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) / (p `1)) / (p `1) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 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) / (p `1)) / (p `1) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 0 ) implies f is continuous )
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) / (p `1)) / (p `1) and
A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 0 ; :: thesis: f is continuous
reconsider g2 = proj2 | K1 as Function of ((TOP-REAL 2) | K1),R^1 by TOPMETR:17;
reconsider g1 = proj1 | K1 as Function of ((TOP-REAL 2) | K1),R^1 by TOPMETR:17;
A3: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1)
.= K1 by PRE_TOPC:def 5 ;
A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q = proj1 . q

proof

let q be Point of ((TOP-REAL 2) | K1); :: thesis: g1 . q = proj1 . q
( q in the carrier of ((TOP-REAL 2) | K1) & dom proj1 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def 1;
then q in (dom proj1) /\ K1 by A3, XBOOLE_0:def 4;
hence g1 . q = proj1 . q by FUNCT_1:48; :: thesis: verum

end;

then A5: g1 is continuous by Th29;
A6: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0

proof

let q be Point of ((TOP-REAL 2) | K1); :: thesis: g1 . q <> 0
q in the carrier of ((TOP-REAL 2) | K1) ;
then reconsider q2 = q as Point of (TOP-REAL 2) by A3;
g1 . q = proj1 . q by A4
.= q2 `1 by PSCOMP_1:def 5 ;
hence g1 . q <> 0 by A2; :: thesis: verum

end;

A7: for q being Point of ((TOP-REAL 2) | K1) holds g2 . q = proj2 . q

proof

let q be Point of ((TOP-REAL 2) | K1); :: thesis: g2 . q = proj2 . q
( q in the carrier of ((TOP-REAL 2) | K1) & dom proj2 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def 1;
then q in (dom proj2) /\ K1 by A3, XBOOLE_0:def 4;
hence g2 . q = proj2 . q by FUNCT_1:48; :: thesis: verum

end;

then g2 is continuous by Th30;
then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that
A8: for q being Point of ((TOP-REAL 2) | K1)
for r1, r2 being Real st g2 . q = r1 & g1 . q = r2 holds
g3 . q = (r1 / r2) / r2 and
A9: g3 is continuous by A5, A6, Th28;
A10: 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 A11: x in dom f ; :: thesis: f . x = g3 . x
then reconsider s = x as Point of ((TOP-REAL 2) | K1) ;
x in [#] ((TOP-REAL 2) | K1) by A11;
then x in K1 by PRE_TOPC:def 5;
then reconsider r = x as Point of (TOP-REAL 2) ;
A12: ( proj2 . r = r `2 & proj1 . r = r `1 ) by PSCOMP_1:def 5, PSCOMP_1:def 6;
A13: ( g2 . s = proj2 . s & g1 . s = proj1 . s ) by A7, A4;
f . r = ((r `2) / (r `1)) / (r `1) by A1, A11;
hence f . x = g3 . x by A8, A13, A12; :: 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 A9, A10, FUNCT_1:2; :: thesis: verum