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;
reserve n for non zero Nat;

theorem Th57:
  for f being Function of Tcircle(0.TOP-REAL(n+1),1),TOP-REAL n holds
  f is without_antipodals implies Sn1->Sn(f) is odd
  proof
    set TC4 = Tcircle(0.TOP-REAL(n+1),1);
    let f be Function of TC4,TOP-REAL n;
    assume
A1: f is without_antipodals;
    set g = Sn1->Sn(f);
    let x, y be complex-valued Function such that
A2: x in dom g and
A3: -x in dom g and
A4: y = g.x;
    reconsider b = -x as Point of TC4 by A3;
    reconsider a = -b as Point of TC4 by A2;
    set p = f.b-f.a;
    set q = f.a-f.b;
A5: p = -q by RLVECT_1:33;
    0.TOP-REAL n = 0*n by EUCLID:70;
    then
A6: -(p qua real-valued Function) <> 0.TOP-REAL n by A1,Th56,Th14;
    thus g.-x = Rn->S1(p) by Def9
    .= p (/) |. p .| by A1,Th56,Def8
    .= p (/) |. -q .| by RLVECT_1:33
    .= (-q) (/) |. -q .| by RLVECT_1:33
    .= -(q qua real-valued Function) (/) |. -q .| by VALUED_2:30
    .= -q (/) |.q.| by TOPRNS_1:26
    .= -Rn->S1(q) by A5,A6,Def8
    .= -y by A4,Def9;
  end;
