:: VALUED_2 semantic presentation

begin

definition
let Y be ( ( functional ) ( functional ) set ) ;
func DOMS Y -> ( ( ) ( ) set ) equals :: VALUED_2:def 1
 union { (dom f : ( ( ) ( Relation-like Function-like ) Element of Y : ( ( functional ) ( functional ) set ) ) ) : ( ( ) ( ) set ) where f is ( ( ) ( Relation-like Function-like ) Element of Y : ( ( Relation-like ) ( Relation-like ) set ) ) : verum } : ( ( ) ( ) set ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
attr X is complex-functions-membered means :: VALUED_2:def 2
 for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in X : ( ( Relation-like ) ( Relation-like ) set ) holds
x : ( ( ) ( ) set ) is ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ;
end;

definition
let X be ( ( ) ( ) set ) ;
attr X is ext-real-functions-membered means :: VALUED_2:def 3
 for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in X : ( ( Relation-like ) ( Relation-like ) set ) holds
x : ( ( ) ( ) set ) is ( ( Relation-like Function-like ext-real-valued ) ( Relation-like Function-like ext-real-valued ) Function) ;
end;

definition
let X be ( ( ) ( ) set ) ;
attr X is real-functions-membered means :: VALUED_2:def 4
 for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in X : ( ( Relation-like ) ( Relation-like ) set ) holds
x : ( ( ) ( ) set ) is ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) Function) ;
end;

definition
let X be ( ( ) ( ) set ) ;
attr X is rational-functions-membered means :: VALUED_2:def 5
 for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in X : ( ( Relation-like ) ( Relation-like ) set ) holds
x : ( ( ) ( ) set ) is ( ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Function) ;
end;

definition
let X be ( ( ) ( ) set ) ;
attr X is integer-functions-membered means :: VALUED_2:def 6
 for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in X : ( ( Relation-like ) ( Relation-like ) set ) holds
x : ( ( ) ( ) set ) is ( ( Relation-like INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like ) ( Relation-like INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Function) ;
end;

definition
let X be ( ( ) ( ) set ) ;
attr X is natural-functions-membered means :: VALUED_2:def 7
 for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in X : ( ( Relation-like ) ( Relation-like ) set ) holds
x : ( ( ) ( ) set ) is ( ( Relation-like Function-like natural-valued ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued ) Function) ;
end;

registration
cluster natural-functions-membered -> integer-functions-membered for ( ( ) ( ) set ) ;
cluster integer-functions-membered -> rational-functions-membered for ( ( ) ( ) set ) ;
cluster rational-functions-membered -> real-functions-membered for ( ( ) ( ) set ) ;
cluster real-functions-membered -> complex-functions-membered for ( ( ) ( ) set ) ;
cluster real-functions-membered -> ext-real-functions-membered for ( ( ) ( ) set ) ;
end;

registration
cluster empty -> natural-functions-membered for ( ( ) ( ) set ) ;
end;

registration
let f be ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ;
cluster {f : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) } : ( ( ) ( functional non empty ) set ) -> complex-functions-membered ;
end;

registration
cluster complex-functions-membered -> functional for ( ( ) ( ) set ) ;
cluster ext-real-functions-membered -> functional for ( ( ) ( ) set ) ;
end;

registration
cluster non empty natural-functions-membered for ( ( ) ( ) set ) ;
end;

registration
let X be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
cluster -> complex-functions-membered for ( ( ) ( ) Element of K19(X : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

registration
let X be ( ( ext-real-functions-membered ) ( functional ext-real-functions-membered ) set ) ;
cluster -> ext-real-functions-membered for ( ( ) ( ) Element of K19(X : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

registration
let X be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
cluster -> real-functions-membered for ( ( ) ( ) Element of K19(X : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

registration
let X be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
cluster -> rational-functions-membered for ( ( ) ( ) Element of K19(X : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

registration
let X be ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) ;
cluster -> integer-functions-membered for ( ( ) ( ) Element of K19(X : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

registration
let X be ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ;
cluster -> natural-functions-membered for ( ( ) ( ) Element of K19(X : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
let D be ( ( ) ( ) set ) ;
func C_PFuncs D -> ( ( ) ( ) set ) means :: VALUED_2:def 8
 for f being ( ( ) ( ) set ) holds
( f : ( ( ) ( ) set ) in it : ( ( ) ( ) set ) iff f : ( ( ) ( ) set ) is ( ( Function-like ) ( Relation-like D : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty complex-membered V59() V60() ) set ) -valued Function-like complex-valued ) PartFunc of ,) );
end;

definition
let D be ( ( ) ( ) set ) ;
func C_Funcs D -> ( ( ) ( ) set ) means :: VALUED_2:def 9
 for f being ( ( ) ( ) set ) holds
( f : ( ( ) ( ) set ) in it : ( ( ) ( ) set ) iff f : ( ( ) ( ) set ) is ( ( Function-like quasi_total ) ( Relation-like D : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty complex-membered V59() V60() ) set ) -valued Function-like quasi_total complex-valued ) Function of D : ( ( ) ( ) set ) , COMPLEX : ( ( ) ( non empty complex-membered V59() V60() ) set ) ) );
end;

definition
let D be ( ( ) ( ) set ) ;
func E_PFuncs D -> ( ( ) ( ) set ) means :: VALUED_2:def 10
 for f being ( ( ) ( ) set ) holds
( f : ( ( ) ( ) set ) in it : ( ( ) ( ) set ) iff f : ( ( ) ( ) set ) is ( ( Function-like ) ( Relation-like D : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) );
end;

definition
let D be ( ( ) ( ) set ) ;
func E_Funcs D -> ( ( ) ( ) set ) means :: VALUED_2:def 11
 for f being ( ( ) ( ) set ) holds
( f : ( ( ) ( ) set ) in it : ( ( ) ( ) set ) iff f : ( ( ) ( ) set ) is ( ( Function-like quasi_total ) ( Relation-like D : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like quasi_total ext-real-valued ) Function of D : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) );
end;

definition
let D be ( ( ) ( ) set ) ;
func R_PFuncs D -> ( ( ) ( ) set ) means :: VALUED_2:def 12
 for f being ( ( ) ( ) set ) holds
( f : ( ( ) ( ) set ) in it : ( ( ) ( ) set ) iff f : ( ( ) ( ) set ) is ( ( Function-like ) ( Relation-like D : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) );
end;

definition
let D be ( ( ) ( ) set ) ;
func R_Funcs D -> ( ( ) ( ) set ) means :: VALUED_2:def 13
 for f being ( ( ) ( ) set ) holds
( f : ( ( ) ( ) set ) in it : ( ( ) ( ) set ) iff f : ( ( ) ( ) set ) is ( ( Function-like quasi_total ) ( Relation-like D : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) -valued Function-like quasi_total complex-valued ext-real-valued real-valued ) Function of D : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) );
end;

definition
let D be ( ( ) ( ) set ) ;
func Q_PFuncs D -> ( ( ) ( ) set ) means :: VALUED_2:def 14
 for f being ( ( ) ( ) set ) holds
( f : ( ( ) ( ) set ) in it : ( ( ) ( ) set ) iff f : ( ( ) ( ) set ) is ( ( Function-like ) ( Relation-like D : ( ( ) ( ) set ) -defined RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) );
end;

definition
let D be ( ( ) ( ) set ) ;
func Q_Funcs D -> ( ( ) ( ) set ) means :: VALUED_2:def 15
 for f being ( ( ) ( ) set ) holds
( f : ( ( ) ( ) set ) in it : ( ( ) ( ) set ) iff f : ( ( ) ( ) set ) is ( ( Function-like quasi_total ) ( Relation-like D : ( ( ) ( ) set ) -defined RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like quasi_total complex-valued ext-real-valued real-valued ) Function of D : ( ( ) ( ) set ) , RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) ) );
end;

definition
let D be ( ( ) ( ) set ) ;
func I_PFuncs D -> ( ( ) ( ) set ) means :: VALUED_2:def 16
 for f being ( ( ) ( ) set ) holds
( f : ( ( ) ( ) set ) in it : ( ( ) ( ) set ) iff f : ( ( ) ( ) set ) is ( ( Function-like ) ( Relation-like D : ( ( ) ( ) set ) -defined INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) );
end;

definition
let D be ( ( ) ( ) set ) ;
func I_Funcs D -> ( ( ) ( ) set ) means :: VALUED_2:def 17
 for f being ( ( ) ( ) set ) holds
( f : ( ( ) ( ) set ) in it : ( ( ) ( ) set ) iff f : ( ( ) ( ) set ) is ( ( Function-like quasi_total ) ( Relation-like D : ( ( ) ( ) set ) -defined INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like quasi_total complex-valued ext-real-valued real-valued ) Function of D : ( ( ) ( ) set ) , INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) ) );
end;

definition
let D be ( ( ) ( ) set ) ;
func N_PFuncs D -> ( ( ) ( ) set ) means :: VALUED_2:def 18
 for f being ( ( ) ( ) set ) holds
( f : ( ( ) ( ) set ) in it : ( ( ) ( ) set ) iff f : ( ( ) ( ) set ) is ( ( Function-like ) ( Relation-like D : ( ( ) ( ) set ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued ) PartFunc of ,) );
end;

definition
let D be ( ( ) ( ) set ) ;
func N_Funcs D -> ( ( ) ( ) set ) means :: VALUED_2:def 19
 for f being ( ( ) ( ) set ) holds
( f : ( ( ) ( ) set ) in it : ( ( ) ( ) set ) iff f : ( ( ) ( ) set ) is ( ( Function-like quasi_total ) ( Relation-like D : ( ( ) ( ) set ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like quasi_total complex-valued ext-real-valued real-valued natural-valued ) Function of D : ( ( ) ( ) set ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) ) );
end;

theorem :: VALUED_2:1
for X being ( ( ) ( ) set ) holds C_Funcs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) is ( ( ) ( ) Subset of ( ( ) ( ) set ) ) ;

theorem :: VALUED_2:2
for X being ( ( ) ( ) set ) holds E_Funcs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) is ( ( ) ( ) Subset of ( ( ) ( ) set ) ) ;

theorem :: VALUED_2:3
for X being ( ( ) ( ) set ) holds R_Funcs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) is ( ( ) ( ) Subset of ( ( ) ( ) set ) ) ;

theorem :: VALUED_2:4
for X being ( ( ) ( ) set ) holds Q_Funcs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) is ( ( ) ( ) Subset of ( ( ) ( ) set ) ) ;

theorem :: VALUED_2:5
for X being ( ( ) ( ) set ) holds I_Funcs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) is ( ( ) ( ) Subset of ( ( ) ( ) set ) ) ;

theorem :: VALUED_2:6
for X being ( ( ) ( ) set ) holds N_Funcs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) is ( ( ) ( ) Subset of ( ( ) ( ) set ) ) ;

registration
let X be ( ( ) ( ) set ) ;
cluster C_PFuncs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) -> complex-functions-membered ;
cluster C_Funcs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) -> complex-functions-membered ;
cluster E_PFuncs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) -> ext-real-functions-membered ;
cluster E_Funcs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) -> ext-real-functions-membered ;
cluster R_PFuncs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) -> real-functions-membered ;
cluster R_Funcs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) -> real-functions-membered ;
cluster Q_PFuncs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) -> rational-functions-membered ;
cluster Q_Funcs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) -> rational-functions-membered ;
cluster I_PFuncs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) -> integer-functions-membered ;
cluster I_Funcs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) -> integer-functions-membered ;
cluster N_PFuncs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) -> natural-functions-membered ;
cluster N_Funcs X : ( ( ) ( ) set ) : ( ( ) ( ) set ) -> natural-functions-membered ;
end;

registration
let X be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
cluster -> complex-valued for ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ;
end;

registration
let X be ( ( ext-real-functions-membered ) ( functional ext-real-functions-membered ) set ) ;
cluster -> ext-real-valued for ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ;
end;

registration
let X be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
cluster -> real-valued for ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ;
end;

registration
let X be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
cluster -> RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued for ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ;
end;

registration
let X be ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) ;
cluster -> INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued for ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ;
end;

registration
let X be ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ;
cluster -> natural-valued for ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ;
end;

registration
let X, x be ( ( ) ( ) set ) ;
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) PartFunc of ,) ;
cluster f : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) Element of K19(K20(X : ( ( ) ( ) set ) ,Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like Function-like ) set ) -> Relation-like Function-like ;
end;

registration
let X, x be ( ( ) ( ) set ) ;
let Y be ( ( ext-real-functions-membered ) ( functional ext-real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( ext-real-functions-membered ) ( functional ext-real-functions-membered ) set ) -valued Function-like ) PartFunc of ,) ;
cluster f : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( ext-real-functions-membered ) ( functional ext-real-functions-membered ) set ) -valued Function-like ) Element of K19(K20(X : ( ( ) ( ) set ) ,Y : ( ( ext-real-functions-membered ) ( functional ext-real-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like Function-like ) set ) -> Relation-like Function-like ;
end;

registration
let X, x be ( ( ) ( ) set ) ;
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) PartFunc of ,) ;
cluster f : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) Element of K19(K20(X : ( ( ) ( ) set ) ,Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like Function-like ) set ) -> complex-valued ;
end;

registration
let X, x be ( ( ) ( ) set ) ;
let Y be ( ( ext-real-functions-membered ) ( functional ext-real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( ext-real-functions-membered ) ( functional ext-real-functions-membered ) set ) -valued Function-like ) PartFunc of ,) ;
cluster f : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( ext-real-functions-membered ) ( functional ext-real-functions-membered ) set ) -valued Function-like ) Element of K19(K20(X : ( ( ) ( ) set ) ,Y : ( ( ext-real-functions-membered ) ( functional ext-real-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like Function-like ) set ) -> ext-real-valued ;
end;

registration
let X, x be ( ( ) ( ) set ) ;
let Y be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like ) PartFunc of ,) ;
cluster f : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like ) Element of K19(K20(X : ( ( ) ( ) set ) ,Y : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like Function-like complex-valued ext-real-valued ) set ) -> real-valued ;
end;

registration
let X, x be ( ( ) ( ) set ) ;
let Y be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like ) PartFunc of ,) ;
cluster f : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like ) Element of K19(K20(X : ( ( ) ( ) set ) ,Y : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) -> RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued ;
end;

registration
let X, x be ( ( ) ( ) set ) ;
let Y be ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like ) PartFunc of ,) ;
cluster f : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like ) Element of K19(K20(X : ( ( ) ( ) set ) ,Y : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) set ) -> INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued ;
end;

registration
let X, x be ( ( ) ( ) set ) ;
let Y be ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like ) PartFunc of ,) ;
cluster f : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like ) Element of K19(K20(X : ( ( ) ( ) set ) ,Y : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) set ) -> natural-valued ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( complex-membered ) ( complex-membered ) set ) ;
cluster PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( complex-membered ) ( complex-membered ) set ) ) : ( ( ) ( ) set ) -> complex-functions-membered ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( ext-real-membered ) ( ext-real-membered ) set ) ;
cluster PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ext-real-membered ) ( ext-real-membered ) set ) ) : ( ( ) ( ) set ) -> ext-real-functions-membered ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set ) ;
cluster PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set ) ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered ) set ) -> real-functions-membered ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( rational-membered ) ( complex-membered ext-real-membered real-membered rational-membered ) set ) ;
cluster PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( rational-membered ) ( complex-membered ext-real-membered real-membered rational-membered ) set ) ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -> rational-functions-membered ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( integer-membered ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered ) set ) ;
cluster PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( integer-membered ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered ) set ) ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -> integer-functions-membered ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( natural-membered ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) set ) ;
cluster PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( natural-membered ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) set ) ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -> natural-functions-membered ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( complex-membered ) ( complex-membered ) set ) ;
cluster Funcs (X : ( ( ) ( ) set ) ,Y : ( ( complex-membered ) ( complex-membered ) set ) ) : ( ( ) ( ) set ) -> complex-functions-membered ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( ext-real-membered ) ( ext-real-membered ) set ) ;
cluster Funcs (X : ( ( ) ( ) set ) ,Y : ( ( ext-real-membered ) ( ext-real-membered ) set ) ) : ( ( ) ( ) set ) -> ext-real-functions-membered ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set ) ;
cluster Funcs (X : ( ( ) ( ) set ) ,Y : ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set ) ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered ) set ) -> real-functions-membered ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( rational-membered ) ( complex-membered ext-real-membered real-membered rational-membered ) set ) ;
cluster Funcs (X : ( ( ) ( ) set ) ,Y : ( ( rational-membered ) ( complex-membered ext-real-membered real-membered rational-membered ) set ) ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -> rational-functions-membered ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( integer-membered ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered ) set ) ;
cluster Funcs (X : ( ( ) ( ) set ) ,Y : ( ( integer-membered ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered ) set ) ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -> integer-functions-membered ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( natural-membered ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) set ) ;
cluster Funcs (X : ( ( ) ( ) set ) ,Y : ( ( natural-membered ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) set ) ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -> natural-functions-membered ;
end;

definition
let R be ( ( Relation-like ) ( Relation-like ) Relation) ;
attr R is complex-functions-valued means :: VALUED_2:def 20
 rng R : ( ( ) ( ) set ) : ( ( ) ( ) set ) is complex-functions-membered ;
attr R is ext-real-functions-valued means :: VALUED_2:def 21
 rng R : ( ( ) ( ) set ) : ( ( ) ( ) set ) is ext-real-functions-membered ;
attr R is real-functions-valued means :: VALUED_2:def 22
 rng R : ( ( ) ( ) set ) : ( ( ) ( ) set ) is real-functions-membered ;
attr R is rational-functions-valued means :: VALUED_2:def 23
 rng R : ( ( ) ( ) set ) : ( ( ) ( ) set ) is rational-functions-membered ;
attr R is integer-functions-valued means :: VALUED_2:def 24
 rng R : ( ( ) ( ) set ) : ( ( ) ( ) set ) is integer-functions-membered ;
attr R is natural-functions-valued means :: VALUED_2:def 25
 rng R : ( ( ) ( ) set ) : ( ( ) ( ) set ) is natural-functions-membered ;
end;

registration
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
cluster Relation-like Y : ( ( ) ( ) set ) -valued Function-like -> Relation-like Y : ( ( ) ( ) set ) -valued Function-like complex-functions-valued for ( ( ) ( ) set ) ;
end;

definition
let f be ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) ;
redefine attr f is complex-functions-valued means :: VALUED_2:def 26
 for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( ) ( ) set ) : ( ( ) ( ) set ) holds
f : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) is ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ;
redefine attr f is ext-real-functions-valued means :: VALUED_2:def 27
 for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( ) ( ) set ) : ( ( ) ( ) set ) holds
f : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) is ( ( Relation-like Function-like ext-real-valued ) ( Relation-like Function-like ext-real-valued ) Function) ;
redefine attr f is real-functions-valued means :: VALUED_2:def 28
 for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( ) ( ) set ) : ( ( ) ( ) set ) holds
f : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) is ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) Function) ;
redefine attr f is rational-functions-valued means :: VALUED_2:def 29
 for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( ) ( ) set ) : ( ( ) ( ) set ) holds
f : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) is ( ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Function) ;
redefine attr f is integer-functions-valued means :: VALUED_2:def 30
 for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( ) ( ) set ) : ( ( ) ( ) set ) holds
f : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) is ( ( Relation-like INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like ) ( Relation-like INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Function) ;
redefine attr f is natural-functions-valued means :: VALUED_2:def 31
 for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( ) ( ) set ) : ( ( ) ( ) set ) holds
f : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) is ( ( Relation-like Function-like natural-valued ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued ) Function) ;
end;

registration
cluster Relation-like natural-functions-valued -> Relation-like integer-functions-valued for ( ( ) ( ) set ) ;
cluster Relation-like integer-functions-valued -> Relation-like rational-functions-valued for ( ( ) ( ) set ) ;
cluster Relation-like rational-functions-valued -> Relation-like real-functions-valued for ( ( ) ( ) set ) ;
cluster Relation-like real-functions-valued -> Relation-like ext-real-functions-valued for ( ( ) ( ) set ) ;
cluster Relation-like real-functions-valued -> Relation-like complex-functions-valued for ( ( ) ( ) set ) ;
end;

registration
cluster Relation-like empty -> Relation-like natural-functions-valued for ( ( ) ( ) set ) ;
end;

registration
cluster Relation-like Function-like natural-functions-valued for ( ( ) ( ) set ) ;
end;

registration
let R be ( ( Relation-like complex-functions-valued ) ( Relation-like complex-functions-valued ) Relation) ;
cluster rng R : ( ( Relation-like complex-functions-valued ) ( Relation-like complex-functions-valued ) set ) : ( ( ) ( ) set ) -> complex-functions-membered ;
end;

registration
let R be ( ( Relation-like ext-real-functions-valued ) ( Relation-like ext-real-functions-valued ) Relation) ;
cluster rng R : ( ( Relation-like ext-real-functions-valued ) ( Relation-like ext-real-functions-valued ) set ) : ( ( ) ( ) set ) -> ext-real-functions-membered ;
end;

registration
let R be ( ( Relation-like real-functions-valued ) ( Relation-like complex-functions-valued ext-real-functions-valued real-functions-valued ) Relation) ;
cluster rng R : ( ( Relation-like real-functions-valued ) ( Relation-like complex-functions-valued ext-real-functions-valued real-functions-valued ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered ) set ) -> real-functions-membered ;
end;

registration
let R be ( ( Relation-like rational-functions-valued ) ( Relation-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) Relation) ;
cluster rng R : ( ( Relation-like rational-functions-valued ) ( Relation-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -> rational-functions-membered ;
end;

registration
let R be ( ( Relation-like integer-functions-valued ) ( Relation-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) Relation) ;
cluster rng R : ( ( Relation-like integer-functions-valued ) ( Relation-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -> integer-functions-membered ;
end;

registration
let R be ( ( Relation-like natural-functions-valued ) ( Relation-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Relation) ;
cluster rng R : ( ( Relation-like natural-functions-valued ) ( Relation-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -> natural-functions-membered ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
cluster Function-like -> Function-like complex-functions-valued for ( ( ) ( ) Element of K19(K20(X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( ext-real-functions-membered ) ( functional ext-real-functions-membered ) set ) ;
cluster Function-like -> Function-like ext-real-functions-valued for ( ( ) ( ) Element of K19(K20(X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
cluster Function-like -> Function-like real-functions-valued for ( ( ) ( ) Element of K19(K20(X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
cluster Function-like -> Function-like rational-functions-valued for ( ( ) ( ) Element of K19(K20(X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) ;
cluster Function-like -> Function-like integer-functions-valued for ( ( ) ( ) Element of K19(K20(X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ;
cluster Function-like -> Function-like natural-functions-valued for ( ( ) ( ) Element of K19(K20(X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;
end;

registration
let f be ( ( Relation-like Function-like complex-functions-valued ) ( Relation-like Function-like complex-functions-valued ) Function) ;
let x be ( ( ) ( ) set ) ;
cluster f : ( ( Relation-like Function-like complex-functions-valued ) ( Relation-like Function-like complex-functions-valued ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) -> Relation-like Function-like ;
end;

registration
let f be ( ( Relation-like Function-like ext-real-functions-valued ) ( Relation-like Function-like ext-real-functions-valued ) Function) ;
let x be ( ( ) ( ) set ) ;
cluster f : ( ( Relation-like Function-like ext-real-functions-valued ) ( Relation-like Function-like ext-real-functions-valued ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) -> Relation-like Function-like ;
end;

registration
let f be ( ( Relation-like Function-like complex-functions-valued ) ( Relation-like Function-like complex-functions-valued ) Function) ;
let x be ( ( ) ( ) set ) ;
cluster f : ( ( Relation-like Function-like complex-functions-valued ) ( Relation-like Function-like complex-functions-valued ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like Function-like ) set ) -> complex-valued ;
end;

registration
let f be ( ( Relation-like Function-like ext-real-functions-valued ) ( Relation-like Function-like ext-real-functions-valued ) Function) ;
let x be ( ( ) ( ) set ) ;
cluster f : ( ( Relation-like Function-like ext-real-functions-valued ) ( Relation-like Function-like ext-real-functions-valued ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like Function-like ) set ) -> ext-real-valued ;
end;

registration
let f be ( ( Relation-like Function-like real-functions-valued ) ( Relation-like Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) Function) ;
let x be ( ( ) ( ) set ) ;
cluster f : ( ( Relation-like Function-like real-functions-valued ) ( Relation-like Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like Function-like complex-valued ext-real-valued ) set ) -> real-valued ;
end;

registration
let f be ( ( Relation-like Function-like rational-functions-valued ) ( Relation-like Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) Function) ;
let x be ( ( ) ( ) set ) ;
cluster f : ( ( Relation-like Function-like rational-functions-valued ) ( Relation-like Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) -> RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued ;
end;

registration
let f be ( ( Relation-like Function-like integer-functions-valued ) ( Relation-like Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) Function) ;
let x be ( ( ) ( ) set ) ;
cluster f : ( ( Relation-like Function-like integer-functions-valued ) ( Relation-like Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) set ) -> INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued ;
end;

registration
let f be ( ( Relation-like Function-like natural-functions-valued ) ( Relation-like Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Function) ;
let x be ( ( ) ( ) set ) ;
cluster f : ( ( Relation-like Function-like natural-functions-valued ) ( Relation-like Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) set ) -> natural-valued ;
end;

begin

theorem :: VALUED_2:7
for c1, c2 being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) st g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) <> {} : ( ( ) ( Relation-like non-empty empty-yielding NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -defined RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like one-to-one constant functional empty complex epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V37() complex-valued ext-real-valued real-valued natural-valued real integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() V60() FinSequence-like FinSubsequence-like FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) set ) & g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) + c1 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) = g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) + c2 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) holds
c1 : ( ( complex ) ( complex ) number ) = c2 : ( ( complex ) ( complex ) number ) ;

theorem :: VALUED_2:8
for c1, c2 being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) st g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) <> {} : ( ( ) ( Relation-like non-empty empty-yielding NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -defined RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like one-to-one constant functional empty complex epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V37() complex-valued ext-real-valued real-valued natural-valued real integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() V60() FinSequence-like FinSubsequence-like FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) set ) & g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) - c1 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) = g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) - c2 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) holds
c1 : ( ( complex ) ( complex ) number ) = c2 : ( ( complex ) ( complex ) number ) ;

theorem :: VALUED_2:9
for c1, c2 being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) st g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) <> {} : ( ( ) ( Relation-like non-empty empty-yielding NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -defined RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like one-to-one constant functional empty complex epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V37() complex-valued ext-real-valued real-valued natural-valued real integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() V60() FinSequence-like FinSubsequence-like FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) set ) & g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) is non-empty & g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) c1 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) = g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) c2 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) holds
c1 : ( ( complex ) ( complex ) number ) = c2 : ( ( complex ) ( complex ) number ) ;

theorem :: VALUED_2:10
for c being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds - (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) + c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) = (- g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) - c : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:11
for c being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds - (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) - c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) = (- g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) + c : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:12
for c1, c2 being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) + c1 : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) + c2 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) = g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) + (c1 : ( ( complex ) ( complex ) number ) + c2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:13
for c1, c2 being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) + c1 : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) - c2 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) = g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) + (c1 : ( ( complex ) ( complex ) number ) - c2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:14
for c1, c2 being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) - c1 : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) + c2 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) = g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) - (c1 : ( ( complex ) ( complex ) number ) - c2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:15
for c1, c2 being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) - c1 : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) - c2 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) = g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) - (c1 : ( ( complex ) ( complex ) number ) + c2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:16
for c1, c2 being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) c1 : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) (#) c2 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) = g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) (c1 : ( ( complex ) ( complex ) number ) * c2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:17
for g, h being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds - (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) + h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) = (- g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) - h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:18
for g, h being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) - h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) = - (h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) - g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:19
for g, h, k being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) /" k : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) = g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) (h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) /" k : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:20
for g, h, k being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) /" h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) (#) k : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) = (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) k : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) /" h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:21
for g, h, k being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) /" h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) /" k : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) = g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) /" (h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) k : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:22
for c being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds c : ( ( complex ) ( complex ) number ) (#) (- g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) = (- c : ( ( complex ) ( complex ) number ) ) : ( ( complex ) ( complex ) set ) (#) g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:23
for c being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds c : ( ( complex ) ( complex ) number ) (#) (- g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) = - (c : ( ( complex ) ( complex ) number ) (#) g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:24
for c being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (- c : ( ( complex ) ( complex ) number ) ) : ( ( complex ) ( complex ) set ) (#) g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) = - (c : ( ( complex ) ( complex ) number ) (#) g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:25
for g, h being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds - (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) = (- g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) (#) h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:26
for g, h being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds - (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) /" h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) = (- g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) /" h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:27
for g, h being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds - (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) /" h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) = g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) /" (- h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

definition
let f be ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ;
let c be ( ( complex ) ( complex ) number ) ;
func f (/) c -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) equals :: VALUED_2:def 32
(1 : ( ( ) ( non empty complex epsilon-transitive epsilon-connected ordinal natural V37() V38() real integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) ) / c : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) (#) f : ( ( ) ( ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ;
end;

registration
let f be ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ;
let c be ( ( complex ) ( complex ) number ) ;
cluster f : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) (/) c : ( ( complex ) ( complex ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) -> Relation-like Function-like complex-valued ;
end;

registration
let f be ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) Function) ;
let r be ( ( real ) ( complex V37() real ) number ) ;
cluster f : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) (/) r : ( ( real ) ( complex V37() real ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) -> Relation-like Function-like real-valued ;
end;

registration
let f be ( ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Function) ;
let r be ( ( rational ) ( complex V37() real rational ) number ) ;
cluster f : ( ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) set ) (/) r : ( ( rational ) ( complex V37() real rational ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) Function) -> Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like ;
end;

registration
let f be ( ( Relation-like Function-like complex-valued FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like complex-valued V60() FinSequence-like FinSubsequence-like ) FinSequence) ;
let c be ( ( complex ) ( complex ) number ) ;
cluster f : ( ( Relation-like Function-like complex-valued FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like complex-valued V60() FinSequence-like FinSubsequence-like ) set ) (/) c : ( ( complex ) ( complex ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) -> Relation-like Function-like FinSequence-like ;
end;

theorem :: VALUED_2:28
for c being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds dom (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (/) c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) : ( ( ) ( ) set ) = dom g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( ) ( ) set ) ;

theorem :: VALUED_2:29
for x being ( ( ) ( ) set )
for c being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (/) c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) . x : ( ( ) ( ) set ) : ( ( ) ( complex ) set ) = (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) . x : ( ( ) ( ) set ) ) : ( ( ) ( complex ) set ) / c : ( ( complex ) ( complex ) number ) : ( ( ) ( complex ) set ) ;

theorem :: VALUED_2:30
for c being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (- g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) (/) c : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) = - (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (/) c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:31
for c being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (/) (- c : ( ( complex ) ( complex ) number ) ) : ( ( complex ) ( complex ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) = - (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (/) c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:32
for c being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (/) (- c : ( ( complex ) ( complex ) number ) ) : ( ( complex ) ( complex ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) = (- g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) (/) c : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) ;

theorem :: VALUED_2:33
for c1, c2 being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) st g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) <> {} : ( ( ) ( Relation-like non-empty empty-yielding NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -defined RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like one-to-one constant functional empty complex epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V37() complex-valued ext-real-valued real-valued natural-valued real integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() V60() FinSequence-like FinSubsequence-like FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) set ) & g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) is non-empty & g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (/) c1 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) = g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (/) c2 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) holds
c1 : ( ( complex ) ( complex ) number ) = c2 : ( ( complex ) ( complex ) number ) ;

theorem :: VALUED_2:34
for c1, c2 being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) c1 : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) (/) c2 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) = g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) (c1 : ( ( complex ) ( complex ) number ) / c2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:35
for c1, c2 being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (/) c1 : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) (#) c2 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) = (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) c2 : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) (/) c1 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) ;

theorem :: VALUED_2:36
for c1, c2 being ( ( complex ) ( complex ) number )
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (/) c1 : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) (/) c2 : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) = g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (/) (c1 : ( ( complex ) ( complex ) number ) * c2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) ;

theorem :: VALUED_2:37
for c being ( ( complex ) ( complex ) number )
for g, h being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) + h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) (/) c : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) = (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (/) c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) + (h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (/) c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:38
for c being ( ( complex ) ( complex ) number )
for g, h being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) - h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) (/) c : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) = (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (/) c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) - (h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (/) c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:39
for c being ( ( complex ) ( complex ) number )
for g, h being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) (/) c : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) = g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) (h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (/) c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

theorem :: VALUED_2:40
for c being ( ( complex ) ( complex ) number )
for g, h being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) /" h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) (/) c : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) Function) = g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) /" (h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

definition
let f be ( ( Relation-like Function-like complex-functions-valued ) ( Relation-like Function-like complex-functions-valued ) Function) ;
func <-> f -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: VALUED_2:def 33
( dom it : ( ( ) ( ) set ) : ( ( ) ( ) set ) = dom f : ( ( ) ( ) set ) : ( ( ) ( ) set ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom it : ( ( ) ( ) set ) : ( ( ) ( ) set ) holds
it : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) = - (f : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) ) );
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
:: original: <->
redefine func <-> f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined C_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
:: original: <->
redefine func <-> f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined R_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
:: original: <->
redefine func <-> f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Q_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
:: original: <->
redefine func <-> f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined I_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
end;

registration
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like V60() FinSequence-like FinSubsequence-like complex-functions-valued ) FinSequence of Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) ;
cluster <-> f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like V60() FinSequence-like FinSubsequence-like complex-functions-valued ) FinSequence of Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) -> Relation-like Function-like FinSequence-like ;
end;

theorem :: VALUED_2:41
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds <-> (<-> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:42
for X1, X2 being ( ( ) ( ) set )
for Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f1 being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) st <-> f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = <-> f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds
f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

definition
let X be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let Y be ( ( ) ( ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -defined Y : ( ( ) ( ) set ) -valued Function-like ) PartFunc of ,) ;
func f (-) -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: VALUED_2:def 34
( dom it : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = dom f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) : ( ( ) ( ) Element of K19(X : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) st x : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) in dom it : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
it : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued ) set ) = f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) . (- x : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) : ( ( ) ( ) set ) ) );
end;

definition
let f be ( ( Relation-like Function-like complex-functions-valued ) ( Relation-like Function-like complex-functions-valued ) Function) ;
func </> f -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: VALUED_2:def 35
( dom it : ( ( ) ( ) set ) : ( ( ) ( ) set ) = dom f : ( ( ) ( ) set ) : ( ( ) ( ) set ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom it : ( ( ) ( ) set ) : ( ( ) ( ) set ) holds
it : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) = (f : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) " : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) ) );
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
:: original: </>
redefine func </> f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined C_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
:: original: </>
redefine func </> f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined R_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
:: original: </>
redefine func </> f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Q_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
end;

registration
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like V60() FinSequence-like FinSubsequence-like complex-functions-valued ) FinSequence of Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) ;
cluster </> f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like V60() FinSequence-like FinSubsequence-like complex-functions-valued ) FinSequence of Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) -> Relation-like Function-like FinSequence-like ;
end;

theorem :: VALUED_2:43
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds </> (</> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

definition
let f be ( ( Relation-like Function-like complex-functions-valued ) ( Relation-like Function-like complex-functions-valued ) Function) ;
func abs f -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: VALUED_2:def 36
( dom it : ( ( ) ( ) set ) : ( ( ) ( ) set ) = dom f : ( ( ) ( ) set ) : ( ( ) ( ) set ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom it : ( ( ) ( ) set ) : ( ( ) ( ) set ) holds
it : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) = abs (f : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) ) );
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
:: original: abs
redefine func abs f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined C_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
:: original: abs
redefine func abs f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined R_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
:: original: abs
redefine func abs f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Q_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
:: original: abs
redefine func abs f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined N_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) PartFunc of ,) ;
end;

registration
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like V60() FinSequence-like FinSubsequence-like complex-functions-valued ) FinSequence of Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) ;
cluster abs f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like V60() FinSequence-like FinSubsequence-like complex-functions-valued ) FinSequence of Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) -> Relation-like Function-like FinSequence-like ;
end;

theorem :: VALUED_2:44
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds abs (abs f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = abs f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

definition
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Relation-like Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) ( Relation-like Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) Function) ;
let c be ( ( complex ) ( complex ) number ) ;
func f [+] c -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: VALUED_2:def 37
( dom it : ( ( Function-like ) ( Relation-like Y : ( ( ) ( ) set ) -defined c : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y : ( ( ) ( ) set ) ,c : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = dom f : ( ( ) ( ) set ) : ( ( ) ( ) set ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom it : ( ( Function-like ) ( Relation-like Y : ( ( ) ( ) set ) -defined c : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y : ( ( ) ( ) set ) ,c : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
it : ( ( Function-like ) ( Relation-like Y : ( ( ) ( ) set ) -defined c : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y : ( ( ) ( ) set ) ,c : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued ) set ) = c : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) + (f : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) );
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
let c be ( ( complex ) ( complex ) number ) ;
:: original: [+]
redefine func f [+] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined C_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
let c be ( ( real ) ( complex V37() real ) number ) ;
:: original: [+]
redefine func f [+] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined R_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
let c be ( ( rational ) ( complex V37() real rational ) number ) ;
:: original: [+]
redefine func f [+] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Q_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
let c be ( ( integer ) ( complex V37() real integer rational ) number ) ;
:: original: [+]
redefine func f [+] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined I_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) PartFunc of ,) ;
let c be ( ( natural ) ( complex epsilon-transitive epsilon-connected ordinal natural V37() real integer rational ) Nat) ;
:: original: [+]
redefine func f [+] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined N_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) PartFunc of ,) ;
end;

theorem :: VALUED_2:45
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for c1, c2 being ( ( complex ) ( complex ) number )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [+] c1 : ( ( complex ) ( complex ) number ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [+] c2 : ( ( complex ) ( complex ) number ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [+] (c1 : ( ( complex ) ( complex ) number ) + c2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:46
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for c1, c2 being ( ( complex ) ( complex ) number )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <> {} : ( ( ) ( Relation-like non-empty empty-yielding NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -defined RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like one-to-one constant functional empty complex epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V37() complex-valued ext-real-valued real-valued natural-valued real integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() V60() FinSequence-like FinSubsequence-like FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) set ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) is non-empty & f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [+] c1 : ( ( complex ) ( complex ) number ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [+] c2 : ( ( complex ) ( complex ) number ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds
c1 : ( ( complex ) ( complex ) number ) = c2 : ( ( complex ) ( complex ) number ) ;

definition
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Relation-like Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) ( Relation-like Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) Function) ;
let c be ( ( complex ) ( complex ) number ) ;
func f [-] c -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) equals :: VALUED_2:def 38
f : ( ( ) ( ) set ) [+] (- c : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( complex ) ( complex ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) ;
end;

theorem :: VALUED_2:47
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for c being ( ( complex ) ( complex ) number )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [-] c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( ) ( ) Element of K19(b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: VALUED_2:48
for X, x being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for c being ( ( complex ) ( complex ) number )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) st x : ( ( ) ( ) set ) in dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [-] c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) holds
(f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [-] c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) = (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) . x : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like Function-like complex-valued ) set ) - c : ( ( complex ) ( complex ) number ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
let c be ( ( complex ) ( complex ) number ) ;
:: original: [-]
redefine func f [-] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined C_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
let c be ( ( real ) ( complex V37() real ) number ) ;
:: original: [-]
redefine func f [-] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined R_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
let c be ( ( rational ) ( complex V37() real rational ) number ) ;
:: original: [-]
redefine func f [-] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Q_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
let c be ( ( integer ) ( complex V37() real integer rational ) number ) ;
:: original: [-]
redefine func f [-] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined I_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
end;

theorem :: VALUED_2:49
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for c1, c2 being ( ( complex ) ( complex ) number )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <> {} : ( ( ) ( Relation-like non-empty empty-yielding NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -defined RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like one-to-one constant functional empty complex epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V37() complex-valued ext-real-valued real-valued natural-valued real integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() V60() FinSequence-like FinSubsequence-like FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) set ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) is non-empty & f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [-] c1 : ( ( complex ) ( complex ) number ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [-] c2 : ( ( complex ) ( complex ) number ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds
c1 : ( ( complex ) ( complex ) number ) = c2 : ( ( complex ) ( complex ) number ) ;

theorem :: VALUED_2:50
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for c1, c2 being ( ( complex ) ( complex ) number )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [+] c1 : ( ( complex ) ( complex ) number ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [-] c2 : ( ( complex ) ( complex ) number ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [+] (c1 : ( ( complex ) ( complex ) number ) - c2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:51
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for c1, c2 being ( ( complex ) ( complex ) number )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [-] c1 : ( ( complex ) ( complex ) number ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [+] c2 : ( ( complex ) ( complex ) number ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [-] (c1 : ( ( complex ) ( complex ) number ) - c2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:52
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for c1, c2 being ( ( complex ) ( complex ) number )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [-] c1 : ( ( complex ) ( complex ) number ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [-] c2 : ( ( complex ) ( complex ) number ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [-] (c1 : ( ( complex ) ( complex ) number ) + c2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

definition
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Relation-like Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) ( Relation-like Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) Function) ;
let c be ( ( complex ) ( complex ) number ) ;
func f [#] c -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: VALUED_2:def 39
( dom it : ( ( Function-like ) ( Relation-like Y : ( ( ) ( ) set ) -defined c : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y : ( ( ) ( ) set ) ,c : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = dom f : ( ( ) ( ) set ) : ( ( ) ( ) set ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom it : ( ( Function-like ) ( Relation-like Y : ( ( ) ( ) set ) -defined c : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y : ( ( ) ( ) set ) ,c : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
it : ( ( Function-like ) ( Relation-like Y : ( ( ) ( ) set ) -defined c : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y : ( ( ) ( ) set ) ,c : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued ) set ) = c : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) (#) (f : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) );
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
let c be ( ( complex ) ( complex ) number ) ;
:: original: [#]
redefine func f [#] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined C_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
let c be ( ( real ) ( complex V37() real ) number ) ;
:: original: [#]
redefine func f [#] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined R_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
let c be ( ( rational ) ( complex V37() real rational ) number ) ;
:: original: [#]
redefine func f [#] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Q_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
let c be ( ( integer ) ( complex V37() real integer rational ) number ) ;
:: original: [#]
redefine func f [#] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined I_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) PartFunc of ,) ;
let c be ( ( natural ) ( complex epsilon-transitive epsilon-connected ordinal natural V37() real integer rational ) Nat) ;
:: original: [#]
redefine func f [#] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined N_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) PartFunc of ,) ;
end;

theorem :: VALUED_2:53
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for c1, c2 being ( ( complex ) ( complex ) number )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [#] c1 : ( ( complex ) ( complex ) number ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [#] c2 : ( ( complex ) ( complex ) number ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [#] (c1 : ( ( complex ) ( complex ) number ) * c2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:54
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for c1, c2 being ( ( complex ) ( complex ) number )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <> {} : ( ( ) ( Relation-like non-empty empty-yielding NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -defined RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like one-to-one constant functional empty complex epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V37() complex-valued ext-real-valued real-valued natural-valued real integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() V60() FinSequence-like FinSubsequence-like FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) set ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) is non-empty & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( ) ( ) Element of K19(b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like Function-like complex-valued ) set ) is non-empty ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [#] c1 : ( ( complex ) ( complex ) number ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [#] c2 : ( ( complex ) ( complex ) number ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds
c1 : ( ( complex ) ( complex ) number ) = c2 : ( ( complex ) ( complex ) number ) ;

definition
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Relation-like Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) ( Relation-like Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) Function) ;
let c be ( ( complex ) ( complex ) number ) ;
func f [/] c -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) equals :: VALUED_2:def 40
f : ( ( ) ( ) set ) [#] (c : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ") : ( ( complex ) ( complex ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) ;
end;

theorem :: VALUED_2:55
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for c being ( ( complex ) ( complex ) number )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [/] c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( ) ( ) Element of K19(b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: VALUED_2:56
for X, x being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for c being ( ( complex ) ( complex ) number )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) st x : ( ( ) ( ) set ) in dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [/] c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) holds
(f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [/] c : ( ( complex ) ( complex ) number ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) = (c : ( ( complex ) ( complex ) number ) ") : ( ( complex ) ( complex ) set ) (#) (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) . x : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
let c be ( ( complex ) ( complex ) number ) ;
:: original: [/]
redefine func f [/] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined C_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
let c be ( ( real ) ( complex V37() real ) number ) ;
:: original: [/]
redefine func f [/] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined R_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
let c be ( ( rational ) ( complex V37() real rational ) number ) ;
:: original: [/]
redefine func f [/] c -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Q_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
end;

theorem :: VALUED_2:57
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for c1, c2 being ( ( complex ) ( complex ) number )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [/] c1 : ( ( complex ) ( complex ) number ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [/] c2 : ( ( complex ) ( complex ) number ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [/] (c1 : ( ( complex ) ( complex ) number ) * c2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:58
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for c1, c2 being ( ( complex ) ( complex ) number )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <> {} : ( ( ) ( Relation-like non-empty empty-yielding NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() ) Element of K19(REAL : ( ( ) ( non empty complex-membered ext-real-membered real-membered V59() V60() ) set ) ) : ( ( ) ( ) set ) ) -defined RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like one-to-one constant functional empty complex epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V37() complex-valued ext-real-valued real-valued natural-valued real integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V59() V60() FinSequence-like FinSubsequence-like FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) set ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) is non-empty & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( ) ( ) Element of K19(b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like Function-like complex-valued ) set ) is non-empty ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [/] c1 : ( ( complex ) ( complex ) number ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) [/] c2 : ( ( complex ) ( complex ) number ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds
c1 : ( ( complex ) ( complex ) number ) = c2 : ( ( complex ) ( complex ) number ) ;

definition
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Relation-like Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) ( Relation-like Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) Function) ;
let g be ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ;
func f <+> g -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: VALUED_2:def 41
( dom it : ( ( Function-like ) ( Relation-like Y : ( ( ) ( ) set ) -defined g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y : ( ( ) ( ) set ) ,g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (dom f : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ (dom g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom it : ( ( Function-like ) ( Relation-like Y : ( ( ) ( ) set ) -defined g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y : ( ( ) ( ) set ) ,g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
it : ( ( Function-like ) ( Relation-like Y : ( ( ) ( ) set ) -defined g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y : ( ( ) ( ) set ) ,g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued ) set ) = (f : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) + (g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) );
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ;
:: original: <+>
redefine func f <+> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) Function) ;
:: original: <+>
redefine func f <+> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined R_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Function) ;
:: original: <+>
redefine func f <+> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined Q_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like ) ( Relation-like INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Function) ;
:: original: <+>
redefine func f <+> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined I_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like Function-like natural-valued ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued ) Function) ;
:: original: <+>
redefine func f <+> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined N_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) PartFunc of ,) ;
end;

theorem :: VALUED_2:59
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for g, h being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <+> g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <+> h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ (dom b5 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <+> (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) + h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom (b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) + b5 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:60
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds <-> (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <+> g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = (<-> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <+> (- g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom (- b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

definition
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Relation-like Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) ( Relation-like Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) Function) ;
let g be ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ;
func f <-> g -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) equals :: VALUED_2:def 42
f : ( ( ) ( ) set ) <+> (- g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) ;
end;

theorem :: VALUED_2:61
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <-> g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( ) ( ) Element of K19(b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /\ (dom g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) Element of K19(b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: VALUED_2:62
for X, x being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) st x : ( ( ) ( ) set ) in dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <-> g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) holds
(f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <-> g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) = (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) . x : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like Function-like complex-valued ) set ) - (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) . x : ( ( ) ( ) set ) ) : ( ( ) ( complex ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ;
:: original: <->
redefine func f <-> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) Function) ;
:: original: <->
redefine func f <-> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined R_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Function) ;
:: original: <->
redefine func f <-> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined Q_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like ) ( Relation-like INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Function) ;
:: original: <->
redefine func f <-> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined I_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
end;

theorem :: VALUED_2:63
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <-> (- g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom (- b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <+> g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:64
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds <-> (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <-> g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = (<-> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <+> g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:65
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for g, h being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <+> g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <-> h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ (dom b5 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <+> (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) - h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom (b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) - b5 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:66
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for g, h being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <-> g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <+> h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ (dom b5 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <-> (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) - h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom (b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) - b5 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:67
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for g, h being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <-> g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <-> h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ (dom b5 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <-> (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) + h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom (b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) + b5 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

definition
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Relation-like Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) ( Relation-like Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) Function) ;
let g be ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ;
func f <#> g -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: VALUED_2:def 43
( dom it : ( ( Function-like ) ( Relation-like Y : ( ( ) ( ) set ) -defined g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y : ( ( ) ( ) set ) ,g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (dom f : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ (dom g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom it : ( ( Function-like ) ( Relation-like Y : ( ( ) ( ) set ) -defined g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y : ( ( ) ( ) set ) ,g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
it : ( ( Function-like ) ( Relation-like Y : ( ( ) ( ) set ) -defined g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y : ( ( ) ( ) set ) ,g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued ) set ) = (f : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) (#) (g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) );
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ;
:: original: <#>
redefine func f <#> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) Function) ;
:: original: <#>
redefine func f <#> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined R_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Function) ;
:: original: <#>
redefine func f <#> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined Q_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like ) ( Relation-like INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Function) ;
:: original: <#>
redefine func f <#> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined I_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like Function-like natural-valued ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued ) Function) ;
:: original: <#>
redefine func f <#> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined N_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) PartFunc of ,) ;
end;

theorem :: VALUED_2:68
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <#> (- g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom (- b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = (<-> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <#> g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:69
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <#> (- g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom (- b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = <-> (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <#> g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:70
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for g, h being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <#> g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <#> h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ (dom b5 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <#> (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom (b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) (#) b5 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

definition
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Relation-like Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) ( Relation-like Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) Function) ;
let g be ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ;
func f </> g -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) equals :: VALUED_2:def 44
f : ( ( ) ( ) set ) <#> (g : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ") : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) ;
end;

theorem :: VALUED_2:71
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for g being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) </> g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( ) ( ) Element of K19(b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /\ (dom g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) Element of K19(b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: VALUED_2:72
errorfrm ;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ;
:: original: </>
redefine func f </> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) Function) ;
:: original: </>
redefine func f </> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined R_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
let g be ( ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Function) ;
:: original: </>
redefine func f </> g -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined Q_PFuncs (DOMS Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
end;

theorem :: VALUED_2:73
for X being ( ( ) ( ) set )
for Y being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for g, h being ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <#> g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) </> h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ (dom b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ (dom b5 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <#> (g : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) /" h : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (dom (b4 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) /" b5 : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

definition
let Y1, Y2 be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Relation-like Y1 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) ( Relation-like Y1 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) Function) ;
let g be ( ( Relation-like Y2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) ( Relation-like Y2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) Function) ;
func f <++> g -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: VALUED_2:def 45
( dom it : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-valued ext-real-valued real-valued complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (dom f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom it : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-valued ext-real-valued real-valued complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
it : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-valued ext-real-valued real-valued complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex V37() complex-valued ext-real-valued real-valued natural-valued real ) set ) = (f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) + (g : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) );
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
:: original: <++>
redefine func f <++> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
:: original: <++>
redefine func f <++> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined R_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
:: original: <++>
redefine func f <++> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined Q_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
:: original: <++>
redefine func f <++> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined I_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) PartFunc of ,) ;
:: original: <++>
redefine func f <++> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined N_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) PartFunc of ,) ;
end;

theorem :: VALUED_2:74
for X1, X2 being ( ( ) ( ) set )
for Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f1 being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:75
for X, X1, X2 being ( ( ) ( ) set )
for Y, Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f1 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:76
for X1, X2 being ( ( ) ( ) set )
for Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f1 being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds <-> (f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = (<-> f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> (<-> f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs (DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

definition
let Y1, Y2 be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Relation-like Y1 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) ( Relation-like Y1 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) Function) ;
let g be ( ( Relation-like Y2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) ( Relation-like Y2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) Function) ;
func f <--> g -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: VALUED_2:def 46
( dom it : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-valued ext-real-valued real-valued complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (dom f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom it : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-valued ext-real-valued real-valued complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
it : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-valued ext-real-valued real-valued complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex V37() complex-valued ext-real-valued real-valued natural-valued real ) set ) = (f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) - (g : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) );
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
:: original: <-->
redefine func f <--> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
:: original: <-->
redefine func f <--> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined R_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
:: original: <-->
redefine func f <--> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined Q_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
:: original: <-->
redefine func f <--> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined I_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
end;

theorem :: VALUED_2:77
for X1, X2 being ( ( ) ( ) set )
for Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f1 being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = <-> (f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:78
for X1, X2 being ( ( ) ( ) set )
for Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f1 being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds <-> (f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = (<-> f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs (DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:79
for X, X1, X2 being ( ( ) ( ) set )
for Y, Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f1 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:80
for X, X1, X2 being ( ( ) ( ) set )
for Y, Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f1 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:81
for X, X1, X2 being ( ( ) ( ) set )
for Y, Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f1 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:82
for X, X1, X2 being ( ( ) ( ) set )
for Y, Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f1 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

definition
let Y1, Y2 be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Relation-like Y1 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) ( Relation-like Y1 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) Function) ;
let g be ( ( Relation-like Y2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) ( Relation-like Y2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) Function) ;
func f <##> g -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: VALUED_2:def 47
( dom it : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-valued ext-real-valued real-valued complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (dom f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom it : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-valued ext-real-valued real-valued complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
it : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-valued ext-real-valued real-valued complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex V37() complex-valued ext-real-valued real-valued natural-valued real ) set ) = (f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) (#) (g : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) );
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
:: original: <##>
redefine func f <##> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
:: original: <##>
redefine func f <##> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined R_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
:: original: <##>
redefine func f <##> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined Q_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( integer-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
:: original: <##>
redefine func f <##> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined I_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued ) PartFunc of ,) ;
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) PartFunc of ,) ;
:: original: <##>
redefine func f <##> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined N_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) PartFunc of ,) ;
end;

theorem :: VALUED_2:83
for X1, X2 being ( ( ) ( ) set )
for Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f1 being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:84
for X, X1, X2 being ( ( ) ( ) set )
for Y, Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f1 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:85
for X1, X2 being ( ( ) ( ) set )
for Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f1 being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (<-> f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs (DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = <-> (f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:86
for X1, X2 being ( ( ) ( ) set )
for Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f1 being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> (<-> f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = <-> (f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:87
for X, X1, X2 being ( ( ) ( ) set )
for Y, Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f1 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ (b1 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:88
for X, X1, X2 being ( ( ) ( ) set )
for Y, Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f1 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> (f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b2 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ (b3 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs ((DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:89
for X, X1, X2 being ( ( ) ( ) set )
for Y, Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f1 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ (b1 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:90
for X, X1, X2 being ( ( ) ( ) set )
for Y, Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f1 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> (f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b2 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ (b3 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs ((DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

definition
let Y1, Y2 be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Relation-like Y1 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) ( Relation-like Y1 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) Function) ;
let g be ( ( Relation-like Y2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like ) ( Relation-like Y2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) Function) ;
func f <//> g -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: VALUED_2:def 48
( dom it : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-valued ext-real-valued real-valued complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (dom f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (dom g : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom it : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-valued ext-real-valued real-valued complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
it : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-valued ext-real-valued real-valued complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex V37() complex-valued ext-real-valued real-valued natural-valued real ) set ) = (f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /" (g : ( ( Function-like ) ( Relation-like Y1 : ( ( ) ( ) set ) -defined f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(Y1 : ( ( ) ( ) set ) ,f : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like RAT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered V59() V60() ) set ) -valued INT : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered V59() V60() ) set ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ) );
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
:: original: <//>
redefine func f <//> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( real-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
:: original: <//>
redefine func f <//> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined R_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
end;

definition
let X1, X2 be ( ( ) ( ) set ) ;
let Y1, Y2 be ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) ;
let f be ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
let g be ( ( Function-like ) ( Relation-like X2 : ( ( ) ( ) set ) -defined Y2 : ( ( rational-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
:: original: <//>
redefine func f <//> g -> ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) /\ X2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined Q_PFuncs ((DOMS Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS Y2 : ( ( Function-like ) ( Relation-like X1 : ( ( ) ( ) set ) -defined Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued ) Element of K19(K20(X1 : ( ( ) ( ) set ) ,Y1 : ( ( natural-functions-membered ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered ) set ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) PartFunc of ,) ;
end;

theorem :: VALUED_2:91
for X1, X2 being ( ( ) ( ) set )
for Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f1 being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (<-> f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs (DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = <-> (f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:92
for X1, X2 being ( ( ) ( ) set )
for Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f1 being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> (<-> f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined C_PFuncs (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = <-> (f1 : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> f2 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs (DOMS (C_PFuncs ((DOMS b3 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:93
for X, X1, X2 being ( ( ) ( ) set )
for Y, Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f1 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:94
for X, X1, X2 being ( ( ) ( ) set )
for Y, Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f1 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:95
for X, X1, X2 being ( ( ) ( ) set )
for Y, Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f1 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b1 : ( ( ) ( ) set ) /\ b2 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <##> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) /\ (b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:96
for X, X1, X2 being ( ( ) ( ) set )
for Y, Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f1 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <++> (f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b2 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ (b3 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs ((DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;

theorem :: VALUED_2:97
for X, X1, X2 being ( ( ) ( ) set )
for Y, Y1, Y2 being ( ( complex-functions-membered ) ( functional complex-functions-membered ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f1 being ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) holds (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b2 : ( ( ) ( ) set ) /\ b3 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) = (f1 : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b2 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <--> (f2 : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) <//> f : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b3 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like (b2 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ (b3 : ( ( ) ( ) set ) /\ b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined C_PFuncs ((DOMS (C_PFuncs ((DOMS b5 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS (C_PFuncs ((DOMS b6 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS b4 : ( ( complex-functions-membered ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ) set ) -valued Function-like complex-functions-valued ) PartFunc of ,) ;