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

theorem Th7:
  i in Seg n & ClosedHypercube(p,R) is non empty implies
    PROJ(n,i).:ClosedHypercube(p,R) = [. p.i - R.i, p.i + R.i .]
proof
  set P=PROJ(n,i),H=ClosedHypercube(p,R),TR=TOP-REAL n;
  assume that
A1: i in Seg n
  and
A2: H is non empty;
  hereby
    let y be object;
    assume y in P.:H;
    then consider x be object such that
A3:   x in dom P
    and
A4:   x in H
    and
A5:   P.x =y by FUNCT_1:def 6;
    reconsider x as Point of TR by A3;
    len x = n by CARD_1:def 7;
    then dom x = Seg n by FINSEQ_1:def 3;
    then
A6:   x/.i = x.i by A1,PARTFUN1:def 6;
    P.x = x/.i by TOPREALC:def 6;
    hence y in [. p.i - R.i, p.i + R.i .] by A6,A1,A4,A5, Def2;
  end;
  let y be object;
  assume
A7:y in [. p.i - R.i, p.i + R.i .];
  then reconsider r=y as Real;
  set pr=p+*(i,r);
A8: dom P=the carrier of TR by FUNCT_2:def 1;
A9: dom pr = Seg len pr by FINSEQ_1:def 3;
  len pr = n by CARD_1:def 7;
  then
A10: pr/.i = pr.i by A9,A1,PARTFUN1:def 6;
A11: dom p = Seg len p by FINSEQ_1:def 3;
  for i st i in Seg n /\ dom R holds R.i >= 0 by A2,Th4;
  then pr in H by A1,Th6, A7,Th5;
  then
A12:P.pr in P.:H by A8,FUNCT_1:def 6;
A13: len p = n by CARD_1:def 7;
  P.pr = pr/.i by TOPREALC:def 6;
  hence thesis by A13,A11,A10,A1, FUNCT_7:31,A12;
end;
