let sn be Real; :: thesis: ( - 1 < sn & sn < 1 implies ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f = sn -FanMorphE & f is being_homeomorphism ) )
assume A1: ( - 1 < sn & sn < 1 ) ; :: thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f = sn -FanMorphE & f is being_homeomorphism )
then A2: ( sn -FanMorphE is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (sn -FanMorphE ) = the carrier of (TOP-REAL 2) ) by Th110;
reconsider f = sn -FanMorphE as Function of (TOP-REAL 2),(TOP-REAL 2) ;
consider h being Function of (TOP-REAL 2),(TOP-REAL 2) such that
A3: ( h = sn -FanMorphE & h is continuous ) by A1, Th108;
A4: f is one-to-one by A1, Th109;
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 A1, Th111;
then f is being_homeomorphism by A2, A3, A4, Th8;
hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f = sn -FanMorphE & f is being_homeomorphism ) ; :: thesis: verum