reserve T,U for non empty TopSpace;
reserve t for Point of T;
reserve n for Nat;

theorem Th3:
  for p being Point of TOP-REAL n st p in Sphere(0.TOP-REAL n,1)
  holds -p in Sphere(0.TOP-REAL n,1) \ {p}
  proof
    let p be Point of TOP-REAL n;
    reconsider n1=n as Element of NAT by ORDINAL1:def 12;
    reconsider p1=p as Point of TOP-REAL n1;
    assume p in Sphere(0.TOP-REAL n,1);
    then |.p1 - 0.TOP-REAL n1.| = 1 by TOPREAL9:9;
    then |.p1 + -0.TOP-REAL n1.| = 1;
    then |.p + (-1)*0.TOP-REAL n.| = 1 by RLVECT_1:16;
    then |.p + 0.TOP-REAL n.| = 1 by RLVECT_1:10;
    then
A1: |. p .| = 1 by RLVECT_1:4;
    reconsider r1 = 1 as Real;
    |. 0.TOP-REAL n .| <> |. p .| by A1,EUCLID_2:39;
    then 0.TOP-REAL n <> (1+1)*p by RLVECT_1:11;
    then 0.TOP-REAL n <> r1*p + r1*p by RLVECT_1:def 6;
    then 0.TOP-REAL n <> r1*p + p by RLVECT_1:def 8;
    then 0.TOP-REAL n <> p + p by RLVECT_1:def 8;
    then p + -p <> p + p by RLVECT_1:5;
    then
A2: not -p in {p} by TARSKI:def 1;
    |. -p .| = 1 by A1,EUCLID:71;
    then |.-p + 0.TOP-REAL n.| = 1 by RLVECT_1:4;
    then |.-p + (-1)*0.TOP-REAL n.| = 1 by RLVECT_1:10;
    then |.-p + -0.TOP-REAL n.| = 1 by RLVECT_1:16;
    then |.-p1 - 0.TOP-REAL n1.| = 1;
    then
    -p1 in Sphere(0.TOP-REAL n1,1) by TOPREAL9:9;
    hence -p in Sphere(0.TOP-REAL n,1) \ {p} by A2,XBOOLE_0:def 5;
  end;
