let cn be Real; :: thesis: for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous
let K1 be non empty Subset of (TOP-REAL 2); :: thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous
let f be Function of ((TOP-REAL 2) | K1),R^1 ; :: thesis: ( - 1 < cn & cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ) implies f is continuous )
assume A1: ( - 1 < cn & cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * ((((p `1 ) / |.p.|) - cn) / (1 - cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ) ) ; :: thesis: f is continuous
set a = cn;
set b = 1 - cn;
A2: 1 - cn > 0 by A1, XREAL_1:151;
reconsider g2 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm2;
reconsider g1 = (2 NormF ) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5;
for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q <> 0. (TOP-REAL 2) by A1;
then for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6;
then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that
A3: ( ( for q being Point of ((TOP-REAL 2) | K1)
for r1, r2 being Real st g2 . q = r1 & g1 . q = r2 holds
g3 . q = r2 * (((r1 / r2) - cn) / (1 - cn)) ) & g3 is continuous ) by A2, Th10;
A4: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def 1;
then A5: dom f = dom g3 by FUNCT_2:def 1;
for x being set st x in dom f holds
f . x = g3 . x

proof

let x be set ; :: thesis: ( x in dom f implies f . x = g3 . x )
assume A6: x in dom f ; :: thesis: f . x = g3 . x
then x in K1 by A4, A5, PRE_TOPC:29;
then reconsider r = x as Point of (TOP-REAL 2) ;
reconsider s = x as Point of ((TOP-REAL 2) | K1) by A6;
A7: f . r = |.r.| * ((((r `1 ) / |.r.|) - cn) / (1 - cn)) by A1, A6;
A8: g2 . s = proj1 . s by Lm2;
A9: g1 . s = (2 NormF ) . s by Lm5;
A10: proj1 . r = r `1 by PSCOMP_1:def 28;
(2 NormF ) . r = |.r.| by Def1;
hence f . x = g3 . x by A3, A7, A8, A9, A10; :: thesis: verum

end;

hence f is continuous by A3, A5, FUNCT_1:9; :: thesis: verum