reserve a for Real;
reserve p,q for Point of TOP-REAL 2;

theorem
  for sn being Real st -1<sn & sn<1 ex f being Function of TOP-REAL 2,
  TOP-REAL 2 st f=(sn-FanMorphE) & f is being_homeomorphism
proof
  let sn be Real;
  reconsider f=(sn-FanMorphE) as Function of TOP-REAL 2,TOP-REAL 2;
  assume
A1: -1<sn & sn<1;
  then
A2: for p2 being Point of TOP-REAL 2 ex K being non empty compact Subset of
  TOP-REAL 2 st K = f.:K & ex V2 being Subset of TOP-REAL 2 st p2 in V2 & V2 is
  open & V2 c= K & f.p2 in V2 by Th104;
  rng (sn-FanMorphE) = the carrier of TOP-REAL 2 & ex h being Function of
( TOP-REAL 2),(TOP-REAL 2) st h=( sn-FanMorphE) & h is continuous by A1,Th101
,Th103;
  then f is being_homeomorphism by A1,A2,Th3,Th102;
  hence thesis;
end;
