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

theorem
  for cn being Real st -1<cn & cn<1 ex f being Function of TOP-REAL 2,
  TOP-REAL 2 st f=cn-FanMorphN & f is being_homeomorphism
proof
  let cn be Real;
  reconsider f=(cn-FanMorphN) as Function of TOP-REAL 2,TOP-REAL 2;
  assume
A1: -1<cn & cn<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 Th73;
  rng (cn-FanMorphN) = the carrier of TOP-REAL 2 & ex h being Function of
  ( TOP-REAL 2),(TOP-REAL 2) st h=( cn-FanMorphN) & h is continuous by A1,Th70
,Th72;
  then f is being_homeomorphism by A1,A2,Th3,Th71;
  hence thesis;
end;
