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 )
reconsider g1 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm7;
reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5;
assume that
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))) and
A2: 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
A3: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;

now :: thesis: for q being Point of ((TOP-REAL 2) | K1) holds g2 . q <> 0

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 A3;
g2 . q = proj2 . q by Lm4
.= q2 `2 by PSCOMP_1:def 6 ;
hence g2 . q <> 0 by A2; :: thesis: verum

end;

then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that
A4: for q being Point of ((TOP-REAL 2) | K1)
for r1, r2 being Real st g1 . q = r1 & g2 . q = r2 holds
g3 . q = r2 / (sqrt (1 + ((r1 / r2) ^2))) and
A5: g3 is continuous by Th10;
A6: for x being object st x in dom f holds
f . x = g3 . x

proof

let x be object ; :: thesis: ( x in dom f implies f . x = g3 . x )
assume A7: x in dom f ; :: thesis: f . x = g3 . x
then reconsider s = x as Point of ((TOP-REAL 2) | K1) ;
x in the carrier of ((TOP-REAL 2) | K1) by A7;
then x in K1 by PRE_TOPC:8;
then reconsider r = x as Point of (TOP-REAL 2) ;
A8: ( proj2 . r = r `2 & proj1 . r = r `1 ) by PSCOMP_1:def 5, PSCOMP_1:def 6;
A9: ( g2 . s = proj2 . s & g1 . s = proj1 . s ) by Lm4, Lm6;
f . r = (r `2) / (sqrt (1 + (((r `1) / (r `2)) ^2))) by A1, A7;
hence f . x = g3 . x by A4, A9, A8; :: thesis: verum

end;

dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def 1;
then dom f = dom g3 by FUNCT_2:def 1;
hence f is continuous by A5, A6, FUNCT_1:2; :: thesis: verum