:: NFCONT_1 semantic presentation

begin


theorem :: NFCONT_1:1
for S being non empty addLoopStr
for seq1, seq2 being sequence of S holds seq1 - seq2 = seq1 + (- seq2)
proof
let S be non empty addLoopStr ; ::_thesis: for seq1, seq2 being sequence of S holds seq1 - seq2 = seq1 + (- seq2)
let seq1, seq2 be sequence of S; ::_thesis: seq1 - seq2 = seq1 + (- seq2)
 for n being Element of NAT holds (seq1 - seq2) . n = (seq1 + (- seq2)) . n
proof
let n be Element of NAT ; ::_thesis: (seq1 - seq2) . n = (seq1 + (- seq2)) . n
thus (seq1 - seq2) . n = (seq1 . n) - (seq2 . n) by NORMSP_1:def_3
.= (seq1 . n) + ((- seq2) . n) by BHSP_1:44
.= (seq1 + (- seq2)) . n by NORMSP_1:def_2 ; ::_thesis: verum
end;
hence  seq1 - seq2 = seq1 + (- seq2) by FUNCT_2:63; ::_thesis: verum
end;

theorem Th2: :: NFCONT_1:2
for S being RealNormSpace
for seq being sequence of S holds - seq = (- 1) * seq
proof
let S be RealNormSpace; ::_thesis: for seq being sequence of S holds - seq = (- 1) * seq
let seq be sequence of S; ::_thesis: - seq = (- 1) * seq
now__::_thesis:_for_n_being_Element_of_NAT_holds_((-_1)_*_seq)_._n_=_(-_seq)_._n
let n be Element of NAT ; ::_thesis: ((- 1) * seq) . n = (- seq) . n
thus ((- 1) * seq) . n = (- 1) * (seq . n) by NORMSP_1:def_5
.= - (seq . n) by RLVECT_1:16
.= (- seq) . n by BHSP_1:44 ; ::_thesis: verum
end;
hence  - seq = (- 1) * seq by FUNCT_2:63; ::_thesis: verum
end;

definition
let S be RealNormSpace;
let x0 be Point of S;
mode Neighbourhood of x0 ->  Subset of S means :Def1: :: NFCONT_1:def 1
 ex g being Real st
( 0 < g &  {  y where y is Point of S : ||.(y - x0).|| < g  }  c= it );
existence
 ex b1 being Subset of S ex g being Real st
( 0 < g &  {  y where y is Point of S : ||.(y - x0).|| < g  }  c= b1 )
proof
set N =  {  y where y is Point of S : ||.(y - x0).|| < 1  }  ;
take   {  y where y is Point of S : ||.(y - x0).|| < 1  }  ; ::_thesis: (  {  y where y is Point of S : ||.(y - x0).|| < 1  }  is Subset of S & ex g being Real st
( 0 < g &  {  y where y is Point of S : ||.(y - x0).|| < g  }  c=  {  y where y is Point of S : ||.(y - x0).|| < 1  }  ) )
  {  y where y is Point of S : ||.(y - x0).|| < 1  }  c= the carrier of S
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  y where y is Point of S : ||.(y - x0).|| < 1  }  or x in the carrier of S )
assume  x in  {  y where y is Point of S : ||.(y - x0).|| < 1  }  ; ::_thesis: x in the carrier of S
then  ex y being Point of S st
( x = y & ||.(y - x0).|| < 1 ) ;
hence  x in the carrier of S ; ::_thesis: verum
end;
hence  (  {  y where y is Point of S : ||.(y - x0).|| < 1  }  is Subset of S & ex g being Real st
( 0 < g &  {  y where y is Point of S : ||.(y - x0).|| < g  }  c=  {  y where y is Point of S : ||.(y - x0).|| < 1  }  ) ) ; ::_thesis: verum
end;
end;

:: deftheorem Def1 defines Neighbourhood NFCONT_1:def_1_:_
 for S being RealNormSpace
for x0 being Point of S
for b3 being Subset of S holds
( b3 is Neighbourhood of x0 iff ex g being Real st
( 0 < g &  {  y where y is Point of S : ||.(y - x0).|| < g  }  c= b3 ) );

theorem Th3: :: NFCONT_1:3
for S being RealNormSpace
for x0 being Point of S
for g being Real st 0 < g holds
{  y where y is Point of S : ||.(y - x0).|| < g  }  is Neighbourhood of x0
proof
let S be RealNormSpace; ::_thesis: for x0 being Point of S
for g being Real st 0 < g holds
{  y where y is Point of S : ||.(y - x0).|| < g  }  is Neighbourhood of x0
let x0 be Point of S; ::_thesis: for g being Real st 0 < g holds
{  y where y is Point of S : ||.(y - x0).|| < g  }  is Neighbourhood of x0
let g be Real; ::_thesis: ( 0 < g implies  {  y where y is Point of S : ||.(y - x0).|| < g  }  is Neighbourhood of x0 )
assume A1: g > 0 ; ::_thesis:  {  y where y is Point of S : ||.(y - x0).|| < g  }  is Neighbourhood of x0
set N =  {  y where y is Point of S : ||.(y - x0).|| < g  }  ;
  {  y where y is Point of S : ||.(y - x0).|| < g  }  c= the carrier of S
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  y where y is Point of S : ||.(y - x0).|| < g  }  or x in the carrier of S )
assume  x in  {  y where y is Point of S : ||.(y - x0).|| < g  }  ; ::_thesis: x in the carrier of S
then  ex y being Point of S st
( x = y & ||.(y - x0).|| < g ) ;
hence  x in the carrier of S ; ::_thesis: verum
end;
hence   {  y where y is Point of S : ||.(y - x0).|| < g  }  is Neighbourhood of x0 by A1, Def1; ::_thesis: verum
end;

theorem Th4: :: NFCONT_1:4
for S being RealNormSpace
for x0 being Point of S
for N being Neighbourhood of x0 holds x0 in N
proof
let S be RealNormSpace; ::_thesis: for x0 being Point of S
for N being Neighbourhood of x0 holds x0 in N
let x0 be Point of S; ::_thesis: for N being Neighbourhood of x0 holds x0 in N
let N be Neighbourhood of x0; ::_thesis: x0 in N
consider g being Real such that
A1: 0 < g and
A2:  {  z where z is Point of S : ||.(z - x0).|| < g  }  c= N by Def1;
||.(x0 - x0).|| = ||.(0. S).|| by RLVECT_1:15
.= 0 by NORMSP_1:1 ;
then  x0 in  {  z where z is Point of S : ||.(z - x0).|| < g  }  by A1;
hence  x0 in N by A2; ::_thesis: verum
end;

definition
let S be RealNormSpace;
let X be Subset of S;
attrX is compact  means :Def2: :: NFCONT_1:def 2
 for s1 being sequence of S st rng s1 c= X holds
ex s2 being sequence of S st
( s2 is subsequence of s1 & s2 is convergent & lim s2 in X );
end;

:: deftheorem Def2 defines compact NFCONT_1:def_2_:_
 for S being RealNormSpace
for X being Subset of S holds
( X is compact iff for s1 being sequence of S st rng s1 c= X holds
ex s2 being sequence of S st
( s2 is subsequence of s1 & s2 is convergent & lim s2 in X ) );

definition
let S be RealNormSpace;
let X be Subset of S;
attrX is closed  means :: NFCONT_1:def 3
 for s1 being sequence of S st rng s1 c= X & s1 is convergent holds
lim s1 in X;
end;

:: deftheorem  defines closed NFCONT_1:def_3_:_
 for S being RealNormSpace
for X being Subset of S holds
( X is closed iff for s1 being sequence of S st rng s1 c= X & s1 is convergent holds
lim s1 in X );

definition
let S be RealNormSpace;
let X be Subset of S;
attrX is open  means :: NFCONT_1:def 4
X ` is closed ;
end;

:: deftheorem  defines open NFCONT_1:def_4_:_
 for S being RealNormSpace
for X being Subset of S holds
( X is open iff X ` is closed );

definition
let S, T be RealNormSpace;
let f be PartFunc of S,T;
let x0 be Point of S;
predf is_continuous_in x0 means :Def5: :: NFCONT_1:def 5
( x0 in dom f & ( for s1 being sequence of S st rng s1 c= dom f & s1 is convergent & lim s1 = x0 holds
( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) );
end;

:: deftheorem Def5 defines is_continuous_in NFCONT_1:def_5_:_
 for S, T being RealNormSpace
for f being PartFunc of S,T
for x0 being Point of S holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for s1 being sequence of S st rng s1 c= dom f & s1 is convergent & lim s1 = x0 holds
( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) );

definition
let S be RealNormSpace;
let f be PartFunc of the carrier of S,REAL;
let x0 be Point of S;
predf is_continuous_in x0 means :Def6: :: NFCONT_1:def 6
( x0 in dom f & ( for s1 being sequence of S st rng s1 c= dom f & s1 is convergent & lim s1 = x0 holds
( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) );
end;

:: deftheorem Def6 defines is_continuous_in NFCONT_1:def_6_:_
 for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL
for x0 being Point of S holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for s1 being sequence of S st rng s1 c= dom f & s1 is convergent & lim s1 = x0 holds
( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) );

theorem Th5: :: NFCONT_1:5
for n being Element of NAT
for S, T being RealNormSpace
for seq being sequence of S
for h being PartFunc of S,T st rng seq c= dom h holds
seq . n in dom h
proof
let n be Element of NAT ; ::_thesis: for S, T being RealNormSpace
for seq being sequence of S
for h being PartFunc of S,T st rng seq c= dom h holds
seq . n in dom h
let S, T be RealNormSpace; ::_thesis: for seq being sequence of S
for h being PartFunc of S,T st rng seq c= dom h holds
seq . n in dom h
let seq be sequence of S; ::_thesis: for h being PartFunc of S,T st rng seq c= dom h holds
seq . n in dom h
 dom seq = NAT by FUNCT_2:def_1;
then  seq . n in rng seq by FUNCT_1:def_3;
hence  for h being PartFunc of S,T st rng seq c= dom h holds
seq . n in dom h ; ::_thesis: verum
end;

theorem Th6: :: NFCONT_1:6
for S being RealNormSpace
for seq being sequence of S
for x being set holds
( x in rng seq iff ex n being Element of NAT st x = seq . n )
proof
let S be RealNormSpace; ::_thesis: for seq being sequence of S
for x being set holds
( x in rng seq iff ex n being Element of NAT st x = seq . n )
let seq be sequence of S; ::_thesis: for x being set holds
( x in rng seq iff ex n being Element of NAT st x = seq . n )
let x be set ; ::_thesis: ( x in rng seq iff ex n being Element of NAT st x = seq . n )
thus  ( x in rng seq implies ex n being Element of NAT st x = seq . n ) ::_thesis: ( ex n being Element of NAT st x = seq . n implies x in rng seq )
proof
assume  x in rng seq ; ::_thesis: ex n being Element of NAT st x = seq . n
then consider y being set  such that
A1: y in dom seq and
A2: x = seq . y by FUNCT_1:def_3;
reconsider m = y as Element of NAT by A1;
take  m ; ::_thesis: x = seq . m
thus  x = seq . m by A2; ::_thesis: verum
end;
given n being Element of NAT  such that A3: x = seq . n ; ::_thesis: x in rng seq
 n in NAT ;
then  n in dom seq by FUNCT_2:def_1;
hence  x in rng seq by A3, FUNCT_1:def_3; ::_thesis: verum
end;

theorem Th7: :: NFCONT_1:7
for T, S being RealNormSpace
for f being PartFunc of S,T
for x0 being Point of S holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
proof
let T, S be RealNormSpace; ::_thesis: for f being PartFunc of S,T
for x0 being Point of S holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let f be PartFunc of S,T; ::_thesis: for x0 being Point of S holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let x0 be Point of S; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
thus  ( f is_continuous_in x0 implies ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) ) ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) implies f is_continuous_in x0 )
proof
assume A1: f is_continuous_in x0 ; ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) )
hence  x0 in dom f by Def5; ::_thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
given r being Real such that A2: 0 < r and
A3: for s being Real holds
( not 0 < s or ex x1 being Point of S st
( x1 in dom f & ||.(x1 - x0).|| < s & not ||.((f /. x1) - (f /. x0)).|| < r ) ) ; ::_thesis: contradiction
defpred S1[ Element of NAT , Point of S] means ( $2 in dom f & ||.($2 - x0).|| < 1 / ($1 + 1) & not ||.((f /. $2) - (f /. x0)).|| < r );
A4: for n being Element of NAT ex p being Point of S st S1[n,p]
proof
let n be Element of NAT ; ::_thesis: ex p being Point of S st S1[n,p]
 0 < n + 1 by NAT_1:3;
then  0 < (n + 1) " by XREAL_1:122;
then  0 < 1 / (n + 1) by XCMPLX_1:215;
then consider p being Point of S such that
A5: ( p in dom f & ||.(p - x0).|| < 1 / (n + 1) & not ||.((f /. p) - (f /. x0)).|| < r ) by A3;
take  p ; ::_thesis: S1[n,p]
thus  S1[n,p] by A5; ::_thesis: verum
end;
consider s1 being Function of NAT, the carrier of S such that
A6: for n being Element of NAT holds S1[n,s1 . n] from FUNCT_2:sch_3(A4);
reconsider s1 = s1 as sequence of S ;
A7: rng s1 c= dom f
proof
A8: dom s1 = NAT by FUNCT_2:def_1;
let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in rng s1 or v in dom f )
assume  v in rng s1 ; ::_thesis: v in dom f
then  ex n being set st
( n in NAT & v = s1 . n ) by A8, FUNCT_1:def_3;
hence  v in dom f by A6; ::_thesis: verum
end;
A9: now__::_thesis:_for_n_being_Element_of_NAT_holds_not_||.(((f_/*_s1)_._n)_-_(f_/._x0)).||_<_r
let n be Element of NAT ; ::_thesis: not ||.(((f /* s1) . n) - (f /. x0)).|| < r
 not ||.((f /. (s1 . n)) - (f /. x0)).|| < r by A6;
hence  not ||.(((f /* s1) . n) - (f /. x0)).|| < r by A7, FUNCT_2:109; ::_thesis: verum
end;
A10: now__::_thesis:_for_s_being_Real_st_0_<_s_holds_
ex_k_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_k_<=_m_holds_
||.((s1_._m)_-_x0).||_<_s
let s be Real; ::_thesis: ( 0 < s implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s )
consider n being Element of NAT  such that
A11: s " < n by SEQ_4:3;
assume  0 < s ; ::_thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s
then A12: 0 < s " by XREAL_1:122;
 (s ") + 0 < n + 1 by A11, XREAL_1:8;
then  1 / (n + 1) < 1 / (s ") by A12, XREAL_1:76;
then A13: 1 / (n + 1) < s by XCMPLX_1:216;
take k = n; ::_thesis: for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s
let m be Element of NAT ; ::_thesis: ( k <= m implies ||.((s1 . m) - x0).|| < s )
assume  k <= m ; ::_thesis: ||.((s1 . m) - x0).|| < s
then  k + 1 <= m + 1 by XREAL_1:6;
then  1 / (m + 1) <= 1 / (k + 1) by NAT_1:3, XREAL_1:118;
then  1 / (m + 1) < s by A13, XXREAL_0:2;
hence  ||.((s1 . m) - x0).|| < s by A6, XXREAL_0:2; ::_thesis: verum
end;
then A14: s1 is convergent by NORMSP_1:def_6;
then  lim s1 = x0 by A10, NORMSP_1:def_7;
then  ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A1, A7, A14, Def5;
then consider n being Element of NAT  such that
A15: for m being Element of NAT st n <= m holds
||.(((f /* s1) . m) - (f /. x0)).|| < r by A2, NORMSP_1:def_7;
 ||.(((f /* s1) . n) - (f /. x0)).|| < r by A15;
hence  contradiction by A9; ::_thesis: verum
end;
assume that
A16: x0 in dom f and
A17: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_s1_being_sequence_of_S_st_rng_s1_c=_dom_f_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_f_/*_s1_is_convergent_&_f_/._x0_=_lim_(f_/*_s1)_)
let s1 be sequence of S; ::_thesis: ( rng s1 c= dom f & s1 is convergent & lim s1 = x0 implies ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) )
assume that
A18: rng s1 c= dom f and
A19: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( f /* s1 is convergent & f /. x0 = lim (f /* s1) )
A20: now__::_thesis:_for_p_being_Real_st_0_<_p_holds_
ex_k_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_k_<=_m_holds_
||.(((f_/*_s1)_._m)_-_(f_/._x0)).||_<_p
let p be Real; ::_thesis: ( 0 < p implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.(((f /* s1) . m) - (f /. x0)).|| < p )
assume  0 < p ; ::_thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.(((f /* s1) . m) - (f /. x0)).|| < p
then consider s being Real such that
A21: 0 < s and
A22: for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < p by A17;
consider n being Element of NAT  such that
A23: for m being Element of NAT st n <= m holds
||.((s1 . m) - x0).|| < s by A19, A21, NORMSP_1:def_7;
take k = n; ::_thesis: for m being Element of NAT st k <= m holds
||.(((f /* s1) . m) - (f /. x0)).|| < p
let m be Element of NAT ; ::_thesis: ( k <= m implies ||.(((f /* s1) . m) - (f /. x0)).|| < p )
assume  k <= m ; ::_thesis: ||.(((f /* s1) . m) - (f /. x0)).|| < p
then A24: ||.((s1 . m) - x0).|| < s by A23;
 dom s1 = NAT by FUNCT_2:def_1;
then  s1 . m in rng s1 by FUNCT_1:3;
then  ||.((f /. (s1 . m)) - (f /. x0)).|| < p by A18, A22, A24;
hence  ||.(((f /* s1) . m) - (f /. x0)).|| < p by A18, FUNCT_2:109; ::_thesis: verum
end;
then  f /* s1 is convergent by NORMSP_1:def_6;
hence  ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A20, NORMSP_1:def_7; ::_thesis: verum
end;
hence  f is_continuous_in x0 by A16, Def5; ::_thesis: verum
end;

theorem Th8: :: NFCONT_1:8
for S being RealNormSpace
for x0 being Point of S
for f being PartFunc of the carrier of S,REAL holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) )
proof
let S be RealNormSpace; ::_thesis: for x0 being Point of S
for f being PartFunc of the carrier of S,REAL holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) )
let x0 be Point of S; ::_thesis: for f being PartFunc of the carrier of S,REAL holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) )
let f be PartFunc of the carrier of S,REAL; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) )
thus  ( f is_continuous_in x0 implies ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) ) ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) implies f is_continuous_in x0 )
proof
assume A1: f is_continuous_in x0 ; ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) )
hence  x0 in dom f by Def6; ::_thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) )
given r being Real such that A2: 0 < r and
A3: for s being Real holds
( not 0 < s or ex x1 being Point of S st
( x1 in dom f & ||.(x1 - x0).|| < s & not abs ((f /. x1) - (f /. x0)) < r ) ) ; ::_thesis: contradiction
defpred S1[ Element of NAT , Point of S] means ( $2 in dom f & ||.($2 - x0).|| < 1 / ($1 + 1) & not abs ((f /. $2) - (f /. x0)) < r );
A4: for n being Element of NAT ex p being Point of S st S1[n,p]
proof
let n be Element of NAT ; ::_thesis: ex p being Point of S st S1[n,p]
 0 < n + 1 by NAT_1:3;
then  0 < (n + 1) " by XREAL_1:122;
then  0 < 1 / (n + 1) by XCMPLX_1:215;
then consider p being Point of S such that
A5: ( p in dom f & ||.(p - x0).|| < 1 / (n + 1) & abs ((f /. p) - (f /. x0)) >= r ) by A3;
take  p ; ::_thesis: S1[n,p]
thus  S1[n,p] by A5; ::_thesis: verum
end;
consider s1 being Function of NAT, the carrier of S such that
A6: for n being Element of NAT holds S1[n,s1 . n] from FUNCT_2:sch_3(A4);
reconsider s1 = s1 as sequence of S ;
A7: rng s1 c= dom f
proof
A8: dom s1 = NAT by FUNCT_2:def_1;
let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in rng s1 or v in dom f )
assume  v in rng s1 ; ::_thesis: v in dom f
then  ex n being set st
( n in NAT & v = s1 . n ) by A8, FUNCT_1:def_3;
hence  v in dom f by A6; ::_thesis: verum
end;
A9: now__::_thesis:_for_n_being_Element_of_NAT_holds_abs_(((f_/*_s1)_._n)_-_(f_/._x0))_>=_r
let n be Element of NAT ; ::_thesis: abs (((f /* s1) . n) - (f /. x0)) >= r
 abs ((f /. (s1 . n)) - (f /. x0)) >= r by A6;
hence  abs (((f /* s1) . n) - (f /. x0)) >= r by A7, FUNCT_2:109; ::_thesis: verum
end;
A10: now__::_thesis:_for_s_being_Real_st_0_<_s_holds_
ex_k_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_k_<=_m_holds_
||.((s1_._m)_-_x0).||_<_s
let s be Real; ::_thesis: ( 0 < s implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s )
consider n being Element of NAT  such that
A11: s " < n by SEQ_4:3;
assume  0 < s ; ::_thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s
then A12: 0 < s " by XREAL_1:122;
 (s ") + 0 < n + 1 by A11, XREAL_1:8;
then  1 / (n + 1) < 1 / (s ") by A12, XREAL_1:76;
then A13: 1 / (n + 1) < s by XCMPLX_1:216;
take k = n; ::_thesis: for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s
let m be Element of NAT ; ::_thesis: ( k <= m implies ||.((s1 . m) - x0).|| < s )
assume  k <= m ; ::_thesis: ||.((s1 . m) - x0).|| < s
then  k + 1 <= m + 1 by XREAL_1:6;
then  1 / (m + 1) <= 1 / (k + 1) by NAT_1:3, XREAL_1:118;
then  1 / (m + 1) < s by A13, XXREAL_0:2;
hence  ||.((s1 . m) - x0).|| < s by A6, XXREAL_0:2; ::_thesis: verum
end;
then A14: s1 is convergent by NORMSP_1:def_6;
then  lim s1 = x0 by A10, NORMSP_1:def_7;
then  ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A1, A7, A14, Def6;
then consider n being Element of NAT  such that
A15: for m being Element of NAT st n <= m holds
abs (((f /* s1) . m) - (f /. x0)) < r by A2, SEQ_2:def_7;
 abs (((f /* s1) . n) - (f /. x0)) < r by A15;
hence  contradiction by A9; ::_thesis: verum
end;
assume that
A16: x0 in dom f and
A17: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_s1_being_sequence_of_S_st_rng_s1_c=_dom_f_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_f_/*_s1_is_convergent_&_f_/._x0_=_lim_(f_/*_s1)_)
let s1 be sequence of S; ::_thesis: ( rng s1 c= dom f & s1 is convergent & lim s1 = x0 implies ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) )
assume that
A18: rng s1 c= dom f and
A19: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( f /* s1 is convergent & f /. x0 = lim (f /* s1) )
A20: now__::_thesis:_for_p_being_real_number_st_0_<_p_holds_
ex_k_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_k_<=_m_holds_
abs_(((f_/*_s1)_._m)_-_(f_/._x0))_<_p
let p be real number ; ::_thesis: ( 0 < p implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
abs (((f /* s1) . m) - (f /. x0)) < p )
reconsider pp = p as Real by XREAL_0:def_1;
assume  0 < p ; ::_thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
abs (((f /* s1) . m) - (f /. x0)) < p
then consider s being Real such that
A21: 0 < s and
A22: for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < pp by A17;
consider n being Element of NAT  such that
A23: for m being Element of NAT st n <= m holds
||.((s1 . m) - x0).|| < s by A19, A21, NORMSP_1:def_7;
take k = n; ::_thesis: for m being Element of NAT st k <= m holds
abs (((f /* s1) . m) - (f /. x0)) < p
let m be Element of NAT ; ::_thesis: ( k <= m implies abs (((f /* s1) . m) - (f /. x0)) < p )
assume  k <= m ; ::_thesis: abs (((f /* s1) . m) - (f /. x0)) < p
then A24: ||.((s1 . m) - x0).|| < s by A23;
 dom s1 = NAT by FUNCT_2:def_1;
then  s1 . m in rng s1 by FUNCT_1:3;
then  abs ((f /. (s1 . m)) - (f /. x0)) < p by A18, A22, A24;
hence  abs (((f /* s1) . m) - (f /. x0)) < p by A18, FUNCT_2:109; ::_thesis: verum
end;
then  f /* s1 is convergent by SEQ_2:def_6;
hence  ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A20, SEQ_2:def_7; ::_thesis: verum
end;
hence  f is_continuous_in x0 by A16, Def6; ::_thesis: verum
end;

theorem Th9: :: NFCONT_1:9
for T, S being RealNormSpace
for f being PartFunc of S,T
for x0 being Point of S holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of S st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
proof
let T, S be RealNormSpace; ::_thesis: for f being PartFunc of S,T
for x0 being Point of S holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of S st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
let f be PartFunc of S,T; ::_thesis: for x0 being Point of S holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of S st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
let x0 be Point of S; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of S st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
thus  ( f is_continuous_in x0 implies ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of S st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) ) ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of S st x1 in dom f & x1 in N holds
f /. x1 in N1 ) implies f is_continuous_in x0 )
proof
assume A1: f is_continuous_in x0 ; ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of S st x1 in dom f & x1 in N holds
f /. x1 in N1 ) )
hence  x0 in dom f by Def5; ::_thesis: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of S st x1 in dom f & x1 in N holds
f /. x1 in N1
let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st
for x1 being Point of S st x1 in dom f & x1 in N holds
f /. x1 in N1
consider r being Real such that
A2: 0 < r and
A3:  {  y where y is Point of T : ||.(y - (f /. x0)).|| < r  }  c= N1 by Def1;
consider s being Real such that
A4: 0 < s and
A5: for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r by A1, A2, Th7;
reconsider N =  {  z where z is Point of S : ||.(z - x0).|| < s  }  as Neighbourhood of x0 by A4, Th3;
take  N ; ::_thesis: for x1 being Point of S st x1 in dom f & x1 in N holds
f /. x1 in N1
let x1 be Point of S; ::_thesis: ( x1 in dom f & x1 in N implies f /. x1 in N1 )
assume that
A6: x1 in dom f and
A7: x1 in N ; ::_thesis: f /. x1 in N1
 ex z being Point of S st
( x1 = z & ||.(z - x0).|| < s ) by A7;
then  ||.((f /. x1) - (f /. x0)).|| < r by A5, A6;
then  f /. x1 in  {  y where y is Point of T : ||.(y - (f /. x0)).|| < r  }  ;
hence  f /. x1 in N1 by A3; ::_thesis: verum
end;
assume that
A8: x0 in dom f and
A9: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of S st x1 in dom f & x1 in N holds
f /. x1 in N1 ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_r_being_Real_st_0_<_r_holds_
ex_s_being_Real_st_
(_0_<_s_&_(_for_x1_being_Point_of_S_st_x1_in_dom_f_&_||.(x1_-_x0).||_<_s_holds_
||.((f_/._x1)_-_(f_/._x0)).||_<_r_)_)
let r be Real; ::_thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) )
assume  0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
then reconsider N1 =  {  y where y is Point of T : ||.(y - (f /. x0)).|| < r  }  as Neighbourhood of f /. x0 by Th3;
consider N2 being Neighbourhood of x0 such that
A10: for x1 being Point of S st x1 in dom f & x1 in N2 holds
f /. x1 in N1 by A9;
consider s being Real such that
A11: 0 < s and
A12:  {  z where z is Point of S : ||.(z - x0).|| < s  }  c= N2 by Def1;
take s = s; ::_thesis: ( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
 for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r
proof
let x1 be Point of S; ::_thesis: ( x1 in dom f & ||.(x1 - x0).|| < s implies ||.((f /. x1) - (f /. x0)).|| < r )
assume that
A13: x1 in dom f and
A14: ||.(x1 - x0).|| < s ; ::_thesis: ||.((f /. x1) - (f /. x0)).|| < r
 x1 in  {  z where z is Point of S : ||.(z - x0).|| < s  }  by A14;
then  f /. x1 in N1 by A10, A12, A13;
then  ex y being Point of T st
( f /. x1 = y & ||.(y - (f /. x0)).|| < r ) ;
hence  ||.((f /. x1) - (f /. x0)).|| < r ; ::_thesis: verum
end;
hence  ( 0 < s & ( for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) by A11; ::_thesis: verum
end;
hence  f is_continuous_in x0 by A8, Th7; ::_thesis: verum
end;

theorem Th10: :: NFCONT_1:10
for T, S being RealNormSpace
for f being PartFunc of S,T
for x0 being Point of S holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
proof
let T, S be RealNormSpace; ::_thesis: for f being PartFunc of S,T
for x0 being Point of S holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
let f be PartFunc of S,T; ::_thesis: for x0 being Point of S holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
let x0 be Point of S; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
thus  ( f is_continuous_in x0 implies ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) ) ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) implies f is_continuous_in x0 )
proof
assume A1: f is_continuous_in x0 ; ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) )
hence  x0 in dom f by Def5; ::_thesis: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1
let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st f .: N c= N1
consider N being Neighbourhood of x0 such that
A2: for x1 being Point of S st x1 in dom f & x1 in N holds
f /. x1 in N1 by A1, Th9;
take  N ; ::_thesis: f .: N c= N1
now__::_thesis:_for_r_being_set_st_r_in_f_.:_N_holds_
r_in_N1
let r be set ; ::_thesis: ( r in f .: N implies r in N1 )
assume  r in f .: N ; ::_thesis: r in N1
then consider x being Point of S such that
A3: x in dom f and
A4: x in N and
A5: r = f . x by PARTFUN2:59;
 r = f /. x by A3, A5, PARTFUN1:def_6;
hence  r in N1 by A2, A3, A4; ::_thesis: verum
end;
hence  f .: N c= N1 by TARSKI:def_3; ::_thesis: verum
end;
assume that
A6: x0 in dom f and
A7: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_N1_being_Neighbourhood_of_f_/._x0_ex_N_being_Neighbourhood_of_x0_st_
for_x1_being_Point_of_S_st_x1_in_dom_f_&_x1_in_N_holds_
f_/._x1_in_N1
let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st
for x1 being Point of S st x1 in dom f & x1 in N holds
f /. x1 in N1
consider N being Neighbourhood of x0 such that
A8: f .: N c= N1 by A7;
take N = N; ::_thesis: for x1 being Point of S st x1 in dom f & x1 in N holds
f /. x1 in N1
let x1 be Point of S; ::_thesis: ( x1 in dom f & x1 in N implies f /. x1 in N1 )
assume that
A9: x1 in dom f and
A10: x1 in N ; ::_thesis: f /. x1 in N1
 f . x1 in f .: N by A9, A10, FUNCT_1:def_6;
then  f . x1 in N1 by A8;
hence  f /. x1 in N1 by A9, PARTFUN1:def_6; ::_thesis: verum
end;
hence  f is_continuous_in x0 by A6, Th9; ::_thesis: verum
end;

theorem :: NFCONT_1:11
for T, S being RealNormSpace
for f being PartFunc of S,T
for x0 being Point of S st x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0
proof
let T, S be RealNormSpace; ::_thesis: for f being PartFunc of S,T
for x0 being Point of S st x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0
let f be PartFunc of S,T; ::_thesis: for x0 being Point of S st x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0
let x0 be Point of S; ::_thesis: ( x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} implies f is_continuous_in x0 )
assume A1: x0 in dom f ; ::_thesis: ( for N being Neighbourhood of x0 holds not (dom f) /\ N = {x0} or f is_continuous_in x0 )
given N being Neighbourhood of x0 such that A2: (dom f) /\ N = {x0} ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_N1_being_Neighbourhood_of_f_/._x0_ex_N_being_Neighbourhood_of_x0_st_f_.:_N_c=_N1
let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st f .: N c= N1
take N = N; ::_thesis: f .: N c= N1
A3: f /. x0 in N1 by Th4;
f .: N = Im (f,x0) by A2, RELAT_1:112
.= {(f . x0)} by A1, FUNCT_1:59
.= {(f /. x0)} by A1, PARTFUN1:def_6 ;
hence  f .: N c= N1 by A3, ZFMISC_1:31; ::_thesis: verum
end;
hence  f is_continuous_in x0 by A1, Th10; ::_thesis: verum
end;

theorem Th12: :: NFCONT_1:12
for S, T being RealNormSpace
for h1, h2 being PartFunc of S,T
for seq being sequence of S st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
proof
let S, T be RealNormSpace; ::_thesis: for h1, h2 being PartFunc of S,T
for seq being sequence of S st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
let h1, h2 be PartFunc of S,T; ::_thesis: for seq being sequence of S st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
let seq be sequence of S; ::_thesis: ( rng seq c= (dom h1) /\ (dom h2) implies ( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) ) )
A1: (dom h1) /\ (dom h2) c= dom h1 by XBOOLE_1:17;
A2: (dom h1) /\ (dom h2) c= dom h2 by XBOOLE_1:17;
assume A3: rng seq c= (dom h1) /\ (dom h2) ; ::_thesis: ( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
then A4: rng seq c= dom (h1 + h2) by VFUNCT_1:def_1;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((h1_+_h2)_/*_seq)_._n_=_((h1_/*_seq)_._n)_+_((h2_/*_seq)_._n)
let n be Element of NAT ; ::_thesis: ((h1 + h2) /* seq) . n = ((h1 /* seq) . n) + ((h2 /* seq) . n)
A5: seq . n in dom (h1 + h2) by A4, Th5;
thus ((h1 + h2) /* seq) . n = (h1 + h2) /. (seq . n) by A4, FUNCT_2:109
.= (h1 /. (seq . n)) + (h2 /. (seq . n)) by A5, VFUNCT_1:def_1
.= ((h1 /* seq) . n) + (h2 /. (seq . n)) by A3, A1, FUNCT_2:109, XBOOLE_1:1
.= ((h1 /* seq) . n) + ((h2 /* seq) . n) by A3, A2, FUNCT_2:109, XBOOLE_1:1 ; ::_thesis: verum
end;
hence  (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) by NORMSP_1:def_2; ::_thesis: (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq)
A6: rng seq c= dom (h1 - h2) by A3, VFUNCT_1:def_2;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((h1_-_h2)_/*_seq)_._n_=_((h1_/*_seq)_._n)_-_((h2_/*_seq)_._n)
let n be Element of NAT ; ::_thesis: ((h1 - h2) /* seq) . n = ((h1 /* seq) . n) - ((h2 /* seq) . n)
A7: seq . n in dom (h1 - h2) by A6, Th5;
thus ((h1 - h2) /* seq) . n = (h1 - h2) /. (seq . n) by A6, FUNCT_2:109
.= (h1 /. (seq . n)) - (h2 /. (seq . n)) by A7, VFUNCT_1:def_2
.= ((h1 /* seq) . n) - (h2 /. (seq . n)) by A3, A1, FUNCT_2:109, XBOOLE_1:1
.= ((h1 /* seq) . n) - ((h2 /* seq) . n) by A3, A2, FUNCT_2:109, XBOOLE_1:1 ; ::_thesis: verum
end;
hence  (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) by NORMSP_1:def_3; ::_thesis: verum
end;

theorem Th13: :: NFCONT_1:13
for S, T being RealNormSpace
for h being PartFunc of S,T
for seq being sequence of S
for r being Real st rng seq c= dom h holds
(r (#) h) /* seq = r * (h /* seq)
proof
let S, T be RealNormSpace; ::_thesis: for h being PartFunc of S,T
for seq being sequence of S
for r being Real st rng seq c= dom h holds
(r (#) h) /* seq = r * (h /* seq)
let h be PartFunc of S,T; ::_thesis: for seq being sequence of S
for r being Real st rng seq c= dom h holds
(r (#) h) /* seq = r * (h /* seq)
let seq be sequence of S; ::_thesis: for r being Real st rng seq c= dom h holds
(r (#) h) /* seq = r * (h /* seq)
let r be Real; ::_thesis: ( rng seq c= dom h implies (r (#) h) /* seq = r * (h /* seq) )
assume A1: rng seq c= dom h ; ::_thesis: (r (#) h) /* seq = r * (h /* seq)
then A2: rng seq c= dom (r (#) h) by VFUNCT_1:def_4;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((r_(#)_h)_/*_seq)_._n_=_r_*_((h_/*_seq)_._n)
let n be Element of NAT ; ::_thesis: ((r (#) h) /* seq) . n = r * ((h /* seq) . n)
A3: seq . n in dom (r (#) h) by A2, Th5;
thus ((r (#) h) /* seq) . n = (r (#) h) /. (seq . n) by A2, FUNCT_2:109
.= r * (h /. (seq . n)) by A3, VFUNCT_1:def_4
.= r * ((h /* seq) . n) by A1, FUNCT_2:109 ; ::_thesis: verum
end;
hence  (r (#) h) /* seq = r * (h /* seq) by NORMSP_1:def_5; ::_thesis: verum
end;

theorem Th14: :: NFCONT_1:14
for S, T being RealNormSpace
for h being PartFunc of S,T
for seq being sequence of S st rng seq c= dom h holds
( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
proof
let S, T be RealNormSpace; ::_thesis: for h being PartFunc of S,T
for seq being sequence of S st rng seq c= dom h holds
( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
let h be PartFunc of S,T; ::_thesis: for seq being sequence of S st rng seq c= dom h holds
( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
let seq be sequence of S; ::_thesis: ( rng seq c= dom h implies ( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq ) )
assume A1: rng seq c= dom h ; ::_thesis: ( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
then A2: rng seq c= dom ||.h.|| by NORMSP_0:def_3;
now__::_thesis:_for_n_being_Element_of_NAT_holds_||.(h_/*_seq).||_._n_=_(||.h.||_/*_seq)_._n
let n be Element of NAT ; ::_thesis: ||.(h /* seq).|| . n = (||.h.|| /* seq) . n
 seq . n in rng seq by Th6;
then  seq . n in dom h by A1;
then A3: seq . n in dom ||.h.|| by NORMSP_0:def_3;
thus ||.(h /* seq).|| . n = ||.((h /* seq) . n).|| by NORMSP_0:def_4
.= ||.(h /. (seq . n)).|| by A1, FUNCT_2:109
.= ||.h.|| . (seq . n) by A3, NORMSP_0:def_3
.= ||.h.|| /. (seq . n) by A3, PARTFUN1:def_6
.= (||.h.|| /* seq) . n by A2, FUNCT_2:109 ; ::_thesis: verum
end;
hence  ||.(h /* seq).|| = ||.h.|| /* seq by FUNCT_2:63; ::_thesis: - (h /* seq) = (- h) /* seq
thus  - (h /* seq) = (- 1) * (h /* seq) by Th2
.= ((- 1) (#) h) /* seq by A1, Th13
.= (- h) /* seq by VFUNCT_1:23 ; ::_thesis: verum
end;

theorem :: NFCONT_1:15
for T, S being RealNormSpace
for f1, f2 being PartFunc of S,T
for x0 being Point of S st f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
proof
let T, S be RealNormSpace; ::_thesis: for f1, f2 being PartFunc of S,T
for x0 being Point of S st f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
let f1, f2 be PartFunc of S,T; ::_thesis: for x0 being Point of S st f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
let x0 be Point of S; ::_thesis: ( f1 is_continuous_in x0 & f2 is_continuous_in x0 implies ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 ) )
assume that
A1: f1 is_continuous_in x0 and
A2: f2 is_continuous_in x0 ; ::_thesis: ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
A3: ( x0 in dom f1 & x0 in dom f2 ) by A1, A2, Def5;
now__::_thesis:_(_x0_in_dom_(f1_+_f2)_&_(_for_s1_being_sequence_of_S_st_rng_s1_c=_dom_(f1_+_f2)_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_(f1_+_f2)_/*_s1_is_convergent_&_(f1_+_f2)_/._x0_=_lim_((f1_+_f2)_/*_s1)_)_)_)
 x0 in (dom f1) /\ (dom f2) by A3, XBOOLE_0:def_4;
hence A4: x0 in dom (f1 + f2) by VFUNCT_1:def_1; ::_thesis: for s1 being sequence of S st rng s1 c= dom (f1 + f2) & s1 is convergent & lim s1 = x0 holds
( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) )
let s1 be sequence of S; ::_thesis: ( rng s1 c= dom (f1 + f2) & s1 is convergent & lim s1 = x0 implies ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) ) )
assume that
A5: rng s1 c= dom (f1 + f2) and
A6: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) )
A7: rng s1 c= (dom f1) /\ (dom f2) by A5, VFUNCT_1:def_1;
 dom (f1 + f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_1;
then  dom (f1 + f2) c= dom f2 by XBOOLE_1:17;
then A8: rng s1 c= dom f2 by A5, XBOOLE_1:1;
then A9: f2 /* s1 is convergent by A2, A6, Def5;
 dom (f1 + f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_1;
then  dom (f1 + f2) c= dom f1 by XBOOLE_1:17;
then A10: rng s1 c= dom f1 by A5, XBOOLE_1:1;
then A11: f1 /* s1 is convergent by A1, A6, Def5;
then  (f1 /* s1) + (f2 /* s1) is convergent by A9, NORMSP_1:19;
hence  (f1 + f2) /* s1 is convergent by A7, Th12; ::_thesis: (f1 + f2) /. x0 = lim ((f1 + f2) /* s1)
A12: f1 /. x0 = lim (f1 /* s1) by A1, A6, A10, Def5;
A13: f2 /. x0 = lim (f2 /* s1) by A2, A6, A8, Def5;
thus (f1 + f2) /. x0 = (f1 /. x0) + (f2 /. x0) by A4, VFUNCT_1:def_1
.= lim ((f1 /* s1) + (f2 /* s1)) by A11, A12, A9, A13, NORMSP_1:25
.= lim ((f1 + f2) /* s1) by A7, Th12 ; ::_thesis: verum
end;
hence  f1 + f2 is_continuous_in x0 by Def5; ::_thesis: f1 - f2 is_continuous_in x0
now__::_thesis:_(_x0_in_dom_(f1_-_f2)_&_(_for_s1_being_sequence_of_S_st_rng_s1_c=_dom_(f1_-_f2)_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_(f1_-_f2)_/*_s1_is_convergent_&_(f1_-_f2)_/._x0_=_lim_((f1_-_f2)_/*_s1)_)_)_)
 x0 in (dom f1) /\ (dom f2) by A3, XBOOLE_0:def_4;
hence A14: x0 in dom (f1 - f2) by VFUNCT_1:def_2; ::_thesis: for s1 being sequence of S st rng s1 c= dom (f1 - f2) & s1 is convergent & lim s1 = x0 holds
( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) )
let s1 be sequence of S; ::_thesis: ( rng s1 c= dom (f1 - f2) & s1 is convergent & lim s1 = x0 implies ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) ) )
assume that
A15: rng s1 c= dom (f1 - f2) and
A16: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) )
A17: rng s1 c= (dom f1) /\ (dom f2) by A15, VFUNCT_1:def_2;
 dom (f1 - f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_2;
then  dom (f1 - f2) c= dom f2 by XBOOLE_1:17;
then A18: rng s1 c= dom f2 by A15, XBOOLE_1:1;
then A19: f2 /* s1 is convergent by A2, A16, Def5;
 dom (f1 - f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_2;
then  dom (f1 - f2) c= dom f1 by XBOOLE_1:17;
then A20: rng s1 c= dom f1 by A15, XBOOLE_1:1;
then A21: f1 /* s1 is convergent by A1, A16, Def5;
then  (f1 /* s1) - (f2 /* s1) is convergent by A19, NORMSP_1:20;
hence  (f1 - f2) /* s1 is convergent by A17, Th12; ::_thesis: (f1 - f2) /. x0 = lim ((f1 - f2) /* s1)
A22: f1 /. x0 = lim (f1 /* s1) by A1, A16, A20, Def5;
A23: f2 /. x0 = lim (f2 /* s1) by A2, A16, A18, Def5;
thus (f1 - f2) /. x0 = (f1 /. x0) - (f2 /. x0) by A14, VFUNCT_1:def_2
.= lim ((f1 /* s1) - (f2 /* s1)) by A21, A22, A19, A23, NORMSP_1:26
.= lim ((f1 - f2) /* s1) by A17, Th12 ; ::_thesis: verum
end;
hence  f1 - f2 is_continuous_in x0 by Def5; ::_thesis: verum
end;

theorem Th16: :: NFCONT_1:16
for r being Real
for T, S being RealNormSpace
for f being PartFunc of S,T
for x0 being Point of S st f is_continuous_in x0 holds
r (#) f is_continuous_in x0
proof
let r be Real; ::_thesis: for T, S being RealNormSpace
for f being PartFunc of S,T
for x0 being Point of S st f is_continuous_in x0 holds
r (#) f is_continuous_in x0
let T, S be RealNormSpace; ::_thesis: for f being PartFunc of S,T
for x0 being Point of S st f is_continuous_in x0 holds
r (#) f is_continuous_in x0
let f be PartFunc of S,T; ::_thesis: for x0 being Point of S st f is_continuous_in x0 holds
r (#) f is_continuous_in x0
let x0 be Point of S; ::_thesis: ( f is_continuous_in x0 implies r (#) f is_continuous_in x0 )
assume A1: f is_continuous_in x0 ; ::_thesis: r (#) f is_continuous_in x0
then  x0 in dom f by Def5;
hence A2: x0 in dom (r (#) f) by VFUNCT_1:def_4; :: according to NFCONT_1:def_5 ::_thesis: for s1 being sequence of S st rng s1 c= dom (r (#) f) & s1 is convergent & lim s1 = x0 holds
( (r (#) f) /* s1 is convergent & (r (#) f) /. x0 = lim ((r (#) f) /* s1) )
let s1 be sequence of S; ::_thesis: ( rng s1 c= dom (r (#) f) & s1 is convergent & lim s1 = x0 implies ( (r (#) f) /* s1 is convergent & (r (#) f) /. x0 = lim ((r (#) f) /* s1) ) )
assume that
A3: rng s1 c= dom (r (#) f) and
A4: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (r (#) f) /* s1 is convergent & (r (#) f) /. x0 = lim ((r (#) f) /* s1) )
A5: rng s1 c= dom f by A3, VFUNCT_1:def_4;
then A6: f /. x0 = lim (f /* s1) by A1, A4, Def5;
A7: f /* s1 is convergent by A1, A4, A5, Def5;
then  r * (f /* s1) is convergent by NORMSP_1:22;
hence  (r (#) f) /* s1 is convergent by A5, Th13; ::_thesis: (r (#) f) /. x0 = lim ((r (#) f) /* s1)
thus (r (#) f) /. x0 = r * (f /. x0) by A2, VFUNCT_1:def_4
.= lim (r * (f /* s1)) by A7, A6, NORMSP_1:28
.= lim ((r (#) f) /* s1) by A5, Th13 ; ::_thesis: verum
end;

theorem :: NFCONT_1:17
for T, S being RealNormSpace
for f being PartFunc of S,T
for x0 being Point of S st f is_continuous_in x0 holds
( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
proof
let T, S be RealNormSpace; ::_thesis: for f being PartFunc of S,T
for x0 being Point of S st f is_continuous_in x0 holds
( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
let f be PartFunc of S,T; ::_thesis: for x0 being Point of S st f is_continuous_in x0 holds
( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
let x0 be Point of S; ::_thesis: ( f is_continuous_in x0 implies ( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 ) )
assume A1: f is_continuous_in x0 ; ::_thesis: ( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
then A2: x0 in dom f by Def5;
now__::_thesis:_(_x0_in_dom_||.f.||_&_(_for_s1_being_sequence_of_S_st_rng_s1_c=_dom_||.f.||_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_||.f.||_/*_s1_is_convergent_&_||.f.||_/._x0_=_lim_(||.f.||_/*_s1)_)_)_)
thus A3: x0 in dom ||.f.|| by A2, NORMSP_0:def_3; ::_thesis: for s1 being sequence of S st rng s1 c= dom ||.f.|| & s1 is convergent & lim s1 = x0 holds
( ||.f.|| /* s1 is convergent & ||.f.|| /. x0 = lim (||.f.|| /* s1) )
let s1 be sequence of S; ::_thesis: ( rng s1 c= dom ||.f.|| & s1 is convergent & lim s1 = x0 implies ( ||.f.|| /* s1 is convergent & ||.f.|| /. x0 = lim (||.f.|| /* s1) ) )
assume that
A4: rng s1 c= dom ||.f.|| and
A5: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( ||.f.|| /* s1 is convergent & ||.f.|| /. x0 = lim (||.f.|| /* s1) )
A6: rng s1 c= dom f by A4, NORMSP_0:def_3;
then A7: f /. x0 = lim (f /* s1) by A1, A5, Def5;
A8: f /* s1 is convergent by A1, A5, A6, Def5;
then  ||.(f /* s1).|| is convergent by NORMSP_1:23;
hence  ||.f.|| /* s1 is convergent by A6, Th14; ::_thesis: ||.f.|| /. x0 = lim (||.f.|| /* s1)
thus ||.f.|| /. x0 = ||.f.|| . x0 by A3, PARTFUN1:def_6
.= ||.(f /. x0).|| by A3, NORMSP_0:def_3
.= lim ||.(f /* s1).|| by A8, A7, LOPBAN_1:41
.= lim (||.f.|| /* s1) by A6, Th14 ; ::_thesis: verum
end;
hence  ||.f.|| is_continuous_in x0 by Def6; ::_thesis: - f is_continuous_in x0
 - f = (- 1) (#) f by VFUNCT_1:23;
hence  - f is_continuous_in x0 by A1, Th16; ::_thesis: verum
end;

definition
let S, T be RealNormSpace;
let f be PartFunc of S,T;
let X be set ;
predf is_continuous_on X means :Def7: :: NFCONT_1:def 7
( X c= dom f & ( for x0 being Point of S st x0 in X holds
f | X is_continuous_in x0 ) );
end;

:: deftheorem Def7 defines is_continuous_on NFCONT_1:def_7_:_
 for S, T being RealNormSpace
for f being PartFunc of S,T
for X being set holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of S st x0 in X holds
f | X is_continuous_in x0 ) ) );

definition
let S be RealNormSpace;
let f be PartFunc of the carrier of S,REAL;
let X be set ;
predf is_continuous_on X means :Def8: :: NFCONT_1:def 8
( X c= dom f & ( for x0 being Point of S st x0 in X holds
f | X is_continuous_in x0 ) );
end;

:: deftheorem Def8 defines is_continuous_on NFCONT_1:def_8_:_
 for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL
for X being set holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of S st x0 in X holds
f | X is_continuous_in x0 ) ) );

theorem Th18: :: NFCONT_1:18
for T, S being RealNormSpace
for X being set
for f being PartFunc of S,T holds
( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of S st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
proof
let T, S be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of S,T holds
( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of S st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
let X be set ; ::_thesis: for f being PartFunc of S,T holds
( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of S st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
let f be PartFunc of S,T; ::_thesis: ( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of S st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
thus  ( f is_continuous_on X implies ( X c= dom f & ( for s1 being sequence of S st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) ) ::_thesis: ( X c= dom f & ( for s1 being sequence of S st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: ( X c= dom f & ( for s1 being sequence of S st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) )
then A2: X c= dom f by Def7;
now__::_thesis:_for_s1_being_sequence_of_S_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_
(_f_/*_s1_is_convergent_&_f_/._(lim_s1)_=_lim_(f_/*_s1)_)
let s1 be sequence of S; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) )
assume that
A3: rng s1 c= X and
A4: s1 is convergent and
A5: lim s1 in X ; ::_thesis: ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) )
A6: f | X is_continuous_in lim s1 by A1, A5, Def7;
A7: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A2, XBOOLE_1:28 ;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_X)_/*_s1)_._n_=_(f_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ((f | X) /* s1) . n = (f /* s1) . n
 dom s1 = NAT by FUNCT_2:def_1;
then A8: s1 . n in rng s1 by FUNCT_1:3;
thus ((f | X) /* s1) . n = (f | X) /. (s1 . n) by A3, A7, FUNCT_2:109
.= f /. (s1 . n) by A3, A7, A8, PARTFUN2:15
.= (f /* s1) . n by A2, A3, FUNCT_2:109, XBOOLE_1:1 ; ::_thesis: verum
end;
then A9: (f | X) /* s1 = f /* s1 by FUNCT_2:63;
f /. (lim s1) = (f | X) /. (lim s1) by A5, A7, PARTFUN2:15
.= lim (f /* s1) by A3, A4, A7, A6, A9, Def5 ;
hence  ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) by A3, A4, A7, A6, A9, Def5; ::_thesis: verum
end;
hence  ( X c= dom f & ( for s1 being sequence of S st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) by A1, Def7; ::_thesis: verum
end;
assume that
A10: X c= dom f and
A11: for s1 being sequence of S st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ; ::_thesis: f is_continuous_on X
now__::_thesis:_for_x1_being_Point_of_S_st_x1_in_X_holds_
f_|_X_is_continuous_in_x1
A12: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A10, XBOOLE_1:28 ;
let x1 be Point of S; ::_thesis: ( x1 in X implies f | X is_continuous_in x1 )
assume A13: x1 in X ; ::_thesis: f | X is_continuous_in x1
now__::_thesis:_for_s1_being_sequence_of_S_st_rng_s1_c=_dom_(f_|_X)_&_s1_is_convergent_&_lim_s1_=_x1_holds_
(_(f_|_X)_/*_s1_is_convergent_&_(f_|_X)_/._x1_=_lim_((f_|_X)_/*_s1)_)
let s1 be sequence of S; ::_thesis: ( rng s1 c= dom (f | X) & s1 is convergent & lim s1 = x1 implies ( (f | X) /* s1 is convergent & (f | X) /. x1 = lim ((f | X) /* s1) ) )
assume that
A14: rng s1 c= dom (f | X) and
A15: s1 is convergent and
A16: lim s1 = x1 ; ::_thesis: ( (f | X) /* s1 is convergent & (f | X) /. x1 = lim ((f | X) /* s1) )
now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_X)_/*_s1)_._n_=_(f_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ((f | X) /* s1) . n = (f /* s1) . n
 dom s1 = NAT by FUNCT_2:def_1;
then A17: s1 . n in rng s1 by FUNCT_1:3;
thus ((f | X) /* s1) . n = (f | X) /. (s1 . n) by A14, FUNCT_2:109
.= f /. (s1 . n) by A14, A17, PARTFUN2:15
.= (f /* s1) . n by A10, A12, A14, FUNCT_2:109, XBOOLE_1:1 ; ::_thesis: verum
end;
then A18: (f | X) /* s1 = f /* s1 by FUNCT_2:63;
(f | X) /. (lim s1) = f /. (lim s1) by A13, A12, A16, PARTFUN2:15
.= lim ((f | X) /* s1) by A11, A13, A12, A14, A15, A16, A18 ;
hence  ( (f | X) /* s1 is convergent & (f | X) /. x1 = lim ((f | X) /* s1) ) by A11, A13, A12, A14, A15, A16, A18; ::_thesis: verum
end;
hence  f | X is_continuous_in x1 by A13, A12, Def5; ::_thesis: verum
end;
hence  f is_continuous_on X by A10, Def7; ::_thesis: verum
end;

theorem Th19: :: NFCONT_1:19
for X being set
for T, S being RealNormSpace
for f being PartFunc of S,T holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
proof
let X be set ; ::_thesis: for T, S being RealNormSpace
for f being PartFunc of S,T holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let T, S be RealNormSpace; ::_thesis: for f being PartFunc of S,T holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let f be PartFunc of S,T; ::_thesis: ( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
thus  ( f is_continuous_on X implies ( X c= dom f & ( for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) ) ::_thesis: ( X c= dom f & ( for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: ( X c= dom f & ( for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) )
hence  X c= dom f by Def7; ::_thesis: for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
A2: X c= dom f by A1, Def7;
let x0 be Point of S; ::_thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) )
assume that
A3: x0 in X and
A4: 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
 f | X is_continuous_in x0 by A1, A3, Def7;
then consider s being Real such that
A5: 0 < s and
A6: for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r by A4, Th7;
take  s ; ::_thesis: ( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
thus  0 < s by A5; ::_thesis: for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r
let x1 be Point of S; ::_thesis: ( x1 in X & ||.(x1 - x0).|| < s implies ||.((f /. x1) - (f /. x0)).|| < r )
assume that
A7: x1 in X and
A8: ||.(x1 - x0).|| < s ; ::_thesis: ||.((f /. x1) - (f /. x0)).|| < r
A9: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A2, XBOOLE_1:28 ;
then ||.((f /. x1) - (f /. x0)).|| = ||.(((f | X) /. x1) - (f /. x0)).|| by A7, PARTFUN2:15
.= ||.(((f | X) /. x1) - ((f | X) /. x0)).|| by A3, A9, PARTFUN2:15 ;
hence  ||.((f /. x1) - (f /. x0)).|| < r by A6, A9, A7, A8; ::_thesis: verum
end;
assume that
A10: X c= dom f and
A11: for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ; ::_thesis: f is_continuous_on X
A12: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A10, XBOOLE_1:28 ;
now__::_thesis:_for_x0_being_Point_of_S_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Point of S; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A13: x0 in X ; ::_thesis: f | X is_continuous_in x0
 for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) )
proof
let r be Real; ::_thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) ) )
assume  0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) )
then consider s being Real such that
A14: 0 < s and
A15: for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r by A11, A13;
take  s ; ::_thesis: ( 0 < s & ( for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) )
thus  0 < s by A14; ::_thesis: for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r
let x1 be Point of S; ::_thesis: ( x1 in dom (f | X) & ||.(x1 - x0).|| < s implies ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r )
assume that
A16: x1 in dom (f | X) and
A17: ||.(x1 - x0).|| < s ; ::_thesis: ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r
||.(((f | X) /. x1) - ((f | X) /. x0)).|| = ||.(((f | X) /. x1) - (f /. x0)).|| by A12, A13, PARTFUN2:15
.= ||.((f /. x1) - (f /. x0)).|| by A16, PARTFUN2:15 ;
hence  ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r by A12, A15, A16, A17; ::_thesis: verum
end;
hence  f | X is_continuous_in x0 by A12, A13, Th7; ::_thesis: verum
end;
hence  f is_continuous_on X by A10, Def7; ::_thesis: verum
end;

theorem :: NFCONT_1:20
for X being set
for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) )
proof
let X be set ; ::_thesis: for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) )
let S be RealNormSpace; ::_thesis: for f being PartFunc of the carrier of S,REAL holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) )
let f be PartFunc of the carrier of S,REAL; ::_thesis: ( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) )
thus  ( f is_continuous_on X implies ( X c= dom f & ( for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) ) ::_thesis: ( X c= dom f & ( for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: ( X c= dom f & ( for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) )
hence  X c= dom f by Def8; ::_thesis: for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) )
A2: X c= dom f by A1, Def8;
let x0 be Point of S; ::_thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) )
let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) )
assume that
A3: x0 in X and
A4: 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) )
 f | X is_continuous_in x0 by A1, A3, Def8;
then consider s being Real such that
A5: 0 < s and
A6: for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < r by A4, Th8;
take  s ; ::_thesis: ( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) )
thus  0 < s by A5; ::_thesis: for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r
let x1 be Point of S; ::_thesis: ( x1 in X & ||.(x1 - x0).|| < s implies abs ((f /. x1) - (f /. x0)) < r )
assume that
A7: x1 in X and
A8: ||.(x1 - x0).|| < s ; ::_thesis: abs ((f /. x1) - (f /. x0)) < r
A9: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A2, XBOOLE_1:28 ;
then  abs ((f /. x1) - (f /. x0)) = abs (((f | X) /. x1) - (f /. x0)) by A7, PARTFUN2:15
.= abs (((f | X) /. x1) - ((f | X) /. x0)) by A3, A9, PARTFUN2:15 ;
hence  abs ((f /. x1) - (f /. x0)) < r by A6, A9, A7, A8; ::_thesis: verum
end;
assume that
A10: X c= dom f and
A11: for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ; ::_thesis: f is_continuous_on X
A12: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A10, XBOOLE_1:28 ;
now__::_thesis:_for_x0_being_Point_of_S_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Point of S; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A13: x0 in X ; ::_thesis: f | X is_continuous_in x0
 for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < r ) )
proof
let r be Real; ::_thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < r ) ) )
assume  0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < r ) )
then consider s being Real such that
A14: 0 < s and
A15: for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r by A11, A13;
take  s ; ::_thesis: ( 0 < s & ( for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < r ) )
thus  0 < s by A14; ::_thesis: for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < r
let x1 be Point of S; ::_thesis: ( x1 in dom (f | X) & ||.(x1 - x0).|| < s implies abs (((f | X) /. x1) - ((f | X) /. x0)) < r )
assume that
A16: x1 in dom (f | X) and
A17: ||.(x1 - x0).|| < s ; ::_thesis: abs (((f | X) /. x1) - ((f | X) /. x0)) < r
abs (((f | X) /. x1) - ((f | X) /. x0)) = abs (((f | X) /. x1) - (f /. x0)) by A12, A13, PARTFUN2:15
.= abs ((f /. x1) - (f /. x0)) by A16, PARTFUN2:15 ;
hence  abs (((f | X) /. x1) - ((f | X) /. x0)) < r by A12, A15, A16, A17; ::_thesis: verum
end;
hence  f | X is_continuous_in x0 by A12, A13, Th8; ::_thesis: verum
end;
hence  f is_continuous_on X by A10, Def8; ::_thesis: verum
end;

theorem :: NFCONT_1:21
for X being set
for S, T being RealNormSpace
for f being PartFunc of S,T holds
( f is_continuous_on X iff f | X is_continuous_on X )
proof
let X be set ; ::_thesis: for S, T being RealNormSpace
for f being PartFunc of S,T holds
( f is_continuous_on X iff f | X is_continuous_on X )
let S, T be RealNormSpace; ::_thesis: for f being PartFunc of S,T holds
( f is_continuous_on X iff f | X is_continuous_on X )
let f be PartFunc of S,T; ::_thesis: ( f is_continuous_on X iff f | X is_continuous_on X )
thus  ( f is_continuous_on X implies f | X is_continuous_on X ) ::_thesis: ( f | X is_continuous_on X implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: f | X is_continuous_on X
then  X c= dom f by Def7;
then  X c= (dom f) /\ X by XBOOLE_1:28;
hence  X c= dom (f | X) by RELAT_1:61; :: according to NFCONT_1:def_7 ::_thesis: for x0 being Point of S st x0 in X holds
(f | X) | X is_continuous_in x0
let r be Point of S; ::_thesis: ( r in X implies (f | X) | X is_continuous_in r )
assume  r in X ; ::_thesis: (f | X) | X is_continuous_in r
then  f | X is_continuous_in r by A1, Def7;
hence  (f | X) | X is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
assume A2: f | X is_continuous_on X ; ::_thesis: f is_continuous_on X
then  X c= dom (f | X) by Def7;
then  ( (dom f) /\ X c= dom f & X c= (dom f) /\ X ) by RELAT_1:61, XBOOLE_1:17;
hence  X c= dom f by XBOOLE_1:1; :: according to NFCONT_1:def_7 ::_thesis: for x0 being Point of S st x0 in X holds
f | X is_continuous_in x0
let r be Point of S; ::_thesis: ( r in X implies f | X is_continuous_in r )
assume  r in X ; ::_thesis: f | X is_continuous_in r
then  (f | X) | X is_continuous_in r by A2, Def7;
hence  f | X is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;

theorem Th22: :: NFCONT_1:22
for X being set
for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL holds
( f is_continuous_on X iff f | X is_continuous_on X )
proof
let X be set ; ::_thesis: for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL holds
( f is_continuous_on X iff f | X is_continuous_on X )
let S be RealNormSpace; ::_thesis: for f being PartFunc of the carrier of S,REAL holds
( f is_continuous_on X iff f | X is_continuous_on X )
let f be PartFunc of the carrier of S,REAL; ::_thesis: ( f is_continuous_on X iff f | X is_continuous_on X )
thus  ( f is_continuous_on X implies f | X is_continuous_on X ) ::_thesis: ( f | X is_continuous_on X implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: f | X is_continuous_on X
then  X c= dom f by Def8;
then  X c= (dom f) /\ X by XBOOLE_1:28;
hence  X c= dom (f | X) by RELAT_1:61; :: according to NFCONT_1:def_8 ::_thesis: for x0 being Point of S st x0 in X holds
(f | X) | X is_continuous_in x0
let r be Point of S; ::_thesis: ( r in X implies (f | X) | X is_continuous_in r )
assume  r in X ; ::_thesis: (f | X) | X is_continuous_in r
then  f | X is_continuous_in r by A1, Def8;
hence  (f | X) | X is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
assume A2: f | X is_continuous_on X ; ::_thesis: f is_continuous_on X
then  X c= dom (f | X) by Def8;
then  ( (dom f) /\ X c= dom f & X c= (dom f) /\ X ) by RELAT_1:61, XBOOLE_1:17;
hence  X c= dom f by XBOOLE_1:1; :: according to NFCONT_1:def_8 ::_thesis: for x0 being Point of S st x0 in X holds
f | X is_continuous_in x0
let r be Point of S; ::_thesis: ( r in X implies f | X is_continuous_in r )
assume  r in X ; ::_thesis: f | X is_continuous_in r
then  (f | X) | X is_continuous_in r by A2, Def8;
hence  f | X is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;

theorem Th23: :: NFCONT_1:23
for X, X1 being set
for S, T being RealNormSpace
for f being PartFunc of S,T st f is_continuous_on X & X1 c= X holds
f is_continuous_on X1
proof
let X, X1 be set ; ::_thesis: for S, T being RealNormSpace
for f being PartFunc of S,T st f is_continuous_on X & X1 c= X holds
f is_continuous_on X1
let S, T be RealNormSpace; ::_thesis: for f being PartFunc of S,T st f is_continuous_on X & X1 c= X holds
f is_continuous_on X1
let f be PartFunc of S,T; ::_thesis: ( f is_continuous_on X & X1 c= X implies f is_continuous_on X1 )
assume that
A1: f is_continuous_on X and
A2: X1 c= X ; ::_thesis: f is_continuous_on X1
 X c= dom f by A1, Def7;
hence A3: X1 c= dom f by A2, XBOOLE_1:1; :: according to NFCONT_1:def_7 ::_thesis: for x0 being Point of S st x0 in X1 holds
f | X1 is_continuous_in x0
let r be Point of S; ::_thesis: ( r in X1 implies f | X1 is_continuous_in r )
assume A4: r in X1 ; ::_thesis: f | X1 is_continuous_in r
then A5: f | X is_continuous_in r by A1, A2, Def7;
thus  f | X1 is_continuous_in r ::_thesis: verum
proof
 (dom f) /\ X1 c= (dom f) /\ X by A2, XBOOLE_1:26;
then  dom (f | X1) c= (dom f) /\ X by RELAT_1:61;
then A6: dom (f | X1) c= dom (f | X) by RELAT_1:61;
 r in (dom f) /\ X1 by A3, A4, XBOOLE_0:def_4;
hence A7: r in dom (f | X1) by RELAT_1:61; :: according to NFCONT_1:def_5 ::_thesis: for s1 being sequence of S st rng s1 c= dom (f | X1) & s1 is convergent & lim s1 = r holds
( (f | X1) /* s1 is convergent & (f | X1) /. r = lim ((f | X1) /* s1) )
then A8: (f | X) /. r = f /. r by A6, PARTFUN2:15
.= (f | X1) /. r by A7, PARTFUN2:15 ;
let s1 be sequence of S; ::_thesis: ( rng s1 c= dom (f | X1) & s1 is convergent & lim s1 = r implies ( (f | X1) /* s1 is convergent & (f | X1) /. r = lim ((f | X1) /* s1) ) )
assume that
A9: rng s1 c= dom (f | X1) and
A10: ( s1 is convergent & lim s1 = r ) ; ::_thesis: ( (f | X1) /* s1 is convergent & (f | X1) /. r = lim ((f | X1) /* s1) )
A11: rng s1 c= dom (f | X) by A9, A6, XBOOLE_1:1;
A12: now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_X)_/*_s1)_._n_=_((f_|_X1)_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ((f | X) /* s1) . n = ((f | X1) /* s1) . n
 dom s1 = NAT by FUNCT_2:def_1;
then A13: s1 . n in rng s1 by FUNCT_1:3;
thus ((f | X) /* s1) . n = (f | X) /. (s1 . n) by A9, A6, FUNCT_2:109, XBOOLE_1:1
.= f /. (s1 . n) by A11, A13, PARTFUN2:15
.= (f | X1) /. (s1 . n) by A9, A13, PARTFUN2:15
.= ((f | X1) /* s1) . n by A9, FUNCT_2:109 ; ::_thesis: verum
end;
 ( (f | X) /* s1 is convergent & (f | X) /. r = lim ((f | X) /* s1) ) by A5, A10, A11, Def5;
hence  ( (f | X1) /* s1 is convergent & (f | X1) /. r = lim ((f | X1) /* s1) ) by A8, A12, FUNCT_2:63; ::_thesis: verum
end;
end;

theorem :: NFCONT_1:24
for T, S being RealNormSpace
for f being PartFunc of S,T
for x0 being Point of S st x0 in dom f holds
f is_continuous_on {x0}
proof
let T, S be RealNormSpace; ::_thesis: for f being PartFunc of S,T
for x0 being Point of S st x0 in dom f holds
f is_continuous_on {x0}
let f be PartFunc of S,T; ::_thesis: for x0 being Point of S st x0 in dom f holds
f is_continuous_on {x0}
let x0 be Point of S; ::_thesis: ( x0 in dom f implies f is_continuous_on {x0} )
assume A1: x0 in dom f ; ::_thesis: f is_continuous_on {x0}
thus  {x0} c= dom f :: according to NFCONT_1:def_7 ::_thesis: for x0 being Point of S st x0 in {x0} holds
f | {x0} is_continuous_in x0
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {x0} or x in dom f )
assume  x in {x0} ; ::_thesis: x in dom f
hence  x in dom f by A1, TARSKI:def_1; ::_thesis: verum
end;
let p be Point of S; ::_thesis: ( p in {x0} implies f | {x0} is_continuous_in p )
assume A2: p in {x0} ; ::_thesis: f | {x0} is_continuous_in p
thus  f | {x0} is_continuous_in p ::_thesis: verum
proof
 p in dom f by A1, A2, TARSKI:def_1;
then  p in (dom f) /\ {x0} by A2, XBOOLE_0:def_4;
hence  p in dom (f | {x0}) by RELAT_1:61; :: according to NFCONT_1:def_5 ::_thesis: for s1 being sequence of S st rng s1 c= dom (f | {x0}) & s1 is convergent & lim s1 = p holds
( (f | {x0}) /* s1 is convergent & (f | {x0}) /. p = lim ((f | {x0}) /* s1) )
let s1 be sequence of S; ::_thesis: ( rng s1 c= dom (f | {x0}) & s1 is convergent & lim s1 = p implies ( (f | {x0}) /* s1 is convergent & (f | {x0}) /. p = lim ((f | {x0}) /* s1) ) )
assume that
A3: rng s1 c= dom (f | {x0}) and
 s1 is convergent and
 lim s1 = p ; ::_thesis: ( (f | {x0}) /* s1 is convergent & (f | {x0}) /. p = lim ((f | {x0}) /* s1) )
A4: (dom f) /\ {x0} c= {x0} by XBOOLE_1:17;
 rng s1 c= (dom f) /\ {x0} by A3, RELAT_1:61;
then A5: rng s1 c= {x0} by A4, XBOOLE_1:1;
A6: now__::_thesis:_for_n_being_Element_of_NAT_holds_s1_._n_=_x0
let n be Element of NAT ; ::_thesis: s1 . n = x0
 dom s1 = NAT by FUNCT_2:def_1;
then  s1 . n in rng s1 by FUNCT_1:3;
hence  s1 . n = x0 by A5, TARSKI:def_1; ::_thesis: verum
end;
A7: p = x0 by A2, TARSKI:def_1;
A8: now__::_thesis:_for_g_being_Real_st_0_<_g_holds_
ex_n_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n_<=_m_holds_
||.((((f_|_{x0})_/*_s1)_._m)_-_((f_|_{x0})_/._p)).||_<_g
let g be Real; ::_thesis: ( 0 < g implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g )
assume A9: 0 < g ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g
take n = 0 ; ::_thesis: for m being Element of NAT st n <= m holds
||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g
let m be Element of NAT ; ::_thesis: ( n <= m implies ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g )
assume  n <= m ; ::_thesis: ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g
||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| = ||.(((f | {x0}) /. (s1 . m)) - ((f | {x0}) /. x0)).|| by A7, A3, FUNCT_2:109
.= ||.(((f | {x0}) /. x0) - ((f | {x0}) /. x0)).|| by A6
.= ||.(0. T).|| by RLVECT_1:15
.= 0 by NORMSP_1:1 ;
hence  ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g by A9; ::_thesis: verum
end;
hence  (f | {x0}) /* s1 is convergent by NORMSP_1:def_6; ::_thesis: (f | {x0}) /. p = lim ((f | {x0}) /* s1)
hence  (f | {x0}) /. p = lim ((f | {x0}) /* s1) by A8, NORMSP_1:def_7; ::_thesis: verum
end;
end;

theorem Th25: :: NFCONT_1:25
for T, S being RealNormSpace
for X being set
for f1, f2 being PartFunc of S,T st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
proof
let T, S be RealNormSpace; ::_thesis: for X being set
for f1, f2 being PartFunc of S,T st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
let X be set ; ::_thesis: for f1, f2 being PartFunc of S,T st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
let f1, f2 be PartFunc of S,T; ::_thesis: ( f1 is_continuous_on X & f2 is_continuous_on X implies ( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X ) )
assume A1: ( f1 is_continuous_on X & f2 is_continuous_on X ) ; ::_thesis: ( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
then  ( X c= dom f1 & X c= dom f2 ) by Th18;
then A2: X c= (dom f1) /\ (dom f2) by XBOOLE_1:19;
then A3: X c= dom (f1 + f2) by VFUNCT_1:def_1;
now__::_thesis:_for_s1_being_sequence_of_S_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_
(_(f1_+_f2)_/*_s1_is_convergent_&_(f1_+_f2)_/._(lim_s1)_=_lim_((f1_+_f2)_/*_s1)_)
let s1 be sequence of S; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) ) )
assume that
A4: rng s1 c= X and
A5: s1 is convergent and
A6: lim s1 in X ; ::_thesis: ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) )
A7: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A1, A4, A5, A6, Th18;
then A8: (f1 /* s1) + (f2 /* s1) is convergent by NORMSP_1:19;
A9: rng s1 c= (dom f1) /\ (dom f2) by A2, A4, XBOOLE_1:1;
 ( f1 /. (lim s1) = lim (f1 /* s1) & f2 /. (lim s1) = lim (f2 /* s1) ) by A1, A4, A5, A6, Th18;
then (f1 + f2) /. (lim s1) = (lim (f1 /* s1)) + (lim (f2 /* s1)) by A3, A6, VFUNCT_1:def_1
.= lim ((f1 /* s1) + (f2 /* s1)) by A7, NORMSP_1:25
.= lim ((f1 + f2) /* s1) by A9, Th12 ;
hence  ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) ) by A9, A8, Th12; ::_thesis: verum
end;
hence  f1 + f2 is_continuous_on X by A3, Th18; ::_thesis: f1 - f2 is_continuous_on X
A10: X c= dom (f1 - f2) by A2, VFUNCT_1:def_2;
now__::_thesis:_for_s1_being_sequence_of_S_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_
(_(f1_-_f2)_/*_s1_is_convergent_&_(f1_-_f2)_/._(lim_s1)_=_lim_((f1_-_f2)_/*_s1)_)
let s1 be sequence of S; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) ) )
assume that
A11: rng s1 c= X and
A12: s1 is convergent and
A13: lim s1 in X ; ::_thesis: ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) )
A14: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A1, A11, A12, A13, Th18;
then A15: (f1 /* s1) - (f2 /* s1) is convergent by NORMSP_1:20;
A16: rng s1 c= (dom f1) /\ (dom f2) by A2, A11, XBOOLE_1:1;
 ( f1 /. (lim s1) = lim (f1 /* s1) & f2 /. (lim s1) = lim (f2 /* s1) ) by A1, A11, A12, A13, Th18;
then (f1 - f2) /. (lim s1) = (lim (f1 /* s1)) - (lim (f2 /* s1)) by A10, A13, VFUNCT_1:def_2
.= lim ((f1 /* s1) - (f2 /* s1)) by A14, NORMSP_1:26
.= lim ((f1 - f2) /* s1) by A16, Th12 ;
hence  ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) ) by A16, A15, Th12; ::_thesis: verum
end;
hence  f1 - f2 is_continuous_on X by A10, Th18; ::_thesis: verum
end;

theorem :: NFCONT_1:26
for T, S being RealNormSpace
for X, X1 being set
for f1, f2 being PartFunc of S,T st f1 is_continuous_on X & f2 is_continuous_on X1 holds
( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
proof
let T, S be RealNormSpace; ::_thesis: for X, X1 being set
for f1, f2 being PartFunc of S,T st f1 is_continuous_on X & f2 is_continuous_on X1 holds
( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
let X, X1 be set ; ::_thesis: for f1, f2 being PartFunc of S,T st f1 is_continuous_on X & f2 is_continuous_on X1 holds
( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
let f1, f2 be PartFunc of S,T; ::_thesis: ( f1 is_continuous_on X & f2 is_continuous_on X1 implies ( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 ) )
assume  ( f1 is_continuous_on X & f2 is_continuous_on X1 ) ; ::_thesis: ( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
then  ( f1 is_continuous_on X /\ X1 & f2 is_continuous_on X /\ X1 ) by Th23, XBOOLE_1:17;
hence  ( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 ) by Th25; ::_thesis: verum
end;

theorem Th27: :: NFCONT_1:27
for T, S being RealNormSpace
for r being Real
for X being set
for f being PartFunc of S,T st f is_continuous_on X holds
r (#) f is_continuous_on X
proof
let T, S be RealNormSpace; ::_thesis: for r being Real
for X being set
for f being PartFunc of S,T st f is_continuous_on X holds
r (#) f is_continuous_on X
let r be Real; ::_thesis: for X being set
for f being PartFunc of S,T st f is_continuous_on X holds
r (#) f is_continuous_on X
let X be set ; ::_thesis: for f being PartFunc of S,T st f is_continuous_on X holds
r (#) f is_continuous_on X
let f be PartFunc of S,T; ::_thesis: ( f is_continuous_on X implies r (#) f is_continuous_on X )
assume A1: f is_continuous_on X ; ::_thesis: r (#) f is_continuous_on X
then A2: X c= dom f by Th18;
then A3: X c= dom (r (#) f) by VFUNCT_1:def_4;
now__::_thesis:_for_s1_being_sequence_of_S_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_
(_(r_(#)_f)_/*_s1_is_convergent_&_(r_(#)_f)_/._(lim_s1)_=_lim_((r_(#)_f)_/*_s1)_)
let s1 be sequence of S; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) ) )
assume that
A4: rng s1 c= X and
A5: s1 is convergent and
A6: lim s1 in X ; ::_thesis: ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) )
A7: f /* s1 is convergent by A1, A4, A5, A6, Th18;
then A8: r * (f /* s1) is convergent by NORMSP_1:22;
 f /. (lim s1) = lim (f /* s1) by A1, A4, A5, A6, Th18;
then (r (#) f) /. (lim s1) = r * (lim (f /* s1)) by A3, A6, VFUNCT_1:def_4
.= lim (r * (f /* s1)) by A7, NORMSP_1:28
.= lim ((r (#) f) /* s1) by A2, A4, Th13, XBOOLE_1:1 ;
hence  ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) ) by A2, A4, A8, Th13, XBOOLE_1:1; ::_thesis: verum
end;
hence  r (#) f is_continuous_on X by A3, Th18; ::_thesis: verum
end;

theorem Th28: :: NFCONT_1:28
for X being set
for S, T being RealNormSpace
for f being PartFunc of S,T st f is_continuous_on X holds
( ||.f.|| is_continuous_on X & - f is_continuous_on X )
proof
let X be set ; ::_thesis: for S, T being RealNormSpace
for f being PartFunc of S,T st f is_continuous_on X holds
( ||.f.|| is_continuous_on X & - f is_continuous_on X )
let S, T be RealNormSpace; ::_thesis: for f being PartFunc of S,T st f is_continuous_on X holds
( ||.f.|| is_continuous_on X & - f is_continuous_on X )
let f be PartFunc of S,T; ::_thesis: ( f is_continuous_on X implies ( ||.f.|| is_continuous_on X & - f is_continuous_on X ) )
assume A1: f is_continuous_on X ; ::_thesis: ( ||.f.|| is_continuous_on X & - f is_continuous_on X )
thus  ||.f.|| is_continuous_on X ::_thesis: - f is_continuous_on X
proof
A2: X c= dom f by A1, Def7;
hence A3: X c= dom ||.f.|| by NORMSP_0:def_3; :: according to NFCONT_1:def_8 ::_thesis: for x0 being Point of S st x0 in X holds
||.f.|| | X is_continuous_in x0
let r be Point of S; ::_thesis: ( r in X implies ||.f.|| | X is_continuous_in r )
assume A4: r in X ; ::_thesis: ||.f.|| | X is_continuous_in r
then A5: f | X is_continuous_in r by A1, Def7;
thus  ||.f.|| | X is_continuous_in r ::_thesis: verum
proof
A6: r in (dom ||.f.||) /\ X by A3, A4, XBOOLE_0:def_4;
hence  r in dom (||.f.|| | X) by RELAT_1:61; :: according to NFCONT_1:def_6 ::_thesis: for s1 being sequence of S st rng s1 c= dom (||.f.|| | X) & s1 is convergent & lim s1 = r holds
( (||.f.|| | X) /* s1 is convergent & (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) )
let s1 be sequence of S; ::_thesis: ( rng s1 c= dom (||.f.|| | X) & s1 is convergent & lim s1 = r implies ( (||.f.|| | X) /* s1 is convergent & (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) ) )
assume that
A7: rng s1 c= dom (||.f.|| | X) and
A8: ( s1 is convergent & lim s1 = r ) ; ::_thesis: ( (||.f.|| | X) /* s1 is convergent & (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) )
 rng s1 c= (dom ||.f.||) /\ X by A7, RELAT_1:61;
then  rng s1 c= (dom f) /\ X by NORMSP_0:def_3;
then A9: rng s1 c= dom (f | X) by RELAT_1:61;
then A10: (f | X) /. r = lim ((f | X) /* s1) by A5, A8, Def5;
now__::_thesis:_for_n_being_Element_of_NAT_holds_||.((f_|_X)_/*_s1).||_._n_=_((||.f.||_|_X)_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ||.((f | X) /* s1).|| . n = ((||.f.|| | X) /* s1) . n
 dom s1 = NAT by FUNCT_2:def_1;
then A11: s1 . n in rng s1 by FUNCT_1:3;
then  s1 . n in dom (f | X) by A9;
then A12: s1 . n in (dom f) /\ X by RELAT_1:61;
then A13: s1 . n in X by XBOOLE_0:def_4;
 s1 . n in dom f by A12, XBOOLE_0:def_4;
then A14: s1 . n in dom ||.f.|| by NORMSP_0:def_3;
thus ||.((f | X) /* s1).|| . n = ||.(((f | X) /* s1) . n).|| by NORMSP_0:def_4
.= ||.((f | X) /. (s1 . n)).|| by A9, FUNCT_2:109
.= ||.(f /. (s1 . n)).|| by A9, A11, PARTFUN2:15
.= ||.f.|| . (s1 . n) by A14, NORMSP_0:def_3
.= (||.f.|| | X) . (s1 . n) by A13, FUNCT_1:49
.= (||.f.|| | X) /. (s1 . n) by A7, A11, PARTFUN1:def_6
.= ((||.f.|| | X) /* s1) . n by A7, FUNCT_2:109 ; ::_thesis: verum
end;
then A15: ||.((f | X) /* s1).|| = (||.f.|| | X) /* s1 by FUNCT_2:63;
A16: (f | X) /* s1 is convergent by A5, A8, A9, Def5;
hence  (||.f.|| | X) /* s1 is convergent by A15, NORMSP_1:23; ::_thesis: (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1)
||.((f | X) /. r).|| = ||.(f /. r).|| by A2, A4, PARTFUN2:17
.= ||.f.|| . r by A3, A4, NORMSP_0:def_3
.= ||.f.|| /. r by A3, A4, PARTFUN1:def_6
.= (||.f.|| | X) /. r by A6, PARTFUN2:16 ;
hence  (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) by A16, A10, A15, LOPBAN_1:20; ::_thesis: verum
end;
end;
 (- 1) (#) f is_continuous_on X by A1, Th27;
hence  - f is_continuous_on X by VFUNCT_1:23; ::_thesis: verum
end;

theorem :: NFCONT_1:29
for T, S being RealNormSpace
for f being PartFunc of S,T st f is total & ( for x1, x2 being Point of S holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Point of S st f is_continuous_in x0 holds
f is_continuous_on the carrier of S
proof
let T, S be RealNormSpace; ::_thesis: for f being PartFunc of S,T st f is total & ( for x1, x2 being Point of S holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Point of S st f is_continuous_in x0 holds
f is_continuous_on the carrier of S
let f be PartFunc of S,T; ::_thesis: ( f is total & ( for x1, x2 being Point of S holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Point of S st f is_continuous_in x0 implies f is_continuous_on the carrier of S )
assume that
A1: f is total and
A2: for x1, x2 being Point of S holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ; ::_thesis: ( for x0 being Point of S holds not f is_continuous_in x0 or f is_continuous_on the carrier of S )
A3: dom f = the carrier of S by A1, PARTFUN1:def_2;
given x0 being Point of S such that A4: f is_continuous_in x0 ; ::_thesis: f is_continuous_on the carrier of S
(f /. x0) + (0. T) = f /. x0 by RLVECT_1:4
.= f /. (x0 + (0. S)) by RLVECT_1:4
.= (f /. x0) + (f /. (0. S)) by A2 ;
then A5: f /. (0. S) = 0. T by RLVECT_1:8;
A6: now__::_thesis:_for_x1_being_Point_of_S_holds_-_(f_/._x1)_=_f_/._(-_x1)
let x1 be Point of S; ::_thesis: - (f /. x1) = f /. (- x1)
0. T = f /. (x1 + (- x1)) by A5, RLVECT_1:5
.= (f /. x1) + (f /. (- x1)) by A2 ;
hence  - (f /. x1) = f /. (- x1) by RLVECT_1:6; ::_thesis: verum
end;
A7: now__::_thesis:_for_x1,_x2_being_Point_of_S_holds_f_/._(x1_-_x2)_=_(f_/._x1)_-_(f_/._x2)
let x1, x2 be Point of S; ::_thesis: f /. (x1 - x2) = (f /. x1) - (f /. x2)
thus f /. (x1 - x2) = (f /. x1) + (f /. (- x2)) by A2
.= (f /. x1) - (f /. x2) by A6 ; ::_thesis: verum
end;
now__::_thesis:_for_x1_being_Point_of_S
for_r_being_Real_st_x1_in_the_carrier_of_S_&_r_>_0_holds_
ex_s_being_Real_st_
(_s_>_0_&_(_for_x2_being_Point_of_S_st_x2_in_the_carrier_of_S_&_||.(x2_-_x1).||_<_s_holds_
||.((f_/._x2)_-_(f_/._x1)).||_<_r_)_)
let x1 be Point of S; ::_thesis: for r being Real st x1 in the carrier of S & r > 0 holds
ex s being Real st
( s > 0 & ( for x2 being Point of S st x2 in the carrier of S & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r ) )
let r be Real; ::_thesis: ( x1 in the carrier of S & r > 0 implies ex s being Real st
( s > 0 & ( for x2 being Point of S st x2 in the carrier of S & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r ) ) )
assume that
 x1 in the carrier of S and
A8: r > 0 ; ::_thesis: ex s being Real st
( s > 0 & ( for x2 being Point of S st x2 in the carrier of S & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r ) )
set y = x1 - x0;
consider s being Real such that
A9: 0 < s and
A10: for x1 being Point of S st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r by A4, A8, Th7;
take s = s; ::_thesis: ( s > 0 & ( for x2 being Point of S st x2 in the carrier of S & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r ) )
thus  s > 0 by A9; ::_thesis: for x2 being Point of S st x2 in the carrier of S & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r
let x2 be Point of S; ::_thesis: ( x2 in the carrier of S & ||.(x2 - x1).|| < s implies ||.((f /. x2) - (f /. x1)).|| < r )
assume that
 x2 in the carrier of S and
A11: ||.(x2 - x1).|| < s ; ::_thesis: ||.((f /. x2) - (f /. x1)).|| < r
A12: (x1 - x0) + x0 = x1 - (x0 - x0) by RLVECT_1:29
.= x1 - (0. S) by RLVECT_1:15
.= x1 by RLVECT_1:13 ;
then A13: ||.((x2 - (x1 - x0)) - x0).|| = ||.(x2 - x1).|| by RLVECT_1:27;
||.((f /. x2) - (f /. x1)).|| = ||.((f /. x2) - ((f /. (x1 - x0)) + (f /. x0))).|| by A2, A12
.= ||.(((f /. x2) - (f /. (x1 - x0))) - (f /. x0)).|| by RLVECT_1:27
.= ||.((f /. (x2 - (x1 - x0))) - (f /. x0)).|| by A7 ;
hence  ||.((f /. x2) - (f /. x1)).|| < r by A3, A10, A11, A13; ::_thesis: verum
end;
hence  f is_continuous_on the carrier of S by A3, Th19; ::_thesis: verum
end;

theorem Th30: :: NFCONT_1:30
for T, S being RealNormSpace
for f being PartFunc of S,T st dom f is compact & f is_continuous_on dom f holds
rng f is compact
proof
let T, S be RealNormSpace; ::_thesis: for f being PartFunc of S,T st dom f is compact & f is_continuous_on dom f holds
rng f is compact
let f be PartFunc of S,T; ::_thesis: ( dom f is compact & f is_continuous_on dom f implies rng f is compact )
assume that
A1: dom f is compact and
A2: f is_continuous_on dom f ; ::_thesis: rng f is compact
now__::_thesis:_for_s1_being_sequence_of_T_st_rng_s1_c=_rng_f_holds_
ex_q2_being_Element_of_K6(K7(NAT,_the_carrier_of_T))_st_
(_q2_is_subsequence_of_s1_&_q2_is_convergent_&_lim_q2_in_rng_f_)
let s1 be sequence of T; ::_thesis: ( rng s1 c= rng f implies ex q2 being Element of K6(K7(NAT, the carrier of T)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) )
assume A3: rng s1 c= rng f ; ::_thesis: ex q2 being Element of K6(K7(NAT, the carrier of T)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
defpred S1[ set , set ] means ( $2 in dom f & f /. $2 = s1 . $1 );
A4: for n being Element of NAT ex p being Point of S st S1[n,p]
proof
let n be Element of NAT ; ::_thesis: ex p being Point of S st S1[n,p]
 dom s1 = NAT by FUNCT_2:def_1;
then  s1 . n in rng s1 by FUNCT_1:3;
then consider p being Point of S such that
A5: ( p in dom f & s1 . n = f . p ) by A3, PARTFUN1:3;
take  p ; ::_thesis: S1[n,p]
thus  S1[n,p] by A5, PARTFUN1:def_6; ::_thesis: verum
end;
consider q1 being sequence of S such that
A6: for n being Element of NAT holds S1[n,q1 . n] from FUNCT_2:sch_3(A4);
now__::_thesis:_for_x_being_set_st_x_in_rng_q1_holds_
x_in_dom_f
let x be set ; ::_thesis: ( x in rng q1 implies x in dom f )
assume  x in rng q1 ; ::_thesis: x in dom f
then  ex n being Element of NAT st x = q1 . n by Th6;
hence  x in dom f by A6; ::_thesis: verum
end;
then A7: rng q1 c= dom f by TARSKI:def_3;
then consider s2 being sequence of S such that
A8: s2 is subsequence of q1 and
A9: s2 is convergent and
A10: lim s2 in dom f by A1, Def2;
take q2 = f /* s2; ::_thesis: ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
 rng s2 c= rng q1 by A8, VALUED_0:21;
then A11: rng s2 c= dom f by A7, XBOOLE_1:1;
now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_q1)_._n_=_s1_._n
let n be Element of NAT ; ::_thesis: (f /* q1) . n = s1 . n
 f /. (q1 . n) = s1 . n by A6;
hence  (f /* q1) . n = s1 . n by A7, FUNCT_2:109; ::_thesis: verum
end;
then A12: f /* q1 = s1 by FUNCT_2:63;
 f | (dom f) is_continuous_in lim s2 by A2, A10, Def7;
then A13: f is_continuous_in lim s2 by RELAT_1:68;
then  f /. (lim s2) = lim (f /* s2) by A9, A11, Def5;
hence  ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) by A7, A12, A8, A9, A10, A13, A11, Def5, PARTFUN2:2, VALUED_0:22; ::_thesis: verum
end;
hence  rng f is compact by Def2; ::_thesis: verum
end;

theorem Th31: :: NFCONT_1:31
for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL st dom f is compact & f is_continuous_on dom f holds
rng f is compact
proof
let S be RealNormSpace; ::_thesis: for f being PartFunc of the carrier of S,REAL st dom f is compact & f is_continuous_on dom f holds
rng f is compact
let f be PartFunc of the carrier of S,REAL; ::_thesis: ( dom f is compact & f is_continuous_on dom f implies rng f is compact )
assume that
A1: dom f is compact and
A2: f is_continuous_on dom f ; ::_thesis: rng f is compact
now__::_thesis:_for_s1_being_Real_Sequence_st_rng_s1_c=_rng_f_holds_
ex_q2_being_Element_of_K6(K7(NAT,REAL))_st_
(_q2_is_subsequence_of_s1_&_q2_is_convergent_&_lim_q2_in_rng_f_)
let s1 be Real_Sequence; ::_thesis: ( rng s1 c= rng f implies ex q2 being Element of K6(K7(NAT,REAL)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) )
assume A3: rng s1 c= rng f ; ::_thesis: ex q2 being Element of K6(K7(NAT,REAL)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
defpred S1[ set , set ] means ( $2 in dom f & f /. $2 = s1 . $1 );
A4: for n being Element of NAT ex p being Point of S st S1[n,p]
proof
let n be Element of NAT ; ::_thesis: ex p being Point of S st S1[n,p]
 dom s1 = NAT by FUNCT_2:def_1;
then  s1 . n in rng s1 by FUNCT_1:3;
then consider p being Point of S such that
A5: ( p in dom f & s1 . n = f . p ) by A3, PARTFUN1:3;
take  p ; ::_thesis: S1[n,p]
thus  S1[n,p] by A5, PARTFUN1:def_6; ::_thesis: verum
end;
consider q1 being sequence of S such that
A6: for n being Element of NAT holds S1[n,q1 . n] from FUNCT_2:sch_3(A4);
now__::_thesis:_for_x_being_set_st_x_in_rng_q1_holds_
x_in_dom_f
let x be set ; ::_thesis: ( x in rng q1 implies x in dom f )
assume  x in rng q1 ; ::_thesis: x in dom f
then  ex n being Element of NAT st x = q1 . n by Th6;
hence  x in dom f by A6; ::_thesis: verum
end;
then A7: rng q1 c= dom f by TARSKI:def_3;
then consider s2 being sequence of S such that
A8: s2 is subsequence of q1 and
A9: s2 is convergent and
A10: lim s2 in dom f by A1, Def2;
take q2 = f /* s2; ::_thesis: ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
 rng s2 c= rng q1 by A8, VALUED_0:21;
then A11: rng s2 c= dom f by A7, XBOOLE_1:1;
now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_q1)_._n_=_s1_._n
let n be Element of NAT ; ::_thesis: (f /* q1) . n = s1 . n
 f /. (q1 . n) = s1 . n by A6;
hence  (f /* q1) . n = s1 . n by A7, FUNCT_2:109; ::_thesis: verum
end;
then A12: f /* q1 = s1 by FUNCT_2:63;
 f | (dom f) is_continuous_in lim s2 by A2, A10, Def8;
then A13: f is_continuous_in lim s2 by RELAT_1:68;
then  f /. (lim s2) = lim (f /* s2) by A9, A11, Def6;
hence  ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) by A7, A12, A8, A9, A10, A13, A11, Def6, PARTFUN2:2, VALUED_0:22; ::_thesis: verum
end;
hence  rng f is compact by RCOMP_1:def_3; ::_thesis: verum
end;

theorem :: NFCONT_1:32
for T, S being RealNormSpace
for f being PartFunc of S,T
for Y being Subset of S st Y c= dom f & Y is compact & f is_continuous_on Y holds
f .: Y is compact
proof
let T, S be RealNormSpace; ::_thesis: for f being PartFunc of S,T
for Y being Subset of S st Y c= dom f & Y is compact & f is_continuous_on Y holds
f .: Y is compact
let f be PartFunc of S,T; ::_thesis: for Y being Subset of S st Y c= dom f & Y is compact & f is_continuous_on Y holds
f .: Y is compact
let Y be Subset of S; ::_thesis: ( Y c= dom f & Y is compact & f is_continuous_on Y implies f .: Y is compact )
assume that
A1: Y c= dom f and
A2: Y is compact and
A3: f is_continuous_on Y ; ::_thesis: f .: Y is compact
A4: dom (f | Y) = (dom f) /\ Y by RELAT_1:61
.= Y by A1, XBOOLE_1:28 ;
 f | Y is_continuous_on Y
proof
thus  Y c= dom (f | Y) by A4; :: according to NFCONT_1:def_7 ::_thesis: for x0 being Point of S st x0 in Y holds
(f | Y) | Y is_continuous_in x0
let r be Point of S; ::_thesis: ( r in Y implies (f | Y) | Y is_continuous_in r )
assume  r in Y ; ::_thesis: (f | Y) | Y is_continuous_in r
then  f | Y is_continuous_in r by A3, Def7;
hence  (f | Y) | Y is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
then  rng (f | Y) is compact by A2, A4, Th30;
hence  f .: Y is compact by RELAT_1:115; ::_thesis: verum
end;

theorem Th33: :: NFCONT_1:33
for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL st dom f <> {} & dom f is compact & f is_continuous_on dom f holds
ex x1, x2 being Point of S st
( x1 in dom f & x2 in dom f & f /. x1 = upper_bound (rng f) & f /. x2 = lower_bound (rng f) )
proof
let S be RealNormSpace; ::_thesis: for f being PartFunc of the carrier of S,REAL st dom f <> {} & dom f is compact & f is_continuous_on dom f holds
ex x1, x2 being Point of S st
( x1 in dom f & x2 in dom f & f /. x1 = upper_bound (rng f) & f /. x2 = lower_bound (rng f) )
let f be PartFunc of the carrier of S,REAL; ::_thesis: ( dom f <> {} & dom f is compact & f is_continuous_on dom f implies ex x1, x2 being Point of S st
( x1 in dom f & x2 in dom f & f /. x1 = upper_bound (rng f) & f /. x2 = lower_bound (rng f) ) )
assume  ( dom f <> {} & dom f is compact & f is_continuous_on dom f ) ; ::_thesis: ex x1, x2 being Point of S st
( x1 in dom f & x2 in dom f & f /. x1 = upper_bound (rng f) & f /. x2 = lower_bound (rng f) )
then A1: ( rng f <> {} & rng f is compact ) by Th31, RELAT_1:42;
then consider x being Element of S such that
A2: ( x in dom f & upper_bound (rng f) = f . x ) by PARTFUN1:3, RCOMP_1:14;
take  x ; ::_thesis: ex x2 being Point of S st
( x in dom f & x2 in dom f & f /. x = upper_bound (rng f) & f /. x2 = lower_bound (rng f) )
consider y being Element of S such that
A3: ( y in dom f & lower_bound (rng f) = f . y ) by A1, PARTFUN1:3, RCOMP_1:14;
take  y ; ::_thesis: ( x in dom f & y in dom f & f /. x = upper_bound (rng f) & f /. y = lower_bound (rng f) )
thus  ( x in dom f & y in dom f & f /. x = upper_bound (rng f) & f /. y = lower_bound (rng f) ) by A2, A3, PARTFUN1:def_6; ::_thesis: verum
end;

theorem Th34: :: NFCONT_1:34
for T, S being RealNormSpace
for f being PartFunc of S,T st dom f <> {} & dom f is compact & f is_continuous_on dom f holds
ex x1, x2 being Point of S st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
proof
let T, S be RealNormSpace; ::_thesis: for f being PartFunc of S,T st dom f <> {} & dom f is compact & f is_continuous_on dom f holds
ex x1, x2 being Point of S st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
let f be PartFunc of S,T; ::_thesis: ( dom f <> {} & dom f is compact & f is_continuous_on dom f implies ex x1, x2 being Point of S st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) ) )
assume that
A1: dom f <> {} and
A2: dom f is compact and
A3: f is_continuous_on dom f ; ::_thesis: ex x1, x2 being Point of S st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
A4: dom f = dom ||.f.|| by NORMSP_0:def_3;
 dom ||.f.|| is compact by A2, NORMSP_0:def_3;
then A5: rng ||.f.|| is compact by A3, A4, Th28, Th31;
A6: rng ||.f.|| <> {} by A1, A4, RELAT_1:42;
then consider x being Element of S such that
A7: ( x in dom ||.f.|| & upper_bound (rng ||.f.||) = ||.f.|| . x ) by A5, PARTFUN1:3, RCOMP_1:14;
consider y being Element of S such that
A8: ( y in dom ||.f.|| & lower_bound (rng ||.f.||) = ||.f.|| . y ) by A6, A5, PARTFUN1:3, RCOMP_1:14;
take  x ; ::_thesis: ex x2 being Point of S st
( x in dom f & x2 in dom f & ||.f.|| /. x = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
take  y ; ::_thesis: ( x in dom f & y in dom f & ||.f.|| /. x = upper_bound (rng ||.f.||) & ||.f.|| /. y = lower_bound (rng ||.f.||) )
thus  ( x in dom f & y in dom f & ||.f.|| /. x = upper_bound (rng ||.f.||) & ||.f.|| /. y = lower_bound (rng ||.f.||) ) by A7, A8, NORMSP_0:def_3, PARTFUN1:def_6; ::_thesis: verum
end;

theorem Th35: :: NFCONT_1:35
for X being set
for S, T being RealNormSpace
for f being PartFunc of S,T holds ||.f.|| | X = ||.(f | X).||
proof
let X be set ; ::_thesis: for S, T being RealNormSpace
for f being PartFunc of S,T holds ||.f.|| | X = ||.(f | X).||
let S, T be RealNormSpace; ::_thesis: for f being PartFunc of S,T holds ||.f.|| | X = ||.(f | X).||
let f be PartFunc of S,T; ::_thesis: ||.f.|| | X = ||.(f | X).||
A1: dom (||.f.|| | X) = (dom ||.f.||) /\ X by RELAT_1:61
.= (dom f) /\ X by NORMSP_0:def_3
.= dom (f | X) by RELAT_1:61
.= dom ||.(f | X).|| by NORMSP_0:def_3 ;
now__::_thesis:_for_c_being_Point_of_S_st_c_in_dom_(||.f.||_|_X)_holds_
(||.f.||_|_X)_._c_=_||.(f_|_X).||_._c
let c be Point of S; ::_thesis: ( c in dom (||.f.|| | X) implies (||.f.|| | X) . c = ||.(f | X).|| . c )
assume A2: c in dom (||.f.|| | X) ; ::_thesis: (||.f.|| | X) . c = ||.(f | X).|| . c
then A3: c in dom (f | X) by A1, NORMSP_0:def_3;
 c in (dom ||.f.||) /\ X by A2, RELAT_1:61;
then A4: c in dom ||.f.|| by XBOOLE_0:def_4;
thus (||.f.|| | X) . c = ||.f.|| . c by A2, FUNCT_1:47
.= ||.(f /. c).|| by A4, NORMSP_0:def_3
.= ||.((f | X) /. c).|| by A3, PARTFUN2:15
.= ||.(f | X).|| . c by A1, A2, NORMSP_0:def_3 ; ::_thesis: verum
end;
hence  ||.f.|| | X = ||.(f | X).|| by A1, PARTFUN1:5; ::_thesis: verum
end;

theorem :: NFCONT_1:36
for T, S being RealNormSpace
for f being PartFunc of S,T
for Y being Subset of S st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of S st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
proof
let T, S be RealNormSpace; ::_thesis: for f being PartFunc of S,T
for Y being Subset of S st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of S st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
let f be PartFunc of S,T; ::_thesis: for Y being Subset of S st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of S st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
let Y be Subset of S; ::_thesis: ( Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y implies ex x1, x2 being Point of S st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) ) )
assume that
A1: Y <> {} and
A2: Y c= dom f and
A3: Y is compact and
A4: f is_continuous_on Y ; ::_thesis: ex x1, x2 being Point of S st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
A5: dom (f | Y) = (dom f) /\ Y by RELAT_1:61
.= Y by A2, XBOOLE_1:28 ;
 f | Y is_continuous_on Y
proof
thus  Y c= dom (f | Y) by A5; :: according to NFCONT_1:def_7 ::_thesis: for x0 being Point of S st x0 in Y holds
(f | Y) | Y is_continuous_in x0
let r be Point of S; ::_thesis: ( r in Y implies (f | Y) | Y is_continuous_in r )
assume  r in Y ; ::_thesis: (f | Y) | Y is_continuous_in r
then  f | Y is_continuous_in r by A4, Def7;
hence  (f | Y) | Y is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
then consider x1, x2 being Point of S such that
A6: x1 in dom (f | Y) and
A7: x2 in dom (f | Y) and
A8: ( ||.(f | Y).|| /. x1 = upper_bound (rng ||.(f | Y).||) & ||.(f | Y).|| /. x2 = lower_bound (rng ||.(f | Y).||) ) by A1, A3, A5, Th34;
A9: dom f = dom ||.f.|| by NORMSP_0:def_3;
take  x1 ; ::_thesis: ex x2 being Point of S st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
take  x2 ; ::_thesis: ( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
thus  ( x1 in Y & x2 in Y ) by A5, A6, A7; ::_thesis: ( ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
A10: ||.f.|| .: Y = rng (||.f.|| | Y) by RELAT_1:115
.= rng ||.(f | Y).|| by Th35 ;
A11: x2 in dom ||.(f | Y).|| by A7, NORMSP_0:def_3;
then A12: ||.(f | Y).|| /. x2 = ||.(f | Y).|| . x2 by PARTFUN1:def_6
.= ||.((f | Y) /. x2).|| by A11, NORMSP_0:def_3
.= ||.(f /. x2).|| by A7, PARTFUN2:15
.= ||.f.|| . x2 by A2, A5, A7, A9, NORMSP_0:def_3
.= ||.f.|| /. x2 by A2, A5, A7, A9, PARTFUN1:def_6 ;
A13: x1 in dom ||.(f | Y).|| by A6, NORMSP_0:def_3;
then ||.(f | Y).|| /. x1 = ||.(f | Y).|| . x1 by PARTFUN1:def_6
.= ||.((f | Y) /. x1).|| by A13, NORMSP_0:def_3
.= ||.(f /. x1).|| by A6, PARTFUN2:15
.= ||.f.|| . x1 by A2, A5, A6, A9, NORMSP_0:def_3
.= ||.f.|| /. x1 by A2, A5, A6, A9, PARTFUN1:def_6 ;
hence  ( ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) ) by A8, A12, A10; ::_thesis: verum
end;

theorem :: NFCONT_1:37
for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL
for Y being Subset of S st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of S st
( x1 in Y & x2 in Y & f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) )
proof
let S be RealNormSpace; ::_thesis: for f being PartFunc of the carrier of S,REAL
for Y being Subset of S st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of S st
( x1 in Y & x2 in Y & f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) )
let f be PartFunc of the carrier of S,REAL; ::_thesis: for Y being Subset of S st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of S st
( x1 in Y & x2 in Y & f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) )
let Y be Subset of S; ::_thesis: ( Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y implies ex x1, x2 being Point of S st
( x1 in Y & x2 in Y & f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) ) )
assume that
A1: Y <> {} and
A2: Y c= dom f and
A3: Y is compact and
A4: f is_continuous_on Y ; ::_thesis: ex x1, x2 being Point of S st
( x1 in Y & x2 in Y & f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) )
A5: dom (f | Y) = (dom f) /\ Y by RELAT_1:61
.= Y by A2, XBOOLE_1:28 ;
 f | Y is_continuous_on Y
proof
thus  Y c= dom (f | Y) by A5; :: according to NFCONT_1:def_8 ::_thesis: for x0 being Point of S st x0 in Y holds
(f | Y) | Y is_continuous_in x0
let r be Point of S; ::_thesis: ( r in Y implies (f | Y) | Y is_continuous_in r )
assume  r in Y ; ::_thesis: (f | Y) | Y is_continuous_in r
then  f | Y is_continuous_in r by A4, Def8;
hence  (f | Y) | Y is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
then consider x1, x2 being Point of S such that
A6: ( x1 in dom (f | Y) & x2 in dom (f | Y) ) and
A7: ( (f | Y) /. x1 = upper_bound (rng (f | Y)) & (f | Y) /. x2 = lower_bound (rng (f | Y)) ) by A1, A3, A5, Th33;
take  x1 ; ::_thesis: ex x2 being Point of S st
( x1 in Y & x2 in Y & f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) )
take  x2 ; ::_thesis: ( x1 in Y & x2 in Y & f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) )
thus  ( x1 in Y & x2 in Y ) by A5, A6; ::_thesis: ( f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) )
 ( (f | Y) /. x1 = f /. x1 & (f | Y) /. x2 = f /. x2 ) by A6, PARTFUN2:15;
hence  ( f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) ) by A7, RELAT_1:115; ::_thesis: verum
end;

definition
let S, T be RealNormSpace;
let X be set ;
let f be PartFunc of S,T;
predf is_Lipschitzian_on X means :Def9: :: NFCONT_1:def 9
( X c= dom f & ex r being Real st
( 0 < r & ( for x1, x2 being Point of S st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| ) ) );
end;

:: deftheorem Def9 defines is_Lipschitzian_on NFCONT_1:def_9_:_
 for S, T being RealNormSpace
for X being set
for f being PartFunc of S,T holds
( f is_Lipschitzian_on X iff ( X c= dom f & ex r being Real st
( 0 < r & ( for x1, x2 being Point of S st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| ) ) ) );

definition
let S be RealNormSpace;
let X be set ;
let f be PartFunc of the carrier of S,REAL;
predf is_Lipschitzian_on X means :Def10: :: NFCONT_1:def 10
( X c= dom f & ex r being Real st
( 0 < r & ( for x1, x2 being Point of S st x1 in X & x2 in X holds
abs ((f /. x1) - (f /. x2)) <= r * ||.(x1 - x2).|| ) ) );
end;

:: deftheorem Def10 defines is_Lipschitzian_on NFCONT_1:def_10_:_
 for S being RealNormSpace
for X being set
for f being PartFunc of the carrier of S,REAL holds
( f is_Lipschitzian_on X iff ( X c= dom f & ex r being Real st
( 0 < r & ( for x1, x2 being Point of S st x1 in X & x2 in X holds
abs ((f /. x1) - (f /. x2)) <= r * ||.(x1 - x2).|| ) ) ) );

theorem Th38: :: NFCONT_1:38
for X, X1 being set
for S, T being RealNormSpace
for f being PartFunc of S,T st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
proof
let X, X1 be set ; ::_thesis: for S, T being RealNormSpace
for f being PartFunc of S,T st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
let S, T be RealNormSpace; ::_thesis: for f being PartFunc of S,T st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
let f be PartFunc of S,T; ::_thesis: ( f is_Lipschitzian_on X & X1 c= X implies f is_Lipschitzian_on X1 )
assume that
A1: f is_Lipschitzian_on X and
A2: X1 c= X ; ::_thesis: f is_Lipschitzian_on X1
 X c= dom f by A1, Def9;
hence  X1 c= dom f by A2, XBOOLE_1:1; :: according to NFCONT_1:def_9 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of S st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| ) )
consider s being Real such that
A3: 0 < s and
A4: for x1, x2 being Point of S st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by A1, Def9;
take  s ; ::_thesis: ( 0 < s & ( for x1, x2 being Point of S st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| ) )
thus  0 < s by A3; ::_thesis: for x1, x2 being Point of S st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).||
let x1, x2 be Point of S; ::_thesis: ( x1 in X1 & x2 in X1 implies ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| )
assume  ( x1 in X1 & x2 in X1 ) ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).||
hence  ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by A2, A4; ::_thesis: verum
end;

theorem :: NFCONT_1:39
for X, X1 being set
for S, T being RealNormSpace
for f1, f2 being PartFunc of S,T st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 + f2 is_Lipschitzian_on X /\ X1
proof
let X, X1 be set ; ::_thesis: for S, T being RealNormSpace
for f1, f2 being PartFunc of S,T st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 + f2 is_Lipschitzian_on X /\ X1
let S, T be RealNormSpace; ::_thesis: for f1, f2 being PartFunc of S,T st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 + f2 is_Lipschitzian_on X /\ X1
let f1, f2 be PartFunc of S,T; ::_thesis: ( f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 implies f1 + f2 is_Lipschitzian_on X /\ X1 )
assume that
A1: f1 is_Lipschitzian_on X and
A2: f2 is_Lipschitzian_on X1 ; ::_thesis: f1 + f2 is_Lipschitzian_on X /\ X1
A3: f1 is_Lipschitzian_on X /\ X1 by A1, Th38, XBOOLE_1:17;
then consider s being Real such that
A4: 0 < s and
A5: for x1, x2 being Point of S st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by Def9;
A6: f2 is_Lipschitzian_on X /\ X1 by A2, Th38, XBOOLE_1:17;
then A7: X /\ X1 c= dom f2 by Def9;
 X /\ X1 c= dom f1 by A3, Def9;
then  X /\ X1 c= (dom f1) /\ (dom f2) by A7, XBOOLE_1:19;
hence A8: X /\ X1 c= dom (f1 + f2) by VFUNCT_1:def_1; :: according to NFCONT_1:def_9 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of S st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
consider g being Real such that
A9: 0 < g and
A10: for x1, x2 being Point of S st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A6, Def9;
take p = s + g; ::_thesis: ( 0 < p & ( for x1, x2 being Point of S st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).|| ) )
 0 + 0 < s + g by A4, A9, XREAL_1:8;
hence  0 < p ; ::_thesis: for x1, x2 being Point of S st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).||
let x1, x2 be Point of S; ::_thesis: ( x1 in X /\ X1 & x2 in X /\ X1 implies ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).|| )
assume that
A11: x1 in X /\ X1 and
A12: x2 in X /\ X1 ; ::_thesis: ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).||
A13: ||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A10, A11, A12;
 ||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by A5, A11, A12;
then A14: ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| <= (s * ||.(x1 - x2).||) + (g * ||.(x1 - x2).||) by A13, XREAL_1:7;
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| = ||.(((f1 /. x1) + (f2 /. x1)) - ((f1 + f2) /. x2)).|| by A8, A11, VFUNCT_1:def_1
.= ||.(((f1 /. x1) + (f2 /. x1)) - ((f1 /. x2) + (f2 /. x2))).|| by A8, A12, VFUNCT_1:def_1
.= ||.((f1 /. x1) + ((f2 /. x1) - ((f1 /. x2) + (f2 /. x2)))).|| by RLVECT_1:def_3
.= ||.((f1 /. x1) + (((f2 /. x1) - (f1 /. x2)) - (f2 /. x2))).|| by RLVECT_1:27
.= ||.((f1 /. x1) + (((- (f1 /. x2)) + (f2 /. x1)) - (f2 /. x2))).||
.= ||.((f1 /. x1) + ((- (f1 /. x2)) + ((f2 /. x1) - (f2 /. x2)))).|| by RLVECT_1:def_3
.= ||.(((f1 /. x1) - (f1 /. x2)) + ((f2 /. x1) - (f2 /. x2))).|| by RLVECT_1:def_3 ;
then  ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| by NORMSP_1:def_1;
hence  ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).|| by A14, XXREAL_0:2; ::_thesis: verum
end;

theorem :: NFCONT_1:40
for X, X1 being set
for S, T being RealNormSpace
for f1, f2 being PartFunc of S,T st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 - f2 is_Lipschitzian_on X /\ X1
proof
let X, X1 be set ; ::_thesis: for S, T being RealNormSpace
for f1, f2 being PartFunc of S,T st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 - f2 is_Lipschitzian_on X /\ X1
let S, T be RealNormSpace; ::_thesis: for f1, f2 being PartFunc of S,T st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 - f2 is_Lipschitzian_on X /\ X1
let f1, f2 be PartFunc of S,T; ::_thesis: ( f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 implies f1 - f2 is_Lipschitzian_on X /\ X1 )
assume that
A1: f1 is_Lipschitzian_on X and
A2: f2 is_Lipschitzian_on X1 ; ::_thesis: f1 - f2 is_Lipschitzian_on X /\ X1
A3: f1 is_Lipschitzian_on X /\ X1 by A1, Th38, XBOOLE_1:17;
then consider s being Real such that
A4: 0 < s and
A5: for x1, x2 being Point of S st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by Def9;
A6: f2 is_Lipschitzian_on X /\ X1 by A2, Th38, XBOOLE_1:17;
then A7: X /\ X1 c= dom f2 by Def9;
 X /\ X1 c= dom f1 by A3, Def9;
then  X /\ X1 c= (dom f1) /\ (dom f2) by A7, XBOOLE_1:19;
hence A8: X /\ X1 c= dom (f1 - f2) by VFUNCT_1:def_2; :: according to NFCONT_1:def_9 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of S st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
consider g being Real such that
A9: 0 < g and
A10: for x1, x2 being Point of S st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A6, Def9;
take p = s + g; ::_thesis: ( 0 < p & ( for x1, x2 being Point of S st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).|| ) )
 0 + 0 < s + g by A4, A9, XREAL_1:8;
hence  0 < p ; ::_thesis: for x1, x2 being Point of S st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).||
let x1, x2 be Point of S; ::_thesis: ( x1 in X /\ X1 & x2 in X /\ X1 implies ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).|| )
assume that
A11: x1 in X /\ X1 and
A12: x2 in X /\ X1 ; ::_thesis: ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).||
A13: ||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A10, A11, A12;
 ||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by A5, A11, A12;
then A14: ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| <= (s * ||.(x1 - x2).||) + (g * ||.(x1 - x2).||) by A13, XREAL_1:7;
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| = ||.(((f1 /. x1) - (f2 /. x1)) - ((f1 - f2) /. x2)).|| by A8, A11, VFUNCT_1:def_2
.= ||.(((f1 /. x1) - (f2 /. x1)) - ((f1 /. x2) - (f2 /. x2))).|| by A8, A12, VFUNCT_1:def_2
.= ||.((f1 /. x1) - ((f2 /. x1) + ((f1 /. x2) - (f2 /. x2)))).|| by RLVECT_1:27
.= ||.((f1 /. x1) - (((f1 /. x2) + (f2 /. x1)) - (f2 /. x2))).|| by RLVECT_1:def_3
.= ||.((f1 /. x1) - ((f1 /. x2) + ((f2 /. x1) - (f2 /. x2)))).|| by RLVECT_1:def_3
.= ||.(((f1 /. x1) - (f1 /. x2)) - ((f2 /. x1) - (f2 /. x2))).|| by RLVECT_1:27 ;
then  ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| by NORMSP_1:3;
hence  ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).|| by A14, XXREAL_0:2; ::_thesis: verum
end;

theorem Th41: :: NFCONT_1:41
for X being set
for p being Real
for S, T being RealNormSpace
for f being PartFunc of S,T st f is_Lipschitzian_on X holds
p (#) f is_Lipschitzian_on X
proof
let X be set ; ::_thesis: for p being Real
for S, T being RealNormSpace
for f being PartFunc of S,T st f is_Lipschitzian_on X holds
p (#) f is_Lipschitzian_on X
let p be Real; ::_thesis: for S, T being RealNormSpace
for f being PartFunc of S,T st f is_Lipschitzian_on X holds
p (#) f is_Lipschitzian_on X
let S, T be RealNormSpace; ::_thesis: for f being PartFunc of S,T st f is_Lipschitzian_on X holds
p (#) f is_Lipschitzian_on X
let f be PartFunc of S,T; ::_thesis: ( f is_Lipschitzian_on X implies p (#) f is_Lipschitzian_on X )
assume A1: f is_Lipschitzian_on X ; ::_thesis: p (#) f is_Lipschitzian_on X
then consider s being Real such that
A2: 0 < s and
A3: for x1, x2 being Point of S st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by Def9;
 X c= dom f by A1, Def9;
hence A4: X c= dom (p (#) f) by VFUNCT_1:def_4; :: according to NFCONT_1:def_9 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of S st x1 in X & x2 in X holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
now__::_thesis:_ex_s_being_Real_st_
(_0_<_s_&_(_for_x1,_x2_being_Point_of_S_st_x1_in_X_&_x2_in_X_holds_
||.(((p_(#)_f)_/._x1)_-_((p_(#)_f)_/._x2)).||_<=_s_*_||.(x1_-_x2).||_)_)
percases ( p = 0 or p <> 0 ) ;
supposeA5: p = 0 ; ::_thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| ) )
take s = s; ::_thesis: ( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| ) )
thus  0 < s by A2; ::_thesis: for x1, x2 being Point of S st x1 in X & x2 in X holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= s * ||.(x1 - x2).||
let x1, x2 be Point of S; ::_thesis: ( x1 in X & x2 in X implies ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| )
assume that
A6: x1 in X and
A7: x2 in X ; ::_thesis: ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= s * ||.(x1 - x2).||
 0 <= ||.(x1 - x2).|| by NORMSP_1:4;
then A8: s * 0 <= s * ||.(x1 - x2).|| by A2, XREAL_1:64;
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| = ||.((p * (f /. x1)) - ((p (#) f) /. x2)).|| by A4, A6, VFUNCT_1:def_4
.= ||.((0. T) - ((p (#) f) /. x2)).|| by A5, RLVECT_1:10
.= ||.((0. T) - (p * (f /. x2))).|| by A4, A7, VFUNCT_1:def_4
.= ||.((0. T) - (0. T)).|| by A5, RLVECT_1:10
.= ||.(0. T).|| by RLVECT_1:13
.= 0 by NORMSP_1:1 ;
hence  ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| by A8; ::_thesis: verum
end;
supposeA9: p <> 0 ; ::_thesis: ex g being Element of REAL st
( 0 < g & ( for x1, x2 being Point of S st x1 in X & x2 in X holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| ) )
take g = (abs p) * s; ::_thesis: ( 0 < g & ( for x1, x2 being Point of S st x1 in X & x2 in X holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| ) )
 0 < abs p by A9, COMPLEX1:47;
then  0 * s < (abs p) * s by A2, XREAL_1:68;
hence  0 < g ; ::_thesis: for x1, x2 being Point of S st x1 in X & x2 in X holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= g * ||.(x1 - x2).||
let x1, x2 be Point of S; ::_thesis: ( x1 in X & x2 in X implies ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| )
assume that
A10: x1 in X and
A11: x2 in X ; ::_thesis: ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= g * ||.(x1 - x2).||
 0 <= abs p by COMPLEX1:46;
then A12: (abs p) * ||.((f /. x1) - (f /. x2)).|| <= (abs p) * (s * ||.(x1 - x2).||) by A3, A10, A11, XREAL_1:64;
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| = ||.((p * (f /. x1)) - ((p (#) f) /. x2)).|| by A4, A10, VFUNCT_1:def_4
.= ||.((p * (f /. x1)) - (p * (f /. x2))).|| by A4, A11, VFUNCT_1:def_4
.= ||.(p * ((f /. x1) - (f /. x2))).|| by RLVECT_1:34
.= (abs p) * ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:def_1 ;
hence  ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| by A12; ::_thesis: verum
end;
end;
end;
hence  ex r being Real st
( 0 < r & ( for x1, x2 being Point of S st x1 in X & x2 in X holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= r * ||.(x1 - x2).|| ) ) ; ::_thesis: verum
end;

theorem :: NFCONT_1:42
for X being set
for S, T being RealNormSpace
for f being PartFunc of S,T st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
proof
let X be set ; ::_thesis: for S, T being RealNormSpace
for f being PartFunc of S,T st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
let S, T be RealNormSpace; ::_thesis: for f being PartFunc of S,T st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
let f be PartFunc of S,T; ::_thesis: ( f is_Lipschitzian_on X implies ( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X ) )
assume A1: f is_Lipschitzian_on X ; ::_thesis: ( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
then consider s being Real such that
A2: 0 < s and
A3: for x1, x2 being Point of S st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by Def9;
 - f = (- 1) (#) f by VFUNCT_1:23;
hence  - f is_Lipschitzian_on X by A1, Th41; ::_thesis: ||.f.|| is_Lipschitzian_on X
 X c= dom f by A1, Def9;
hence A4: X c= dom ||.f.|| by NORMSP_0:def_3; :: according to NFCONT_1:def_10 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of S st x1 in X & x2 in X holds
abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= r * ||.(x1 - x2).|| ) )
take  s ; ::_thesis: ( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X holds
abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).|| ) )
thus  0 < s by A2; ::_thesis: for x1, x2 being Point of S st x1 in X & x2 in X holds
abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).||
let x1, x2 be Point of S; ::_thesis: ( x1 in X & x2 in X implies abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).|| )
assume that
A5: x1 in X and
A6: x2 in X ; ::_thesis: abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).||
abs ((||.f.|| /. x1) - (||.f.|| /. x2)) = abs ((||.f.|| . x1) - (||.f.|| /. x2)) by A4, A5, PARTFUN1:def_6
.= abs ((||.f.|| . x1) - (||.f.|| . x2)) by A4, A6, PARTFUN1:def_6
.= abs (||.(f /. x1).|| - (||.f.|| . x2)) by A4, A5, NORMSP_0:def_3
.= abs (||.(f /. x1).|| - ||.(f /. x2).||) by A4, A6, NORMSP_0:def_3 ;
then A7: abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:9;
 ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by A3, A5, A6;
hence  abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).|| by A7, XXREAL_0:2; ::_thesis: verum
end;

theorem Th43: :: NFCONT_1:43
for X being set
for S, T being RealNormSpace
for f being PartFunc of S,T st X c= dom f & f | X is V23() holds
f is_Lipschitzian_on X
proof
let X be set ; ::_thesis: for S, T being RealNormSpace
for f being PartFunc of S,T st X c= dom f & f | X is V23() holds
f is_Lipschitzian_on X
let S, T be RealNormSpace; ::_thesis: for f being PartFunc of S,T st X c= dom f & f | X is V23() holds
f is_Lipschitzian_on X
let f be PartFunc of S,T; ::_thesis: ( X c= dom f & f | X is V23() implies f is_Lipschitzian_on X )
assume that
A1: X c= dom f and
A2: f | X is V23() ; ::_thesis: f is_Lipschitzian_on X
now__::_thesis:_for_x1,_x2_being_Point_of_S_st_x1_in_X_&_x2_in_X_holds_
||.((f_/._x1)_-_(f_/._x2)).||_<=_1_*_||.(x1_-_x2).||
let x1, x2 be Point of S; ::_thesis: ( x1 in X & x2 in X implies ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| )
assume that
A3: x1 in X and
A4: x2 in X ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).||
A5: ( x1 in X /\ (dom f) & x2 in X /\ (dom f) ) by A1, A3, A4, XBOOLE_0:def_4;
f /. x1 = f . x1 by A1, A3, PARTFUN1:def_6
.= f . x2 by A2, A5, PARTFUN2:58
.= f /. x2 by A1, A4, PARTFUN1:def_6 ;
then ||.((f /. x1) - (f /. x2)).|| = ||.(0. T).|| by RLVECT_1:15
.= 0 by NORMSP_1:1 ;
hence  ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| by NORMSP_1:4; ::_thesis: verum
end;
hence  f is_Lipschitzian_on X by A1, Def9; ::_thesis: verum
end;

theorem :: NFCONT_1:44
for S being RealNormSpace
for Y being Subset of S holds id Y is_Lipschitzian_on Y
proof
let S be RealNormSpace; ::_thesis: for Y being Subset of S holds id Y is_Lipschitzian_on Y
let Y be Subset of S; ::_thesis: id Y is_Lipschitzian_on Y
reconsider r = 1 as Real ;
thus  Y c= dom (id Y) by RELAT_1:45; :: according to NFCONT_1:def_9 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of S st x1 in Y & x2 in Y holds
||.(((id Y) /. x1) - ((id Y) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
take  r ; ::_thesis: ( 0 < r & ( for x1, x2 being Point of S st x1 in Y & x2 in Y holds
||.(((id Y) /. x1) - ((id Y) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
thus  r > 0 ; ::_thesis: for x1, x2 being Point of S st x1 in Y & x2 in Y holds
||.(((id Y) /. x1) - ((id Y) /. x2)).|| <= r * ||.(x1 - x2).||
let x1, x2 be Point of S; ::_thesis: ( x1 in Y & x2 in Y implies ||.(((id Y) /. x1) - ((id Y) /. x2)).|| <= r * ||.(x1 - x2).|| )
assume that
A1: x1 in Y and
A2: x2 in Y ; ::_thesis: ||.(((id Y) /. x1) - ((id Y) /. x2)).|| <= r * ||.(x1 - x2).||
||.(((id Y) /. x1) - ((id Y) /. x2)).|| = ||.(x1 - ((id Y) /. x2)).|| by A1, PARTFUN2:6
.= r * ||.(x1 - x2).|| by A2, PARTFUN2:6 ;
hence  ||.(((id Y) /. x1) - ((id Y) /. x2)).|| <= r * ||.(x1 - x2).|| ; ::_thesis: verum
end;

theorem Th45: :: NFCONT_1:45
for X being set
for S, T being RealNormSpace
for f being PartFunc of S,T st f is_Lipschitzian_on X holds
f is_continuous_on X
proof
let X be set ; ::_thesis: for S, T being RealNormSpace
for f being PartFunc of S,T st f is_Lipschitzian_on X holds
f is_continuous_on X
let S, T be RealNormSpace; ::_thesis: for f being PartFunc of S,T st f is_Lipschitzian_on X holds
f is_continuous_on X
let f be PartFunc of S,T; ::_thesis: ( f is_Lipschitzian_on X implies f is_continuous_on X )
assume A1: f is_Lipschitzian_on X ; ::_thesis: f is_continuous_on X
then consider r being Real such that
A2: 0 < r and
A3: for x1, x2 being Point of S st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| by Def9;
A4: X c= dom f by A1, Def9;
then A5: dom (f | X) = X by RELAT_1:62;
now__::_thesis:_for_x0_being_Point_of_S_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Point of S; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A6: x0 in X ; ::_thesis: f | X is_continuous_in x0
now__::_thesis:_for_g_being_Real_st_0_<_g_holds_
ex_s9_being_Element_of_REAL_st_
(_0_<_s9_&_(_for_x1_being_Point_of_S_st_x1_in_dom_(f_|_X)_&_||.(x1_-_x0).||_<_s9_holds_
||.(((f_|_X)_/._x1)_-_((f_|_X)_/._x0)).||_<_g_)_)
let g be Real; ::_thesis: ( 0 < g implies ex s9 being Element of REAL st
( 0 < s9 & ( for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g ) ) )
assume A7: 0 < g ; ::_thesis: ex s9 being Element of REAL st
( 0 < s9 & ( for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g ) )
set s = g / r;
take s9 = g / r; ::_thesis: ( 0 < s9 & ( for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g ) )
A8: now__::_thesis:_for_x1_being_Point_of_S_st_x1_in_dom_(f_|_X)_&_||.(x1_-_x0).||_<_g_/_r_holds_
||.(((f_|_X)_/._x1)_-_((f_|_X)_/._x0)).||_<_g
let x1 be Point of S; ::_thesis: ( x1 in dom (f | X) & ||.(x1 - x0).|| < g / r implies ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g )
assume that
A9: x1 in dom (f | X) and
A10: ||.(x1 - x0).|| < g / r ; ::_thesis: ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g
 r * ||.(x1 - x0).|| < (g / r) * r by A2, A10, XREAL_1:68;
then A11: r * ||.(x1 - x0).|| < g by A2, XCMPLX_1:87;
 ||.((f /. x1) - (f /. x0)).|| <= r * ||.(x1 - x0).|| by A3, A5, A6, A9;
then  ||.((f /. x1) - (f /. x0)).|| < g by A11, XXREAL_0:2;
then  ||.(((f | X) /. x1) - (f /. x0)).|| < g by A9, PARTFUN2:15;
hence  ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g by A5, A6, PARTFUN2:15; ::_thesis: verum
end;
 ( 0 < r " & s9 = g * (r ") ) by A2, XCMPLX_0:def_9, XREAL_1:122;
hence  ( 0 < s9 & ( for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g ) ) by A7, A8, XREAL_1:129; ::_thesis: verum
end;
hence  f | X is_continuous_in x0 by A5, A6, Th7; ::_thesis: verum
end;
hence  f is_continuous_on X by A4, Def7; ::_thesis: verum
end;

theorem Th46: :: NFCONT_1:46
for X being set
for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL st f is_Lipschitzian_on X holds
f is_continuous_on X
proof
let X be set ; ::_thesis: for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL st f is_Lipschitzian_on X holds
f is_continuous_on X
let S be RealNormSpace; ::_thesis: for f being PartFunc of the carrier of S,REAL st f is_Lipschitzian_on X holds
f is_continuous_on X
let f be PartFunc of the carrier of S,REAL; ::_thesis: ( f is_Lipschitzian_on X implies f is_continuous_on X )
assume A1: f is_Lipschitzian_on X ; ::_thesis: f is_continuous_on X
then consider r being Real such that
A2: 0 < r and
A3: for x1, x2 being Point of S st x1 in X & x2 in X holds
abs ((f /. x1) - (f /. x2)) <= r * ||.(x1 - x2).|| by Def10;
A4: X c= dom f by A1, Def10;
then A5: dom (f | X) = X by RELAT_1:62;
now__::_thesis:_for_x0_being_Point_of_S_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Point of S; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A6: x0 in X ; ::_thesis: f | X is_continuous_in x0
now__::_thesis:_for_g_being_Real_st_0_<_g_holds_
ex_s9_being_Element_of_REAL_st_
(_0_<_s9_&_(_for_x1_being_Point_of_S_st_x1_in_dom_(f_|_X)_&_||.(x1_-_x0).||_<_s9_holds_
abs_(((f_|_X)_/._x1)_-_((f_|_X)_/._x0))_<_g_)_)
let g be Real; ::_thesis: ( 0 < g implies ex s9 being Element of REAL st
( 0 < s9 & ( for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < g ) ) )
assume A7: 0 < g ; ::_thesis: ex s9 being Element of REAL st
( 0 < s9 & ( for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < g ) )
set s = g / r;
take s9 = g / r; ::_thesis: ( 0 < s9 & ( for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < g ) )
A8: now__::_thesis:_for_x1_being_Point_of_S_st_x1_in_dom_(f_|_X)_&_||.(x1_-_x0).||_<_g_/_r_holds_
abs_(((f_|_X)_/._x1)_-_((f_|_X)_/._x0))_<_g
let x1 be Point of S; ::_thesis: ( x1 in dom (f | X) & ||.(x1 - x0).|| < g / r implies abs (((f | X) /. x1) - ((f | X) /. x0)) < g )
assume that
A9: x1 in dom (f | X) and
A10: ||.(x1 - x0).|| < g / r ; ::_thesis: abs (((f | X) /. x1) - ((f | X) /. x0)) < g
 r * ||.(x1 - x0).|| < (g / r) * r by A2, A10, XREAL_1:68;
then A11: r * ||.(x1 - x0).|| < g by A2, XCMPLX_1:87;
 abs ((f /. x1) - (f /. x0)) <= r * ||.(x1 - x0).|| by A3, A5, A6, A9;
then  abs ((f /. x1) - (f /. x0)) < g by A11, XXREAL_0:2;
then  abs (((f | X) /. x1) - (f /. x0)) < g by A9, PARTFUN2:15;
hence  abs (((f | X) /. x1) - ((f | X) /. x0)) < g by A5, A6, PARTFUN2:15; ::_thesis: verum
end;
 ( 0 < r " & s9 = g * (r ") ) by A2, XCMPLX_0:def_9, XREAL_1:122;
hence  ( 0 < s9 & ( for x1 being Point of S st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < g ) ) by A7, A8, XREAL_1:129; ::_thesis: verum
end;
hence  f | X is_continuous_in x0 by A5, A6, Th8; ::_thesis: verum
end;
hence  f is_continuous_on X by A4, Def8; ::_thesis: verum
end;

theorem :: NFCONT_1:47
for T, S being RealNormSpace
for f being PartFunc of S,T st ex r being Point of T st rng f = {r} holds
f is_continuous_on dom f
proof
let T, S be RealNormSpace; ::_thesis: for f being PartFunc of S,T st ex r being Point of T st rng f = {r} holds
f is_continuous_on dom f
let f be PartFunc of S,T; ::_thesis: ( ex r being Point of T st rng f = {r} implies f is_continuous_on dom f )
given r being Point of T such that A1: rng f = {r} ; ::_thesis: f is_continuous_on dom f
now__::_thesis:_for_x1,_x2_being_Point_of_S_st_x1_in_dom_f_&_x2_in_dom_f_holds_
||.((f_/._x1)_-_(f_/._x2)).||_<=_1_*_||.(x1_-_x2).||
let x1, x2 be Point of S; ::_thesis: ( x1 in dom f & x2 in dom f implies ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| )
assume that
A2: x1 in dom f and
A3: x2 in dom f ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).||
 f . x2 in rng f by A3, FUNCT_1:def_3;
then  f /. x2 in rng f by A3, PARTFUN1:def_6;
then A4: f /. x2 = r by A1, TARSKI:def_1;
 f . x1 in rng f by A2, FUNCT_1:def_3;
then  f /. x1 in rng f by A2, PARTFUN1:def_6;
then  f /. x1 = r by A1, TARSKI:def_1;
then ||.((f /. x1) - (f /. x2)).|| = ||.(0. T).|| by A4, RLVECT_1:15
.= 0 by NORMSP_1:1 ;
hence  ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| by NORMSP_1:4; ::_thesis: verum
end;
then  f is_Lipschitzian_on dom f by Def9;
hence  f is_continuous_on dom f by Th45; ::_thesis: verum
end;

theorem :: NFCONT_1:48
for X being set
for S, T being RealNormSpace
for f being PartFunc of S,T st X c= dom f & f | X is V23() holds
f is_continuous_on X by Th43, Th45;

theorem Th49: :: NFCONT_1:49
for S being RealNormSpace
for f being PartFunc of S,S st ( for x0 being Point of S st x0 in dom f holds
f /. x0 = x0 ) holds
f is_continuous_on dom f
proof
let S be RealNormSpace; ::_thesis: for f being PartFunc of S,S st ( for x0 being Point of S st x0 in dom f holds
f /. x0 = x0 ) holds
f is_continuous_on dom f
let f be PartFunc of S,S; ::_thesis: ( ( for x0 being Point of S st x0 in dom f holds
f /. x0 = x0 ) implies f is_continuous_on dom f )
assume A1: for x0 being Point of S st x0 in dom f holds
f /. x0 = x0 ; ::_thesis: f is_continuous_on dom f
now__::_thesis:_for_x1,_x2_being_Point_of_S_st_x1_in_dom_f_&_x2_in_dom_f_holds_
||.((f_/._x1)_-_(f_/._x2)).||_<=_1_*_||.(x1_-_x2).||
let x1, x2 be Point of S; ::_thesis: ( x1 in dom f & x2 in dom f implies ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| )
assume that
A2: x1 in dom f and
A3: x2 in dom f ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).||
 f /. x1 = x1 by A1, A2;
hence  ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| by A1, A3; ::_thesis: verum
end;
then  f is_Lipschitzian_on dom f by Def9;
hence  f is_continuous_on dom f by Th45; ::_thesis: verum
end;

theorem :: NFCONT_1:50
for S being RealNormSpace
for f being PartFunc of S,S st f = id (dom f) holds
f is_continuous_on dom f
proof
let S be RealNormSpace; ::_thesis: for f being PartFunc of S,S st f = id (dom f) holds
f is_continuous_on dom f
let f be PartFunc of S,S; ::_thesis: ( f = id (dom f) implies f is_continuous_on dom f )
assume A1: f = id (dom f) ; ::_thesis: f is_continuous_on dom f
now__::_thesis:_for_x0_being_Point_of_S_st_x0_in_dom_f_holds_
f_/._x0_=_x0
let x0 be Point of S; ::_thesis: ( x0 in dom f implies f /. x0 = x0 )
assume A2: x0 in dom f ; ::_thesis: f /. x0 = x0
thus f /. x0 = f . x0 by A2, PARTFUN1:def_6
.= x0 by A1, A2, FUNCT_1:17 ; ::_thesis: verum
end;
hence  f is_continuous_on dom f by Th49; ::_thesis: verum
end;

theorem :: NFCONT_1:51
for S being RealNormSpace
for Y being Subset of S
for f being PartFunc of S,S st Y c= dom f & f | Y = id Y holds
f is_continuous_on Y
proof
let S be RealNormSpace; ::_thesis: for Y being Subset of S
for f being PartFunc of S,S st Y c= dom f & f | Y = id Y holds
f is_continuous_on Y
let Y be Subset of S; ::_thesis: for f being PartFunc of S,S st Y c= dom f & f | Y = id Y holds
f is_continuous_on Y
let f be PartFunc of S,S; ::_thesis: ( Y c= dom f & f | Y = id Y implies f is_continuous_on Y )
assume that
A1: Y c= dom f and
A2: f | Y = id Y ; ::_thesis: f is_continuous_on Y
now__::_thesis:_for_x1,_x2_being_Point_of_S_st_x1_in_Y_&_x2_in_Y_holds_
||.((f_/._x1)_-_(f_/._x2)).||_<=_1_*_||.(x1_-_x2).||
let x1, x2 be Point of S; ::_thesis: ( x1 in Y & x2 in Y implies ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| )
assume that
A3: x1 in Y and
A4: x2 in Y ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).||
 x1 in (dom f) /\ Y by A1, A3, XBOOLE_0:def_4;
then A5: x1 in dom (f | Y) by RELAT_1:61;
 (f | Y) . x1 = x1 by A2, A3, FUNCT_1:17;
then  f . x1 = x1 by A5, FUNCT_1:47;
then A6: f /. x1 = x1 by A1, A3, PARTFUN1:def_6;
 x2 in (dom f) /\ Y by A1, A4, XBOOLE_0:def_4;
then A7: x2 in dom (f | Y) by RELAT_1:61;
 (f | Y) . x2 = x2 by A2, A4, FUNCT_1:17;
then  f . x2 = x2 by A7, FUNCT_1:47;
hence  ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| by A1, A4, A6, PARTFUN1:def_6; ::_thesis: verum
end;
then  f is_Lipschitzian_on Y by A1, Def9;
hence  f is_continuous_on Y by Th45; ::_thesis: verum
end;

theorem :: NFCONT_1:52
for X being set
for S being RealNormSpace
for f being PartFunc of S,S
for r being Real
for p being Point of S st X c= dom f & ( for x0 being Point of S st x0 in X holds
f /. x0 = (r * x0) + p ) holds
f is_continuous_on X
proof
let X be set ; ::_thesis: for S being RealNormSpace
for f being PartFunc of S,S
for r being Real
for p being Point of S st X c= dom f & ( for x0 being Point of S st x0 in X holds
f /. x0 = (r * x0) + p ) holds
f is_continuous_on X
let S be RealNormSpace; ::_thesis: for f being PartFunc of S,S
for r being Real
for p being Point of S st X c= dom f & ( for x0 being Point of S st x0 in X holds
f /. x0 = (r * x0) + p ) holds
f is_continuous_on X
let f be PartFunc of S,S; ::_thesis: for r being Real
for p being Point of S st X c= dom f & ( for x0 being Point of S st x0 in X holds
f /. x0 = (r * x0) + p ) holds
f is_continuous_on X
let r be Real; ::_thesis: for p being Point of S st X c= dom f & ( for x0 being Point of S st x0 in X holds
f /. x0 = (r * x0) + p ) holds
f is_continuous_on X
let p be Point of S; ::_thesis: ( X c= dom f & ( for x0 being Point of S st x0 in X holds
f /. x0 = (r * x0) + p ) implies f is_continuous_on X )
assume that
A1: X c= dom f and
A2: for x0 being Point of S st x0 in X holds
f /. x0 = (r * x0) + p ; ::_thesis: f is_continuous_on X
now__::_thesis:_(_0_<_(abs_r)_+_1_&_(_for_x1,_x2_being_Point_of_S_st_x1_in_X_&_x2_in_X_holds_
||.((f_/._x1)_-_(f_/._x2)).||_<=_((abs_r)_+_1)_*_||.(x1_-_x2).||_)_)
 0 + 0 < (abs r) + 1 by COMPLEX1:46, XREAL_1:8;
hence  0 < (abs r) + 1 ; ::_thesis: for x1, x2 being Point of S st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= ((abs r) + 1) * ||.(x1 - x2).||
let x1, x2 be Point of S; ::_thesis: ( x1 in X & x2 in X implies ||.((f /. x1) - (f /. x2)).|| <= ((abs r) + 1) * ||.(x1 - x2).|| )
assume  ( x1 in X & x2 in X ) ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= ((abs r) + 1) * ||.(x1 - x2).||
then  ( f /. x1 = (r * x1) + p & f /. x2 = (r * x2) + p ) by A2;
then A3: ||.((f /. x1) - (f /. x2)).|| = ||.((r * x1) + (p - (p + (r * x2)))).|| by RLVECT_1:def_3
.= ||.((r * x1) + ((p - p) - (r * x2))).|| by RLVECT_1:27
.= ||.((r * x1) + ((0. S) - (r * x2))).|| by RLVECT_1:15
.= ||.((r * x1) - (r * x2)).|| by RLVECT_1:14
.= ||.(r * (x1 - x2)).|| by RLVECT_1:34
.= (abs r) * ||.(x1 - x2).|| by NORMSP_1:def_1 ;
 0 <= ||.(x1 - x2).|| by NORMSP_1:4;
then  ||.((f /. x1) - (f /. x2)).|| + 0 <= ((abs r) * ||.(x1 - x2).||) + (1 * ||.(x1 - x2).||) by A3, XREAL_1:7;
hence  ||.((f /. x1) - (f /. x2)).|| <= ((abs r) + 1) * ||.(x1 - x2).|| ; ::_thesis: verum
end;
then  f is_Lipschitzian_on X by A1, Def9;
hence  f is_continuous_on X by Th45; ::_thesis: verum
end;

theorem Th53: :: NFCONT_1:53
for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL st ( for x0 being Point of S st x0 in dom f holds
f /. x0 = ||.x0.|| ) holds
f is_continuous_on dom f
proof
let S be RealNormSpace; ::_thesis: for f being PartFunc of the carrier of S,REAL st ( for x0 being Point of S st x0 in dom f holds
f /. x0 = ||.x0.|| ) holds
f is_continuous_on dom f
let f be PartFunc of the carrier of S,REAL; ::_thesis: ( ( for x0 being Point of S st x0 in dom f holds
f /. x0 = ||.x0.|| ) implies f is_continuous_on dom f )
assume A1: for x0 being Point of S st x0 in dom f holds
f /. x0 = ||.x0.|| ; ::_thesis: f is_continuous_on dom f
now__::_thesis:_for_x1,_x2_being_Point_of_S_st_x1_in_dom_f_&_x2_in_dom_f_holds_
abs_((f_/._x1)_-_(f_/._x2))_<=_1_*_||.(x1_-_x2).||
let x1, x2 be Point of S; ::_thesis: ( x1 in dom f & x2 in dom f implies abs ((f /. x1) - (f /. x2)) <= 1 * ||.(x1 - x2).|| )
assume  ( x1 in dom f & x2 in dom f ) ; ::_thesis: abs ((f /. x1) - (f /. x2)) <= 1 * ||.(x1 - x2).||
then  ( f /. x1 = ||.x1.|| & f /. x2 = ||.x2.|| ) by A1;
hence  abs ((f /. x1) - (f /. x2)) <= 1 * ||.(x1 - x2).|| by NORMSP_1:9; ::_thesis: verum
end;
then  f is_Lipschitzian_on dom f by Def10;
hence  f is_continuous_on dom f by Th46; ::_thesis: verum
end;

theorem :: NFCONT_1:54
for X being set
for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL st X c= dom f & ( for x0 being Point of S st x0 in X holds
f /. x0 = ||.x0.|| ) holds
f is_continuous_on X
proof
let X be set ; ::_thesis: for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL st X c= dom f & ( for x0 being Point of S st x0 in X holds
f /. x0 = ||.x0.|| ) holds
f is_continuous_on X
let S be RealNormSpace; ::_thesis: for f being PartFunc of the carrier of S,REAL st X c= dom f & ( for x0 being Point of S st x0 in X holds
f /. x0 = ||.x0.|| ) holds
f is_continuous_on X
let f be PartFunc of the carrier of S,REAL; ::_thesis: ( X c= dom f & ( for x0 being Point of S st x0 in X holds
f /. x0 = ||.x0.|| ) implies f is_continuous_on X )
assume that
A1: X c= dom f and
A2: for x0 being Point of S st x0 in X holds
f /. x0 = ||.x0.|| ; ::_thesis: f is_continuous_on X
 X = (dom f) /\ X by A1, XBOOLE_1:28;
then A3: X = dom (f | X) by RELAT_1:61;
now__::_thesis:_for_x0_being_Point_of_S_st_x0_in_dom_(f_|_X)_holds_
(f_|_X)_/._x0_=_||.x0.||
let x0 be Point of S; ::_thesis: ( x0 in dom (f | X) implies (f | X) /. x0 = ||.x0.|| )
assume A4: x0 in dom (f | X) ; ::_thesis: (f | X) /. x0 = ||.x0.||
hence (f | X) /. x0 = f /. x0 by PARTFUN2:15
.= ||.x0.|| by A2, A3, A4 ;
::_thesis: verum
end;
then  f | X is_continuous_on X by A3, Th53;
hence  f is_continuous_on X by Th22; ::_thesis: verum
end;