reconsider O = 0. (TOP-REAL 2) as Point of (Euclid 2) by EUCLID:71;
let sn be Real; :: thesis: for p2 being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 holds
ex K being non empty compact Subset of (TOP-REAL 2) st
( K = (sn -FanMorphE ) .: K & ex V2 being Subset of (TOP-REAL 2) st
( p2 in V2 & V2 is open & V2 c= K & (sn -FanMorphE ) . p2 in V2 ) )
let p2 be Point of (TOP-REAL 2); :: thesis: ( - 1 < sn & sn < 1 implies ex K being non empty compact Subset of (TOP-REAL 2) st
( K = (sn -FanMorphE ) .: K & ex V2 being Subset of (TOP-REAL 2) st
( p2 in V2 & V2 is open & V2 c= K & (sn -FanMorphE ) . p2 in V2 ) ) )
A1: TopStruct(# the carrier of (TOP-REAL 2),the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def 8;
TopStruct(# the carrier of (TOP-REAL 2),the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def 8;
then reconsider V0 = Ball O,(|.p2.| + 1) as Subset of (TOP-REAL 2) ;
( O in V0 & V0 c= cl_Ball O,(|.p2.| + 1) ) by GOBOARD6:4, METRIC_1:15;
then reconsider K0 = cl_Ball O,(|.p2.| + 1) as non empty compact Subset of (TOP-REAL 2) by A1, Th22;
set q3 = (sn -FanMorphE ) . p2;
reconsider VV0 = V0 as Subset of (TopSpaceMetr (Euclid 2)) ;
reconsider u2 = p2 as Point of (Euclid 2) by EUCLID:71;
reconsider u3 = (sn -FanMorphE ) . p2 as Point of (Euclid 2) by EUCLID:71;
A2: (sn -FanMorphE ) .: K0 c= K0

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in (sn -FanMorphE ) .: K0 or y in K0 )
assume y in (sn -FanMorphE ) .: K0 ; :: thesis: y in K0
then consider x being set such that
A3: x in dom (sn -FanMorphE ) and
A4: x in K0 and
A5: y = (sn -FanMorphE ) . x by FUNCT_1:def 12;
reconsider q = x as Point of (TOP-REAL 2) by A3;
reconsider uq = q as Point of (Euclid 2) by EUCLID:71;
dist O,uq <= |.p2.| + 1 by A4, METRIC_1:13;
then |.((0. (TOP-REAL 2)) - q).| <= |.p2.| + 1 by JGRAPH_1:45;
then |.(- q).| <= |.p2.| + 1 by EUCLID:31;
then A6: |.q.| <= |.p2.| + 1 by TOPRNS_1:27;
A7: y in rng (sn -FanMorphE ) by A3, A5, FUNCT_1:def 5;
then reconsider u = y as Point of (Euclid 2) by EUCLID:71;
reconsider q4 = y as Point of (TOP-REAL 2) by A7;
|.q4.| = |.q.| by A5, Th104;
then |.(- q4).| <= |.p2.| + 1 by A6, TOPRNS_1:27;
then |.((0. (TOP-REAL 2)) - q4).| <= |.p2.| + 1 by EUCLID:31;
then dist O,u <= |.p2.| + 1 by JGRAPH_1:45;
hence y in K0 by METRIC_1:13; :: thesis: verum

end;

VV0 is open by TOPMETR:21;
then A8: V0 is open by Lm11, PRE_TOPC:60;
A9: |.p2.| < |.p2.| + 1 by XREAL_1:31;
then |.(- p2).| < |.p2.| + 1 by TOPRNS_1:27;
then |.((0. (TOP-REAL 2)) - p2).| < |.p2.| + 1 by EUCLID:31;
then dist O,u2 < |.p2.| + 1 by JGRAPH_1:45;
then A10: p2 in V0 by METRIC_1:12;
|.((sn -FanMorphE ) . p2).| = |.p2.| by Th104;
then |.(- ((sn -FanMorphE ) . p2)).| < |.p2.| + 1 by A9, TOPRNS_1:27;
then |.((0. (TOP-REAL 2)) - ((sn -FanMorphE ) . p2)).| < |.p2.| + 1 by EUCLID:31;
then dist O,u3 < |.p2.| + 1 by JGRAPH_1:45;
then A11: (sn -FanMorphE ) . p2 in V0 by METRIC_1:12;
assume A12: ( - 1 < sn & sn < 1 ) ; :: thesis: ex K being non empty compact Subset of (TOP-REAL 2) st
( K = (sn -FanMorphE ) .: K & ex V2 being Subset of (TOP-REAL 2) st
( p2 in V2 & V2 is open & V2 c= K & (sn -FanMorphE ) . p2 in V2 ) )
K0 c= (sn -FanMorphE ) .: K0

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in K0 or y in (sn -FanMorphE ) .: K0 )
assume A13: y in K0 ; :: thesis: y in (sn -FanMorphE ) .: K0
then reconsider q4 = y as Point of (TOP-REAL 2) ;
reconsider y = y as Point of (Euclid 2) by A13;
the carrier of (TOP-REAL 2) c= rng (sn -FanMorphE ) by A12, Th110;
then q4 in rng (sn -FanMorphE ) by TARSKI:def 3;
then consider x being set such that
A14: x in dom (sn -FanMorphE ) and
A15: y = (sn -FanMorphE ) . x by FUNCT_1:def 5;
reconsider x = x as Point of (Euclid 2) by A14, Lm11;
reconsider q = x as Point of (TOP-REAL 2) by A14;
|.q4.| = |.q.| by A15, Th104;
then q in K0 by A13, Lm12;
hence y in (sn -FanMorphE ) .: K0 by A14, A15, FUNCT_1:def 12; :: thesis: verum

end;

then K0 = (sn -FanMorphE ) .: K0 by A2, XBOOLE_0:def 10;
hence ex K being non empty compact Subset of (TOP-REAL 2) st
( K = (sn -FanMorphE ) .: K & ex V2 being Subset of (TOP-REAL 2) st
( p2 in V2 & V2 is open & V2 c= K & (sn -FanMorphE ) . p2 in V2 ) ) by A10, A8, A11, METRIC_1:15; :: thesis: verum