let o be Point of (TOP-REAL 2); :: thesis: diffX2_1 o is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1
reconsider Xo = diffX2_1 o as Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1 by TOPMETR:24;
for p being Point of [:(TOP-REAL 2),(TOP-REAL 2):]
for V being Subset of R^1 st Xo . p in V & V is open holds
ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Xo .: W c= V )

proof

let p be Point of [:(TOP-REAL 2),(TOP-REAL 2):]; :: thesis: for V being Subset of R^1 st Xo . p in V & V is open holds
ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Xo .: W c= V )
let V be Subset of R^1 ; :: thesis: ( Xo . p in V & V is open implies ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Xo .: W c= V ) )
assume that
A1: Xo . p in V and
A2: V is open ; :: thesis: ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Xo .: W c= V )
A3: Xo . p = ((p `2 ) `1 ) - (o `1 ) by Def1;
set r = ((p `2 ) `1 ) - (o `1 );
reconsider V1 = V as open Subset of REAL by A2, BORSUK_5:62, TOPMETR:24;
consider g being real number such that
A4: 0 < g and
A5: ].((((p `2 ) `1 ) - (o `1 )) - g),((((p `2 ) `1 ) - (o `1 )) + g).[ c= V1 by A1, A3, RCOMP_1:40;
reconsider g = g as Element of REAL by XREAL_0:def 1;
set W2 = { |[x,y]| where x, y is Element of REAL : ( ((p `2 ) `1 ) - g < x & x < ((p `2 ) `1 ) + g ) } ;
{ |[x,y]| where x, y is Element of REAL : ( ((p `2 ) `1 ) - g < x & x < ((p `2 ) `1 ) + g ) } c= the carrier of (TOP-REAL 2)

proof

let a be set ; :: according to TARSKI:def 3 :: thesis: ( not a in { |[x,y]| where x, y is Element of REAL : ( ((p `2 ) `1 ) - g < x & x < ((p `2 ) `1 ) + g ) } or a in the carrier of (TOP-REAL 2) )
assume a in { |[x,y]| where x, y is Element of REAL : ( ((p `2 ) `1 ) - g < x & x < ((p `2 ) `1 ) + g ) } ; :: thesis: a in the carrier of (TOP-REAL 2)
then ex x, y being Element of REAL st
( a = |[x,y]| & ((p `2 ) `1 ) - g < x & x < ((p `2 ) `1 ) + g ) ;
hence a in the carrier of (TOP-REAL 2) ; :: thesis: verum

end;

then reconsider W2 = { |[x,y]| where x, y is Element of REAL : ( ((p `2 ) `1 ) - g < x & x < ((p `2 ) `1 ) + g ) } as Subset of (TOP-REAL 2) ;
take [:([#] (TOP-REAL 2)),W2:] ; :: thesis: ( p in [:([#] (TOP-REAL 2)),W2:] & [:([#] (TOP-REAL 2)),W2:] is open & Xo .: [:([#] (TOP-REAL 2)),W2:] c= V )
A6: p `2 = |[((p `2 ) `1 ),((p `2 ) `2 )]| by EUCLID:57;
A7: p = [(p `1 ),(p `2 )] by Lm5, MCART_1:23;
A8: ((p `2 ) `1 ) - g < ((p `2 ) `1 ) - 0 by A4, XREAL_1:17;
((p `2 ) `1 ) + 0 < ((p `2 ) `1 ) + g by A4, XREAL_1:8;
then p `2 in W2 by A6, A8;
hence p in [:([#] (TOP-REAL 2)),W2:] by A7, ZFMISC_1:def 2; :: thesis: ( [:([#] (TOP-REAL 2)),W2:] is open & Xo .: [:([#] (TOP-REAL 2)),W2:] c= V )
W2 is open by PSCOMP_1:66;
hence [:([#] (TOP-REAL 2)),W2:] is open by BORSUK_1:46; :: thesis: Xo .: [:([#] (TOP-REAL 2)),W2:] c= V
let b be set ; :: according to TARSKI:def 3 :: thesis: ( not b in Xo .: [:([#] (TOP-REAL 2)),W2:] or b in V )
assume b in Xo .: [:([#] (TOP-REAL 2)),W2:] ; :: thesis: b in V
then consider a being Point of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A9: a in [:([#] (TOP-REAL 2)),W2:] and
A10: Xo . a = b by FUNCT_2:116;
A11: a = [(a `1 ),(a `2 )] by Lm5, MCART_1:23;
A12: (diffX2_1 o) . a = ((a `2 ) `1 ) - (o `1 ) by Def1;
a `2 in W2 by A9, A11, ZFMISC_1:106;
then consider x2, y2 being Element of REAL such that
A13: a `2 = |[x2,y2]| and
A14: ((p `2 ) `1 ) - g < x2 and
A15: x2 < ((p `2 ) `1 ) + g ;
A16: (a `2 ) `1 = x2 by A13, EUCLID:56;
then A17: (((p `2 ) `1 ) - g) - (o `1 ) < ((a `2 ) `1 ) - (o `1 ) by A14, XREAL_1:11;
((a `2 ) `1 ) - (o `1 ) < (((p `2 ) `1 ) + g) - (o `1 ) by A15, A16, XREAL_1:11;
then ((a `2 ) `1 ) - (o `1 ) in ].((((p `2 ) `1 ) - (o `1 )) - g),((((p `2 ) `1 ) - (o `1 )) + g).[ by A17, XXREAL_1:4;
hence b in V by A5, A10, A12; :: thesis: verum

end;

hence diffX2_1 o is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1 by JGRAPH_2:20; :: thesis: verum