:: NFCONT_1 semantic presentation

begin

theorem :: NFCONT_1:1
for S being ( ( non empty ) ( non empty ) addLoopStr )
for seq1, seq2 being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) holds seq1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) - seq2 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = seq1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) + (- seq2 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NFCONT_1:2
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) holds - seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non positive ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) * seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

definition
let S be ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) ;
let x0 be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
mode Neighbourhood of x0 -> ( ( ) ( ) Subset of ) means :: NFCONT_1:def 1
 ex g being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ) < g : ( ( ) ( V11() real ext-real ) Real) & { y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) where y is ( ( ) ( ) Point of ( ( ) ( ) set ) ) : ||.(y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) ) : ( ( ) ( ) Element of the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) < g : ( ( ) ( V11() real ext-real ) Real) } c= it : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) );
end;

theorem :: NFCONT_1:3
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for g being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ) < g : ( ( ) ( V11() real ext-real ) Real) holds
{ y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) where y is ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ||.(y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) < g : ( ( ) ( V11() real ext-real ) Real) } is ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: NFCONT_1:4
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) holds x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in N : ( ( ) ( ) Neighbourhood of b2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

definition
let S be ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) ;
let X be ( ( ) ( ) Subset of ) ;
attr X is compact means :: NFCONT_1:def 2
 for s1 being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) st rng s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= X : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) holds
ex s2 being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) st
( s2 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) subsequence of s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) & s2 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim s2 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) in X : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) );
end;

definition
let S be ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) ;
let X be ( ( ) ( ) Subset of ) ;
attr X is closed means :: NFCONT_1:def 3
 for s1 being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) st rng s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= X : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) & s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent holds
lim s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) in X : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) ;
end;

definition
let S be ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) ;
let X be ( ( ) ( ) Subset of ) ;
attr X is open means :: NFCONT_1:def 4
X : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) ` : ( ( ) ( ) Element of K6( the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is closed ;
end;

definition
let S, T be ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( Relation-like the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of T : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ;
let x0 be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
pred f is_continuous_in x0 means :: NFCONT_1:def 5
( x0 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in dom f : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for s1 being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) st rng s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) = x0 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
( f : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of T : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of T : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is convergent & f : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x0 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of T : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) = lim (f : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of T : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of T : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of T : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) ) ) );
end;

definition
let S be ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( Relation-like the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let x0 be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
pred f is_continuous_in x0 means :: NFCONT_1:def 6
( x0 : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) in dom f : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) Element of K6( the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for s1 being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) st rng s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) Element of K6( the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) = x0 : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
( f : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) /* s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is convergent & f : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) /. x0 : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) = lim (f : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) /* s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) ) ) );
end;

theorem :: NFCONT_1:5
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) )
for S, T being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) )
for h being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= dom h : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) in dom h : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NFCONT_1:6
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) )
for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) iff ex n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ) st x : ( ( ) ( ) set ) = seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: NFCONT_1:7
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
||.((f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) - (f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: NFCONT_1:8
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
abs ((f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) - (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: NFCONT_1:9
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for N1 being ( ( ) ( ) Neighbourhood of f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ex N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in N : ( ( ) ( ) Neighbourhood of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) in N1 : ( ( ) ( ) Neighbourhood of b3 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ) ) ) ;

theorem :: NFCONT_1:10
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for N1 being ( ( ) ( ) Neighbourhood of f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ex N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .: N : ( ( ) ( ) Neighbourhood of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= N1 : ( ( ) ( ) Neighbourhood of b3 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ) ) ) ;

theorem :: NFCONT_1:11
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ex N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st (dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ N : ( ( ) ( ) Neighbourhood of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: NFCONT_1:12
for S, T being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for h1, h2 being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= (dom h1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (dom h2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
( (h1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + h2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (h1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) + (h2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & (h1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - h2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (h1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (h2 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: NFCONT_1:13
for S, T being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for h being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= dom h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
(r : ( ( ) ( V11() real ext-real ) Real) (#) h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) * (h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NFCONT_1:14
for S, T being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for h being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) st rng seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= dom h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
( ||.(h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = ||.h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & - (h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (- h : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: NFCONT_1:15
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: NFCONT_1:16
for r being ( ( ) ( V11() real ext-real ) Real)
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: NFCONT_1:17
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & - f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

definition
let S, T be ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( Relation-like the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of T : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
pred f is_continuous_on X means :: NFCONT_1:def 7
( X : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) | X : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -defined the carrier of T : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -valued Function-like ) Element of K6(K7( the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) , the carrier of T : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) );
end;

definition
let S be ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( Relation-like the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
pred f is_continuous_on X means :: NFCONT_1:def 8
( X : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) Element of K6( the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) | X : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) );
end;

theorem :: NFCONT_1:18
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff ( X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for s1 being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) st rng s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= X : ( ( ) ( ) set ) & s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is convergent & f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. (lim s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) = lim (f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /* s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ) ) ) ;

theorem :: NFCONT_1:19
for X being ( ( ) ( ) set )
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff ( X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) & 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) & ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
||.((f : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) - (f : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: NFCONT_1:20
for X being ( ( ) ( ) set )
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff ( X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) & 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) & ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
abs ((f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) - (f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: NFCONT_1:21
for X being ( ( ) ( ) set )
for S, T being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: NFCONT_1:22
for X being ( ( ) ( ) set )
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: NFCONT_1:23
for X, X1 being ( ( ) ( ) set )
for S, T being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) & X1 : ( ( ) ( ) set ) c= X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X1 : ( ( ) ( ) set ) ;

theorem :: NFCONT_1:24
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on {x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) ;

theorem :: NFCONT_1:25
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) holds
( f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) & f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: NFCONT_1:26
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for X, X1 being ( ( ) ( ) set )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X1 : ( ( ) ( ) set ) holds
( f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) /\ X1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) & f1 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) /\ X1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;

theorem :: NFCONT_1:27
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for r being ( ( ) ( V11() real ext-real ) Real)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) holds
r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NFCONT_1:28
for X being ( ( ) ( ) set )
for S, T being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) holds
( ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) & - f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: NFCONT_1:29
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is total & ( for x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. (x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) + x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) = (f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) + (f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) & ex x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ;

theorem :: NFCONT_1:30
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact ;

theorem :: NFCONT_1:31
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
rng f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( V47() V48() V49() ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) is compact ;

theorem :: NFCONT_1:32
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for Y being ( ( ) ( ) Subset of ) st Y : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Y : ( ( ) ( ) Subset of ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on Y : ( ( ) ( ) Subset of ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .: Y : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact ;

theorem :: NFCONT_1:33
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) <> {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative ) set ) & dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
ex x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st
( x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) = upper_bound (rng f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( V47() V48() V49() ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) = lower_bound (rng f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( V47() V48() V49() ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) ) ;

theorem :: NFCONT_1:34
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) <> {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative ) set ) & dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
ex x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st
( x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) = upper_bound (rng ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V47() V48() V49() ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) = lower_bound (rng ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V47() V48() V49() ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) ) ;

theorem :: NFCONT_1:35
for X being ( ( ) ( ) set )
for S, T being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = ||.(f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NFCONT_1:36
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for Y being ( ( ) ( ) Subset of ) st Y : ( ( ) ( ) Subset of ) <> {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative ) set ) & Y : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Y : ( ( ) ( ) Subset of ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on Y : ( ( ) ( ) Subset of ) holds
ex x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st
( x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in Y : ( ( ) ( ) Subset of ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in Y : ( ( ) ( ) Subset of ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) = upper_bound (||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .: Y : ( ( ) ( ) Subset of ) ) : ( ( ) ( V47() V48() V49() ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) = lower_bound (||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .: Y : ( ( ) ( ) Subset of ) ) : ( ( ) ( V47() V48() V49() ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) ) ;

theorem :: NFCONT_1:37
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for Y being ( ( ) ( ) Subset of ) st Y : ( ( ) ( ) Subset of ) <> {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative ) set ) & Y : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Y : ( ( ) ( ) Subset of ) is compact & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_on Y : ( ( ) ( ) Subset of ) holds
ex x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st
( x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in Y : ( ( ) ( ) Subset of ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in Y : ( ( ) ( ) Subset of ) & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) = upper_bound (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) .: Y : ( ( ) ( ) Subset of ) ) : ( ( ) ( V47() V48() V49() ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) = lower_bound (f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) .: Y : ( ( ) ( ) Subset of ) ) : ( ( ) ( V47() V48() V49() ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) ) ;

definition
let S, T be ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) ;
let X be ( ( ) ( ) set ) ;
let f be ( ( Function-like ) ( Relation-like the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of T : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ;
pred f is_Lipschitzian_on X means :: NFCONT_1:def 9
( X : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ( for x1, x2 being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
||.((f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of T : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) - (f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V54() ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of T : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of T : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) * ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) ) ) );
end;

definition
let S be ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) ;
let X be ( ( ) ( ) set ) ;
let f be ( ( Function-like ) ( Relation-like the carrier of S : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
pred f is_Lipschitzian_on X means :: NFCONT_1:def 10
( X : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non negative ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal ) Element of K6(REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ( for x1, x2 being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) Element of S : ( ( ) ( ) NORMSTR ) ) holds
abs ((f : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) - (f : ( ( Function-like quasi_total ) ( Relation-like K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined S : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) NORMSTR ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) * ||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of S : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) ) ) );
end;

theorem :: NFCONT_1:38
for X, X1 being ( ( ) ( ) set )
for S, T being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) & X1 : ( ( ) ( ) set ) c= X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X1 : ( ( ) ( ) set ) ;

theorem :: NFCONT_1:39
for X, X1 being ( ( ) ( ) set )
for S, T being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X1 : ( ( ) ( ) set ) holds
f1 : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) /\ X1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) ;

theorem :: NFCONT_1:40
for X, X1 being ( ( ) ( ) set )
for S, T being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) & f2 : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X1 : ( ( ) ( ) set ) holds
f1 : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) /\ X1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) ;

theorem :: NFCONT_1:41
for X being ( ( ) ( ) set )
for p being ( ( ) ( V11() real ext-real ) Real)
for S, T being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) holds
p : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) ;

theorem :: NFCONT_1:42
for X being ( ( ) ( ) set )
for S, T being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) holds
( - f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) & ||.f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) .|| : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on X : ( ( ) ( ) set ) ) ;

theorem :: NFCONT_1:43
for X being ( ( ) ( ) set )
for S, T being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is V23() holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) ;

theorem :: NFCONT_1:44
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for Y being ( ( ) ( ) Subset of ) holds id Y : ( ( ) ( ) Subset of ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_Lipschitzian_on Y : ( ( ) ( ) Subset of ) ;

theorem :: NFCONT_1:45
for X being ( ( ) ( ) set )
for S, T being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NFCONT_1:46
for X being ( ( ) ( ) set )
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_Lipschitzian_on X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NFCONT_1:47
for T, S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st ex r being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st rng f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {r : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NFCONT_1:48
for X being ( ( ) ( ) set )
for S, T being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is V23() holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NFCONT_1:49
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) = x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NFCONT_1:50
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) = id (dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NFCONT_1:51
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for Y being ( ( ) ( ) Subset of )
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st Y : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) | Y : ( ( ) ( ) Subset of ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = id Y : ( ( ) ( ) Subset of ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on Y : ( ( ) ( ) Subset of ) ;

theorem :: NFCONT_1:52
for X being ( ( ) ( ) set )
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for r being ( ( ) ( V11() real ext-real ) Real)
for p being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) = (r : ( ( ) ( V11() real ext-real ) Real) * x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) + p : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NFCONT_1:53
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) = ||.x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_on dom f : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NFCONT_1:54
for X being ( ( ) ( ) set )
for S being ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) = ||.x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty right_complementable V135() V136() V137() V138() V139() V140() V141() V145() V146() RealNormSpace-like ) ( non empty left_complementable right_complementable V135() V136() V137() V138() V139() V140() V141() zeroed V145() V146() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V54() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;