let K1 be non empty Subset of (TOP-REAL 2); :: thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = (p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0 ) holds
f is continuous
let f be Function of ((TOP-REAL 2) | K1),R^1 ; :: thesis: ( ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = (p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0 ) implies f is continuous )
assume A1: ( ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = (p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0 ) ) ; :: thesis: f is continuous
A2: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:29;
reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm4;
reconsider g1 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm6;

now

let q be Point of ((TOP-REAL 2) | K1); :: thesis: g2 . q <> 0
q in the carrier of ((TOP-REAL 2) | K1) ;
then reconsider q2 = q as Point of (TOP-REAL 2) by A2;
g2 . q = proj2 . q by Lm3
.= q2 `2 by PSCOMP_1:def 29 ;
hence g2 . q <> 0 by A1; :: thesis: verum

end;

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 number st g1 . q = r1 & g2 . q = r2 holds
g3 . q = r2 / (sqrt (1 + ((r1 / r2) ^2 ))) ) & g3 is continuous ) by Th20;
dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def 1;
then A4: 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 A5: x in dom f ; :: thesis: f . x = g3 . x
then x in the carrier of ((TOP-REAL 2) | K1) ;
then x in K1 by PRE_TOPC:29;
then reconsider r = x as Point of (TOP-REAL 2) ;
reconsider s = x as Point of ((TOP-REAL 2) | K1) by A5;
A6: f . r = (r `2 ) / (sqrt (1 + (((r `1 ) / (r `2 )) ^2 ))) by A1, A5;
A7: g2 . s = proj2 . s by Lm3;
A8: g1 . s = proj1 . s by Lm5;
A9: proj2 . r = r `2 by PSCOMP_1:def 29;
proj1 . r = r `1 by PSCOMP_1:def 28;
hence f . x = g3 . x by A3, A6, A7, A8, A9; :: thesis: verum

end;

hence f is continuous by A3, A4, FUNCT_1:9; :: thesis: verum