let K1 be non empty Subset of (); :: thesis: for f being Function of (() | K1),R^1 st ( for p being Point of () st p in the carrier of (() | K1) holds
f . p = (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2))) ) & ( for q being Point of () st q in the carrier of (() | K1) holds
q `1 <> 0 ) holds
f is continuous

let f be Function of (() | K1),R^1; :: thesis: ( ( for p being Point of () st p in the carrier of (() | K1) holds
f . p = (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2))) ) & ( for q being Point of () st q in the carrier of (() | K1) holds
q `1 <> 0 ) implies f is continuous )

reconsider g1 = proj1 | K1 as continuous Function of (() | K1),R^1 by Lm7;
reconsider g2 = proj2 | K1 as continuous Function of (() | K1),R^1 by Lm5;
assume that
A1: for p being Point of () st p in the carrier of (() | K1) holds
f . p = (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2))) and
A2: for q being Point of () st q in the carrier of (() | K1) holds
q `1 <> 0 ; :: thesis: f is continuous
A3: the carrier of (() | K1) = K1 by PRE_TOPC:8;
now :: thesis: for q being Point of (() | K1) holds g1 . q <> 0
let q be Point of (() | K1); :: thesis: g1 . q <> 0
q in the carrier of (() | K1) ;
then reconsider q2 = q as Point of () by A3;
g1 . q = proj1 . q by Lm6
.= q2 `1 by PSCOMP_1:def 5 ;
hence g1 . q <> 0 by A2; :: thesis: verum
end;
then consider g3 being Function of (() | K1),R^1 such that
A4: for q being Point of (() | K1)
for r1, r2 being Real st g2 . q = r1 & g1 . 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 (() | K1) ;
x in the carrier of (() | K1) by A7;
then x in K1 by PRE_TOPC:8;
then reconsider r = x as Point of () ;
A8: ( proj2 . r = r `2 & proj1 . r = r `1 ) by ;
A9: ( g2 . s = proj2 . s & g1 . s = proj1 . s ) by ;
f . r = (r `1) / (sqrt (1 + (((r `2) / (r `1)) ^2))) by A1, A7;
hence f . x = g3 . x by A4, A9, A8; :: thesis: verum
end;
dom g3 = the carrier of (() | K1) by FUNCT_2:def 1;
then dom f = dom g3 by FUNCT_2:def 1;
hence f is continuous by ; :: thesis: verum