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 Th58:
  for f being Function of Tcircle(0.TOP-REAL(n+1),1),TOP-REAL n
  for g, B1 being Function of Tcircle(0.TOP-REAL(n+1),1),TOP-REAL n
  st g = f(-) & B1 = f<-->g & f is without_antipodals holds
  Sn1->Sn(f) = B1 </> (n NormF * B1)
  proof
    let f be Function of Tcircle(0.TOP-REAL(n+1),1),TOP-REAL n;
    let g, B1 be Function of Tcircle(0.TOP-REAL(n+1),1),TOP-REAL n such that
A1: g = f(-) and
A2: B1 = f<-->g and
A3: f is without_antipodals;
    set T = Tcircle(0.TOP-REAL(n+1),1);
    set B = Sn1->Sn(f);
    set B2 = (n NormF)*B1;
    set BB = B1 </> B2;
    set TC3 = Tunit_circle(n+1);
A4: dom B1 = the carrier of TC3 by FUNCT_2:def 1;
    dom(n NormF) = the carrier of TOP-REAL n by FUNCT_2:def 1;
    then rng B1 c= dom(n NormF);
    then
A5: dom B2 = dom B1 by RELAT_1:27;
A6: dom BB = dom B1 /\ dom B2 by VALUED_2:71
    .= the carrier of TC3 by A5,FUNCT_2:def 1;
    hence dom B = dom BB by FUNCT_2:def 1;
    let x be object;
    assume x in dom B;
    then reconsider x1 = x as Point of T;
    reconsider y1 = -x1 as Point of T by TOPREALC:60;
    set p = f.x1-f.y1;
A7: dom g = the carrier of T by FUNCT_2:def 1;
A8: B1.x1 = (f.x1) qua real-valued Function - g.x1 by A4,A2,VALUED_2:def 46
    .= p by A7,A1,VALUED_2:def 34;
A9: B2.x1 = (n NormF).(B1.x1) by FUNCT_2:15
    .= |.p.| by A8,JGRAPH_4:def 1;
    B.x1 = Rn->S1(p) by Def9
    .= B1.x1 (/) B2.x1 by A8,A9,A3,Th56,Def8
    .= B1.x1 (#) (B2 qua complex-valued Function").x1 by VALUED_1:10
    .= BB.x1 by A6,VALUED_2:def 43;
    hence thesis;
  end;
