defpred S₁[ Point of (TOP-REAL 2)] means ( $1 `1 <= 0 & $1 <> 0. (TOP-REAL 2) );
let sn be Real; :: thesis: for B0 being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | B0)
for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous
let B0 be Subset of (TOP-REAL 2); :: thesis: for K0 being Subset of ((TOP-REAL 2) | B0)
for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous
let K0 be Subset of ((TOP-REAL 2) | B0); :: thesis: for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous
let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); :: thesis: ( - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous )
reconsider K1 = { p where p is Point of (TOP-REAL 2) : S<sub>1</sub>[p] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch 1();
assume A1: ( - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; :: thesis: f is continuous
K0 c= B0 then ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by A1, PRE_TOPC:7;
hence f is continuous by A1, Th27; :: thesis: verum