let D be non empty set ; :: thesis: for Y being RealNormSpace
for H being Functional_Sequence of D, the carrier of Y
for X being set holds
( H is_point_conv_on X iff ( X common_on_dom H & ex f being PartFunc of D, the carrier of Y st
( X = dom f & ( for x being Element of D st x in X holds
( H # x is convergent & lim (H # x) = f . x ) ) ) ) )
let Y be RealNormSpace; :: thesis: for H being Functional_Sequence of D, the carrier of Y
for X being set holds
( H is_point_conv_on X iff ( X common_on_dom H & ex f being PartFunc of D, the carrier of Y st
( X = dom f & ( for x being Element of D st x in X holds
( H # x is convergent & lim (H # x) = f . x ) ) ) ) )
let H be Functional_Sequence of D, the carrier of Y; :: thesis: for X being set holds
( H is_point_conv_on X iff ( X common_on_dom H & ex f being PartFunc of D, the carrier of Y st
( X = dom f & ( for x being Element of D st x in X holds
( H # x is convergent & lim (H # x) = f . x ) ) ) ) )
let X be set ; :: thesis: ( H is_point_conv_on X iff ( X common_on_dom H & ex f being PartFunc of D, the carrier of Y st
( X = dom f & ( for x being Element of D st x in X holds
( H # x is convergent & lim (H # x) = f . x ) ) ) ) )
thus ( H is_point_conv_on X implies ( X common_on_dom H & ex f being PartFunc of D, the carrier of Y st
( X = dom f & ( for x being Element of D st x in X holds
( H # x is convergent & lim (H # x) = f . x ) ) ) ) ) :: thesis: ( X common_on_dom H & ex f being PartFunc of D, the carrier of Y st
( X = dom f & ( for x being Element of D st x in X holds
( H # x is convergent & lim (H # x) = f . x ) ) ) implies H is_point_conv_on X )

proof

assume A1: H is_point_conv_on X ; :: thesis: ( X common_on_dom H & ex f being PartFunc of D, the carrier of Y st
( X = dom f & ( for x being Element of D st x in X holds
( H # x is convergent & lim (H # x) = f . x ) ) ) )
hence X common_on_dom H ; :: thesis: ex f being PartFunc of D, the carrier of Y st
( X = dom f & ( for x being Element of D st x in X holds
( H # x is convergent & lim (H # x) = f . x ) ) )
consider f being PartFunc of D, the carrier of Y such that
A2: X = dom f and
A3: for x being Element of D st x in X holds
for p being Real st p > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p by A1;
take f ; :: thesis: ( X = dom f & ( for x being Element of D st x in X holds
( H # x is convergent & lim (H # x) = f . x ) ) )
thus X = dom f by A2; :: thesis: for x being Element of D st x in X holds
( H # x is convergent & lim (H # x) = f . x )
let x be Element of D; :: thesis: ( x in X implies ( H # x is convergent & lim (H # x) = f . x ) )
assume A4: x in X ; :: thesis: ( H # x is convergent & lim (H # x) = f . x )
then X4: f /. x = f . x by A2, PARTFUN1:def 6;

A5: now :: thesis: for p being Real st p > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
||.(((H # x) . n) - (f /. x)).|| < p

let p be Real; :: thesis: ( p > 0 implies ex k being Nat st
for n being Nat st n >= k holds
||.(((H # x) . n) - (f /. x)).|| < p )
assume A6: p > 0 ; :: thesis: ex k being Nat st
for n being Nat st n >= k holds
||.(((H # x) . n) - (f /. x)).|| < p
consider k being Nat such that
A7: for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p by A3, A4, A6;
take k = k; :: thesis: for n being Nat st n >= k holds
||.(((H # x) . n) - (f /. x)).|| < p
let n be Nat; :: thesis: ( n >= k implies ||.(((H # x) . n) - (f /. x)).|| < p )
assume n >= k ; :: thesis: ||.(((H # x) . n) - (f /. x)).|| < p
then ||.(((H . n) /. x) - (f /. x)).|| < p by A7;
hence ||.(((H # x) . n) - (f /. x)).|| < p by Def10; :: thesis: verum

end;

hence H # x is convergent by NORMSP_1:def 6; :: thesis: lim (H # x) = f . x
hence lim (H # x) = f . x by X4, A5, NORMSP_1:def 7; :: thesis: verum

end;

assume A8: X common_on_dom H ; :: thesis: ( for f being PartFunc of D, the carrier of Y holds
( not X = dom f or ex x being Element of D st
( x in X & not ( H # x is convergent & lim (H # x) = f . x ) ) ) or H is_point_conv_on X )
given f being PartFunc of D, the carrier of Y such that A9: X = dom f and
A10: for x being Element of D st x in X holds
( H # x is convergent & lim (H # x) = f . x ) ; :: thesis: H is_point_conv_on X
ex f being PartFunc of D, the carrier of Y st
( X = dom f & ( for x being Element of D st x in X holds
for p being Real st p > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p ) )

proof

take f ; :: thesis: ( X = dom f & ( for x being Element of D st x in X holds
for p being Real st p > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p ) )
thus X = dom f by A9; :: thesis: for x being Element of D st x in X holds
for p being Real st p > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p
let x be Element of D; :: thesis: ( x in X implies for p being Real st p > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p )
assume X10: x in X ; :: thesis: for p being Real st p > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p
then A11: ( H # x is convergent & lim (H # x) = f . x ) by A10;
X11: f /. x = f . x by PARTFUN1:def 6, X10, A9;
let p be Real; :: thesis: ( p > 0 implies ex k being Nat st
for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p )
assume p > 0 ; :: thesis: ex k being Nat st
for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p
then consider k being Nat such that
A12: for n being Nat st n >= k holds
||.(((H # x) . n) - (f /. x)).|| < p by A11, NORMSP_1:def 7, X11;
take k ; :: thesis: for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p
let n be Nat; :: thesis: ( n >= k implies ||.(((H . n) /. x) - (f /. x)).|| < p )
assume n >= k ; :: thesis: ||.(((H . n) /. x) - (f /. x)).|| < p
then ||.(((H # x) . n) - (f /. x)).|| < p by A12;
hence ||.(((H . n) /. x) - (f /. x)).|| < p by Def10; :: thesis: verum

end;

hence H is_point_conv_on X by A8; :: thesis: verum