reserve T1,T2,T3 for TopSpace,
  A1 for Subset of T1, A2 for Subset of T2, A3 for Subset of T3;
reserve n,k for Nat;
reserve M,N for non empty TopSpace;
reserve p,q,p1,p2 for Point of TOP-REAL n;
reserve r for Real;

theorem Th29:
  for p,q being Point of TOP-REAL(n+1) st p<>0.TOP-REAL(n+1)
  holds TOP-REAL n, TPlane(p,q) are_homeomorphic
proof
  set T1 = TOP-REAL(n+1);
  let p,q be Point of T1;
  assume
A1: p <> 0.T1;
  reconsider p0 = Base_FinSeq(n+1,n+1) as Point of T1 by EUCLID:22;
A2: p0 <> 0.T1
  proof
    assume
A3: p0 = 0.T1;
    0+1 <= n+1 by XREAL_1:6;
    then |. p0 .| = 1 by EUCLID_7:28;
    hence contradiction by A3,EUCLID_2:39;
  end;
A4: TOP-REAL n, TPlane(p0, 0.TOP-REAL(n+1)) are_homeomorphic by Lm3;
  ex R being Function of T1,T1 st R is being_homeomorphism &
  R .: Plane(p0,0.T1) = Plane(p,0.T1) by A1,A2,Th27;
  then TPlane(p0, 0.T1), TPlane(p,0.T1) are_homeomorphic
  by METRIZTS:3,def 1;
  then
A5: TOP-REAL n, TPlane(p,0.T1) are_homeomorphic by A4,BORSUK_3:3;
  transl(q,T1) .: Plane(p,0.T1) = Plane(p,0.T1+q) by Th25
  .= Plane(p,q) by RLVECT_1:4;
  then T1|Plane(p, 0.T1), T1|Plane(p,q) are_homeomorphic by METRIZTS:3,def 1;
  hence thesis by A5,BORSUK_3:3;
end;
