def 3;
A18: dom PROJ(n,i)=the carrier of TOP-REAL n by FUNCT_2:def 1;
  PROJ(n,i).ps = ps/.i by TOPREALC:def 6
              .= ps.i by A16,A1,PARTFUN1:def 6
              .= s by A17,A1,FUNCT_7:31;
  hence thesis by A3, A18,A15, FUNCT_1:def 6;
end;
