:: VALUED_2 semantic presentation

begin


Lm1: now__::_thesis:_for_X1,_X2,_X3_being_set_holds_X1_/\_(X2_/\_X3)_=_(X1_/\_X2)_/\_(X1_/\_X3)
let X1, X2, X3 be set ; ::_thesis: X1 /\ (X2 /\ X3) = (X1 /\ X2) /\ (X1 /\ X3)
thus X1 /\ (X2 /\ X3) = ((X1 /\ X1) /\ X2) /\ X3 by XBOOLE_1:16
.= (X1 /\ (X1 /\ X2)) /\ X3 by XBOOLE_1:16
.= (X1 /\ X2) /\ (X1 /\ X3) by XBOOLE_1:16 ; ::_thesis: verum
end;

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

:: deftheorem  defines DOMS VALUED_2:def_1_:_
 for Y being functional set holds DOMS Y = union  {  (dom f) where f is Element of Y : verum  }  ;

definition
let X be set ;
attrX is complex-functions-membered  means :Def2: :: VALUED_2:def 2
 for x being set st x in X holds
x is complex-valued Function;
end;

:: deftheorem Def2 defines complex-functions-membered VALUED_2:def_2_:_
 for X being set holds
( X is complex-functions-membered iff for x being set st x in X holds
x is complex-valued Function );

definition
let X be set ;
attrX is ext-real-functions-membered  means :Def3: :: VALUED_2:def 3
 for x being set st x in X holds
x is ext-real-valued Function;
end;

:: deftheorem Def3 defines ext-real-functions-membered VALUED_2:def_3_:_
 for X being set holds
( X is ext-real-functions-membered iff for x being set st x in X holds
x is ext-real-valued Function );

definition
let X be set ;
attrX is real-functions-membered  means :Def4: :: VALUED_2:def 4
 for x being set st x in X holds
x is real-valued Function;
end;

:: deftheorem Def4 defines real-functions-membered VALUED_2:def_4_:_
 for X being set holds
( X is real-functions-membered iff for x being set st x in X holds
x is real-valued Function );

definition
let X be set ;
attrX is rational-functions-membered  means :Def5: :: VALUED_2:def 5
 for x being set st x in X holds
x is RAT -valued Function;
end;

:: deftheorem Def5 defines rational-functions-membered VALUED_2:def_5_:_
 for X being set holds
( X is rational-functions-membered iff for x being set st x in X holds
x is RAT -valued Function );

definition
let X be set ;
attrX is integer-functions-membered  means :Def6: :: VALUED_2:def 6
 for x being set st x in X holds
x is INT -valued Function;
end;

:: deftheorem Def6 defines integer-functions-membered VALUED_2:def_6_:_
 for X being set holds
( X is integer-functions-membered iff for x being set st x in X holds
x is INT -valued Function );

definition
let X be set ;
attrX is natural-functions-membered  means :Def7: :: VALUED_2:def 7
 for x being set st x in X holds
x is natural-valued Function;
end;

:: deftheorem Def7 defines natural-functions-membered VALUED_2:def_7_:_
 for X being set holds
( X is natural-functions-membered iff for x being set st x in X holds
x is natural-valued Function );

registration
cluster natural-functions-membered  ->  integer-functions-membered for set ;
coherence
 for b1 being set st b1 is natural-functions-membered holds
b1 is integer-functions-membered
proof
let X be set ; ::_thesis: ( X is natural-functions-membered implies X is integer-functions-membered )
assume A1: for x being set st x in X holds
x is natural-valued Function ; :: according to VALUED_2:def_7 ::_thesis: X is integer-functions-membered
let x be set ; :: according to VALUED_2:def_6 ::_thesis: ( x in X implies x is INT -valued Function )
assume  x in X ; ::_thesis: x is INT -valued Function
then  x is natural-valued Function by A1;
hence  x is INT -valued Function ; ::_thesis: verum
end;
cluster integer-functions-membered  ->  rational-functions-membered for set ;
coherence
 for b1 being set st b1 is integer-functions-membered holds
b1 is rational-functions-membered
proof
let X be set ; ::_thesis: ( X is integer-functions-membered implies X is rational-functions-membered )
assume A2: for x being set st x in X holds
x is INT -valued Function ; :: according to VALUED_2:def_6 ::_thesis: X is rational-functions-membered
let x be set ; :: according to VALUED_2:def_5 ::_thesis: ( x in X implies x is RAT -valued Function )
assume  x in X ; ::_thesis: x is RAT -valued Function
then  x is INT -valued Function by A2;
hence  x is RAT -valued Function ; ::_thesis: verum
end;
cluster rational-functions-membered  ->  real-functions-membered for set ;
coherence
 for b1 being set st b1 is rational-functions-membered holds
b1 is real-functions-membered
proof
let X be set ; ::_thesis: ( X is rational-functions-membered implies X is real-functions-membered )
assume A3: for x being set st x in X holds
x is RAT -valued Function ; :: according to VALUED_2:def_5 ::_thesis: X is real-functions-membered
let x be set ; :: according to VALUED_2:def_4 ::_thesis: ( x in X implies x is real-valued Function )
thus  ( x in X implies x is real-valued Function ) by A3; ::_thesis: verum
end;
cluster real-functions-membered  ->  complex-functions-membered for set ;
coherence
 for b1 being set st b1 is real-functions-membered holds
b1 is complex-functions-membered
proof
let X be set ; ::_thesis: ( X is real-functions-membered implies X is complex-functions-membered )
assume A4: for x being set st x in X holds
x is real-valued Function ; :: according to VALUED_2:def_4 ::_thesis: X is complex-functions-membered
let x be set ; :: according to VALUED_2:def_2 ::_thesis: ( x in X implies x is complex-valued Function )
thus  ( x in X implies x is complex-valued Function ) by A4; ::_thesis: verum
end;
cluster real-functions-membered  ->  ext-real-functions-membered for set ;
coherence
 for b1 being set st b1 is real-functions-membered holds
b1 is ext-real-functions-membered
proof
let X be set ; ::_thesis: ( X is real-functions-membered implies X is ext-real-functions-membered )
assume A5: for x being set st x in X holds
x is real-valued Function ; :: according to VALUED_2:def_4 ::_thesis: X is ext-real-functions-membered
let x be set ; :: according to VALUED_2:def_3 ::_thesis: ( x in X implies x is ext-real-valued Function )
thus  ( x in X implies x is ext-real-valued Function ) by A5; ::_thesis: verum
end;
end;

registration
cluster empty  ->  natural-functions-membered for set ;
coherence
 for b1 being set st b1 is empty holds
b1 is natural-functions-membered
proof
let X be set ; ::_thesis: ( X is empty implies X is natural-functions-membered )
assume A1: X is empty ; ::_thesis: X is natural-functions-membered
let x be set ; :: according to VALUED_2:def_7 ::_thesis: ( x in X implies x is natural-valued Function )
thus  ( x in X implies x is natural-valued Function ) by A1; ::_thesis: verum
end;
end;

registration
let f be complex-valued Function;
cluster{f} ->  complex-functions-membered ;
coherence
 {f} is complex-functions-membered
proof
let x be set ; :: according to VALUED_2:def_2 ::_thesis: ( x in {f} implies x is complex-valued Function )
thus  ( x in {f} implies x is complex-valued Function ) by TARSKI:def_1; ::_thesis: verum
end;
end;

registration
cluster complex-functions-membered  ->  functional for set ;
coherence
 for b1 being set st b1 is complex-functions-membered holds
b1 is functional
proof
let X be set ; ::_thesis: ( X is complex-functions-membered implies X is functional )
assume A1: X is complex-functions-membered ; ::_thesis: X is functional
let x be set ; :: according to FUNCT_1:def_13 ::_thesis: ( not x in X or x is set )
thus  ( not x in X or x is set ) by A1, Def2; ::_thesis: verum
end;
cluster ext-real-functions-membered  ->  functional for set ;
coherence
 for b1 being set st b1 is ext-real-functions-membered holds
b1 is functional
proof
let X be set ; ::_thesis: ( X is ext-real-functions-membered implies X is functional )
assume A2: X is ext-real-functions-membered ; ::_thesis: X is functional
let x be set ; :: according to FUNCT_1:def_13 ::_thesis: ( not x in X or x is set )
thus  ( not x in X or x is set ) by A2, Def3; ::_thesis: verum
end;
end;

set ff = the natural-valued Function;

registration
cluster non empty natural-functions-membered for set ;
existence
 ex b1 being set st
( b1 is natural-functions-membered & not b1 is empty )
proof
take  { the natural-valued Function} ; ::_thesis: ( { the natural-valued Function} is natural-functions-membered & not { the natural-valued Function} is empty )
thus  for x being set st x in { the natural-valued Function} holds
x is natural-valued Function by TARSKI:def_1; :: according to VALUED_2:def_7 ::_thesis: not { the natural-valued Function} is empty
thus  not { the natural-valued Function} is empty ; ::_thesis: verum
end;
end;

registration
let X be complex-functions-membered set ;
cluster  ->  complex-functions-membered for Element of K19(X);
coherence
 for b1 being Subset of X holds b1 is complex-functions-membered
proof
let S be Subset of X; ::_thesis: S is complex-functions-membered
let x be set ; :: according to VALUED_2:def_2 ::_thesis: ( x in S implies x is complex-valued Function )
thus  ( x in S implies x is complex-valued Function ) by Def2; ::_thesis: verum
end;
end;

registration
let X be ext-real-functions-membered set ;
cluster  ->  ext-real-functions-membered for Element of K19(X);
coherence
 for b1 being Subset of X holds b1 is ext-real-functions-membered
proof
let S be Subset of X; ::_thesis: S is ext-real-functions-membered
let x be set ; :: according to VALUED_2:def_3 ::_thesis: ( x in S implies x is ext-real-valued Function )
thus  ( x in S implies x is ext-real-valued Function ) by Def3; ::_thesis: verum
end;
end;

registration
let X be real-functions-membered set ;
cluster  ->  real-functions-membered for Element of K19(X);
coherence
 for b1 being Subset of X holds b1 is real-functions-membered
proof
let S be Subset of X; ::_thesis: S is real-functions-membered
let x be set ; :: according to VALUED_2:def_4 ::_thesis: ( x in S implies x is real-valued Function )
thus  ( x in S implies x is real-valued Function ) by Def4; ::_thesis: verum
end;
end;

registration
let X be rational-functions-membered set ;
cluster  ->  rational-functions-membered for Element of K19(X);
coherence
 for b1 being Subset of X holds b1 is rational-functions-membered
proof
let S be Subset of X; ::_thesis: S is rational-functions-membered
let x be set ; :: according to VALUED_2:def_5 ::_thesis: ( x in S implies x is RAT -valued Function )
thus  ( x in S implies x is RAT -valued Function ) by Def5; ::_thesis: verum
end;
end;

registration
let X be integer-functions-membered set ;
cluster  ->  integer-functions-membered for Element of K19(X);
coherence
 for b1 being Subset of X holds b1 is integer-functions-membered
proof
let S be Subset of X; ::_thesis: S is integer-functions-membered
let x be set ; :: according to VALUED_2:def_6 ::_thesis: ( x in S implies x is INT -valued Function )
thus  ( x in S implies x is INT -valued Function ) by Def6; ::_thesis: verum
end;
end;

registration
let X be natural-functions-membered set ;
cluster  ->  natural-functions-membered for Element of K19(X);
coherence
 for b1 being Subset of X holds b1 is natural-functions-membered
proof
let S be Subset of X; ::_thesis: S is natural-functions-membered
let x be set ; :: according to VALUED_2:def_7 ::_thesis: ( x in S implies x is natural-valued Function )
thus  ( x in S implies x is natural-valued Function ) by Def7; ::_thesis: verum
end;
end;

definition
set A = COMPLEX ;
let D be set ;
defpred S1[ set ] means $1 is PartFunc of D,COMPLEX;
func C_PFuncs D ->  set  means :Def8: :: VALUED_2:def 8
 for f being set holds
( f in it iff f is PartFunc of D,COMPLEX );
existence
 ex b1 being set st
for f being set holds
( f in b1 iff f is PartFunc of D,COMPLEX )
proof
consider X being set  such that
A1: for x being set holds
( x in X iff ( x in PFuncs (D,COMPLEX) & S1[x] ) ) from XBOOLE_0:sch_1();
take  X ; ::_thesis: for f being set holds
( f in X iff f is PartFunc of D,COMPLEX )
let f be set ; ::_thesis: ( f in X iff f is PartFunc of D,COMPLEX )
thus  ( f in X implies f is PartFunc of D,COMPLEX ) by A1; ::_thesis: ( f is PartFunc of D,COMPLEX implies f in X )
assume A2: f is PartFunc of D,COMPLEX ; ::_thesis: f in X
then  f in PFuncs (D,COMPLEX) by PARTFUN1:45;
hence  f in X by A1, A2; ::_thesis: verum
end;
uniqueness
 for b1, b2 being set st ( for f being set holds
( f in b1 iff f is PartFunc of D,COMPLEX ) ) & ( for f being set holds
( f in b2 iff f is PartFunc of D,COMPLEX ) ) holds
b1 = b2
proof
let P, Q be set ; ::_thesis: ( ( for f being set holds
( f in P iff f is PartFunc of D,COMPLEX ) ) & ( for f being set holds
( f in Q iff f is PartFunc of D,COMPLEX ) ) implies P = Q )
assume  for f being set holds
( f in P iff f is PartFunc of D,COMPLEX ) ; ::_thesis: ( ex f being set st
( ( f in Q implies f is PartFunc of D,COMPLEX ) implies ( f is PartFunc of D,COMPLEX & not f in Q ) ) or P = Q )
then A3: for f being set holds
( f in P iff S1[f] ) ;
assume  for f being set holds
( f in Q iff f is PartFunc of D,COMPLEX ) ; ::_thesis: P = Q
then A4: for f being set holds
( f in Q iff S1[f] ) ;
thus  P = Q from XBOOLE_0:sch_2(A3, A4); ::_thesis: verum
end;
end;

:: deftheorem Def8 defines C_PFuncs VALUED_2:def_8_:_
 for D, b2 being set holds
( b2 = C_PFuncs D iff for f being set holds
( f in b2 iff f is PartFunc of D,COMPLEX ) );

definition
set A = COMPLEX ;
let D be set ;
defpred S1[ set ] means $1 is Function of D,COMPLEX;
func C_Funcs D ->  set  means :Def9: :: VALUED_2:def 9
 for f being set holds
( f in it iff f is Function of D,COMPLEX );
existence
 ex b1 being set st
for f being set holds
( f in b1 iff f is Function of D,COMPLEX )
proof
consider X being set  such that
A1: for x being set holds
( x in X iff ( x in Funcs (D,COMPLEX) & S1[x] ) ) from XBOOLE_0:sch_1();
take  X ; ::_thesis: for f being set holds
( f in X iff f is Function of D,COMPLEX )
let f be set ; ::_thesis: ( f in X iff f is Function of D,COMPLEX )
thus  ( f in X implies f is Function of D,COMPLEX ) by A1; ::_thesis: ( f is Function of D,COMPLEX implies f in X )
assume A2: f is Function of D,COMPLEX ; ::_thesis: f in X
then  f in Funcs (D,COMPLEX) by FUNCT_2:8;
hence  f in X by A1, A2; ::_thesis: verum
end;
uniqueness
 for b1, b2 being set st ( for f being set holds
( f in b1 iff f is Function of D,COMPLEX ) ) & ( for f being set holds
( f in b2 iff f is Function of D,COMPLEX ) ) holds
b1 = b2
proof
let P, Q be set ; ::_thesis: ( ( for f being set holds
( f in P iff f is Function of D,COMPLEX ) ) & ( for f being set holds
( f in Q iff f is Function of D,COMPLEX ) ) implies P = Q )
assume  for f being set holds
( f in P iff f is Function of D,COMPLEX ) ; ::_thesis: ( ex f being set st
( ( f in Q implies f is Function of D,COMPLEX ) implies ( f is Function of D,COMPLEX & not f in Q ) ) or P = Q )
then A3: for f being set holds
( f in P iff S1[f] ) ;
assume  for f being set holds
( f in Q iff f is Function of D,COMPLEX ) ; ::_thesis: P = Q
then A4: for f being set holds
( f in Q iff S1[f] ) ;
thus  P = Q from XBOOLE_0:sch_2(A3, A4); ::_thesis: verum
end;
end;

:: deftheorem Def9 defines C_Funcs VALUED_2:def_9_:_
 for D, b2 being set holds
( b2 = C_Funcs D iff for f being set holds
( f in b2 iff f is Function of D,COMPLEX ) );

definition
set A = ExtREAL ;
let D be set ;
defpred S1[ set ] means $1 is PartFunc of D,ExtREAL;
func E_PFuncs D ->  set  means :Def10: :: VALUED_2:def 10
 for f being set holds
( f in it iff f is PartFunc of D,ExtREAL );
existence
 ex b1 being set st
for f being set holds
( f in b1 iff f is PartFunc of D,ExtREAL )
proof
consider X being set  such that
A1: for x being set holds
( x in X iff ( x in PFuncs (D,ExtREAL) & S1[x] ) ) from XBOOLE_0:sch_1();
take  X ; ::_thesis: for f being set holds
( f in X iff f is PartFunc of D,ExtREAL )
let f be set ; ::_thesis: ( f in X iff f is PartFunc of D,ExtREAL )
thus  ( f in X implies f is PartFunc of D,ExtREAL ) by A1; ::_thesis: ( f is PartFunc of D,ExtREAL implies f in X )
assume A2: f is PartFunc of D,ExtREAL ; ::_thesis: f in X
then  f in PFuncs (D,ExtREAL) by PARTFUN1:45;
hence  f in X by A1, A2; ::_thesis: verum
end;
uniqueness
 for b1, b2 being set st ( for f being set holds
( f in b1 iff f is PartFunc of D,ExtREAL ) ) & ( for f being set holds
( f in b2 iff f is PartFunc of D,ExtREAL ) ) holds
b1 = b2
proof
let P, Q be set ; ::_thesis: ( ( for f being set holds
( f in P iff f is PartFunc of D,ExtREAL ) ) & ( for f being set holds
( f in Q iff f is PartFunc of D,ExtREAL ) ) implies P = Q )
assume  for f being set holds
( f in P iff f is PartFunc of D,ExtREAL ) ; ::_thesis: ( ex f being set st
( ( f in Q implies f is PartFunc of D,ExtREAL ) implies ( f is PartFunc of D,ExtREAL & not f in Q ) ) or P = Q )
then A3: for f being set holds
( f in P iff S1[f] ) ;
assume  for f being set holds
( f in Q iff f is PartFunc of D,ExtREAL ) ; ::_thesis: P = Q
then A4: for f being set holds
( f in Q iff S1[f] ) ;
thus  P = Q from XBOOLE_0:sch_2(A3, A4); ::_thesis: verum
end;
end;

:: deftheorem Def10 defines E_PFuncs VALUED_2:def_10_:_
 for D, b2 being set holds
( b2 = E_PFuncs D iff for f being set holds
( f in b2 iff f is PartFunc of D,ExtREAL ) );

definition
set A = ExtREAL ;
let D be set ;
defpred S1[ set ] means $1 is Function of D,ExtREAL;
func E_Funcs D ->  set  means :Def11: :: VALUED_2:def 11
 for f being set holds
( f in it iff f is Function of D,ExtREAL );
existence
 ex b1 being set st
for f being set holds
( f in b1 iff f is Function of D,ExtREAL )
proof
consider X being set  such that
A1: for x being set holds
( x in X iff ( x in Funcs (D,ExtREAL) & S1[x] ) ) from XBOOLE_0:sch_1();
take  X ; ::_thesis: for f being set holds
( f in X iff f is Function of D,ExtREAL )
let f be set ; ::_thesis: ( f in X iff f is Function of D,ExtREAL )
thus  ( f in X implies f is Function of D,ExtREAL ) by A1; ::_thesis: ( f is Function of D,ExtREAL implies f in X )
assume A2: f is Function of D,ExtREAL ; ::_thesis: f in X
then  f in Funcs (D,ExtREAL) by FUNCT_2:8;
hence  f in X by A1, A2; ::_thesis: verum
end;
uniqueness
 for b1, b2 being set st ( for f being set holds
( f in b1 iff f is Function of D,ExtREAL ) ) & ( for f being set holds
( f in b2 iff f is Function of D,ExtREAL ) ) holds
b1 = b2
proof
let P, Q be set ; ::_thesis: ( ( for f being set holds
( f in P iff f is Function of D,ExtREAL ) ) & ( for f being set holds
( f in Q iff f is Function of D,ExtREAL ) ) implies P = Q )
assume  for f being set holds
( f in P iff f is Function of D,ExtREAL ) ; ::_thesis: ( ex f being set st
( ( f in Q implies f is Function of D,ExtREAL ) implies ( f is Function of D,ExtREAL & not f in Q ) ) or P = Q )
then A3: for f being set holds
( f in P iff S1[f] ) ;
assume  for f being set holds
( f in Q iff f is Function of D,ExtREAL ) ; ::_thesis: P = Q
then A4: for f being set holds
( f in Q iff S1[f] ) ;
thus  P = Q from XBOOLE_0:sch_2(A3, A4); ::_thesis: verum
end;
end;

:: deftheorem Def11 defines E_Funcs VALUED_2:def_11_:_
 for D, b2 being set holds
( b2 = E_Funcs D iff for f being set holds
( f in b2 iff f is Function of D,ExtREAL ) );

definition
set A = REAL ;
let D be set ;
defpred S1[ set ] means $1 is PartFunc of D,REAL;
func R_PFuncs D ->  set  means :Def12: :: VALUED_2:def 12
 for f being set holds
( f in it iff f is PartFunc of D,REAL );
existence
 ex b1 being set st
for f being set holds
( f in b1 iff f is PartFunc of D,REAL )
proof
consider X being set  such that
A1: for x being set holds
( x in X iff ( x in PFuncs (D,REAL) & S1[x] ) ) from XBOOLE_0:sch_1();
take  X ; ::_thesis: for f being set holds
( f in X iff f is PartFunc of D,REAL )
let f be set ; ::_thesis: ( f in X iff f is PartFunc of D,REAL )
thus  ( f in X implies f is PartFunc of D,REAL ) by A1; ::_thesis: ( f is PartFunc of D,REAL implies f in X )
assume A2: f is PartFunc of D,REAL ; ::_thesis: f in X
then  f in PFuncs (D,REAL) by PARTFUN1:45;
hence  f in X by A1, A2; ::_thesis: verum
end;
uniqueness
 for b1, b2 being set st ( for f being set holds
( f in b1 iff f is PartFunc of D,REAL ) ) & ( for f being set holds
( f in b2 iff f is PartFunc of D,REAL ) ) holds
b1 = b2
proof
let P, Q be set ; ::_thesis: ( ( for f being set holds
( f in P iff f is PartFunc of D,REAL ) ) & ( for f being set holds
( f in Q iff f is PartFunc of D,REAL ) ) implies P = Q )
assume  for f being set holds
( f in P iff f is PartFunc of D,REAL ) ; ::_thesis: ( ex f being set st
( ( f in Q implies f is PartFunc of D,REAL ) implies ( f is PartFunc of D,REAL & not f in Q ) ) or P = Q )
then A3: for f being set holds
( f in P iff S1[f] ) ;
assume  for f being set holds
( f in Q iff f is PartFunc of D,REAL ) ; ::_thesis: P = Q
then A4: for f being set holds
( f in Q iff S1[f] ) ;
thus  P = Q from XBOOLE_0:sch_2(A3, A4); ::_thesis: verum
end;
end;

:: deftheorem Def12 defines R_PFuncs VALUED_2:def_12_:_
 for D, b2 being set holds
( b2 = R_PFuncs D iff for f being set holds
( f in b2 iff f is PartFunc of D,REAL ) );

definition
set A = REAL ;
let D be set ;
defpred S1[ set ] means $1 is Function of D,REAL;
func R_Funcs D ->  set  means :Def13: :: VALUED_2:def 13
 for f being set holds
( f in it iff f is Function of D,REAL );
existence
 ex b1 being set st
for f being set holds
( f in b1 iff f is Function of D,REAL )
proof
consider X being set  such that
A1: for x being set holds
( x in X iff ( x in Funcs (D,REAL) & S1[x] ) ) from XBOOLE_0:sch_1();
take  X ; ::_thesis: for f being set holds
( f in X iff f is Function of D,REAL )
let f be set ; ::_thesis: ( f in X iff f is Function of D,REAL )
thus  ( f in X implies f is Function of D,REAL ) by A1; ::_thesis: ( f is Function of D,REAL implies f in X )
assume A2: f is Function of D,REAL ; ::_thesis: f in X
then  f in Funcs (D,REAL) by FUNCT_2:8;
hence  f in X by A1, A2; ::_thesis: verum
end;
uniqueness
 for b1, b2 being set st ( for f being set holds
( f in b1 iff f is Function of D,REAL ) ) & ( for f being set holds
( f in b2 iff f is Function of D,REAL ) ) holds
b1 = b2
proof
let P, Q be set ; ::_thesis: ( ( for f being set holds
( f in P iff f is Function of D,REAL ) ) & ( for f being set holds
( f in Q iff f is Function of D,REAL ) ) implies P = Q )
assume  for f being set holds
( f in P iff f is Function of D,REAL ) ; ::_thesis: ( ex f being set st
( ( f in Q implies f is Function of D,REAL ) implies ( f is Function of D,REAL & not f in Q ) ) or P = Q )
then A3: for f being set holds
( f in P iff S1[f] ) ;
assume  for f being set holds
( f in Q iff f is Function of D,REAL ) ; ::_thesis: P = Q
then A4: for f being set holds
( f in Q iff S1[f] ) ;
thus  P = Q from XBOOLE_0:sch_2(A3, A4); ::_thesis: verum
end;
end;

:: deftheorem Def13 defines R_Funcs VALUED_2:def_13_:_
 for D, b2 being set holds
( b2 = R_Funcs D iff for f being set holds
( f in b2 iff f is Function of D,REAL ) );

definition
set A = RAT ;
let D be set ;
defpred S1[ set ] means $1 is PartFunc of D,RAT;
func Q_PFuncs D ->  set  means :Def14: :: VALUED_2:def 14
 for f being set holds
( f in it iff f is PartFunc of D,RAT );
existence
 ex b1 being set st
for f being set holds
( f in b1 iff f is PartFunc of D,RAT )
proof
consider X being set  such that
A1: for x being set holds
( x in X iff ( x in PFuncs (D,RAT) & S1[x] ) ) from XBOOLE_0:sch_1();
take  X ; ::_thesis: for f being set holds
( f in X iff f is PartFunc of D,RAT )
let f be set ; ::_thesis: ( f in X iff f is PartFunc of D,RAT )
thus  ( f in X implies f is PartFunc of D,RAT ) by A1; ::_thesis: ( f is PartFunc of D,RAT implies f in X )
assume A2: f is PartFunc of D,RAT ; ::_thesis: f in X
then  f in PFuncs (D,RAT) by PARTFUN1:45;
hence  f in X by A1, A2; ::_thesis: verum
end;
uniqueness
 for b1, b2 being set st ( for f being set holds
( f in b1 iff f is PartFunc of D,RAT ) ) & ( for f being set holds
( f in b2 iff f is PartFunc of D,RAT ) ) holds
b1 = b2
proof
let P, Q be set ; ::_thesis: ( ( for f being set holds
( f in P iff f is PartFunc of D,RAT ) ) & ( for f being set holds
( f in Q iff f is PartFunc of D,RAT ) ) implies P = Q )
assume  for f being set holds
( f in P iff f is PartFunc of D,RAT ) ; ::_thesis: ( ex f being set st
( ( f in Q implies f is PartFunc of D,RAT ) implies ( f is PartFunc of D,RAT & not f in Q ) ) or P = Q )
then A3: for f being set holds
( f in P iff S1[f] ) ;
assume  for f being set holds
( f in Q iff f is PartFunc of D,RAT ) ; ::_thesis: P = Q
then A4: for f being set holds
( f in Q iff S1[f] ) ;
thus  P = Q from XBOOLE_0:sch_2(A3, A4); ::_thesis: verum
end;
end;

:: deftheorem Def14 defines Q_PFuncs VALUED_2:def_14_:_
 for D, b2 being set holds
( b2 = Q_PFuncs D iff for f being set holds
( f in b2 iff f is PartFunc of D,RAT ) );

definition
set A = RAT ;
let D be set ;
defpred S1[ set ] means $1 is Function of D,RAT;
func Q_Funcs D ->  set  means :Def15: :: VALUED_2:def 15
 for f being set holds
( f in it iff f is Function of D,RAT );
existence
 ex b1 being set st
for f being set holds
( f in b1 iff f is Function of D,RAT )
proof
consider X being set  such that
A1: for x being set holds
( x in X iff ( x in Funcs (D,RAT) & S1[x] ) ) from XBOOLE_0:sch_1();
take  X ; ::_thesis: for f being set holds
( f in X iff f is Function of D,RAT )
let f be set ; ::_thesis: ( f in X iff f is Function of D,RAT )
thus  ( f in X implies f is Function of D,RAT ) by A1; ::_thesis: ( f is Function of D,RAT implies f in X )
assume A2: f is Function of D,RAT ; ::_thesis: f in X
then  f in Funcs (D,RAT) by FUNCT_2:8;
hence  f in X by A1, A2; ::_thesis: verum
end;
uniqueness
 for b1, b2 being set st ( for f being set holds
( f in b1 iff f is Function of D,RAT ) ) & ( for f being set holds
( f in b2 iff f is Function of D,RAT ) ) holds
b1 = b2
proof
let P, Q be set ; ::_thesis: ( ( for f being set holds
( f in P iff f is Function of D,RAT ) ) & ( for f being set holds
( f in Q iff f is Function of D,RAT ) ) implies P = Q )
assume  for f being set holds
( f in P iff f is Function of D,RAT ) ; ::_thesis: ( ex f being set st
( ( f in Q implies f is Function of D,RAT ) implies ( f is Function of D,RAT & not f in Q ) ) or P = Q )
then A3: for f being set holds
( f in P iff S1[f] ) ;
assume  for f being set holds
( f in Q iff f is Function of D,RAT ) ; ::_thesis: P = Q
then A4: for f being set holds
( f in Q iff S1[f] ) ;
thus  P = Q from XBOOLE_0:sch_2(A3, A4); ::_thesis: verum
end;
end;

:: deftheorem Def15 defines Q_Funcs VALUED_2:def_15_:_
 for D, b2 being set holds
( b2 = Q_Funcs D iff for f being set holds
( f in b2 iff f is Function of D,RAT ) );

definition
set A = INT ;
let D be set ;
defpred S1[ set ] means $1 is PartFunc of D,INT;
func I_PFuncs D ->  set  means :Def16: :: VALUED_2:def 16
 for f being set holds
( f in it iff f is PartFunc of D,INT );
existence
 ex b1 being set st
for f being set holds
( f in b1 iff f is PartFunc of D,INT )
proof
consider X being set  such that
A1: for x being set holds
( x in X iff ( x in PFuncs (D,INT) & S1[x] ) ) from XBOOLE_0:sch_1();
take  X ; ::_thesis: for f being set holds
( f in X iff f is PartFunc of D,INT )
let f be set ; ::_thesis: ( f in X iff f is PartFunc of D,INT )
thus  ( f in X implies f is PartFunc of D,INT ) by A1; ::_thesis: ( f is PartFunc of D,INT implies f in X )
assume A2: f is PartFunc of D,INT ; ::_thesis: f in X
then  f in PFuncs (D,INT) by PARTFUN1:45;
hence  f in X by A1, A2; ::_thesis: verum
end;
uniqueness
 for b1, b2 being set st ( for f being set holds
( f in b1 iff f is PartFunc of D,INT ) ) & ( for f being set holds
( f in b2 iff f is PartFunc of D,INT ) ) holds
b1 = b2
proof
let P, Q be set ; ::_thesis: ( ( for f being set holds
( f in P iff f is PartFunc of D,INT ) ) & ( for f being set holds
( f in Q iff f is PartFunc of D,INT ) ) implies P = Q )
assume  for f being set holds
( f in P iff f is PartFunc of D,INT ) ; ::_thesis: ( ex f being set st
( ( f in Q implies f is PartFunc of D,INT ) implies ( f is PartFunc of D,INT & not f in Q ) ) or P = Q )
then A3: for f being set holds
( f in P iff S1[f] ) ;
assume  for f being set holds
( f in Q iff f is PartFunc of D,INT ) ; ::_thesis: P = Q
then A4: for f being set holds
( f in Q iff S1[f] ) ;
thus  P = Q from XBOOLE_0:sch_2(A3, A4); ::_thesis: verum
end;
end;

:: deftheorem Def16 defines I_PFuncs VALUED_2:def_16_:_
 for D, b2 being set holds
( b2 = I_PFuncs D iff for f being set holds
( f in b2 iff f is PartFunc of D,INT ) );

definition
set A = INT ;
let D be set ;
defpred S1[ set ] means $1 is Function of D,INT;
func I_Funcs D ->  set  means :Def17: :: VALUED_2:def 17
 for f being set holds
( f in it iff f is Function of D,INT );
existence
 ex b1 being set st
for f being set holds
( f in b1 iff f is Function of D,INT )
proof
consider X being set  such that
A1: for x being set holds
( x in X iff ( x in Funcs (D,INT) & S1[x] ) ) from XBOOLE_0:sch_1();
take  X ; ::_thesis: for f being set holds
( f in X iff f is Function of D,INT )
let f be set ; ::_thesis: ( f in X iff f is Function of D,INT )
thus  ( f in X implies f is Function of D,INT ) by A1; ::_thesis: ( f is Function of D,INT implies f in X )
assume A2: f is Function of D,INT ; ::_thesis: f in X
then  f in Funcs (D,INT) by FUNCT_2:8;
hence  f in X by A1, A2; ::_thesis: verum
end;
uniqueness
 for b1, b2 being set st ( for f being set holds
( f in b1 iff f is Function of D,INT ) ) & ( for f being set holds
( f in b2 iff f is Function of D,INT ) ) holds
b1 = b2
proof
let P, Q be set ; ::_thesis: ( ( for f being set holds
( f in P iff f is Function of D,INT ) ) & ( for f being set holds
( f in Q iff f is Function of D,INT ) ) implies P = Q )
assume  for f being set holds
( f in P iff f is Function of D,INT ) ; ::_thesis: ( ex f being set st
( ( f in Q implies f is Function of D,INT ) implies ( f is Function of D,INT & not f in Q ) ) or P = Q )
then A3: for f being set holds
( f in P iff S1[f] ) ;
assume  for f being set holds
( f in Q iff f is Function of D,INT ) ; ::_thesis: P = Q
then A4: for f being set holds
( f in Q iff S1[f] ) ;
thus  P = Q from XBOOLE_0:sch_2(A3, A4); ::_thesis: verum
end;
end;

:: deftheorem Def17 defines I_Funcs VALUED_2:def_17_:_
 for D, b2 being set holds
( b2 = I_Funcs D iff for f being set holds
( f in b2 iff f is Function of D,INT ) );

definition
set A = NAT ;
let D be set ;
defpred S1[ set ] means $1 is PartFunc of D,NAT;
func N_PFuncs D ->  set  means :Def18: :: VALUED_2:def 18
 for f being set holds
( f in it iff f is PartFunc of D,NAT );
existence
 ex b1 being set st
for f being set holds
( f in b1 iff f is PartFunc of D,NAT )
proof
consider X being set  such that
A1: for x being set holds
( x in X iff ( x in PFuncs (D,NAT) & S1[x] ) ) from XBOOLE_0:sch_1();
take  X ; ::_thesis: for f being set holds
( f in X iff f is PartFunc of D,NAT )
let f be set ; ::_thesis: ( f in X iff f is PartFunc of D,NAT )
thus  ( f in X implies f is PartFunc of D,NAT ) by A1; ::_thesis: ( f is PartFunc of D,NAT implies f in X )
assume A2: f is PartFunc of D,NAT ; ::_thesis: f in X
then  f in PFuncs (D,NAT) by PARTFUN1:45;
hence  f in X by A1, A2; ::_thesis: verum
end;
uniqueness
 for b1, b2 being set st ( for f being set holds
( f in b1 iff f is PartFunc of D,NAT ) ) & ( for f being set holds
( f in b2 iff f is PartFunc of D,NAT ) ) holds
b1 = b2
proof
let P, Q be set ; ::_thesis: ( ( for f being set holds
( f in P iff f is PartFunc of D,NAT ) ) & ( for f being set holds
( f in Q iff f is PartFunc of D,NAT ) ) implies P = Q )
assume  for f being set holds
( f in P iff f is PartFunc of D,NAT ) ; ::_thesis: ( ex f being set st
( ( f in Q implies f is PartFunc of D,NAT ) implies ( f is PartFunc of D,NAT & not f in Q ) ) or P = Q )
then A3: for f being set holds
( f in P iff S1[f] ) ;
assume  for f being set holds
( f in Q iff f is PartFunc of D,NAT ) ; ::_thesis: P = Q
then A4: for f being set holds
( f in Q iff S1[f] ) ;
thus  P = Q from XBOOLE_0:sch_2(A3, A4); ::_thesis: verum
end;
end;

:: deftheorem Def18 defines N_PFuncs VALUED_2:def_18_:_
 for D, b2 being set holds
( b2 = N_PFuncs D iff for f being set holds
( f in b2 iff f is PartFunc of D,NAT ) );

definition
set A = NAT ;
let D be set ;
defpred S1[ set ] means $1 is Function of D,NAT;
func N_Funcs D ->  set  means :Def19: :: VALUED_2:def 19
 for f being set holds
( f in it iff f is Function of D,NAT );
existence
 ex b1 being set st
for f being set holds
( f in b1 iff f is Function of D,NAT )
proof
consider X being set  such that
A1: for x being set holds
( x in X iff ( x in Funcs (D,NAT) & S1[x] ) ) from XBOOLE_0:sch_1();
take  X ; ::_thesis: for f being set holds
( f in X iff f is Function of D,NAT )
let f be set ; ::_thesis: ( f in X iff f is Function of D,NAT )
thus  ( f in X implies f is Function of D,NAT ) by A1; ::_thesis: ( f is Function of D,NAT implies f in X )
assume A2: f is Function of D,NAT ; ::_thesis: f in X
then  f in Funcs (D,NAT) by FUNCT_2:8;
hence  f in X by A1, A2; ::_thesis: verum
end;
uniqueness
 for b1, b2 being set st ( for f being set holds
( f in b1 iff f is Function of D,NAT ) ) & ( for f being set holds
( f in b2 iff f is Function of D,NAT ) ) holds
b1 = b2
proof
let P, Q be set ; ::_thesis: ( ( for f being set holds
( f in P iff f is Function of D,NAT ) ) & ( for f being set holds
( f in Q iff f is Function of D,NAT ) ) implies P = Q )
assume  for f being set holds
( f in P iff f is Function of D,NAT ) ; ::_thesis: ( ex f being set st
( ( f in Q implies f is Function of D,NAT ) implies ( f is Function of D,NAT & not f in Q ) ) or P = Q )
then A3: for f being set holds
( f in P iff S1[f] ) ;
assume  for f being set holds
( f in Q iff f is Function of D,NAT ) ; ::_thesis: P = Q
then A4: for f being set holds
( f in Q iff S1[f] ) ;
thus  P = Q from XBOOLE_0:sch_2(A3, A4); ::_thesis: verum
end;
end;

:: deftheorem Def19 defines N_Funcs VALUED_2:def_19_:_
 for D, b2 being set holds
( b2 = N_Funcs D iff for f being set holds
( f in b2 iff f is Function of D,NAT ) );

theorem Th1: :: VALUED_2:1
for X being set holds C_Funcs X is Subset of (C_PFuncs X)
proof
let X be set ; ::_thesis: C_Funcs X is Subset of (C_PFuncs X)
 C_Funcs X c= C_PFuncs X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in C_Funcs X or x in C_PFuncs X )
assume  x in C_Funcs X ; ::_thesis: x in C_PFuncs X
then  x is Function of X,COMPLEX by Def9;
hence  x in C_PFuncs X by Def8; ::_thesis: verum
end;
hence  C_Funcs X is Subset of (C_PFuncs X) ; ::_thesis: verum
end;

theorem Th2: :: VALUED_2:2
for X being set holds E_Funcs X is Subset of (E_PFuncs X)
proof
let X be set ; ::_thesis: E_Funcs X is Subset of (E_PFuncs X)
 E_Funcs X c= E_PFuncs X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in E_Funcs X or x in E_PFuncs X )
assume  x in E_Funcs X ; ::_thesis: x in E_PFuncs X
then  x is Function of X,ExtREAL by Def11;
hence  x in E_PFuncs X by Def10; ::_thesis: verum
end;
hence  E_Funcs X is Subset of (E_PFuncs X) ; ::_thesis: verum
end;

theorem Th3: :: VALUED_2:3
for X being set holds R_Funcs X is Subset of (R_PFuncs X)
proof
let X be set ; ::_thesis: R_Funcs X is Subset of (R_PFuncs X)
 R_Funcs X c= R_PFuncs X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in R_Funcs X or x in R_PFuncs X )
assume  x in R_Funcs X ; ::_thesis: x in R_PFuncs X
then  x is Function of X,REAL by Def13;
hence  x in R_PFuncs X by Def12; ::_thesis: verum
end;
hence  R_Funcs X is Subset of (R_PFuncs X) ; ::_thesis: verum
end;

theorem Th4: :: VALUED_2:4
for X being set holds Q_Funcs X is Subset of (Q_PFuncs X)
proof
let X be set ; ::_thesis: Q_Funcs X is Subset of (Q_PFuncs X)
 Q_Funcs X c= Q_PFuncs X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Q_Funcs X or x in Q_PFuncs X )
assume  x in Q_Funcs X ; ::_thesis: x in Q_PFuncs X
then  x is Function of X,RAT by Def15;
hence  x in Q_PFuncs X by Def14; ::_thesis: verum
end;
hence  Q_Funcs X is Subset of (Q_PFuncs X) ; ::_thesis: verum
end;

theorem Th5: :: VALUED_2:5
for X being set holds I_Funcs X is Subset of (I_PFuncs X)
proof
let X be set ; ::_thesis: I_Funcs X is Subset of (I_PFuncs X)
 I_Funcs X c= I_PFuncs X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in I_Funcs X or x in I_PFuncs X )
assume  x in I_Funcs X ; ::_thesis: x in I_PFuncs X
then  x is Function of X,INT by Def17;
hence  x in I_PFuncs X by Def16; ::_thesis: verum
end;
hence  I_Funcs X is Subset of (I_PFuncs X) ; ::_thesis: verum
end;

theorem Th6: :: VALUED_2:6
for X being set holds N_Funcs X is Subset of (N_PFuncs X)
proof
let X be set ; ::_thesis: N_Funcs X is Subset of (N_PFuncs X)
 N_Funcs X c= N_PFuncs X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in N_Funcs X or x in N_PFuncs X )
assume  x in N_Funcs X ; ::_thesis: x in N_PFuncs X
then  x is Function of X,NAT by Def19;
hence  x in N_PFuncs X by Def18; ::_thesis: verum
end;
hence  N_Funcs X is Subset of (N_PFuncs X) ; ::_thesis: verum
end;

registration
let X be set ;
cluster C_PFuncs X ->  complex-functions-membered ;
coherence
 C_PFuncs X is complex-functions-membered
proof
let x be set ; :: according to VALUED_2:def_2 ::_thesis: ( x in C_PFuncs X implies x is complex-valued Function )
thus  ( x in C_PFuncs X implies x is complex-valued Function ) by Def8; ::_thesis: verum
end;
cluster C_Funcs X ->  complex-functions-membered ;
coherence
 C_Funcs X is complex-functions-membered
proof
reconsider C = C_Funcs X as Subset of (C_PFuncs X) by Th1;
 C is complex-functions-membered ;
hence  C_Funcs X is complex-functions-membered ; ::_thesis: verum
end;
cluster E_PFuncs X ->  ext-real-functions-membered ;
coherence
 E_PFuncs X is ext-real-functions-membered
proof
let x be set ; :: according to VALUED_2:def_3 ::_thesis: ( x in E_PFuncs X implies x is ext-real-valued Function )
thus  ( x in E_PFuncs X implies x is ext-real-valued Function ) by Def10; ::_thesis: verum
end;
cluster E_Funcs X ->  ext-real-functions-membered ;
coherence
 E_Funcs X is ext-real-functions-membered
proof
reconsider C = E_Funcs X as Subset of (E_PFuncs X) by Th2;
 C is ext-real-functions-membered ;
hence  E_Funcs X is ext-real-functions-membered ; ::_thesis: verum
end;
cluster R_PFuncs X ->  real-functions-membered ;
coherence
 R_PFuncs X is real-functions-membered
proof
let x be set ; :: according to VALUED_2:def_4 ::_thesis: ( x in R_PFuncs X implies x is real-valued Function )
thus  ( x in R_PFuncs X implies x is real-valued Function ) by Def12; ::_thesis: verum
end;
cluster R_Funcs X ->  real-functions-membered ;
coherence
 R_Funcs X is real-functions-membered
proof
reconsider C = R_Funcs X as Subset of (R_PFuncs X) by Th3;
 C is real-functions-membered ;
hence  R_Funcs X is real-functions-membered ; ::_thesis: verum
end;
cluster Q_PFuncs X ->  rational-functions-membered ;
coherence
 Q_PFuncs X is rational-functions-membered
proof
let x be set ; :: according to VALUED_2:def_5 ::_thesis: ( x in Q_PFuncs X implies x is RAT -valued Function )
thus  ( x in Q_PFuncs X implies x is RAT -valued Function ) by Def14; ::_thesis: verum
end;
cluster Q_Funcs X ->  rational-functions-membered ;
coherence
 Q_Funcs X is rational-functions-membered
proof
reconsider C = Q_Funcs X as Subset of (Q_PFuncs X) by Th4;
 C is rational-functions-membered ;
hence  Q_Funcs X is rational-functions-membered ; ::_thesis: verum
end;
cluster I_PFuncs X ->  integer-functions-membered ;
coherence
 I_PFuncs X is integer-functions-membered
proof
let x be set ; :: according to VALUED_2:def_6 ::_thesis: ( x in I_PFuncs X implies x is INT -valued Function )
thus  ( x in I_PFuncs X implies x is INT -valued Function ) by Def16; ::_thesis: verum
end;
cluster I_Funcs X ->  integer-functions-membered ;
coherence
 I_Funcs X is integer-functions-membered
proof
reconsider C = I_Funcs X as Subset of (I_PFuncs X) by Th5;
 C is integer-functions-membered ;
hence  I_Funcs X is integer-functions-membered ; ::_thesis: verum
end;
cluster N_PFuncs X ->  natural-functions-membered ;
coherence
 N_PFuncs X is natural-functions-membered
proof
let x be set ; :: according to VALUED_2:def_7 ::_thesis: ( x in N_PFuncs X implies x is natural-valued Function )
thus  ( x in N_PFuncs X implies x is natural-valued Function ) by Def18; ::_thesis: verum
end;
cluster N_Funcs X ->  natural-functions-membered ;
coherence
 N_Funcs X is natural-functions-membered
proof
reconsider C = N_Funcs X as Subset of (N_PFuncs X) by Th6;
 C is natural-functions-membered ;
hence  N_Funcs X is natural-functions-membered ; ::_thesis: verum
end;
end;

registration
let X be complex-functions-membered set ;
cluster  ->  complex-valued for Element of X;
coherence
 for b1 being Element of X holds b1 is complex-valued
proof
 ( X is empty or not X is empty ) ;
hence  for b1 being Element of X holds b1 is complex-valued by Def2, SUBSET_1:def_1; ::_thesis: verum
end;
end;

registration
let X be ext-real-functions-membered set ;
cluster  ->  ext-real-valued for Element of X;
coherence
 for b1 being Element of X holds b1 is ext-real-valued
proof
 ( X is empty or not X is empty ) ;
hence  for b1 being Element of X holds b1 is ext-real-valued by Def3, SUBSET_1:def_1; ::_thesis: verum
end;
end;

registration
let X be real-functions-membered set ;
cluster  ->  real-valued for Element of X;
coherence
 for b1 being Element of X holds b1 is real-valued
proof
 ( X is empty or not X is empty ) ;
hence  for b1 being Element of X holds b1 is real-valued by Def4, SUBSET_1:def_1; ::_thesis: verum
end;
end;

registration
let X be rational-functions-membered set ;
cluster  ->  RAT -valued for Element of X;
coherence
 for b1 being Element of X holds b1 is RAT -valued
proof
 ( X is empty or not X is empty ) ;
hence  for b1 being Element of X holds b1 is RAT -valued by Def5, SUBSET_1:def_1; ::_thesis: verum
end;
end;

registration
let X be integer-functions-membered set ;
cluster  ->  INT -valued for Element of X;
coherence
 for b1 being Element of X holds b1 is INT -valued
proof
 ( X is empty or not X is empty ) ;
hence  for b1 being Element of X holds b1 is INT -valued by Def6, SUBSET_1:def_1; ::_thesis: verum
end;
end;

registration
let X be natural-functions-membered set ;
cluster  ->  natural-valued for Element of X;
coherence
 for b1 being Element of X holds b1 is natural-valued
proof
 ( X is empty or not X is empty ) ;
hence  for b1 being Element of X holds b1 is natural-valued by Def7, SUBSET_1:def_1; ::_thesis: verum
end;
end;

registration
let X, x be set ;
let Y be complex-functions-membered set ;
let f be PartFunc of X,Y;
clusterf . x ->  Relation-like Function-like ;
coherence
 ( f . x is Function-like & f . x is Relation-like ) ;
end;

registration
let X, x be set ;
let Y be ext-real-functions-membered set ;
let f be PartFunc of X,Y;
clusterf . x ->  Relation-like Function-like ;
coherence
 ( f . x is Function-like & f . x is Relation-like ) ;
end;

registration
let X, x be set ;
let Y be complex-functions-membered set ;
let f be PartFunc of X,Y;
clusterf . x ->  complex-valued ;
coherence
 f . x is complex-valued
proof
percases ( x in dom f or not x in dom f ) ;
suppose x in dom f ; ::_thesis: f . x is complex-valued
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is complex-valued ; ::_thesis: verum
end;
suppose not x in dom f ; ::_thesis: f . x is complex-valued
hence  f . x is complex-valued by FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
end;

registration
let X, x be set ;
let Y be ext-real-functions-membered set ;
let f be PartFunc of X,Y;
clusterf . x ->  ext-real-valued ;
coherence
 f . x is ext-real-valued
proof
percases ( x in dom f or not x in dom f ) ;
suppose x in dom f ; ::_thesis: f . x is ext-real-valued
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is ext-real-valued ; ::_thesis: verum
end;
suppose not x in dom f ; ::_thesis: f . x is ext-real-valued
hence  f . x is ext-real-valued by FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
end;

registration
let X, x be set ;
let Y be real-functions-membered set ;
let f be PartFunc of X,Y;
clusterf . x ->  real-valued ;
coherence
 f . x is real-valued
proof
percases ( x in dom f or not x in dom f ) ;
suppose x in dom f ; ::_thesis: f . x is real-valued
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is real-valued ; ::_thesis: verum
end;
suppose not x in dom f ; ::_thesis: f . x is real-valued
hence  f . x is real-valued by FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
end;

registration
let X, x be set ;
let Y be rational-functions-membered set ;
let f be PartFunc of X,Y;
clusterf . x ->  RAT -valued ;
coherence
 f . x is RAT -valued
proof
percases ( x in dom f or not x in dom f ) ;
suppose x in dom f ; ::_thesis: f . x is RAT -valued
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is RAT -valued ; ::_thesis: verum
end;
suppose not x in dom f ; ::_thesis: f . x is RAT -valued
hence  f . x is RAT -valued by FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
end;

registration
let X, x be set ;
let Y be integer-functions-membered set ;
let f be PartFunc of X,Y;
clusterf . x ->  INT -valued ;
coherence
 f . x is INT -valued
proof
percases ( x in dom f or not x in dom f ) ;
suppose x in dom f ; ::_thesis: f . x is INT -valued
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is INT -valued ; ::_thesis: verum
end;
suppose not x in dom f ; ::_thesis: f . x is INT -valued
hence  f . x is INT -valued by FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
end;

registration
let X, x be set ;
let Y be natural-functions-membered set ;
let f be PartFunc of X,Y;
clusterf . x ->  natural-valued ;
coherence
 f . x is natural-valued
proof
percases ( x in dom f or not x in dom f ) ;
suppose x in dom f ; ::_thesis: f . x is natural-valued
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is natural-valued ; ::_thesis: verum
end;
suppose not x in dom f ; ::_thesis: f . x is natural-valued
hence  f . x is natural-valued by FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
end;

registration
let X be set ;
let Y be complex-membered set ;
cluster PFuncs (X,Y) ->  complex-functions-membered ;
coherence
 PFuncs (X,Y) is complex-functions-membered
proof
let x be set ; :: according to VALUED_2:def_2 ::_thesis: ( x in PFuncs (X,Y) implies x is complex-valued Function )
assume  x in PFuncs (X,Y) ; ::_thesis: x is complex-valued Function
then consider f being Function such that
A1: x = f and
A2: ( dom f c= X & rng f c= Y ) by PARTFUN1:def_3;
reconsider f = f as PartFunc of X,Y by A2, RELSET_1:4;
 f is set ;
hence  x is complex-valued Function by A1; ::_thesis: verum
end;
end;

registration
let X be set ;
let Y be ext-real-membered set ;
cluster PFuncs (X,Y) ->  ext-real-functions-membered ;
coherence
 PFuncs (X,Y) is ext-real-functions-membered
proof
let x be set ; :: according to VALUED_2:def_3 ::_thesis: ( x in PFuncs (X,Y) implies x is ext-real-valued Function )
assume  x in PFuncs (X,Y) ; ::_thesis: x is ext-real-valued Function
then consider f being Function such that
A1: x = f and
A2: ( dom f c= X & rng f c= Y ) by PARTFUN1:def_3;
reconsider f = f as PartFunc of X,Y by A2, RELSET_1:4;
 f is set ;
hence  x is ext-real-valued Function by A1; ::_thesis: verum
end;
end;

registration
let X be set ;
let Y be real-membered set ;
cluster PFuncs (X,Y) ->  real-functions-membered ;
coherence
 PFuncs (X,Y) is real-functions-membered
proof
let x be set ; :: according to VALUED_2:def_4 ::_thesis: ( x in PFuncs (X,Y) implies x is real-valued Function )
assume  x in PFuncs (X,Y) ; ::_thesis: x is real-valued Function
then consider f being Function such that
A1: x = f and
A2: ( dom f c= X & rng f c= Y ) by PARTFUN1:def_3;
reconsider f = f as PartFunc of X,Y by A2, RELSET_1:4;
 f is set ;
hence  x is real-valued Function by A1; ::_thesis: verum
end;
end;

registration
let X be set ;
let Y be rational-membered set ;
cluster PFuncs (X,Y) ->  rational-functions-membered ;
coherence
 PFuncs (X,Y) is rational-functions-membered
proof
let x be set ; :: according to VALUED_2:def_5 ::_thesis: ( x in PFuncs (X,Y) implies x is RAT -valued Function )
assume  x in PFuncs (X,Y) ; ::_thesis: x is RAT -valued Function
then consider f being Function such that
A1: x = f and
A2: ( dom f c= X & rng f c= Y ) by PARTFUN1:def_3;
reconsider f = f as PartFunc of X,Y by A2, RELSET_1:4;
 f is set ;
hence  x is RAT -valued Function by A1; ::_thesis: verum
end;
end;

registration
let X be set ;
let Y be integer-membered set ;
cluster PFuncs (X,Y) ->  integer-functions-membered ;
coherence
 PFuncs (X,Y) is integer-functions-membered
proof
let x be set ; :: according to VALUED_2:def_6 ::_thesis: ( x in PFuncs (X,Y) implies x is INT -valued Function )
assume  x in PFuncs (X,Y) ; ::_thesis: x is INT -valued Function
then consider f being Function such that
A1: x = f and
A2: ( dom f c= X & rng f c= Y ) by PARTFUN1:def_3;
reconsider f = f as PartFunc of X,Y by A2, RELSET_1:4;
 f is set ;
hence  x is INT -valued Function by A1; ::_thesis: verum
end;
end;

registration
let X be set ;
let Y be natural-membered set ;
cluster PFuncs (X,Y) ->  natural-functions-membered ;
coherence
 PFuncs (X,Y) is natural-functions-membered
proof
let x be set ; :: according to VALUED_2:def_7 ::_thesis: ( x in PFuncs (X,Y) implies x is natural-valued Function )
assume  x in PFuncs (X,Y) ; ::_thesis: x is natural-valued Function
then consider f being Function such that
A1: x = f and
A2: ( dom f c= X & rng f c= Y ) by PARTFUN1:def_3;
reconsider f = f as PartFunc of X,Y by A2, RELSET_1:4;
 f is set ;
hence  x is natural-valued Function by A1; ::_thesis: verum
end;
end;

registration
let X be set ;
let Y be complex-membered set ;
cluster Funcs (X,Y) ->  complex-functions-membered ;
coherence
 Funcs (X,Y) is complex-functions-membered
proof
let x be set ; :: according to VALUED_2:def_2 ::_thesis: ( x in Funcs (X,Y) implies x is complex-valued Function )
assume  x in Funcs (X,Y) ; ::_thesis: x is complex-valued Function
then consider f being Function such that
A1: x = f and
A2: ( dom f = X & rng f c= Y ) by FUNCT_2:def_2;
reconsider f = f as PartFunc of X,Y by A2, RELSET_1:4;
 f is set ;
hence  x is complex-valued Function by A1; ::_thesis: verum
end;
end;

registration
let X be set ;
let Y be ext-real-membered set ;
cluster Funcs (X,Y) ->  ext-real-functions-membered ;
coherence
 Funcs (X,Y) is ext-real-functions-membered
proof
let x be set ; :: according to VALUED_2:def_3 ::_thesis: ( x in Funcs (X,Y) implies x is ext-real-valued Function )
assume  x in Funcs (X,Y) ; ::_thesis: x is ext-real-valued Function
then consider f being Function such that
A1: x = f and
A2: ( dom f = X & rng f c= Y ) by FUNCT_2:def_2;
reconsider f = f as PartFunc of X,Y by A2, RELSET_1:4;
 f is set ;
hence  x is ext-real-valued Function by A1; ::_thesis: verum
end;
end;

registration
let X be set ;
let Y be real-membered set ;
cluster Funcs (X,Y) ->  real-functions-membered ;
coherence
 Funcs (X,Y) is real-functions-membered
proof
let x be set ; :: according to VALUED_2:def_4 ::_thesis: ( x in Funcs (X,Y) implies x is real-valued Function )
assume  x in Funcs (X,Y) ; ::_thesis: x is real-valued Function
then consider f being Function such that
A1: x = f and
A2: ( dom f = X & rng f c= Y ) by FUNCT_2:def_2;
reconsider f = f as PartFunc of X,Y by A2, RELSET_1:4;
 f is set ;
hence  x is real-valued Function by A1; ::_thesis: verum
end;
end;

registration
let X be set ;
let Y be rational-membered set ;
cluster Funcs (X,Y) ->  rational-functions-membered ;
coherence
 Funcs (X,Y) is rational-functions-membered
proof
let x be set ; :: according to VALUED_2:def_5 ::_thesis: ( x in Funcs (X,Y) implies x is RAT -valued Function )
assume  x in Funcs (X,Y) ; ::_thesis: x is RAT -valued Function
then consider f being Function such that
A1: x = f and
A2: ( dom f = X & rng f c= Y ) by FUNCT_2:def_2;
reconsider f = f as PartFunc of X,Y by A2, RELSET_1:4;
 f is set ;
hence  x is RAT -valued Function by A1; ::_thesis: verum
end;
end;

registration
let X be set ;
let Y be integer-membered set ;
cluster Funcs (X,Y) ->  integer-functions-membered ;
coherence
 Funcs (X,Y) is integer-functions-membered
proof
let x be set ; :: according to VALUED_2:def_6 ::_thesis: ( x in Funcs (X,Y) implies x is INT -valued Function )
assume  x in Funcs (X,Y) ; ::_thesis: x is INT -valued Function
then consider f being Function such that
A1: x = f and
A2: ( dom f = X & rng f c= Y ) by FUNCT_2:def_2;
reconsider f = f as PartFunc of X,Y by A2, RELSET_1:4;
 f is set ;
hence  x is INT -valued Function by A1; ::_thesis: verum
end;
end;

registration
let X be set ;
let Y be natural-membered set ;
cluster Funcs (X,Y) ->  natural-functions-membered ;
coherence
 Funcs (X,Y) is natural-functions-membered
proof
let x be set ; :: according to VALUED_2:def_7 ::_thesis: ( x in Funcs (X,Y) implies x is natural-valued Function )
assume  x in Funcs (X,Y) ; ::_thesis: x is natural-valued Function
then consider f being Function such that
A1: x = f and
A2: ( dom f = X & rng f c= Y ) by FUNCT_2:def_2;
reconsider f = f as PartFunc of X,Y by A2, RELSET_1:4;
 f is set ;
hence  x is natural-valued Function by A1; ::_thesis: verum
end;
end;

definition
let R be Relation;
attrR is complex-functions-valued  means :Def20: :: VALUED_2:def 20
 rng R is complex-functions-membered ;
attrR is ext-real-functions-valued  means :Def21: :: VALUED_2:def 21
 rng R is ext-real-functions-membered ;
attrR is real-functions-valued  means :Def22: :: VALUED_2:def 22
 rng R is real-functions-membered ;
attrR is rational-functions-valued  means :Def23: :: VALUED_2:def 23
 rng R is rational-functions-membered ;
attrR is integer-functions-valued  means :Def24: :: VALUED_2:def 24
 rng R is integer-functions-membered ;
attrR is natural-functions-valued  means :Def25: :: VALUED_2:def 25
 rng R is natural-functions-membered ;
end;

:: deftheorem Def20 defines complex-functions-valued VALUED_2:def_20_:_
 for R being Relation holds
( R is complex-functions-valued iff rng R is complex-functions-membered );

:: deftheorem Def21 defines ext-real-functions-valued VALUED_2:def_21_:_
 for R being Relation holds
( R is ext-real-functions-valued iff rng R is ext-real-functions-membered );

:: deftheorem Def22 defines real-functions-valued VALUED_2:def_22_:_
 for R being Relation holds
( R is real-functions-valued iff rng R is real-functions-membered );

:: deftheorem Def23 defines rational-functions-valued VALUED_2:def_23_:_
 for R being Relation holds
( R is rational-functions-valued iff rng R is rational-functions-membered );

:: deftheorem Def24 defines integer-functions-valued VALUED_2:def_24_:_
 for R being Relation holds
( R is integer-functions-valued iff rng R is integer-functions-membered );

:: deftheorem Def25 defines natural-functions-valued VALUED_2:def_25_:_
 for R being Relation holds
( R is natural-functions-valued iff rng R is natural-functions-membered );

registration
let Y be complex-functions-membered set ;
cluster Relation-like Y -valued Function-like  -> Y -valued complex-functions-valued for set ;
coherence
 for b1 being Y -valued Function holds b1 is complex-functions-valued
proof
let f be Y -valued Function; ::_thesis: f is complex-functions-valued
thus  rng f is complex-functions-membered ; :: according to VALUED_2:def_20 ::_thesis: verum
end;
end;

definition
let f be Function;
redefine attr f is complex-functions-valued  means :: VALUED_2:def 26
 for x being set st x in dom f holds
f . x is complex-valued Function;
compatibility
 ( f is complex-functions-valued iff for x being set st x in dom f holds
f . x is complex-valued Function )
proof
thus  ( f is complex-functions-valued implies for x being set st x in dom f holds
f . x is complex-valued Function ) ::_thesis: ( ( for x being set st x in dom f holds
f . x is complex-valued Function ) implies f is complex-functions-valued )
proof
assume A1: rng f is complex-functions-membered ; :: according to VALUED_2:def_20 ::_thesis: for x being set st x in dom f holds
f . x is complex-valued Function
let x be set ; ::_thesis: ( x in dom f implies f . x is complex-valued Function )
assume  x in dom f ; ::_thesis: f . x is complex-valued Function
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is complex-valued Function by A1; ::_thesis: verum
end;
assume A2: for x being set st x in dom f holds
f . x is complex-valued Function ; ::_thesis: f is complex-functions-valued
let y be set ; :: according to VALUED_2:def_2,VALUED_2:def_20 ::_thesis: ( y in rng f implies y is complex-valued Function )
assume  y in rng f ; ::_thesis: y is complex-valued Function
then  ex x being set st
( x in dom f & f . x = y ) by FUNCT_1:def_3;
hence  y is complex-valued Function by A2; ::_thesis: verum
end;
redefine attr f is ext-real-functions-valued  means :: VALUED_2:def 27
 for x being set st x in dom f holds
f . x is ext-real-valued Function;
compatibility
 ( f is ext-real-functions-valued iff for x being set st x in dom f holds
f . x is ext-real-valued Function )
proof
thus  ( f is ext-real-functions-valued implies for x being set st x in dom f holds
f . x is ext-real-valued Function ) ::_thesis: ( ( for x being set st x in dom f holds
f . x is ext-real-valued Function ) implies f is ext-real-functions-valued )
proof
assume A3: rng f is ext-real-functions-membered ; :: according to VALUED_2:def_21 ::_thesis: for x being set st x in dom f holds
f . x is ext-real-valued Function
let x be set ; ::_thesis: ( x in dom f implies f . x is ext-real-valued Function )
assume  x in dom f ; ::_thesis: f . x is ext-real-valued Function
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is ext-real-valued Function by A3; ::_thesis: verum
end;
assume A4: for x being set st x in dom f holds
f . x is ext-real-valued Function ; ::_thesis: f is ext-real-functions-valued
let y be set ; :: according to VALUED_2:def_3,VALUED_2:def_21 ::_thesis: ( y in rng f implies y is ext-real-valued Function )
assume  y in rng f ; ::_thesis: y is ext-real-valued Function
then  ex x being set st
( x in dom f & f . x = y ) by FUNCT_1:def_3;
hence  y is ext-real-valued Function by A4; ::_thesis: verum
end;
redefine attr f is real-functions-valued  means :: VALUED_2:def 28
 for x being set st x in dom f holds
f . x is real-valued Function;
compatibility
 ( f is real-functions-valued iff for x being set st x in dom f holds
f . x is real-valued Function )
proof
thus  ( f is real-functions-valued implies for x being set st x in dom f holds
f . x is real-valued Function ) ::_thesis: ( ( for x being set st x in dom f holds
f . x is real-valued Function ) implies f is real-functions-valued )
proof
assume A5: rng f is real-functions-membered ; :: according to VALUED_2:def_22 ::_thesis: for x being set st x in dom f holds
f . x is real-valued Function
let x be set ; ::_thesis: ( x in dom f implies f . x is real-valued Function )
assume  x in dom f ; ::_thesis: f . x is real-valued Function
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is real-valued Function by A5; ::_thesis: verum
end;
assume A6: for x being set st x in dom f holds
f . x is real-valued Function ; ::_thesis: f is real-functions-valued
let y be set ; :: according to VALUED_2:def_4,VALUED_2:def_22 ::_thesis: ( y in rng f implies y is real-valued Function )
assume  y in rng f ; ::_thesis: y is real-valued Function
then  ex x being set st
( x in dom f & f . x = y ) by FUNCT_1:def_3;
hence  y is real-valued Function by A6; ::_thesis: verum
end;
redefine attr f is rational-functions-valued  means :: VALUED_2:def 29
 for x being set st x in dom f holds
f . x is RAT -valued Function;
compatibility
 ( f is rational-functions-valued iff for x being set st x in dom f holds
f . x is RAT -valued Function )
proof
thus  ( f is rational-functions-valued implies for x being set st x in dom f holds
f . x is RAT -valued Function ) ::_thesis: ( ( for x being set st x in dom f holds
f . x is RAT -valued Function ) implies f is rational-functions-valued )
proof
assume A7: rng f is rational-functions-membered ; :: according to VALUED_2:def_23 ::_thesis: for x being set st x in dom f holds
f . x is RAT -valued Function
let x be set ; ::_thesis: ( x in dom f implies f . x is RAT -valued Function )
assume  x in dom f ; ::_thesis: f . x is RAT -valued Function
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is RAT -valued Function by A7; ::_thesis: verum
end;
assume A8: for x being set st x in dom f holds
f . x is RAT -valued Function ; ::_thesis: f is rational-functions-valued
let y be set ; :: according to VALUED_2:def_5,VALUED_2:def_23 ::_thesis: ( y in rng f implies y is RAT -valued Function )
assume  y in rng f ; ::_thesis: y is RAT -valued Function
then  ex x being set st
( x in dom f & f . x = y ) by FUNCT_1:def_3;
hence  y is RAT -valued Function by A8; ::_thesis: verum
end;
redefine attr f is integer-functions-valued  means :: VALUED_2:def 30
 for x being set st x in dom f holds
f . x is INT -valued Function;
compatibility
 ( f is integer-functions-valued iff for x being set st x in dom f holds
f . x is INT -valued Function )
proof
thus  ( f is integer-functions-valued implies for x being set st x in dom f holds
f . x is INT -valued Function ) ::_thesis: ( ( for x being set st x in dom f holds
f . x is INT -valued Function ) implies f is integer-functions-valued )
proof
assume A9: rng f is integer-functions-membered ; :: according to VALUED_2:def_24 ::_thesis: for x being set st x in dom f holds
f . x is INT -valued Function
let x be set ; ::_thesis: ( x in dom f implies f . x is INT -valued Function )
assume  x in dom f ; ::_thesis: f . x is INT -valued Function
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is INT -valued Function by A9; ::_thesis: verum
end;
assume A10: for x being set st x in dom f holds
f . x is INT -valued Function ; ::_thesis: f is integer-functions-valued
let y be set ; :: according to VALUED_2:def_6,VALUED_2:def_24 ::_thesis: ( y in rng f implies y is INT -valued Function )
assume  y in rng f ; ::_thesis: y is INT -valued Function
then  ex x being set st
( x in dom f & f . x = y ) by FUNCT_1:def_3;
hence  y is INT -valued Function by A10; ::_thesis: verum
end;
redefine attr f is natural-functions-valued  means :: VALUED_2:def 31
 for x being set st x in dom f holds
f . x is natural-valued Function;
compatibility
 ( f is natural-functions-valued iff for x being set st x in dom f holds
f . x is natural-valued Function )
proof
thus  ( f is natural-functions-valued implies for x being set st x in dom f holds
f . x is natural-valued Function ) ::_thesis: ( ( for x being set st x in dom f holds
f . x is natural-valued Function ) implies f is natural-functions-valued )
proof
assume A11: rng f is natural-functions-membered ; :: according to VALUED_2:def_25 ::_thesis: for x being set st x in dom f holds
f . x is natural-valued Function
let x be set ; ::_thesis: ( x in dom f implies f . x is natural-valued Function )
assume  x in dom f ; ::_thesis: f . x is natural-valued Function
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is natural-valued Function by A11; ::_thesis: verum
end;
assume A12: for x being set st x in dom f holds
f . x is natural-valued Function ; ::_thesis: f is natural-functions-valued
let y be set ; :: according to VALUED_2:def_7,VALUED_2:def_25 ::_thesis: ( y in rng f implies y is natural-valued Function )
assume  y in rng f ; ::_thesis: y is natural-valued Function
then  ex x being set st
( x in dom f & f . x = y ) by FUNCT_1:def_3;
hence  y is natural-valued Function by A12; ::_thesis: verum
end;
end;

:: deftheorem  defines complex-functions-valued VALUED_2:def_26_:_
 for f being Function holds
( f is complex-functions-valued iff for x being set st x in dom f holds
f . x is complex-valued Function );

:: deftheorem  defines ext-real-functions-valued VALUED_2:def_27_:_
 for f being Function holds
( f is ext-real-functions-valued iff for x being set st x in dom f holds
f . x is ext-real-valued Function );

:: deftheorem  defines real-functions-valued VALUED_2:def_28_:_
 for f being Function holds
( f is real-functions-valued iff for x being set st x in dom f holds
f . x is real-valued Function );

:: deftheorem  defines rational-functions-valued VALUED_2:def_29_:_
 for f being Function holds
( f is rational-functions-valued iff for x being set st x in dom f holds
f . x is RAT -valued Function );

:: deftheorem  defines integer-functions-valued VALUED_2:def_30_:_
 for f being Function holds
( f is integer-functions-valued iff for x being set st x in dom f holds
f . x is INT -valued Function );

:: deftheorem  defines natural-functions-valued VALUED_2:def_31_:_
 for f being Function holds
( f is natural-functions-valued iff for x being set st x in dom f holds
f . x is natural-valued Function );

registration
cluster Relation-like natural-functions-valued  ->  integer-functions-valued for set ;
coherence
 for b1 being Relation st b1 is natural-functions-valued holds
b1 is integer-functions-valued
proof
let R be Relation; ::_thesis: ( R is natural-functions-valued implies R is integer-functions-valued )
assume A1: rng R is natural-functions-membered ; :: according to VALUED_2:def_25 ::_thesis: R is integer-functions-valued
let y be set ; :: according to VALUED_2:def_6,VALUED_2:def_24 ::_thesis: ( y in rng R implies y is INT -valued Function )
thus  ( y in rng R implies y is INT -valued Function ) by A1; ::_thesis: verum
end;
cluster Relation-like integer-functions-valued  ->  rational-functions-valued for set ;
coherence
 for b1 being Relation st b1 is integer-functions-valued holds
b1 is rational-functions-valued
proof
let R be Relation; ::_thesis: ( R is integer-functions-valued implies R is rational-functions-valued )
assume A2: rng R is integer-functions-membered ; :: according to VALUED_2:def_24 ::_thesis: R is rational-functions-valued
let y be set ; :: according to VALUED_2:def_5,VALUED_2:def_23 ::_thesis: ( y in rng R implies y is RAT -valued Function )
thus  ( y in rng R implies y is RAT -valued Function ) by A2; ::_thesis: verum
end;
cluster Relation-like rational-functions-valued  ->  real-functions-valued for set ;
coherence
 for b1 being Relation st b1 is rational-functions-valued holds
b1 is real-functions-valued
proof
let R be Relation; ::_thesis: ( R is rational-functions-valued implies R is real-functions-valued )
assume A3: rng R is rational-functions-membered ; :: according to VALUED_2:def_23 ::_thesis: R is real-functions-valued
let y be set ; :: according to VALUED_2:def_4,VALUED_2:def_22 ::_thesis: ( y in rng R implies y is real-valued Function )
thus  ( y in rng R implies y is real-valued Function ) by A3; ::_thesis: verum
end;
cluster Relation-like real-functions-valued  ->  ext-real-functions-valued for set ;
coherence
 for b1 being Relation st b1 is real-functions-valued holds
b1 is ext-real-functions-valued
proof
let R be Relation; ::_thesis: ( R is real-functions-valued implies R is ext-real-functions-valued )
assume A4: rng R is real-functions-membered ; :: according to VALUED_2:def_22 ::_thesis: R is ext-real-functions-valued
let y be set ; :: according to VALUED_2:def_3,VALUED_2:def_21 ::_thesis: ( y in rng R implies y is ext-real-valued Function )
thus  ( y in rng R implies y is ext-real-valued Function ) by A4; ::_thesis: verum
end;
cluster Relation-like real-functions-valued  ->  complex-functions-valued for set ;
coherence
 for b1 being Relation st b1 is real-functions-valued holds
b1 is complex-functions-valued
proof
let R be Relation; ::_thesis: ( R is real-functions-valued implies R is complex-functions-valued )
assume A5: rng R is real-functions-membered ; :: according to VALUED_2:def_22 ::_thesis: R is complex-functions-valued
let y be set ; :: according to VALUED_2:def_2,VALUED_2:def_20 ::_thesis: ( y in rng R implies y is complex-valued Function )
thus  ( y in rng R implies y is complex-valued Function ) by A5; ::_thesis: verum
end;
end;

registration
cluster Relation-like empty  ->  natural-functions-valued for set ;
coherence
 for b1 being Relation st b1 is empty holds
b1 is natural-functions-valued
proof
let X be Relation; ::_thesis: ( X is empty implies X is natural-functions-valued )
assume A1: X is empty ; ::_thesis: X is natural-functions-valued
let x be set ; :: according to VALUED_2:def_7,VALUED_2:def_25 ::_thesis: ( x in rng X implies x is natural-valued Function )
thus  ( x in rng X implies x is natural-valued Function ) by A1; ::_thesis: verum
end;
end;

registration
cluster Relation-like Function-like natural-functions-valued for set ;
existence
 ex b1 being Function st b1 is natural-functions-valued
proof
take  {} ; ::_thesis: {} is natural-functions-valued
thus  {} is natural-functions-valued ; ::_thesis: verum
end;
end;

registration
let R be complex-functions-valued Relation;
cluster rng R ->  complex-functions-membered ;
coherence
 rng R is complex-functions-membered by Def20;
end;

registration
let R be ext-real-functions-valued Relation;
cluster rng R ->  ext-real-functions-membered ;
coherence
 rng R is ext-real-functions-membered by Def21;
end;

registration
let R be real-functions-valued Relation;
cluster rng R ->  real-functions-membered ;
coherence
 rng R is real-functions-membered by Def22;
end;

registration
let R be rational-functions-valued Relation;
cluster rng R ->  rational-functions-membered ;
coherence
 rng R is rational-functions-membered by Def23;
end;

registration
let R be integer-functions-valued Relation;
cluster rng R ->  integer-functions-membered ;
coherence
 rng R is integer-functions-membered by Def24;
end;

registration
let R be natural-functions-valued Relation;
cluster rng R ->  natural-functions-membered ;
coherence
 rng R is natural-functions-membered by Def25;
end;

registration
let X be set ;
let Y be complex-functions-membered set ;
cluster Function-like  ->  complex-functions-valued for Element of K19(K20(X,Y));
coherence
 for b1 being PartFunc of X,Y holds b1 is complex-functions-valued ;
end;

registration
let X be set ;
let Y be ext-real-functions-membered set ;
cluster Function-like  ->  ext-real-functions-valued for Element of K19(K20(X,Y));
coherence
 for b1 being PartFunc of X,Y holds b1 is ext-real-functions-valued
proof
let f be PartFunc of X,Y; ::_thesis: f is ext-real-functions-valued
let x be set ; :: according to VALUED_2:def_27 ::_thesis: ( x in dom f implies f . x is ext-real-valued Function )
thus  ( x in dom f implies f . x is ext-real-valued Function ) ; ::_thesis: verum
end;
end;

registration
let X be set ;
let Y be real-functions-membered set ;
cluster Function-like  ->  real-functions-valued for Element of K19(K20(X,Y));
coherence
 for b1 being PartFunc of X,Y holds b1 is real-functions-valued
proof
let f be PartFunc of X,Y; ::_thesis: f is real-functions-valued
let x be set ; :: according to VALUED_2:def_28 ::_thesis: ( x in dom f implies f . x is real-valued Function )
thus  ( x in dom f implies f . x is real-valued Function ) ; ::_thesis: verum
end;
end;

registration
let X be set ;
let Y be rational-functions-membered set ;
cluster Function-like  ->  rational-functions-valued for Element of K19(K20(X,Y));
coherence
 for b1 being PartFunc of X,Y holds b1 is rational-functions-valued
proof
let f be PartFunc of X,Y; ::_thesis: f is rational-functions-valued
let x be set ; :: according to VALUED_2:def_29 ::_thesis: ( x in dom f implies f . x is RAT -valued Function )
thus  ( x in dom f implies f . x is RAT -valued Function ) ; ::_thesis: verum
end;
end;

registration
let X be set ;
let Y be integer-functions-membered set ;
cluster Function-like  ->  integer-functions-valued for Element of K19(K20(X,Y));
coherence
 for b1 being PartFunc of X,Y holds b1 is integer-functions-valued
proof
let f be PartFunc of X,Y; ::_thesis: f is integer-functions-valued
let x be set ; :: according to VALUED_2:def_30 ::_thesis: ( x in dom f implies f . x is INT -valued Function )
thus  ( x in dom f implies f . x is INT -valued Function ) ; ::_thesis: verum
end;
end;

registration
let X be set ;
let Y be natural-functions-membered set ;
cluster Function-like  ->  natural-functions-valued for Element of K19(K20(X,Y));
coherence
 for b1 being PartFunc of X,Y holds b1 is natural-functions-valued
proof
let f be PartFunc of X,Y; ::_thesis: f is natural-functions-valued
let x be set ; :: according to VALUED_2:def_31 ::_thesis: ( x in dom f implies f . x is natural-valued Function )
thus  ( x in dom f implies f . x is natural-valued Function ) ; ::_thesis: verum
end;
end;

registration
let f be complex-functions-valued Function;
let x be set ;
clusterf . x ->  Relation-like Function-like ;
coherence
 ( f . x is Function-like & f . x is Relation-like )
proof
percases ( x in dom f or not x in dom f ) ;
suppose x in dom f ; ::_thesis: ( f . x is Function-like & f . x is Relation-like )
then  f . x in rng f by FUNCT_1:def_3;
hence  ( f . x is Function-like & f . x is Relation-like ) ; ::_thesis: verum
end;
suppose not x in dom f ; ::_thesis: ( f . x is Function-like & f . x is Relation-like )
hence  ( f . x is Function-like & f . x is Relation-like ) by FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
end;

registration
let f be ext-real-functions-valued Function;
let x be set ;
clusterf . x ->  Relation-like Function-like ;
coherence
 ( f . x is Function-like & f . x is Relation-like )
proof
percases ( x in dom f or not x in dom f ) ;
suppose x in dom f ; ::_thesis: ( f . x is Function-like & f . x is Relation-like )
then  f . x in rng f by FUNCT_1:def_3;
hence  ( f . x is Function-like & f . x is Relation-like ) ; ::_thesis: verum
end;
suppose not x in dom f ; ::_thesis: ( f . x is Function-like & f . x is Relation-like )
hence  ( f . x is Function-like & f . x is Relation-like ) by FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
end;

registration
let f be complex-functions-valued Function;
let x be set ;
clusterf . x ->  complex-valued ;
coherence
 f . x is complex-valued
proof
percases ( x in dom f or not x in dom f ) ;
suppose x in dom f ; ::_thesis: f . x is complex-valued
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is complex-valued ; ::_thesis: verum
end;
suppose not x in dom f ; ::_thesis: f . x is complex-valued
hence  f . x is complex-valued by FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
end;

registration
let f be ext-real-functions-valued Function;
let x be set ;
clusterf . x ->  ext-real-valued ;
coherence
 f . x is ext-real-valued
proof
percases ( x in dom f or not x in dom f ) ;
suppose x in dom f ; ::_thesis: f . x is ext-real-valued
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is ext-real-valued ; ::_thesis: verum
end;
suppose not x in dom f ; ::_thesis: f . x is ext-real-valued
hence  f . x is ext-real-valued by FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
end;

registration
let f be real-functions-valued Function;
let x be set ;
clusterf . x ->  real-valued ;
coherence
 f . x is real-valued
proof
percases ( x in dom f or not x in dom f ) ;
suppose x in dom f ; ::_thesis: f . x is real-valued
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is real-valued ; ::_thesis: verum
end;
suppose not x in dom f ; ::_thesis: f . x is real-valued
hence  f . x is real-valued by FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
end;

registration
let f be rational-functions-valued Function;
let x be set ;
clusterf . x ->  RAT -valued ;
coherence
 f . x is RAT -valued
proof
percases ( x in dom f or not x in dom f ) ;
suppose x in dom f ; ::_thesis: f . x is RAT -valued
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is RAT -valued ; ::_thesis: verum
end;
suppose not x in dom f ; ::_thesis: f . x is RAT -valued
hence  f . x is RAT -valued by FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
end;

registration
let f be integer-functions-valued Function;
let x be set ;
clusterf . x ->  INT -valued ;
coherence
 f . x is INT -valued
proof
percases ( x in dom f or not x in dom f ) ;
suppose x in dom f ; ::_thesis: f . x is INT -valued
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is INT -valued ; ::_thesis: verum
end;
suppose not x in dom f ; ::_thesis: f . x is INT -valued
hence  f . x is INT -valued by FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
end;

registration
let f be natural-functions-valued Function;
let x be set ;
clusterf . x ->  natural-valued ;
coherence
 f . x is natural-valued
proof
percases ( x in dom f or not x in dom f ) ;
suppose x in dom f ; ::_thesis: f . x is natural-valued
then  f . x in rng f by FUNCT_1:def_3;
hence  f . x is natural-valued ; ::_thesis: verum
end;
suppose not x in dom f ; ::_thesis: f . x is natural-valued
hence  f . x is natural-valued by FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
end;

begin


theorem Th7: :: VALUED_2:7
for c1, c2 being complex number
for g being complex-valued Function st g <> {} & g + c1 = g + c2 holds
c1 = c2
proof
let c1, c2 be complex number ; ::_thesis: for g being complex-valued Function st g <> {} & g + c1 = g + c2 holds
c1 = c2
let g be complex-valued Function; ::_thesis: ( g <> {} & g + c1 = g + c2 implies c1 = c2 )
assume that
A1: g <> {} and
A2: g + c1 = g + c2 ; ::_thesis: c1 = c2
consider x being set  such that
A3: x in dom g by A1, XBOOLE_0:def_1;
 dom g = dom (g + c2) by VALUED_1:def_2;
then A4: (g + c2) . x = (g . x) + c2 by A3, VALUED_1:def_2;
 dom g = dom (g + c1) by VALUED_1:def_2;
then  (g + c1) . x = (g . x) + c1 by A3, VALUED_1:def_2;
hence  c1 = c2 by A2, A4; ::_thesis: verum
end;

theorem Th8: :: VALUED_2:8
for c1, c2 being complex number
for g being complex-valued Function st g <> {} & g - c1 = g - c2 holds
c1 = c2
proof
let c1, c2 be complex number ; ::_thesis: for g being complex-valued Function st g <> {} & g - c1 = g - c2 holds
c1 = c2
let g be complex-valued Function; ::_thesis: ( g <> {} & g - c1 = g - c2 implies c1 = c2 )
assume that
A1: g <> {} and
A2: g - c1 = g - c2 ; ::_thesis: c1 = c2
consider x being set  such that
A3: x in dom g by A1, XBOOLE_0:def_1;
 dom g = dom (g - c2) by VALUED_1:def_2;
then A4: (g - c2) . x = (g . x) - c2 by A3, VALUED_1:def_2;
 dom g = dom (g - c1) by VALUED_1:def_2;
then  (g - c1) . x = (g . x) - c1 by A3, VALUED_1:def_2;
hence  c1 = c2 by A2, A4; ::_thesis: verum
end;

theorem Th9: :: VALUED_2:9
for c1, c2 being complex number
for g being complex-valued Function st g <> {} & g is non-empty & g (#) c1 = g (#) c2 holds
c1 = c2
proof
let c1, c2 be complex number ; ::_thesis: for g being complex-valued Function st g <> {} & g is non-empty & g (#) c1 = g (#) c2 holds
c1 = c2
let g be complex-valued Function; ::_thesis: ( g <> {} & g is non-empty & g (#) c1 = g (#) c2 implies c1 = c2 )
assume that
A1: g <> {} and
A2: g is non-empty and
A3: g (#) c1 = g (#) c2 ; ::_thesis: c1 = c2
consider x being set  such that
A4: x in dom g by A1, XBOOLE_0:def_1;
 g . x in rng g by A4, FUNCT_1:def_3;
then A5: g . x <> {} by A2, RELAT_1:def_9;
 ( (g (#) c1) . x = (g . x) * c1 & (g (#) c2) . x = (g . x) * c2 ) by VALUED_1:6;
hence  c1 = c2 by A3, A5, XCMPLX_1:5; ::_thesis: verum
end;

theorem Th10: :: VALUED_2:10
for c being complex number
for g being complex-valued Function holds - (g + c) = (- g) - c
proof
let c be complex number ; ::_thesis: for g being complex-valued Function holds - (g + c) = (- g) - c
let g be complex-valued Function; ::_thesis: - (g + c) = (- g) - c
A1: dom (- (g + c)) = dom (g + c) by VALUED_1:8;
A2: ( dom (g + c) = dom g & dom ((- g) - c) = dom (- g) ) by VALUED_1:def_2;
hence  dom (- (g + c)) = dom ((- g) - c) by A1, VALUED_1:8; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (- (g + c)) or (- (g + c)) . b1 = ((- g) - c) . b1 )
let x be set ; ::_thesis: ( not x in dom (- (g + c)) or (- (g + c)) . x = ((- g) - c) . x )
assume A3: x in dom (- (g + c)) ; ::_thesis: (- (g + c)) . x = ((- g) - c) . x
A4: dom (- g) = dom g by VALUED_1:8;
thus (- (g + c)) . x = - ((g + c) . x) by VALUED_1:8
.= - ((g . x) + c) by A1, A3, VALUED_1:def_2
.= (- (g . x)) - c
.= ((- g) . x) - c by VALUED_1:8
.= ((- g) - c) . x by A2, A1, A4, A3, VALUED_1:def_2 ; ::_thesis: verum
end;

theorem Th11: :: VALUED_2:11
for c being complex number
for g being complex-valued Function holds - (g - c) = (- g) + c
proof
let c be complex number ; ::_thesis: for g being complex-valued Function holds - (g - c) = (- g) + c
let g be complex-valued Function; ::_thesis: - (g - c) = (- g) + c
A1: dom (- (g - c)) = dom (g - c) by VALUED_1:8;
A2: ( dom (g - c) = dom g & dom ((- g) + c) = dom (- g) ) by VALUED_1:def_2;
hence  dom (- (g - c)) = dom ((- g) + c) by A1, VALUED_1:8; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (- (g - c)) or (- (g - c)) . b1 = ((- g) + c) . b1 )
let x be set ; ::_thesis: ( not x in dom (- (g - c)) or (- (g - c)) . x = ((- g) + c) . x )
assume A3: x in dom (- (g - c)) ; ::_thesis: (- (g - c)) . x = ((- g) + c) . x
A4: dom (- g) = dom g by VALUED_1:8;
thus (- (g - c)) . x = - ((g - c) . x) by VALUED_1:8
.= - ((g . x) - c) by A1, A3, VALUED_1:def_2
.= (- (g . x)) + c
.= ((- g) . x) + c by VALUED_1:8
.= ((- g) + c) . x by A2, A1, A4, A3, VALUED_1:def_2 ; ::_thesis: verum
end;

theorem Th12: :: VALUED_2:12
for c1, c2 being complex number
for g being complex-valued Function holds (g + c1) + c2 = g + (c1 + c2)
proof
let c1, c2 be complex number ; ::_thesis: for g being complex-valued Function holds (g + c1) + c2 = g + (c1 + c2)
let g be complex-valued Function; ::_thesis: (g + c1) + c2 = g + (c1 + c2)
A1: dom ((g + c1) + c2) = dom (g + c1) by VALUED_1:def_2;
A2: dom (g + c1) = dom g by VALUED_1:def_2;
hence  dom ((g + c1) + c2) = dom (g + (c1 + c2)) by A1, VALUED_1:def_2; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((g + c1) + c2) or ((g + c1) + c2) . b1 = (g + (c1 + c2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((g + c1) + c2) or ((g + c1) + c2) . x = (g + (c1 + c2)) . x )
A3: dom (g + (c1 + c2)) = dom g by VALUED_1:def_2;
assume A4: x in dom ((g + c1) + c2) ; ::_thesis: ((g + c1) + c2) . x = (g + (c1 + c2)) . x
hence ((g + c1) + c2) . x = ((g + c1) . x) + c2 by VALUED_1:def_2
.= ((g . x) + c1) + c2 by A1, A4, VALUED_1:def_2
.= (g . x) + (c1 + c2)
.= (g + (c1 + c2)) . x by A1, A2, A3, A4, VALUED_1:def_2 ;
::_thesis: verum
end;

theorem Th13: :: VALUED_2:13
for c1, c2 being complex number
for g being complex-valued Function holds (g + c1) - c2 = g + (c1 - c2)
proof
let c1, c2 be complex number ; ::_thesis: for g being complex-valued Function holds (g + c1) - c2 = g + (c1 - c2)
let g be complex-valued Function; ::_thesis: (g + c1) - c2 = g + (c1 - c2)
A1: dom ((g + c1) - c2) = dom (g + c1) by VALUED_1:def_2;
A2: dom (g + c1) = dom g by VALUED_1:def_2;
hence  dom ((g + c1) - c2) = dom (g + (c1 - c2)) by A1, VALUED_1:def_2; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((g + c1) - c2) or ((g + c1) - c2) . b1 = (g + (c1 - c2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((g + c1) - c2) or ((g + c1) - c2) . x = (g + (c1 - c2)) . x )
A3: dom (g + (c1 - c2)) = dom g by VALUED_1:def_2;
assume A4: x in dom ((g + c1) - c2) ; ::_thesis: ((g + c1) - c2) . x = (g + (c1 - c2)) . x
hence ((g + c1) - c2) . x = ((g + c1) . x) - c2 by VALUED_1:def_2
.= ((g . x) + c1) - c2 by A1, A4, VALUED_1:def_2
.= (g . x) + (c1 - c2)
.= (g + (c1 - c2)) . x by A1, A2, A3, A4, VALUED_1:def_2 ;
::_thesis: verum
end;

theorem Th14: :: VALUED_2:14
for c1, c2 being complex number
for g being complex-valued Function holds (g - c1) + c2 = g - (c1 - c2)
proof
let c1, c2 be complex number ; ::_thesis: for g being complex-valued Function holds (g - c1) + c2 = g - (c1 - c2)
let g be complex-valued Function; ::_thesis: (g - c1) + c2 = g - (c1 - c2)
A1: dom ((g - c1) + c2) = dom (g - c1) by VALUED_1:def_2;
A2: dom (g - c1) = dom g by VALUED_1:def_2;
hence  dom ((g - c1) + c2) = dom (g - (c1 - c2)) by A1, VALUED_1:def_2; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((g - c1) + c2) or ((g - c1) + c2) . b1 = (g - (c1 - c2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((g - c1) + c2) or ((g - c1) + c2) . x = (g - (c1 - c2)) . x )
A3: dom (g - (c1 - c2)) = dom g by VALUED_1:def_2;
assume A4: x in dom ((g - c1) + c2) ; ::_thesis: ((g - c1) + c2) . x = (g - (c1 - c2)) . x
hence ((g - c1) + c2) . x = ((g - c1) . x) + c2 by VALUED_1:def_2
.= ((g . x) - c1) + c2 by A1, A4, VALUED_1:def_2
.= (g . x) - (c1 - c2)
.= (g - (c1 - c2)) . x by A1, A2, A3, A4, VALUED_1:def_2 ;
::_thesis: verum
end;

theorem Th15: :: VALUED_2:15
for c1, c2 being complex number
for g being complex-valued Function holds (g - c1) - c2 = g - (c1 + c2)
proof
let c1, c2 be complex number ; ::_thesis: for g being complex-valued Function holds (g - c1) - c2 = g - (c1 + c2)
let g be complex-valued Function; ::_thesis: (g - c1) - c2 = g - (c1 + c2)
A1: dom ((g - c1) - c2) = dom (g - c1) by VALUED_1:def_2;
A2: dom (g - c1) = dom g by VALUED_1:def_2;
hence  dom ((g - c1) - c2) = dom (g - (c1 + c2)) by A1, VALUED_1:def_2; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((g - c1) - c2) or ((g - c1) - c2) . b1 = (g - (c1 + c2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((g - c1) - c2) or ((g - c1) - c2) . x = (g - (c1 + c2)) . x )
A3: dom (g - (c1 + c2)) = dom g by VALUED_1:def_2;
assume A4: x in dom ((g - c1) - c2) ; ::_thesis: ((g - c1) - c2) . x = (g - (c1 + c2)) . x
hence ((g - c1) - c2) . x = ((g - c1) . x) - c2 by VALUED_1:def_2
.= ((g . x) - c1) - c2 by A1, A4, VALUED_1:def_2
.= (g . x) - (c1 + c2)
.= (g - (c1 + c2)) . x by A1, A2, A3, A4, VALUED_1:def_2 ;
::_thesis: verum
end;

theorem Th16: :: VALUED_2:16
for c1, c2 being complex number
for g being complex-valued Function holds (g (#) c1) (#) c2 = g (#) (c1 * c2)
proof
let c1, c2 be complex number ; ::_thesis: for g being complex-valued Function holds (g (#) c1) (#) c2 = g (#) (c1 * c2)
let g be complex-valued Function; ::_thesis: (g (#) c1) (#) c2 = g (#) (c1 * c2)
 ( dom ((g (#) c1) (#) c2) = dom (g (#) c1) & dom (g (#) c1) = dom g ) by VALUED_1:def_5;
hence  dom ((g (#) c1) (#) c2) = dom (g (#) (c1 * c2)) by VALUED_1:def_5; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((g (#) c1) (#) c2) or ((g (#) c1) (#) c2) . b1 = (g (#) (c1 * c2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((g (#) c1) (#) c2) or ((g (#) c1) (#) c2) . x = (g (#) (c1 * c2)) . x )
assume  x in dom ((g (#) c1) (#) c2) ; ::_thesis: ((g (#) c1) (#) c2) . x = (g (#) (c1 * c2)) . x
thus ((g (#) c1) (#) c2) . x = ((g (#) c1) . x) * c2 by VALUED_1:6
.= ((g . x) * c1) * c2 by VALUED_1:6
.= (g . x) * (c1 * c2)
.= (g (#) (c1 * c2)) . x by VALUED_1:6 ; ::_thesis: verum
end;

theorem Th17: :: VALUED_2:17
for g, h being complex-valued Function holds - (g + h) = (- g) - h
proof
let g, h be complex-valued Function; ::_thesis: - (g + h) = (- g) - h
A1: dom (- (g + h)) = dom (g + h) by VALUED_1:8;
 ( dom (g + h) = (dom g) /\ (dom h) & dom ((- g) - h) = (dom (- g)) /\ (dom h) ) by VALUED_1:12, VALUED_1:def_1;
hence A2: dom (- (g + h)) = dom ((- g) - h) by A1, VALUED_1:8; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (- (g + h)) or (- (g + h)) . b1 = ((- g) - h) . b1 )
let x be set ; ::_thesis: ( not x in dom (- (g + h)) or (- (g + h)) . x = ((- g) - h) . x )
assume A3: x in dom (- (g + h)) ; ::_thesis: (- (g + h)) . x = ((- g) - h) . x
thus (- (g + h)) . x = - ((g + h) . x) by VALUED_1:8
.= - ((g . x) + (h . x)) by A1, A3, VALUED_1:def_1
.= (- (g . x)) - (h . x)
.= ((- g) . x) - (h . x) by VALUED_1:8
.= ((- g) - h) . x by A2, A3, VALUED_1:13 ; ::_thesis: verum
end;

theorem Th18: :: VALUED_2:18
for g, h being complex-valued Function holds g - h = - (h - g)
proof
let g, h be complex-valued Function; ::_thesis: g - h = - (h - g)
A1: dom (- (h - g)) = dom (h - g) by VALUED_1:8;
 dom (g - h) = (dom g) /\ (dom h) by VALUED_1:12;
hence A2: dom (g - h) = dom (- (h - g)) by A1, VALUED_1:12; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (g - h) or (g - h) . b1 = (- (h - g)) . b1 )
let x be set ; ::_thesis: ( not x in dom (g - h) or (g - h) . x = (- (h - g)) . x )
assume A3: x in dom (g - h) ; ::_thesis: (g - h) . x = (- (h - g)) . x
hence (g - h) . x = (g . x) - (h . x) by VALUED_1:13
.= - ((h . x) - (g . x))
.= - ((h - g) . x) by A1, A2, A3, VALUED_1:13
.= (- (h - g)) . x by VALUED_1:8 ;
::_thesis: verum
end;

theorem Th19: :: VALUED_2:19
for g, h, k being complex-valued Function holds (g (#) h) /" k = g (#) (h /" k)
proof
let g, h, k be complex-valued Function; ::_thesis: (g (#) h) /" k = g (#) (h /" k)
A1: ( dom (g (#) (h /" k)) = (dom g) /\ (dom (h /" k)) & dom ((g (#) h) /" k) = (dom (g (#) h)) /\ (dom k) ) by VALUED_1:16, VALUED_1:def_4;
 ( dom (g (#) h) = (dom g) /\ (dom h) & dom (h /" k) = (dom h) /\ (dom k) ) by VALUED_1:16, VALUED_1:def_4;
hence  dom ((g (#) h) /" k) = dom (g (#) (h /" k)) by A1, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((g (#) h) /" k) or ((g (#) h) /" k) . b1 = (g (#) (h /" k)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((g (#) h) /" k) or ((g (#) h) /" k) . x = (g (#) (h /" k)) . x )
assume  x in dom ((g (#) h) /" k) ; ::_thesis: ((g (#) h) /" k) . x = (g (#) (h /" k)) . x
thus ((g (#) h) /" k) . x = ((g (#) h) . x) / (k . x) by VALUED_1:17
.= ((g . x) * (h . x)) / (k . x) by VALUED_1:5
.= (g . x) * ((h . x) / (k . x))
.= (g . x) * ((h /" k) . x) by VALUED_1:17
.= (g (#) (h /" k)) . x by VALUED_1:5 ; ::_thesis: verum
end;

theorem Th20: :: VALUED_2:20
for g, h, k being complex-valued Function holds (g /" h) (#) k = (g (#) k) /" h
proof
let g, h, k be complex-valued Function; ::_thesis: (g /" h) (#) k = (g (#) k) /" h
A1: ( dom ((g /" h) (#) k) = (dom (g /" h)) /\ (dom k) & dom ((g (#) k) /" h) = (dom (g (#) k)) /\ (dom h) ) by VALUED_1:16, VALUED_1:def_4;
 ( dom (g /" h) = (dom g) /\ (dom h) & dom (g (#) k) = (dom g) /\ (dom k) ) by VALUED_1:16, VALUED_1:def_4;
hence  dom ((g /" h) (#) k) = dom ((g (#) k) /" h) by A1, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((g /" h) (#) k) or ((g /" h) (#) k) . b1 = ((g (#) k) /" h) . b1 )
let x be set ; ::_thesis: ( not x in dom ((g /" h) (#) k) or ((g /" h) (#) k) . x = ((g (#) k) /" h) . x )
assume  x in dom ((g /" h) (#) k) ; ::_thesis: ((g /" h) (#) k) . x = ((g (#) k) /" h) . x
thus ((g /" h) (#) k) . x = ((g /" h) . x) * (k . x) by VALUED_1:5
.= ((g . x) / (h . x)) * (k . x) by VALUED_1:17
.= ((g . x) * (k . x)) / (h . x)
.= ((g (#) k) . x) / (h . x) by VALUED_1:5
.= ((g (#) k) /" h) . x by VALUED_1:17 ; ::_thesis: verum
end;

theorem Th21: :: VALUED_2:21
for g, h, k being complex-valued Function holds (g /" h) /" k = g /" (h (#) k)
proof
let g, h, k be complex-valued Function; ::_thesis: (g /" h) /" k = g /" (h (#) k)
A1: ( dom ((g /" h) /" k) = (dom (g /" h)) /\ (dom k) & dom (g /" (h (#) k)) = (dom g) /\ (dom (h (#) k)) ) by VALUED_1:16;
 ( dom (g /" h) = (dom g) /\ (dom h) & dom (h (#) k) = (dom h) /\ (dom k) ) by VALUED_1:16, VALUED_1:def_4;
hence  dom ((g /" h) /" k) = dom (g /" (h (#) k)) by A1, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((g /" h) /" k) or ((g /" h) /" k) . b1 = (g /" (h (#) k)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((g /" h) /" k) or ((g /" h) /" k) . x = (g /" (h (#) k)) . x )
assume  x in dom ((g /" h) /" k) ; ::_thesis: ((g /" h) /" k) . x = (g /" (h (#) k)) . x
thus ((g /" h) /" k) . x = ((g /" h) . x) / (k . x) by VALUED_1:17
.= ((g . x) / (h . x)) / (k . x) by VALUED_1:17
.= (g . x) / ((h . x) * (k . x)) by XCMPLX_1:78
.= (g . x) / ((h (#) k) . x) by VALUED_1:5
.= (g /" (h (#) k)) . x by VALUED_1:17 ; ::_thesis: verum
end;

theorem Th22: :: VALUED_2:22
for c being complex number
for g being complex-valued Function holds c (#) (- g) = (- c) (#) g
proof
let c be complex number ; ::_thesis: for g being complex-valued Function holds c (#) (- g) = (- c) (#) g
let g be complex-valued Function; ::_thesis: c (#) (- g) = (- c) (#) g
dom (c (#) (- g)) = dom (- g) by VALUED_1:def_5
.= dom g by VALUED_1:8 ;
hence  dom (c (#) (- g)) = dom ((- c) (#) g) by VALUED_1:def_5; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (c (#) (- g)) or (c (#) (- g)) . b1 = ((- c) (#) g) . b1 )
let x be set ; ::_thesis: ( not x in dom (c (#) (- g)) or (c (#) (- g)) . x = ((- c) (#) g) . x )
assume  x in dom (c (#) (- g)) ; ::_thesis: (c (#) (- g)) . x = ((- c) (#) g) . x
thus (c (#) (- g)) . x = c * ((- g) . x) by VALUED_1:6
.= c * (- (g . x)) by VALUED_1:8
.= (- c) * (g . x)
.= ((- c) (#) g) . x by VALUED_1:6 ; ::_thesis: verum
end;

theorem Th23: :: VALUED_2:23
for c being complex number
for g being complex-valued Function holds c (#) (- g) = - (c (#) g)
proof
let c be complex number ; ::_thesis: for g being complex-valued Function holds c (#) (- g) = - (c (#) g)
let g be complex-valued Function; ::_thesis: c (#) (- g) = - (c (#) g)
A1: dom (- (c (#) g)) = dom (c (#) g) by VALUED_1:8
.= dom g by VALUED_1:def_5 ;
dom (c (#) (- g)) = dom (- g) by VALUED_1:def_5
.= dom g by VALUED_1:8 ;
hence  dom (c (#) (- g)) = dom (- (c (#) g)) by A1; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (c (#) (- g)) or (c (#) (- g)) . b1 = (- (c (#) g)) . b1 )
let x be set ; ::_thesis: ( not x in dom (c (#) (- g)) or (c (#) (- g)) . x = (- (c (#) g)) . x )
assume  x in dom (c (#) (- g)) ; ::_thesis: (c (#) (- g)) . x = (- (c (#) g)) . x
thus (c (#) (- g)) . x = c * ((- g) . x) by VALUED_1:6
.= c * (- (g . x)) by VALUED_1:8
.= - (c * (g . x))
.= - ((c (#) g) . x) by VALUED_1:6
.= (- (c (#) g)) . x by VALUED_1:8 ; ::_thesis: verum
end;

theorem Th24: :: VALUED_2:24
for c being complex number
for g being complex-valued Function holds (- c) (#) g = - (c (#) g)
proof
let c be complex number ; ::_thesis: for g being complex-valued Function holds (- c) (#) g = - (c (#) g)
let g be complex-valued Function; ::_thesis: (- c) (#) g = - (c (#) g)
thus (- c) (#) g = c (#) (- g) by Th22
.= - (c (#) g) by Th23 ; ::_thesis: verum
end;

theorem Th25: :: VALUED_2:25
for g, h being complex-valued Function holds - (g (#) h) = (- g) (#) h
proof
let g, h be complex-valued Function; ::_thesis: - (g (#) h) = (- g) (#) h
A1: dom (- (g (#) h)) = dom (g (#) h) by VALUED_1:8;
 ( dom (g (#) h) = (dom g) /\ (dom h) & dom ((- g) (#) h) = (dom (- g)) /\ (dom h) ) by VALUED_1:def_4;
hence  dom (- (g (#) h)) = dom ((- g) (#) h) by A1, VALUED_1:8; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (- (g (#) h)) or (- (g (#) h)) . b1 = ((- g) (#) h) . b1 )
let x be set ; ::_thesis: ( not x in dom (- (g (#) h)) or (- (g (#) h)) . x = ((- g) (#) h) . x )
assume  x in dom (- (g (#) h)) ; ::_thesis: (- (g (#) h)) . x = ((- g) (#) h) . x
thus (- (g (#) h)) . x = - ((g (#) h) . x) by VALUED_1:8
.= - ((g . x) * (h . x)) by VALUED_1:5
.= (- (g . x)) * (h . x)
.= ((- g) . x) * (h . x) by VALUED_1:8
.= ((- g) (#) h) . x by VALUED_1:5 ; ::_thesis: verum
end;

theorem :: VALUED_2:26
for g, h being complex-valued Function holds - (g /" h) = (- g) /" h
proof
let g, h be complex-valued Function; ::_thesis: - (g /" h) = (- g) /" h
A1: dom (- (g /" h)) = dom (g /" h) by VALUED_1:8;
 ( dom (g /" h) = (dom g) /\ (dom h) & dom ((- g) /" h) = (dom (- g)) /\ (dom h) ) by VALUED_1:16;
hence  dom (- (g /" h)) = dom ((- g) /" h) by A1, VALUED_1:8; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (- (g /" h)) or (- (g /" h)) . b1 = ((- g) /" h) . b1 )
let x be set ; ::_thesis: ( not x in dom (- (g /" h)) or (- (g /" h)) . x = ((- g) /" h) . x )
assume  x in dom (- (g /" h)) ; ::_thesis: (- (g /" h)) . x = ((- g) /" h) . x
thus (- (g /" h)) . x = - ((g /" h) . x) by VALUED_1:8
.= - ((g . x) / (h . x)) by VALUED_1:17
.= (- (g . x)) / (h . x)
.= ((- g) . x) / (h . x) by VALUED_1:8
.= ((- g) /" h) . x by VALUED_1:17 ; ::_thesis: verum
end;

theorem Th27: :: VALUED_2:27
for g, h being complex-valued Function holds - (g /" h) = g /" (- h)
proof
let g, h be complex-valued Function; ::_thesis: - (g /" h) = g /" (- h)
A1: dom (- h) = dom h by VALUED_1:8;
 ( dom (g /" h) = (dom g) /\ (dom h) & dom (g /" (- h)) = (dom g) /\ (dom (- h)) ) by VALUED_1:16;
hence  dom (- (g /" h)) = dom (g /" (- h)) by A1, VALUED_1:8; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (- (g /" h)) or (- (g /" h)) . b1 = (g /" (- h)) . b1 )
let x be set ; ::_thesis: ( not x in dom (- (g /" h)) or (- (g /" h)) . x = (g /" (- h)) . x )
assume  x in dom (- (g /" h)) ; ::_thesis: (- (g /" h)) . x = (g /" (- h)) . x
thus (- (g /" h)) . x = - ((g /" h) . x) by VALUED_1:8
.= - ((g . x) / (h . x)) by VALUED_1:17
.= (g . x) / (- (h . x)) by XCMPLX_1:188
.= (g . x) / ((- h) . x) by VALUED_1:8
.= (g /" (- h)) . x by VALUED_1:17 ; ::_thesis: verum
end;

definition
let f be complex-valued Function;
let c be complex number ;
funcf (/) c ->  Function equals :: VALUED_2:def 32
(1 / c) (#) f;
coherence
 (1 / c) (#) f is Function ;
end;

:: deftheorem  defines (/) VALUED_2:def_32_:_
 for f being complex-valued Function
for c being complex number holds f (/) c = (1 / c) (#) f;

registration
let f be complex-valued Function;
let c be complex number ;
clusterf (/) c ->  complex-valued ;
coherence
 f (/) c is complex-valued ;
end;

registration
let f be real-valued Function;
let r be real number ;
clusterf (/) r ->  real-valued ;
coherence
 f (/) r is real-valued ;
end;

registration
let f be RAT -valued Function;
let r be rational number ;
clusterf (/) r ->  RAT -valued ;
coherence
 f (/) r is RAT -valued ;
end;

registration
let f be complex-valued FinSequence;
let c be complex number ;
clusterf (/) c ->  FinSequence-like ;
coherence
 f (/) c is FinSequence-like ;
end;

theorem :: VALUED_2:28
for c being complex number
for g being complex-valued Function holds dom (g (/) c) = dom g by VALUED_1:def_5;

theorem :: VALUED_2:29
for x being set
for c being complex number
for g being complex-valued Function holds (g (/) c) . x = (g . x) / c by VALUED_1:6;

theorem Th30: :: VALUED_2:30
for c being complex number
for g being complex-valued Function holds (- g) (/) c = - (g (/) c)
proof
let c be complex number ; ::_thesis: for g being complex-valued Function holds (- g) (/) c = - (g (/) c)
let g be complex-valued Function; ::_thesis: (- g) (/) c = - (g (/) c)
thus (- g) (/) c = (- (1 / c)) (#) g by Th22
.= - (g (/) c) by Th24 ; ::_thesis: verum
end;

theorem Th31: :: VALUED_2:31
for c being complex number
for g being complex-valued Function holds g (/) (- c) = - (g (/) c)
proof
let c be complex number ; ::_thesis: for g being complex-valued Function holds g (/) (- c) = - (g (/) c)
let g be complex-valued Function; ::_thesis: g (/) (- c) = - (g (/) c)
thus g (/) (- c) = (- (1 / c)) (#) g by XCMPLX_1:188
.= - (g (/) c) by Th24 ; ::_thesis: verum
end;

theorem :: VALUED_2:32
for c being complex number
for g being complex-valued Function holds g (/) (- c) = (- g) (/) c
proof
let c be complex number ; ::_thesis: for g being complex-valued Function holds g (/) (- c) = (- g) (/) c
let g be complex-valued Function; ::_thesis: g (/) (- c) = (- g) (/) c
thus g (/) (- c) = - (g (/) c) by Th31
.= (- g) (/) c by Th30 ; ::_thesis: verum
end;

theorem Th33: :: VALUED_2:33
for c1, c2 being complex number
for g being complex-valued Function st g <> {} & g is non-empty & g (/) c1 = g (/) c2 holds
c1 = c2
proof
let c1, c2 be complex number ; ::_thesis: for g being complex-valued Function st g <> {} & g is non-empty & g (/) c1 = g (/) c2 holds
c1 = c2
let g be complex-valued Function; ::_thesis: ( g <> {} & g is non-empty & g (/) c1 = g (/) c2 implies c1 = c2 )
assume that
A1: g <> {} and
A2: g is non-empty and
A3: g (/) c1 = g (/) c2 ; ::_thesis: c1 = c2
consider x being set  such that
A4: x in dom g by A1, XBOOLE_0:def_1;
 g . x in rng g by A4, FUNCT_1:def_3;
then A5: g . x <> {} by A2, RELAT_1:def_9;
 ( (g (/) c1) . x = (g . x) / c1 & (g (/) c2) . x = (g . x) / c2 ) by VALUED_1:6;
then  c1 " = c2 " by A3, A5, XCMPLX_1:5;
hence  c1 = c2 by XCMPLX_1:201; ::_thesis: verum
end;

theorem :: VALUED_2:34
for c1, c2 being complex number
for g being complex-valued Function holds (g (#) c1) (/) c2 = g (#) (c1 / c2)
proof
let c1, c2 be complex number ; ::_thesis: for g being complex-valued Function holds (g (#) c1) (/) c2 = g (#) (c1 / c2)
let g be complex-valued Function; ::_thesis: (g (#) c1) (/) c2 = g (#) (c1 / c2)
 ( dom (g (#) c1) = dom g & dom ((g (#) c1) (/) c2) = dom (g (#) c1) ) by VALUED_1:def_5;
hence  dom ((g (#) c1) (/) c2) = dom (g (#) (c1 / c2)) by VALUED_1:def_5; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((g (#) c1) (/) c2) or ((g (#) c1) (/) c2) . b1 = (g (#) (c1 / c2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((g (#) c1) (/) c2) or ((g (#) c1) (/) c2) . x = (g (#) (c1 / c2)) . x )
assume  x in dom ((g (#) c1) (/) c2) ; ::_thesis: ((g (#) c1) (/) c2) . x = (g (#) (c1 / c2)) . x
thus ((g (#) c1) (/) c2) . x = ((g (#) c1) . x) * (c2 ") by VALUED_1:6
.= ((g . x) * c1) * (c2 ") by VALUED_1:6
.= (g . x) * (c1 / c2)
.= (g (#) (c1 / c2)) . x by VALUED_1:6 ; ::_thesis: verum
end;

theorem :: VALUED_2:35
for c1, c2 being complex number
for g being complex-valued Function holds (g (/) c1) (#) c2 = (g (#) c2) (/) c1
proof
let c1, c2 be complex number ; ::_thesis: for g being complex-valued Function holds (g (/) c1) (#) c2 = (g (#) c2) (/) c1
let g be complex-valued Function; ::_thesis: (g (/) c1) (#) c2 = (g (#) c2) (/) c1
A1: dom ((g (/) c1) (#) c2) = dom (g (/) c1) by VALUED_1:def_5;
 ( dom (g (/) c1) = dom g & dom (g (#) c2) = dom g ) by VALUED_1:def_5;
hence  dom ((g (/) c1) (#) c2) = dom ((g (#) c2) (/) c1) by A1, VALUED_1:def_5; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((g (/) c1) (#) c2) or ((g (/) c1) (#) c2) . b1 = ((g (#) c2) (/) c1) . b1 )
let x be set ; ::_thesis: ( not x in dom ((g (/) c1) (#) c2) or ((g (/) c1) (#) c2) . x = ((g (#) c2) (/) c1) . x )
assume  x in dom ((g (/) c1) (#) c2) ; ::_thesis: ((g (/) c1) (#) c2) . x = ((g (#) c2) (/) c1) . x
thus ((g (/) c1) (#) c2) . x = ((g (/) c1) . x) * c2 by VALUED_1:6
.= ((g . x) * (c1 ")) * c2 by VALUED_1:6
.= ((g . x) * c2) * (c1 ")
.= ((g (#) c2) . x) * (c1 ") by VALUED_1:6
.= ((g (#) c2) (/) c1) . x by VALUED_1:6 ; ::_thesis: verum
end;

theorem :: VALUED_2:36
for c1, c2 being complex number
for g being complex-valued Function holds (g (/) c1) (/) c2 = g (/) (c1 * c2)
proof
let c1, c2 be complex number ; ::_thesis: for g being complex-valued Function holds (g (/) c1) (/) c2 = g (/) (c1 * c2)
let g be complex-valued Function; ::_thesis: (g (/) c1) (/) c2 = g (/) (c1 * c2)
 ( dom (g (/) c1) = dom g & dom (g (/) (c1 * c2)) = dom g ) by VALUED_1:def_5;
hence  dom ((g (/) c1) (/) c2) = dom (g (/) (c1 * c2)) by VALUED_1:def_5; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((g (/) c1) (/) c2) or ((g (/) c1) (/) c2) . b1 = (g (/) (c1 * c2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((g (/) c1) (/) c2) or ((g (/) c1) (/) c2) . x = (g (/) (c1 * c2)) . x )
assume  x in dom ((g (/) c1) (/) c2) ; ::_thesis: ((g (/) c1) (/) c2) . x = (g (/) (c1 * c2)) . x
thus ((g (/) c1) (/) c2) . x = ((g (/) c1) . x) * (c2 ") by VALUED_1:6
.= ((g . x) * (c1 ")) * (c2 ") by VALUED_1:6
.= (g . x) * ((c1 ") * (c2 "))
.= (g . x) * ((c1 * c2) ") by XCMPLX_1:204
.= (g (/) (c1 * c2)) . x by VALUED_1:6 ; ::_thesis: verum
end;

theorem :: VALUED_2:37
for c being complex number
for g, h being complex-valued Function holds (g + h) (/) c = (g (/) c) + (h (/) c)
proof
let c be complex number ; ::_thesis: for g, h being complex-valued Function holds (g + h) (/) c = (g (/) c) + (h (/) c)
let g, h be complex-valued Function; ::_thesis: (g + h) (/) c = (g (/) c) + (h (/) c)
A1: dom ((g + h) (/) c) = dom (g + h) by VALUED_1:def_5;
A2: dom (g + h) = (dom g) /\ (dom h) by VALUED_1:def_1;
 ( dom (g (/) c) = dom g & dom (h (/) c) = dom h ) by VALUED_1:def_5;
hence A3: dom ((g + h) (/) c) = dom ((g (/) c) + (h (/) c)) by A1, A2, VALUED_1:def_1; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((g + h) (/) c) or ((g + h) (/) c) . b1 = ((g (/) c) + (h (/) c)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((g + h) (/) c) or ((g + h) (/) c) . x = ((g (/) c) + (h (/) c)) . x )
assume A4: x in dom ((g + h) (/) c) ; ::_thesis: ((g + h) (/) c) . x = ((g (/) c) + (h (/) c)) . x
thus ((g + h) (/) c) . x = ((g + h) . x) * (c ") by VALUED_1:6
.= ((g . x) + (h . x)) * (c ") by A1, A4, VALUED_1:def_1
.= ((g . x) * (c ")) + ((h . x) * (c "))
.= ((g (/) c) . x) + ((h . x) * (c ")) by VALUED_1:6
.= ((g (/) c) . x) + ((h (/) c) . x) by VALUED_1:6
.= ((g (/) c) + (h (/) c)) . x by A3, A4, VALUED_1:def_1 ; ::_thesis: verum
end;

theorem :: VALUED_2:38
for c being complex number
for g, h being complex-valued Function holds (g - h) (/) c = (g (/) c) - (h (/) c)
proof
let c be complex number ; ::_thesis: for g, h being complex-valued Function holds (g - h) (/) c = (g (/) c) - (h (/) c)
let g, h be complex-valued Function; ::_thesis: (g - h) (/) c = (g (/) c) - (h (/) c)
A1: dom ((g - h) (/) c) = dom (g - h) by VALUED_1:def_5;
A2: dom (g - h) = (dom g) /\ (dom h) by VALUED_1:12;
 ( dom (g (/) c) = dom g & dom (h (/) c) = dom h ) by VALUED_1:def_5;
hence A3: dom ((g - h) (/) c) = dom ((g (/) c) - (h (/) c)) by A1, A2, VALUED_1:12; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((g - h) (/) c) or ((g - h) (/) c) . b1 = ((g (/) c) - (h (/) c)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((g - h) (/) c) or ((g - h) (/) c) . x = ((g (/) c) - (h (/) c)) . x )
assume A4: x in dom ((g - h) (/) c) ; ::_thesis: ((g - h) (/) c) . x = ((g (/) c) - (h (/) c)) . x
thus ((g - h) (/) c) . x = ((g - h) . x) * (c ") by VALUED_1:6
.= ((g . x) - (h . x)) * (c ") by A1, A4, VALUED_1:13
.= ((g . x) * (c ")) - ((h . x) * (c "))
.= ((g (/) c) . x) - ((h . x) * (c ")) by VALUED_1:6
.= ((g (/) c) . x) - ((h (/) c) . x) by VALUED_1:6
.= ((g (/) c) - (h (/) c)) . x by A3, A4, VALUED_1:13 ; ::_thesis: verum
end;

theorem :: VALUED_2:39
for c being complex number
for g, h being complex-valued Function holds (g (#) h) (/) c = g (#) (h (/) c)
proof
let c be complex number ; ::_thesis: for g, h being complex-valued Function holds (g (#) h) (/) c = g (#) (h (/) c)
let g, h be complex-valued Function; ::_thesis: (g (#) h) (/) c = g (#) (h (/) c)
A1: dom ((g (#) h) (/) c) = dom (g (#) h) by VALUED_1:def_5;
 ( dom (g (#) h) = (dom g) /\ (dom h) & dom (h (/) c) = dom h ) by VALUED_1:def_4, VALUED_1:def_5;
hence  dom ((g (#) h) (/) c) = dom (g (#) (h (/) c)) by A1, VALUED_1:def_4; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((g (#) h) (/) c) or ((g (#) h) (/) c) . b1 = (g (#) (h (/) c)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((g (#) h) (/) c) or ((g (#) h) (/) c) . x = (g (#) (h (/) c)) . x )
assume  x in dom ((g (#) h) (/) c) ; ::_thesis: ((g (#) h) (/) c) . x = (g (#) (h (/) c)) . x
thus ((g (#) h) (/) c) . x = ((g (#) h) . x) * (c ") by VALUED_1:6
.= ((g . x) * (h . x)) * (c ") by VALUED_1:5
.= (g . x) * ((h . x) * (c "))
.= (g . x) * ((h (/) c) . x) by VALUED_1:6
.= (g (#) (h (/) c)) . x by VALUED_1:5 ; ::_thesis: verum
end;

theorem :: VALUED_2:40
for c being complex number
for g, h being complex-valued Function holds (g /" h) (/) c = g /" (h (#) c)
proof
let c be complex number ; ::_thesis: for g, h being complex-valued Function holds (g /" h) (/) c = g /" (h (#) c)
let g, h be complex-valued Function; ::_thesis: (g /" h) (/) c = g /" (h (#) c)
A1: dom ((g /" h) (/) c) = dom (g /" h) by VALUED_1:def_5;
 ( dom (g /" h) = (dom g) /\ (dom h) & dom (h (#) c) = dom h ) by VALUED_1:16, VALUED_1:def_5;
hence  dom ((g /" h) (/) c) = dom (g /" (h (#) c)) by A1, VALUED_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((g /" h) (/) c) or ((g /" h) (/) c) . b1 = (g /" (h (#) c)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((g /" h) (/) c) or ((g /" h) (/) c) . x = (g /" (h (#) c)) . x )
assume  x in dom ((g /" h) (/) c) ; ::_thesis: ((g /" h) (/) c) . x = (g /" (h (#) c)) . x
thus ((g /" h) (/) c) . x = ((g /" h) . x) * (c ") by VALUED_1:6
.= ((g . x) / (h . x)) / c by VALUED_1:17
.= (g . x) / ((h . x) * c) by XCMPLX_1:78
.= (g . x) / ((h (#) c) . x) by VALUED_1:6
.= (g /" (h (#) c)) . x by VALUED_1:17 ; ::_thesis: verum
end;

definition
let f be complex-functions-valued Function;
deffunc H1( set ) ->  set = - (f . $1);
func <-> f ->  Function means :Def33: :: VALUED_2:def 33
( dom it = dom f & ( for x being set st x in dom it holds
it . x = - (f . x) ) );
existence
 ex b1 being Function st
( dom b1 = dom f & ( for x being set st x in dom b1 holds
b1 . x = - (f . x) ) )
proof
 ex F being Function st
( dom F = dom f & ( for x being set st x in dom f holds
F . x = H1(x) ) ) from FUNCT_1:sch_3();
hence  ex b1 being Function st
( dom b1 = dom f & ( for x being set st x in dom b1 holds
b1 . x = - (f . x) ) ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function st dom b1 = dom f & ( for x being set st x in dom b1 holds
b1 . x = - (f . x) ) & dom b2 = dom f & ( for x being set st x in dom b2 holds
b2 . x = - (f . x) ) holds
b1 = b2
proof
let F, G be Function; ::_thesis: ( dom F = dom f & ( for x being set st x in dom F holds
F . x = - (f . x) ) & dom G = dom f & ( for x being set st x in dom G holds
G . x = - (f . x) ) implies F = G )
assume that
A1: dom F = dom f and
A2: for x being set st x in dom F holds
F . x = H1(x) and
A3: dom G = dom f and
A4: for x being set st x in dom G holds
G . x = H1(x) ; ::_thesis: F = G
thus  dom F = dom G by A1, A3; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom F or F . b1 = G . b1 )
let x be set ; ::_thesis: ( not x in dom F or F . x = G . x )
assume A5: x in dom F ; ::_thesis: F . x = G . x
hence F . x = H1(x) by A2
.= G . x by A1, A3, A4, A5 ;
::_thesis: verum
end;
end;

:: deftheorem Def33 defines <-> VALUED_2:def_33_:_
 for f being complex-functions-valued Function
for b2 being Function holds
( b2 = <-> f iff ( dom b2 = dom f & ( for x being set st x in dom b2 holds
b2 . x = - (f . x) ) ) );

definition
let X be set ;
let Y be complex-functions-membered set ;
let f be PartFunc of X,Y;
:: original: <->
redefine func <-> f ->  PartFunc of X,(C_PFuncs (DOMS Y));
coherence
 <-> f is PartFunc of X,(C_PFuncs (DOMS Y))
proof
set h = <-> f;
A1: dom (<-> f) = dom f by Def33;
 rng (<-> f) c= C_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (<-> f) or y in C_PFuncs (DOMS Y) )
assume  y in rng (<-> f) ; ::_thesis: y in C_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (<-> f) and
A3: (<-> f) . x = y by FUNCT_1:def_3;
A4: (<-> f) . x = - (f . x) by A2, Def33;
then reconsider y = y as Function by A3;
A5: rng y c= COMPLEX
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in COMPLEX )
thus  ( not b in rng y or b in COMPLEX ) by A3, A4, XCMPLX_0:def_2; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:8;
then  y is PartFunc of (DOMS Y),COMPLEX by A6, A5, RELSET_1:4;
hence  y in C_PFuncs (DOMS Y) by Def8; ::_thesis: verum
end;
hence  <-> f is PartFunc of X,(C_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be real-functions-membered set ;
let f be PartFunc of X,Y;
:: original: <->
redefine func <-> f ->  PartFunc of X,(R_PFuncs (DOMS Y));
coherence
 <-> f is PartFunc of X,(R_PFuncs (DOMS Y))
proof
set h = <-> f;
A1: dom (<-> f) = dom f by Def33;
 rng (<-> f) c= R_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (<-> f) or y in R_PFuncs (DOMS Y) )
assume  y in rng (<-> f) ; ::_thesis: y in R_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (<-> f) and
A3: (<-> f) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (<-> f) . x = - (f . x) by A2, Def33;
A5: rng y c= REAL
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in REAL )
thus  ( not b in rng y or b in REAL ) by A3, A4, XREAL_0:def_1; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:8;
then  y is PartFunc of (DOMS Y),REAL by A6, A5, RELSET_1:4;
hence  y in R_PFuncs (DOMS Y) by Def12; ::_thesis: verum
end;
hence  <-> f is PartFunc of X,(R_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be rational-functions-membered set ;
let f be PartFunc of X,Y;
:: original: <->
redefine func <-> f ->  PartFunc of X,(Q_PFuncs (DOMS Y));
coherence
 <-> f is PartFunc of X,(Q_PFuncs (DOMS Y))
proof
set h = <-> f;
A1: dom (<-> f) = dom f by Def33;
 rng (<-> f) c= Q_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (<-> f) or y in Q_PFuncs (DOMS Y) )
assume  y in rng (<-> f) ; ::_thesis: y in Q_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (<-> f) and
A3: (<-> f) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (<-> f) . x = - (f . x) by A2, Def33;
A5: rng y c= RAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in RAT )
thus  ( not b in rng y or b in RAT ) by A3, A4, RAT_1:def_2; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:8;
then  y is PartFunc of (DOMS Y),RAT by A6, A5, RELSET_1:4;
hence  y in Q_PFuncs (DOMS Y) by Def14; ::_thesis: verum
end;
hence  <-> f is PartFunc of X,(Q_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be integer-functions-membered set ;
let f be PartFunc of X,Y;
:: original: <->
redefine func <-> f ->  PartFunc of X,(I_PFuncs (DOMS Y));
coherence
 <-> f is PartFunc of X,(I_PFuncs (DOMS Y))
proof
set h = <-> f;
A1: dom (<-> f) = dom f by Def33;
 rng (<-> f) c= I_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (<-> f) or y in I_PFuncs (DOMS Y) )
assume  y in rng (<-> f) ; ::_thesis: y in I_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (<-> f) and
A3: (<-> f) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (<-> f) . x = - (f . x) by A2, Def33;
A5: rng y c= INT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in INT )
thus  ( not b in rng y or b in INT ) by A3, A4, INT_1:def_2; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:8;
then  y is PartFunc of (DOMS Y),INT by A6, A5, RELSET_1:4;
hence  y in I_PFuncs (DOMS Y) by Def16; ::_thesis: verum
end;
hence  <-> f is PartFunc of X,(I_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

registration
let Y be complex-functions-membered set ;
let f be FinSequence of Y;
cluster <-> f ->  FinSequence-like ;
coherence
 <-> f is FinSequence-like
proof
 ( dom (<-> f) = dom f & ex n being Nat st dom f = Seg n ) by Def33, FINSEQ_1:def_2;
hence  <-> f is FinSequence-like by FINSEQ_1:def_2; ::_thesis: verum
end;
end;

theorem :: VALUED_2:41
for X being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y holds <-> (<-> f) = f
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y holds <-> (<-> f) = f
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y holds <-> (<-> f) = f
let f be PartFunc of X,Y; ::_thesis: <-> (<-> f) = f
set f1 = <-> f;
A1: dom (<-> f) = dom f by Def33;
hence A2: dom (<-> (<-> f)) = dom f by Def33; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (<-> (<-> f)) or (<-> (<-> f)) . b1 = f . b1 )
let x be set ; ::_thesis: ( not x in dom (<-> (<-> f)) or (<-> (<-> f)) . x = f . x )
assume A3: x in dom (<-> (<-> f)) ; ::_thesis: (<-> (<-> f)) . x = f . x
hence (<-> (<-> f)) . x = - ((<-> f) . x) by Def33
.= - (- (f . x)) by A1, A2, A3, Def33
.= f . x ;
::_thesis: verum
end;

theorem :: VALUED_2:42
for X1, X2 being set
for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 st <-> f1 = <-> f2 holds
f1 = f2
proof
let X1, X2 be set ; ::_thesis: for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 st <-> f1 = <-> f2 holds
f1 = f2
let Y1, Y2 be complex-functions-membered set ; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 st <-> f1 = <-> f2 holds
f1 = f2
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 st <-> f1 = <-> f2 holds
f1 = f2
let f2 be PartFunc of X2,Y2; ::_thesis: ( <-> f1 = <-> f2 implies f1 = f2 )
A1: dom (<-> f1) = dom f1 by Def33;
assume A2: <-> f1 = <-> f2 ; ::_thesis: f1 = f2
hence  dom f1 = dom f2 by A1, Def33; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom f1 or f1 . b1 = f2 . b1 )
let x be set ; ::_thesis: ( not x in dom f1 or f1 . x = f2 . x )
assume A3: x in dom f1 ; ::_thesis: f1 . x = f2 . x
thus f1 . x = - (- (f1 . x))
.= - ((<-> f1) . x) by A1, A3, Def33
.= - (- (f2 . x)) by A2, A1, A3, Def33
.= f2 . x ; ::_thesis: verum
end;

definition
let X be complex-functions-membered set ;
let Y be set ;
let f be PartFunc of X,Y;
defpred S1[ set , set ] means  ex a being complex-valued Function st
( $1 = a & $2 = f . (- a) );
funcf (-)  ->  Function means :: VALUED_2:def 34
( dom it = dom f & ( for x being complex-valued Function st x in dom it holds
it . x = f . (- x) ) );
existence
 ex b1 being Function st
( dom b1 = dom f & ( for x being complex-valued Function st x in dom b1 holds
b1 . x = f . (- x) ) )
proof
A1: for x being set st x in dom f holds
ex y being set st S1[x,y]
proof
let x be set ; ::_thesis: ( x in dom f implies ex y being set st S1[x,y] )
assume  x in dom f ; ::_thesis: ex y being set st S1[x,y]
then reconsider a = x as complex-valued Function ;
take  f . (- a) ; ::_thesis: S1[x,f . (- a)]
take  a ; ::_thesis: ( x = a & f . (- a) = f . (- a) )
thus  ( x = a & f . (- a) = f . (- a) ) ; ::_thesis: verum
end;
consider F being Function such that
A2: dom F = dom f and
A3: for x being set st x in dom f holds
S1[x,F . x] from CLASSES1:sch_1(A1);
take  F ; ::_thesis: ( dom F = dom f & ( for x being complex-valued Function st x in dom F holds
F . x = f . (- x) ) )
thus  dom F = dom f by A2; ::_thesis: for x being complex-valued Function st x in dom F holds
F . x = f . (- x)
let x be complex-valued Function; ::_thesis: ( x in dom F implies F . x = f . (- x) )
assume  x in dom F ; ::_thesis: F . x = f . (- x)
then  S1[x,F . x] by A2, A3;
hence  F . x = f . (- x) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function st dom b1 = dom f & ( for x being complex-valued Function st x in dom b1 holds
b1 . x = f . (- x) ) & dom b2 = dom f & ( for x being complex-valued Function st x in dom b2 holds
b2 . x = f . (- x) ) holds
b1 = b2
proof
let F, G be Function; ::_thesis: ( dom F = dom f & ( for x being complex-valued Function st x in dom F holds
F . x = f . (- x) ) & dom G = dom f & ( for x being complex-valued Function st x in dom G holds
G . x = f . (- x) ) implies F = G )
assume that
A4: dom F = dom f and
A5: for x being complex-valued Function st x in dom F holds
F . x = f . (- x) and
A6: dom G = dom f and
A7: for x being complex-valued Function st x in dom G holds
G . x = f . (- x) ; ::_thesis: F = G
thus  dom F = dom G by A4, A6; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom F or F . b1 = G . b1 )
let x be set ; ::_thesis: ( not x in dom F or F . x = G . x )
assume A8: x in dom F ; ::_thesis: F . x = G . x
then reconsider y = x as complex-valued Function by A4;
thus F . x = f . (- y) by A5, A8
.= G . x by A4, A6, A7, A8 ; ::_thesis: verum
end;
end;

:: deftheorem  defines (-) VALUED_2:def_34_:_
 for X being complex-functions-membered set
for Y being set
for f being PartFunc of X,Y
for b4 being Function holds
( b4 = f (-) iff ( dom b4 = dom f & ( for x being complex-valued Function st x in dom b4 holds
b4 . x = f . (- x) ) ) );

definition
let f be complex-functions-valued Function;
deffunc H1( set ) ->  set = (f . $1) " ;
func </> f ->  Function means :Def35: :: VALUED_2:def 35
( dom it = dom f & ( for x being set st x in dom it holds
it . x = (f . x) " ) );
existence
 ex b1 being Function st
( dom b1 = dom f & ( for x being set st x in dom b1 holds
b1 . x = (f . x) " ) )
proof
 ex F being Function st
( dom F = dom f & ( for x being set st x in dom f holds
F . x = H1(x) ) ) from FUNCT_1:sch_3();
hence  ex b1 being Function st
( dom b1 = dom f & ( for x being set st x in dom b1 holds
b1 . x = (f . x) " ) ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function st dom b1 = dom f & ( for x being set st x in dom b1 holds
b1 . x = (f . x) " ) & dom b2 = dom f & ( for x being set st x in dom b2 holds
b2 . x = (f . x) " ) holds
b1 = b2
proof
let F, G be Function; ::_thesis: ( dom F = dom f & ( for x being set st x in dom F holds
F . x = (f . x) " ) & dom G = dom f & ( for x being set st x in dom G holds
G . x = (f . x) " ) implies F = G )
assume that
A1: dom F = dom f and
A2: for x being set st x in dom F holds
F . x = H1(x) and
A3: dom G = dom f and
A4: for x being set st x in dom G holds
G . x = H1(x) ; ::_thesis: F = G
thus  dom F = dom G by A1, A3; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom F or F . b1 = G . b1 )
let x be set ; ::_thesis: ( not x in dom F or F . x = G . x )
assume A5: x in dom F ; ::_thesis: F . x = G . x
hence F . x = H1(x) by A2
.= G . x by A1, A3, A4, A5 ;
::_thesis: verum
end;
end;

:: deftheorem Def35 defines </> VALUED_2:def_35_:_
 for f being complex-functions-valued Function
for b2 being Function holds
( b2 = </> f iff ( dom b2 = dom f & ( for x being set st x in dom b2 holds
b2 . x = (f . x) " ) ) );

definition
let X be set ;
let Y be complex-functions-membered set ;
let f be PartFunc of X,Y;
:: original: </>
redefine func </> f ->  PartFunc of X,(C_PFuncs (DOMS Y));
coherence
 </> f is PartFunc of X,(C_PFuncs (DOMS Y))
proof
set h = </> f;
A1: dom (</> f) = dom f by Def35;
 rng (</> f) c= C_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (</> f) or y in C_PFuncs (DOMS Y) )
assume  y in rng (</> f) ; ::_thesis: y in C_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (</> f) and
A3: (</> f) . x = y by FUNCT_1:def_3;
A4: (</> f) . x = (f . x) " by A2, Def35;
then reconsider y = y as Function by A3;
A5: rng y c= COMPLEX
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in COMPLEX )
thus  ( not b in rng y or b in COMPLEX ) by A3, A4, XCMPLX_0:def_2; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_7;
then  y is PartFunc of (DOMS Y),COMPLEX by A6, A5, RELSET_1:4;
hence  y in C_PFuncs (DOMS Y) by Def8; ::_thesis: verum
end;
hence  </> f is PartFunc of X,(C_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be real-functions-membered set ;
let f be PartFunc of X,Y;
:: original: </>
redefine func </> f ->  PartFunc of X,(R_PFuncs (DOMS Y));
coherence
 </> f is PartFunc of X,(R_PFuncs (DOMS Y))
proof
set h = </> f;
A1: dom (</> f) = dom f by Def35;
 rng (</> f) c= R_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (</> f) or y in R_PFuncs (DOMS Y) )
assume  y in rng (</> f) ; ::_thesis: y in R_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (</> f) and
A3: (</> f) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (</> f) . x = (f . x) " by A2, Def35;
A5: rng y c= REAL
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in REAL )
thus  ( not b in rng y or b in REAL ) by A3, A4, XREAL_0:def_1; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_7;
then  y is PartFunc of (DOMS Y),REAL by A6, A5, RELSET_1:4;
hence  y in R_PFuncs (DOMS Y) by Def12; ::_thesis: verum
end;
hence  </> f is PartFunc of X,(R_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be rational-functions-membered set ;
let f be PartFunc of X,Y;
:: original: </>
redefine func </> f ->  PartFunc of X,(Q_PFuncs (DOMS Y));
coherence
 </> f is PartFunc of X,(Q_PFuncs (DOMS Y))
proof
set h = </> f;
A1: dom (</> f) = dom f by Def35;
 rng (</> f) c= Q_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (</> f) or y in Q_PFuncs (DOMS Y) )
assume  y in rng (</> f) ; ::_thesis: y in Q_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (</> f) and
A3: (</> f) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (</> f) . x = (f . x) " by A2, Def35;
A5: rng y c= RAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in RAT )
thus  ( not b in rng y or b in RAT ) by A3, A4, RAT_1:def_2; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_7;
then  y is PartFunc of (DOMS Y),RAT by A6, A5, RELSET_1:4;
hence  y in Q_PFuncs (DOMS Y) by Def14; ::_thesis: verum
end;
hence  </> f is PartFunc of X,(Q_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

registration
let Y be complex-functions-membered set ;
let f be FinSequence of Y;
cluster </> f ->  FinSequence-like ;
coherence
 </> f is FinSequence-like
proof
 ( dom (</> f) = dom f & ex n being Nat st dom f = Seg n ) by Def35, FINSEQ_1:def_2;
hence  </> f is FinSequence-like by FINSEQ_1:def_2; ::_thesis: verum
end;
end;

theorem :: VALUED_2:43
for X being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y holds </> (</> f) = f
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y holds </> (</> f) = f
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y holds </> (</> f) = f
let f be PartFunc of X,Y; ::_thesis: </> (</> f) = f
set f1 = </> f;
A1: dom (</> f) = dom f by Def35;
hence A2: dom (</> (</> f)) = dom f by Def35; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (</> (</> f)) or (</> (</> f)) . b1 = f . b1 )
let x be set ; ::_thesis: ( not x in dom (</> (</> f)) or (</> (</> f)) . x = f . x )
assume A3: x in dom (</> (</> f)) ; ::_thesis: (</> (</> f)) . x = f . x
hence (</> (</> f)) . x = ((</> f) . x) " by Def35
.= ((f . x) ") " by A1, A2, A3, Def35
.= f . x ;
::_thesis: verum
end;

definition
let f be complex-functions-valued Function;
deffunc H1( set ) ->  set = abs (f . $1);
func abs f ->  Function means :Def36: :: VALUED_2:def 36
( dom it = dom f & ( for x being set st x in dom it holds
it . x = abs (f . x) ) );
existence
 ex b1 being Function st
( dom b1 = dom f & ( for x being set st x in dom b1 holds
b1 . x = abs (f . x) ) )
proof
 ex F being Function st
( dom F = dom f & ( for x being set st x in dom f holds
F . x = H1(x) ) ) from FUNCT_1:sch_3();
hence  ex b1 being Function st
( dom b1 = dom f & ( for x being set st x in dom b1 holds
b1 . x = abs (f . x) ) ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function st dom b1 = dom f & ( for x being set st x in dom b1 holds
b1 . x = abs (f . x) ) & dom b2 = dom f & ( for x being set st x in dom b2 holds
b2 . x = abs (f . x) ) holds
b1 = b2
proof
let F, G be Function; ::_thesis: ( dom F = dom f & ( for x being set st x in dom F holds
F . x = abs (f . x) ) & dom G = dom f & ( for x being set st x in dom G holds
G . x = abs (f . x) ) implies F = G )
assume that
A1: dom F = dom f and
A2: for x being set st x in dom F holds
F . x = H1(x) and
A3: dom G = dom f and
A4: for x being set st x in dom G holds
G . x = H1(x) ; ::_thesis: F = G
thus  dom F = dom G by A1, A3; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom F or F . b1 = G . b1 )
let x be set ; ::_thesis: ( not x in dom F or F . x = G . x )
assume A5: x in dom F ; ::_thesis: F . x = G . x
hence F . x = H1(x) by A2
.= G . x by A1, A3, A4, A5 ;
::_thesis: verum
end;
end;

:: deftheorem Def36 defines abs VALUED_2:def_36_:_
 for f being complex-functions-valued Function
for b2 being Function holds
( b2 = abs f iff ( dom b2 = dom f & ( for x being set st x in dom b2 holds
b2 . x = abs (f . x) ) ) );

definition
let X be set ;
let Y be complex-functions-membered set ;
let f be PartFunc of X,Y;
:: original: abs
redefine func abs f ->  PartFunc of X,(C_PFuncs (DOMS Y));
coherence
 abs f is PartFunc of X,(C_PFuncs (DOMS Y))
proof
set h = abs f;
A1: dom (abs f) = dom f by Def36;
 rng (abs f) c= C_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (abs f) or y in C_PFuncs (DOMS Y) )
assume  y in rng (abs f) ; ::_thesis: y in C_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (abs f) and
A3: (abs f) . x = y by FUNCT_1:def_3;
A4: (abs f) . x = abs (f . x) by A2, Def36;
then reconsider y = y as Function by A3;
A5: rng y c= COMPLEX
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in COMPLEX )
thus  ( not b in rng y or b in COMPLEX ) by A3, A4, XCMPLX_0:def_2; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_11;
then  y is PartFunc of (DOMS Y),COMPLEX by A6, A5, RELSET_1:4;
hence  y in C_PFuncs (DOMS Y) by Def8; ::_thesis: verum
end;
hence  abs f is PartFunc of X,(C_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be real-functions-membered set ;
let f be PartFunc of X,Y;
:: original: abs
redefine func abs f ->  PartFunc of X,(R_PFuncs (DOMS Y));
coherence
 abs f is PartFunc of X,(R_PFuncs (DOMS Y))
proof
set h = abs f;
A1: dom (abs f) = dom f by Def36;
 rng (abs f) c= R_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (abs f) or y in R_PFuncs (DOMS Y) )
assume  y in rng (abs f) ; ::_thesis: y in R_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (abs f) and
A3: (abs f) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (abs f) . x = abs (f . x) by A2, Def36;
A5: rng y c= REAL
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in REAL )
thus  ( not b in rng y or b in REAL ) by A3, A4, XREAL_0:def_1; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_11;
then  y is PartFunc of (DOMS Y),REAL by A6, A5, RELSET_1:4;
hence  y in R_PFuncs (DOMS Y) by Def12; ::_thesis: verum
end;
hence  abs f is PartFunc of X,(R_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be rational-functions-membered set ;
let f be PartFunc of X,Y;
:: original: abs
redefine func abs f ->  PartFunc of X,(Q_PFuncs (DOMS Y));
coherence
 abs f is PartFunc of X,(Q_PFuncs (DOMS Y))
proof
set h = abs f;
A1: dom (abs f) = dom f by Def36;
 rng (abs f) c= Q_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (abs f) or y in Q_PFuncs (DOMS Y) )
assume  y in rng (abs f) ; ::_thesis: y in Q_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (abs f) and
A3: (abs f) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (abs f) . x = abs (f . x) by A2, Def36;
A5: rng y c= RAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in RAT )
thus  ( not b in rng y or b in RAT ) by A3, A4, RAT_1:def_2; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_11;
then  y is PartFunc of (DOMS Y),RAT by A6, A5, RELSET_1:4;
hence  y in Q_PFuncs (DOMS Y) by Def14; ::_thesis: verum
end;
hence  abs f is PartFunc of X,(Q_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be integer-functions-membered set ;
let f be PartFunc of X,Y;
:: original: abs
redefine func abs f ->  PartFunc of X,(N_PFuncs (DOMS Y));
coherence
 abs f is PartFunc of X,(N_PFuncs (DOMS Y))
proof
set h = abs f;
A1: dom (abs f) = dom f by Def36;
 rng (abs f) c= N_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (abs f) or y in N_PFuncs (DOMS Y) )
assume  y in rng (abs f) ; ::_thesis: y in N_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (abs f) and
A3: (abs f) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (abs f) . x = abs (f . x) by A2, Def36;
A5: rng y c= NAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in NAT )
thus  ( not b in rng y or b in NAT ) by A3, A4, ORDINAL1:def_12; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_11;
then  y is PartFunc of (DOMS Y),NAT by A6, A5, RELSET_1:4;
hence  y in N_PFuncs (DOMS Y) by Def18; ::_thesis: verum
end;
hence  abs f is PartFunc of X,(N_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

registration
let Y be complex-functions-membered set ;
let f be FinSequence of Y;
cluster abs f ->  FinSequence-like ;
coherence
 abs f is FinSequence-like
proof
 ( dom (abs f) = dom f & ex n being Nat st dom f = Seg n ) by Def36, FINSEQ_1:def_2;
hence  abs f is FinSequence-like by FINSEQ_1:def_2; ::_thesis: verum
end;
end;

theorem :: VALUED_2:44
for X being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y holds abs (abs f) = abs f
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y holds abs (abs f) = abs f
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y holds abs (abs f) = abs f
let f be PartFunc of X,Y; ::_thesis: abs (abs f) = abs f
set f1 = abs f;
thus A1: dom (abs (abs f)) = dom (abs f) by Def36; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (abs (abs f)) or (abs (abs f)) . b1 = (abs f) . b1 )
let x be set ; ::_thesis: ( not x in dom (abs (abs f)) or (abs (abs f)) . x = (abs f) . x )
assume A2: x in dom (abs (abs f)) ; ::_thesis: (abs (abs f)) . x = (abs f) . x
hence (abs (abs f)) . x = abs ((abs f) . x) by Def36
.= abs (abs (f . x)) by A1, A2, Def36
.= (abs f) . x by A1, A2, Def36 ;
::_thesis: verum
end;

definition
let Y be complex-functions-membered set ;
let f be Y -valued Function;
let c be complex number ;
deffunc H1( set ) ->  set = c + (f . $1);
funcf [+] c ->  Function means :Def37: :: VALUED_2:def 37
( dom it = dom f & ( for x being set st x in dom it holds
it . x = c + (f . x) ) );
existence
 ex b1 being Function st
( dom b1 = dom f & ( for x being set st x in dom b1 holds
b1 . x = c + (f . x) ) )
proof
 ex F being Function st
( dom F = dom f & ( for x being set st x in dom f holds
F . x = H1(x) ) ) from FUNCT_1:sch_3();
hence  ex b1 being Function st
( dom b1 = dom f & ( for x being set st x in dom b1 holds
b1 . x = c + (f . x) ) ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function st dom b1 = dom f & ( for x being set st x in dom b1 holds
b1 . x = c + (f . x) ) & dom b2 = dom f & ( for x being set st x in dom b2 holds
b2 . x = c + (f . x) ) holds
b1 = b2
proof
let F, G be Function; ::_thesis: ( dom F = dom f & ( for x being set st x in dom F holds
F . x = c + (f . x) ) & dom G = dom f & ( for x being set st x in dom G holds
G . x = c + (f . x) ) implies F = G )
assume that
A1: dom F = dom f and
A2: for x being set st x in dom F holds
F . x = H1(x) and
A3: dom G = dom f and
A4: for x being set st x in dom G holds
G . x = H1(x) ; ::_thesis: F = G
thus  dom F = dom G by A1, A3; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom F or F . b1 = G . b1 )
let x be set ; ::_thesis: ( not x in dom F or F . x = G . x )
assume A5: x in dom F ; ::_thesis: F . x = G . x
hence F . x = H1(x) by A2
.= G . x by A1, A3, A4, A5 ;
::_thesis: verum
end;
end;

:: deftheorem Def37 defines [+] VALUED_2:def_37_:_
 for Y being complex-functions-membered set
for f being b1 -valued Function
for c being complex number
for b4 being Function holds
( b4 = f [+] c iff ( dom b4 = dom f & ( for x being set st x in dom b4 holds
b4 . x = c + (f . x) ) ) );

definition
let X be set ;
let Y be complex-functions-membered set ;
let f be PartFunc of X,Y;
let c be complex number ;
:: original: [+]
redefine funcf [+] c ->  PartFunc of X,(C_PFuncs (DOMS Y));
coherence
 f [+] c is PartFunc of X,(C_PFuncs (DOMS Y))
proof
set h = f [+] c;
A1: dom (f [+] c) = dom f by Def37;
 rng (f [+] c) c= C_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f [+] c) or y in C_PFuncs (DOMS Y) )
assume  y in rng (f [+] c) ; ::_thesis: y in C_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f [+] c) and
A3: (f [+] c) . x = y by FUNCT_1:def_3;
A4: (f [+] c) . x = (f . x) + c by A2, Def37;
then reconsider y = y as Function by A3;
A5: rng y c= COMPLEX
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in COMPLEX )
thus  ( not b in rng y or b in COMPLEX ) by A3, A4, XCMPLX_0:def_2; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_2;
then  y is PartFunc of (DOMS Y),COMPLEX by A6, A5, RELSET_1:4;
hence  y in C_PFuncs (DOMS Y) by Def8; ::_thesis: verum
end;
hence  f [+] c is PartFunc of X,(C_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be real-functions-membered set ;
let f be PartFunc of X,Y;
let c be real number ;
:: original: [+]
redefine funcf [+] c ->  PartFunc of X,(R_PFuncs (DOMS Y));
coherence
 f [+] c is PartFunc of X,(R_PFuncs (DOMS Y))
proof
set h = f [+] c;
A1: dom (f [+] c) = dom f by Def37;
 rng (f [+] c) c= R_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f [+] c) or y in R_PFuncs (DOMS Y) )
assume  y in rng (f [+] c) ; ::_thesis: y in R_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f [+] c) and
A3: (f [+] c) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f [+] c) . x = (f . x) + c by A2, Def37;
A5: rng y c= REAL
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in REAL )
thus  ( not b in rng y or b in REAL ) by A3, A4, XREAL_0:def_1; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_2;
then  y is PartFunc of (DOMS Y),REAL by A6, A5, RELSET_1:4;
hence  y in R_PFuncs (DOMS Y) by Def12; ::_thesis: verum
end;
hence  f [+] c is PartFunc of X,(R_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be rational-functions-membered set ;
let f be PartFunc of X,Y;
let c be rational number ;
:: original: [+]
redefine funcf [+] c ->  PartFunc of X,(Q_PFuncs (DOMS Y));
coherence
 f [+] c is PartFunc of X,(Q_PFuncs (DOMS Y))
proof
set h = f [+] c;
A1: dom (f [+] c) = dom f by Def37;
 rng (f [+] c) c= Q_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f [+] c) or y in Q_PFuncs (DOMS Y) )
assume  y in rng (f [+] c) ; ::_thesis: y in Q_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f [+] c) and
A3: (f [+] c) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f [+] c) . x = (f . x) + c by A2, Def37;
A5: rng y c= RAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in RAT )
thus  ( not b in rng y or b in RAT ) by A3, A4, RAT_1:def_2; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_2;
then  y is PartFunc of (DOMS Y),RAT by A6, A5, RELSET_1:4;
hence  y in Q_PFuncs (DOMS Y) by Def14; ::_thesis: verum
end;
hence  f [+] c is PartFunc of X,(Q_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be integer-functions-membered set ;
let f be PartFunc of X,Y;
let c be integer number ;
:: original: [+]
redefine funcf [+] c ->  PartFunc of X,(I_PFuncs (DOMS Y));
coherence
 f [+] c is PartFunc of X,(I_PFuncs (DOMS Y))
proof
set h = f [+] c;
A1: dom (f [+] c) = dom f by Def37;
 rng (f [+] c) c= I_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f [+] c) or y in I_PFuncs (DOMS Y) )
assume  y in rng (f [+] c) ; ::_thesis: y in I_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f [+] c) and
A3: (f [+] c) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f [+] c) . x = (f . x) + c by A2, Def37;
A5: rng y c= INT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in INT )
thus  ( not b in rng y or b in INT ) by A3, A4, INT_1:def_2; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_2;
then  y is PartFunc of (DOMS Y),INT by A6, A5, RELSET_1:4;
hence  y in I_PFuncs (DOMS Y) by Def16; ::_thesis: verum
end;
hence  f [+] c is PartFunc of X,(I_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be natural-functions-membered set ;
let f be PartFunc of X,Y;
let c be Nat;
:: original: [+]
redefine funcf [+] c ->  PartFunc of X,(N_PFuncs (DOMS Y));
coherence
 f [+] c is PartFunc of X,(N_PFuncs (DOMS Y))
proof
set h = f [+] c;
A1: dom (f [+] c) = dom f by Def37;
 rng (f [+] c) c= N_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f [+] c) or y in N_PFuncs (DOMS Y) )
assume  y in rng (f [+] c) ; ::_thesis: y in N_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f [+] c) and
A3: (f [+] c) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f [+] c) . x = (f . x) + c by A2, Def37;
A5: rng y c= NAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in NAT )
thus  ( not b in rng y or b in NAT ) by A3, A4, ORDINAL1:def_12; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_2;
then  y is PartFunc of (DOMS Y),NAT by A6, A5, RELSET_1:4;
hence  y in N_PFuncs (DOMS Y) by Def18; ::_thesis: verum
end;
hence  f [+] c is PartFunc of X,(N_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

theorem :: VALUED_2:45
for X being set
for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [+] c1) [+] c2 = f [+] (c1 + c2)
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [+] c1) [+] c2 = f [+] (c1 + c2)
let Y be complex-functions-membered set ; ::_thesis: for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [+] c1) [+] c2 = f [+] (c1 + c2)
let c1, c2 be complex number ; ::_thesis: for f being PartFunc of X,Y holds (f [+] c1) [+] c2 = f [+] (c1 + c2)
let f be PartFunc of X,Y; ::_thesis: (f [+] c1) [+] c2 = f [+] (c1 + c2)
set f1 = f [+] c1;
A1: dom ((f [+] c1) [+] c2) = dom (f [+] c1) by Def37;
 dom (f [+] c1) = dom f by Def37;
hence A2: dom ((f [+] c1) [+] c2) = dom (f [+] (c1 + c2)) by A1, Def37; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f [+] c1) [+] c2) or ((f [+] c1) [+] c2) . b1 = (f [+] (c1 + c2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f [+] c1) [+] c2) or ((f [+] c1) [+] c2) . x = (f [+] (c1 + c2)) . x )
assume A3: x in dom ((f [+] c1) [+] c2) ; ::_thesis: ((f [+] c1) [+] c2) . x = (f [+] (c1 + c2)) . x
hence ((f [+] c1) [+] c2) . x = ((f [+] c1) . x) + c2 by Def37
.= ((f . x) + c1) + c2 by A1, A3, Def37
.= (f . x) + (c1 + c2) by Th12
.= (f [+] (c1 + c2)) . x by A2, A3, Def37 ;
::_thesis: verum
end;

theorem :: VALUED_2:46
for X being set
for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y st f <> {} & f is non-empty & f [+] c1 = f [+] c2 holds
c1 = c2
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y st f <> {} & f is non-empty & f [+] c1 = f [+] c2 holds
c1 = c2
let Y be complex-functions-membered set ; ::_thesis: for c1, c2 being complex number
for f being PartFunc of X,Y st f <> {} & f is non-empty & f [+] c1 = f [+] c2 holds
c1 = c2
let c1, c2 be complex number ; ::_thesis: for f being PartFunc of X,Y st f <> {} & f is non-empty & f [+] c1 = f [+] c2 holds
c1 = c2
let f be PartFunc of X,Y; ::_thesis: ( f <> {} & f is non-empty & f [+] c1 = f [+] c2 implies c1 = c2 )
assume that
A1: f <> {} and
A2: f is non-empty and
A3: f [+] c1 = f [+] c2 ; ::_thesis: c1 = c2
consider x being set  such that
A4: x in dom f by A1, XBOOLE_0:def_1;
 f . x in rng f by A4, FUNCT_1:def_3;
then A5: f . x <> {} by A2, RELAT_1:def_9;
 dom f = dom (f [+] c2) by Def37;
then A6: (f [+] c2) . x = (f . x) + c2 by A4, Def37;
 dom f = dom (f [+] c1) by Def37;
then  (f [+] c1) . x = (f . x) + c1 by A4, Def37;
hence  c1 = c2 by A3, A5, A6, Th7; ::_thesis: verum
end;

definition
let Y be complex-functions-membered set ;
let f be Y -valued Function;
let c be complex number ;
funcf [-] c ->  Function equals :: VALUED_2:def 38
f [+] (- c);
coherence
 f [+] (- c) is Function ;
end;

:: deftheorem  defines [-] VALUED_2:def_38_:_
 for Y being complex-functions-membered set
for f being b1 -valued Function
for c being complex number holds f [-] c = f [+] (- c);

theorem :: VALUED_2:47
for X being set
for Y being complex-functions-membered set
for c being complex number
for f being PartFunc of X,Y holds dom (f [-] c) = dom f by Def37;

theorem :: VALUED_2:48
for X, x being set
for Y being complex-functions-membered set
for c being complex number
for f being PartFunc of X,Y st x in dom (f [-] c) holds
(f [-] c) . x = (f . x) - c by Def37;

definition
let X be set ;
let Y be complex-functions-membered set ;
let f be PartFunc of X,Y;
let c be complex number ;
:: original: [-]
redefine funcf [-] c ->  PartFunc of X,(C_PFuncs (DOMS Y));
coherence
 f [-] c is PartFunc of X,(C_PFuncs (DOMS Y))
proof
 f [-] c = f [+] (- c) ;
hence  f [-] c is PartFunc of X,(C_PFuncs (DOMS Y)) ; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be real-functions-membered set ;
let f be PartFunc of X,Y;
let c be real number ;
:: original: [-]
redefine funcf [-] c ->  PartFunc of X,(R_PFuncs (DOMS Y));
coherence
 f [-] c is PartFunc of X,(R_PFuncs (DOMS Y))
proof
 f [-] c = f [+] (- c) ;
hence  f [-] c is PartFunc of X,(R_PFuncs (DOMS Y)) ; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be rational-functions-membered set ;
let f be PartFunc of X,Y;
let c be rational number ;
:: original: [-]
redefine funcf [-] c ->  PartFunc of X,(Q_PFuncs (DOMS Y));
coherence
 f [-] c is PartFunc of X,(Q_PFuncs (DOMS Y))
proof
 f [-] c = f [+] (- c) ;
hence  f [-] c is PartFunc of X,(Q_PFuncs (DOMS Y)) ; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be integer-functions-membered set ;
let f be PartFunc of X,Y;
let c be integer number ;
:: original: [-]
redefine funcf [-] c ->  PartFunc of X,(I_PFuncs (DOMS Y));
coherence
 f [-] c is PartFunc of X,(I_PFuncs (DOMS Y))
proof
 f [-] c = f [+] (- c) ;
hence  f [-] c is PartFunc of X,(I_PFuncs (DOMS Y)) ; ::_thesis: verum
end;
end;

theorem :: VALUED_2:49
for X being set
for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y st f <> {} & f is non-empty & f [-] c1 = f [-] c2 holds
c1 = c2
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y st f <> {} & f is non-empty & f [-] c1 = f [-] c2 holds
c1 = c2
let Y be complex-functions-membered set ; ::_thesis: for c1, c2 being complex number
for f being PartFunc of X,Y st f <> {} & f is non-empty & f [-] c1 = f [-] c2 holds
c1 = c2
let c1, c2 be complex number ; ::_thesis: for f being PartFunc of X,Y st f <> {} & f is non-empty & f [-] c1 = f [-] c2 holds
c1 = c2
let f be PartFunc of X,Y; ::_thesis: ( f <> {} & f is non-empty & f [-] c1 = f [-] c2 implies c1 = c2 )
assume that
A1: f <> {} and
A2: f is non-empty and
A3: f [-] c1 = f [-] c2 ; ::_thesis: c1 = c2
consider x being set  such that
A4: x in dom f by A1, XBOOLE_0:def_1;
 f . x in rng f by A4, FUNCT_1:def_3;
then A5: f . x <> {} by A2, RELAT_1:def_9;
 dom f = dom (f [-] c2) by Def37;
then A6: (f [-] c2) . x = (f . x) - c2 by A4, Def37;
 dom f = dom (f [-] c1) by Def37;
then  (f [-] c1) . x = (f . x) - c1 by A4, Def37;
hence  c1 = c2 by A3, A5, A6, Th8; ::_thesis: verum
end;

theorem :: VALUED_2:50
for X being set
for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [+] c1) [-] c2 = f [+] (c1 - c2)
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [+] c1) [-] c2 = f [+] (c1 - c2)
let Y be complex-functions-membered set ; ::_thesis: for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [+] c1) [-] c2 = f [+] (c1 - c2)
let c1, c2 be complex number ; ::_thesis: for f being PartFunc of X,Y holds (f [+] c1) [-] c2 = f [+] (c1 - c2)
let f be PartFunc of X,Y; ::_thesis: (f [+] c1) [-] c2 = f [+] (c1 - c2)
set f1 = f [+] c1;
A1: dom ((f [+] c1) [-] c2) = dom (f [+] c1) by Def37;
 dom (f [+] c1) = dom f by Def37;
hence A2: dom ((f [+] c1) [-] c2) = dom (f [+] (c1 - c2)) by A1, Def37; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f [+] c1) [-] c2) or ((f [+] c1) [-] c2) . b1 = (f [+] (c1 - c2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f [+] c1) [-] c2) or ((f [+] c1) [-] c2) . x = (f [+] (c1 - c2)) . x )
assume A3: x in dom ((f [+] c1) [-] c2) ; ::_thesis: ((f [+] c1) [-] c2) . x = (f [+] (c1 - c2)) . x
hence ((f [+] c1) [-] c2) . x = ((f [+] c1) . x) - c2 by Def37
.= ((f . x) + c1) - c2 by A1, A3, Def37
.= (f . x) + (c1 - c2) by Th12
.= (f [+] (c1 - c2)) . x by A2, A3, Def37 ;
::_thesis: verum
end;

theorem :: VALUED_2:51
for X being set
for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [-] c1) [+] c2 = f [-] (c1 - c2)
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [-] c1) [+] c2 = f [-] (c1 - c2)
let Y be complex-functions-membered set ; ::_thesis: for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [-] c1) [+] c2 = f [-] (c1 - c2)
let c1, c2 be complex number ; ::_thesis: for f being PartFunc of X,Y holds (f [-] c1) [+] c2 = f [-] (c1 - c2)
let f be PartFunc of X,Y; ::_thesis: (f [-] c1) [+] c2 = f [-] (c1 - c2)
set f1 = f [-] c1;
A1: dom ((f [-] c1) [+] c2) = dom (f [-] c1) by Def37;
 dom (f [-] c1) = dom f by Def37;
hence A2: dom ((f [-] c1) [+] c2) = dom (f [-] (c1 - c2)) by A1, Def37; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f [-] c1) [+] c2) or ((f [-] c1) [+] c2) . b1 = (f [-] (c1 - c2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f [-] c1) [+] c2) or ((f [-] c1) [+] c2) . x = (f [-] (c1 - c2)) . x )
assume A3: x in dom ((f [-] c1) [+] c2) ; ::_thesis: ((f [-] c1) [+] c2) . x = (f [-] (c1 - c2)) . x
hence ((f [-] c1) [+] c2) . x = ((f [-] c1) . x) + c2 by Def37
.= ((f . x) - c1) + c2 by A1, A3, Def37
.= (f . x) - (c1 - c2) by Th14
.= (f [-] (c1 - c2)) . x by A2, A3, Def37 ;
::_thesis: verum
end;

theorem :: VALUED_2:52
for X being set
for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [-] c1) [-] c2 = f [-] (c1 + c2)
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [-] c1) [-] c2 = f [-] (c1 + c2)
let Y be complex-functions-membered set ; ::_thesis: for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [-] c1) [-] c2 = f [-] (c1 + c2)
let c1, c2 be complex number ; ::_thesis: for f being PartFunc of X,Y holds (f [-] c1) [-] c2 = f [-] (c1 + c2)
let f be PartFunc of X,Y; ::_thesis: (f [-] c1) [-] c2 = f [-] (c1 + c2)
set f1 = f [-] c1;
A1: dom ((f [-] c1) [-] c2) = dom (f [-] c1) by Def37;
 dom (f [-] c1) = dom f by Def37;
hence A2: dom ((f [-] c1) [-] c2) = dom (f [-] (c1 + c2)) by A1, Def37; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f [-] c1) [-] c2) or ((f [-] c1) [-] c2) . b1 = (f [-] (c1 + c2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f [-] c1) [-] c2) or ((f [-] c1) [-] c2) . x = (f [-] (c1 + c2)) . x )
assume A3: x in dom ((f [-] c1) [-] c2) ; ::_thesis: ((f [-] c1) [-] c2) . x = (f [-] (c1 + c2)) . x
hence ((f [-] c1) [-] c2) . x = ((f [-] c1) . x) - c2 by Def37
.= ((f . x) - c1) - c2 by A1, A3, Def37
.= (f . x) - (c1 + c2) by Th15
.= (f [-] (c1 + c2)) . x by A2, A3, Def37 ;
::_thesis: verum
end;

definition
let Y be complex-functions-membered set ;
let f be Y -valued Function;
let c be complex number ;
deffunc H1( set ) ->  set = c (#) (f . $1);
funcf [#] c ->  Function means :Def39: :: VALUED_2:def 39
( dom it = dom f & ( for x being set st x in dom it holds
it . x = c (#) (f . x) ) );
existence
 ex b1 being Function st
( dom b1 = dom f & ( for x being set st x in dom b1 holds
b1 . x = c (#) (f . x) ) )
proof
 ex F being Function st
( dom F = dom f & ( for x being set st x in dom f holds
F . x = H1(x) ) ) from FUNCT_1:sch_3();
hence  ex b1 being Function st
( dom b1 = dom f & ( for x being set st x in dom b1 holds
b1 . x = c (#) (f . x) ) ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function st dom b1 = dom f & ( for x being set st x in dom b1 holds
b1 . x = c (#) (f . x) ) & dom b2 = dom f & ( for x being set st x in dom b2 holds
b2 . x = c (#) (f . x) ) holds
b1 = b2
proof
let F, G be Function; ::_thesis: ( dom F = dom f & ( for x being set st x in dom F holds
F . x = c (#) (f . x) ) & dom G = dom f & ( for x being set st x in dom G holds
G . x = c (#) (f . x) ) implies F = G )
assume that
A1: dom F = dom f and
A2: for x being set st x in dom F holds
F . x = H1(x) and
A3: dom G = dom f and
A4: for x being set st x in dom G holds
G . x = H1(x) ; ::_thesis: F = G
thus  dom F = dom G by A1, A3; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom F or F . b1 = G . b1 )
let x be set ; ::_thesis: ( not x in dom F or F . x = G . x )
assume A5: x in dom F ; ::_thesis: F . x = G . x
hence F . x = H1(x) by A2
.= G . x by A1, A3, A4, A5 ;
::_thesis: verum
end;
end;

:: deftheorem Def39 defines [#] VALUED_2:def_39_:_
 for Y being complex-functions-membered set
for f being b1 -valued Function
for c being complex number
for b4 being Function holds
( b4 = f [#] c iff ( dom b4 = dom f & ( for x being set st x in dom b4 holds
b4 . x = c (#) (f . x) ) ) );

definition
let X be set ;
let Y be complex-functions-membered set ;
let f be PartFunc of X,Y;
let c be complex number ;
:: original: [#]
redefine funcf [#] c ->  PartFunc of X,(C_PFuncs (DOMS Y));
coherence
 f [#] c is PartFunc of X,(C_PFuncs (DOMS Y))
proof
set h = f [#] c;
A1: dom (f [#] c) = dom f by Def39;
 rng (f [#] c) c= C_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f [#] c) or y in C_PFuncs (DOMS Y) )
assume  y in rng (f [#] c) ; ::_thesis: y in C_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f [#] c) and
A3: (f [#] c) . x = y by FUNCT_1:def_3;
A4: (f [#] c) . x = c (#) (f . x) by A2, Def39;
then reconsider y = y as Function by A3;
A5: rng y c= COMPLEX
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in COMPLEX )
thus  ( not b in rng y or b in COMPLEX ) by A3, A4, XCMPLX_0:def_2; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_5;
then  y is PartFunc of (DOMS Y),COMPLEX by A6, A5, RELSET_1:4;
hence  y in C_PFuncs (DOMS Y) by Def8; ::_thesis: verum
end;
hence  f [#] c is PartFunc of X,(C_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be real-functions-membered set ;
let f be PartFunc of X,Y;
let c be real number ;
:: original: [#]
redefine funcf [#] c ->  PartFunc of X,(R_PFuncs (DOMS Y));
coherence
 f [#] c is PartFunc of X,(R_PFuncs (DOMS Y))
proof
set h = f [#] c;
A1: dom (f [#] c) = dom f by Def39;
 rng (f [#] c) c= R_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f [#] c) or y in R_PFuncs (DOMS Y) )
assume  y in rng (f [#] c) ; ::_thesis: y in R_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f [#] c) and
A3: (f [#] c) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f [#] c) . x = c (#) (f . x) by A2, Def39;
A5: rng y c= REAL
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in REAL )
thus  ( not b in rng y or b in REAL ) by A3, A4, XREAL_0:def_1; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_5;
then  y is PartFunc of (DOMS Y),REAL by A6, A5, RELSET_1:4;
hence  y in R_PFuncs (DOMS Y) by Def12; ::_thesis: verum
end;
hence  f [#] c is PartFunc of X,(R_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be rational-functions-membered set ;
let f be PartFunc of X,Y;
let c be rational number ;
:: original: [#]
redefine funcf [#] c ->  PartFunc of X,(Q_PFuncs (DOMS Y));
coherence
 f [#] c is PartFunc of X,(Q_PFuncs (DOMS Y))
proof
set h = f [#] c;
A1: dom (f [#] c) = dom f by Def39;
 rng (f [#] c) c= Q_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f [#] c) or y in Q_PFuncs (DOMS Y) )
assume  y in rng (f [#] c) ; ::_thesis: y in Q_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f [#] c) and
A3: (f [#] c) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f [#] c) . x = c (#) (f . x) by A2, Def39;
A5: rng y c= RAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in RAT )
thus  ( not b in rng y or b in RAT ) by A3, A4, RAT_1:def_2; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_5;
then  y is PartFunc of (DOMS Y),RAT by A6, A5, RELSET_1:4;
hence  y in Q_PFuncs (DOMS Y) by Def14; ::_thesis: verum
end;
hence  f [#] c is PartFunc of X,(Q_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be integer-functions-membered set ;
let f be PartFunc of X,Y;
let c be integer number ;
:: original: [#]
redefine funcf [#] c ->  PartFunc of X,(I_PFuncs (DOMS Y));
coherence
 f [#] c is PartFunc of X,(I_PFuncs (DOMS Y))
proof
set h = f [#] c;
A1: dom (f [#] c) = dom f by Def39;
 rng (f [#] c) c= I_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f [#] c) or y in I_PFuncs (DOMS Y) )
assume  y in rng (f [#] c) ; ::_thesis: y in I_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f [#] c) and
A3: (f [#] c) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f [#] c) . x = c (#) (f . x) by A2, Def39;
A5: rng y c= INT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in INT )
thus  ( not b in rng y or b in INT ) by A3, A4, INT_1:def_2; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_5;
then  y is PartFunc of (DOMS Y),INT by A6, A5, RELSET_1:4;
hence  y in I_PFuncs (DOMS Y) by Def16; ::_thesis: verum
end;
hence  f [#] c is PartFunc of X,(I_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be natural-functions-membered set ;
let f be PartFunc of X,Y;
let c be Nat;
:: original: [#]
redefine funcf [#] c ->  PartFunc of X,(N_PFuncs (DOMS Y));
coherence
 f [#] c is PartFunc of X,(N_PFuncs (DOMS Y))
proof
set h = f [#] c;
A1: dom (f [#] c) = dom f by Def39;
 rng (f [#] c) c= N_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f [#] c) or y in N_PFuncs (DOMS Y) )
assume  y in rng (f [#] c) ; ::_thesis: y in N_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f [#] c) and
A3: (f [#] c) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f [#] c) . x = c (#) (f . x) by A2, Def39;
A5: rng y c= NAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in NAT )
thus  ( not b in rng y or b in NAT ) by A3, A4, ORDINAL1:def_12; ::_thesis: verum
end;
 f . x in Y by A1, A2, PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_5;
then  y is PartFunc of (DOMS Y),NAT by A6, A5, RELSET_1:4;
hence  y in N_PFuncs (DOMS Y) by Def18; ::_thesis: verum
end;
hence  f [#] c is PartFunc of X,(N_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

theorem :: VALUED_2:53
for X being set
for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [#] c1) [#] c2 = f [#] (c1 * c2)
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [#] c1) [#] c2 = f [#] (c1 * c2)
let Y be complex-functions-membered set ; ::_thesis: for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [#] c1) [#] c2 = f [#] (c1 * c2)
let c1, c2 be complex number ; ::_thesis: for f being PartFunc of X,Y holds (f [#] c1) [#] c2 = f [#] (c1 * c2)
let f be PartFunc of X,Y; ::_thesis: (f [#] c1) [#] c2 = f [#] (c1 * c2)
set f1 = f [#] c1;
A1: dom ((f [#] c1) [#] c2) = dom (f [#] c1) by Def39;
 dom (f [#] c1) = dom f by Def39;
hence A2: dom ((f [#] c1) [#] c2) = dom (f [#] (c1 * c2)) by A1, Def39; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f [#] c1) [#] c2) or ((f [#] c1) [#] c2) . b1 = (f [#] (c1 * c2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f [#] c1) [#] c2) or ((f [#] c1) [#] c2) . x = (f [#] (c1 * c2)) . x )
assume A3: x in dom ((f [#] c1) [#] c2) ; ::_thesis: ((f [#] c1) [#] c2) . x = (f [#] (c1 * c2)) . x
hence ((f [#] c1) [#] c2) . x = ((f [#] c1) . x) (#) c2 by Def39
.= ((f . x) (#) c1) (#) c2 by A1, A3, Def39
.= (f . x) (#) (c1 * c2) by Th16
.= (f [#] (c1 * c2)) . x by A2, A3, Def39 ;
::_thesis: verum
end;

theorem :: VALUED_2:54
for X being set
for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y st f <> {} & f is non-empty & ( for x being set st x in dom f holds
f . x is non-empty ) & f [#] c1 = f [#] c2 holds
c1 = c2
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y st f <> {} & f is non-empty & ( for x being set st x in dom f holds
f . x is non-empty ) & f [#] c1 = f [#] c2 holds
c1 = c2
let Y be complex-functions-membered set ; ::_thesis: for c1, c2 being complex number
for f being PartFunc of X,Y st f <> {} & f is non-empty & ( for x being set st x in dom f holds
f . x is non-empty ) & f [#] c1 = f [#] c2 holds
c1 = c2
let c1, c2 be complex number ; ::_thesis: for f being PartFunc of X,Y st f <> {} & f is non-empty & ( for x being set st x in dom f holds
f . x is non-empty ) & f [#] c1 = f [#] c2 holds
c1 = c2
let f be PartFunc of X,Y; ::_thesis: ( f <> {} & f is non-empty & ( for x being set st x in dom f holds
f . x is non-empty ) & f [#] c1 = f [#] c2 implies c1 = c2 )
assume that
A1: f <> {} and
A2: f is non-empty and
A3: for x being set st x in dom f holds
f . x is non-empty and
A4: f [#] c1 = f [#] c2 ; ::_thesis: c1 = c2
consider x being set  such that
A5: x in dom f by A1, XBOOLE_0:def_1;
 dom f = dom (f [#] c2) by Def39;
then A6: (f [#] c2) . x = (f . x) (#) c2 by A5, Def39;
 dom f = dom (f [#] c1) by Def39;
then A7: (f [#] c1) . x = (f . x) (#) c1 by A5, Def39;
 f . x in rng f by A5, FUNCT_1:def_3;
then A8: f . x <> {} by A2, RELAT_1:def_9;
 f . x is non-empty by A3, A5;
hence  c1 = c2 by A4, A8, A7, A6, Th9; ::_thesis: verum
end;

definition
let Y be complex-functions-membered set ;
let f be Y -valued Function;
let c be complex number ;
funcf [/] c ->  Function equals :: VALUED_2:def 40
f [#] (c ");
coherence
 f [#] (c ") is Function ;
end;

:: deftheorem  defines [/] VALUED_2:def_40_:_
 for Y being complex-functions-membered set
for f being b1 -valued Function
for c being complex number holds f [/] c = f [#] (c ");

theorem :: VALUED_2:55
for X being set
for Y being complex-functions-membered set
for c being complex number
for f being PartFunc of X,Y holds dom (f [/] c) = dom f by Def39;

theorem :: VALUED_2:56
for X, x being set
for Y being complex-functions-membered set
for c being complex number
for f being PartFunc of X,Y st x in dom (f [/] c) holds
(f [/] c) . x = (c ") (#) (f . x) by Def39;

definition
let X be set ;
let Y be complex-functions-membered set ;
let f be PartFunc of X,Y;
let c be complex number ;
:: original: [/]
redefine funcf [/] c ->  PartFunc of X,(C_PFuncs (DOMS Y));
coherence
 f [/] c is PartFunc of X,(C_PFuncs (DOMS Y))
proof
 f [/] c = f [#] (c ") ;
hence  f [/] c is PartFunc of X,(C_PFuncs (DOMS Y)) ; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be real-functions-membered set ;
let f be PartFunc of X,Y;
let c be real number ;
:: original: [/]
redefine funcf [/] c ->  PartFunc of X,(R_PFuncs (DOMS Y));
coherence
 f [/] c is PartFunc of X,(R_PFuncs (DOMS Y))
proof
 f [/] c = f [#] (c ") ;
hence  f [/] c is PartFunc of X,(R_PFuncs (DOMS Y)) ; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be rational-functions-membered set ;
let f be PartFunc of X,Y;
let c be rational number ;
:: original: [/]
redefine funcf [/] c ->  PartFunc of X,(Q_PFuncs (DOMS Y));
coherence
 f [/] c is PartFunc of X,(Q_PFuncs (DOMS Y))
proof
 f [/] c = f [#] (c ") ;
hence  f [/] c is PartFunc of X,(Q_PFuncs (DOMS Y)) ; ::_thesis: verum
end;
end;

theorem :: VALUED_2:57
for X being set
for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [/] c1) [/] c2 = f [/] (c1 * c2)
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [/] c1) [/] c2 = f [/] (c1 * c2)
let Y be complex-functions-membered set ; ::_thesis: for c1, c2 being complex number
for f being PartFunc of X,Y holds (f [/] c1) [/] c2 = f [/] (c1 * c2)
let c1, c2 be complex number ; ::_thesis: for f being PartFunc of X,Y holds (f [/] c1) [/] c2 = f [/] (c1 * c2)
let f be PartFunc of X,Y; ::_thesis: (f [/] c1) [/] c2 = f [/] (c1 * c2)
set f1 = f [/] c1;
A1: dom ((f [/] c1) [/] c2) = dom (f [/] c1) by Def39;
 dom (f [/] c1) = dom f by Def39;
hence A2: dom ((f [/] c1) [/] c2) = dom (f [/] (c1 * c2)) by A1, Def39; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f [/] c1) [/] c2) or ((f [/] c1) [/] c2) . b1 = (f [/] (c1 * c2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f [/] c1) [/] c2) or ((f [/] c1) [/] c2) . x = (f [/] (c1 * c2)) . x )
assume A3: x in dom ((f [/] c1) [/] c2) ; ::_thesis: ((f [/] c1) [/] c2) . x = (f [/] (c1 * c2)) . x
hence ((f [/] c1) [/] c2) . x = ((f [/] c1) . x) (#) (c2 ") by Def39
.= ((f . x) (#) (c1 ")) (#) (c2 ") by A1, A3, Def39
.= (f . x) (#) ((c1 ") * (c2 ")) by Th16
.= (f . x) (#) ((c1 * c2) ") by XCMPLX_1:204
.= (f [/] (c1 * c2)) . x by A2, A3, Def39 ;
::_thesis: verum
end;

theorem :: VALUED_2:58
for X being set
for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y st f <> {} & f is non-empty & ( for x being set st x in dom f holds
f . x is non-empty ) & f [/] c1 = f [/] c2 holds
c1 = c2
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for c1, c2 being complex number
for f being PartFunc of X,Y st f <> {} & f is non-empty & ( for x being set st x in dom f holds
f . x is non-empty ) & f [/] c1 = f [/] c2 holds
c1 = c2
let Y be complex-functions-membered set ; ::_thesis: for c1, c2 being complex number
for f being PartFunc of X,Y st f <> {} & f is non-empty & ( for x being set st x in dom f holds
f . x is non-empty ) & f [/] c1 = f [/] c2 holds
c1 = c2
let c1, c2 be complex number ; ::_thesis: for f being PartFunc of X,Y st f <> {} & f is non-empty & ( for x being set st x in dom f holds
f . x is non-empty ) & f [/] c1 = f [/] c2 holds
c1 = c2
let f be PartFunc of X,Y; ::_thesis: ( f <> {} & f is non-empty & ( for x being set st x in dom f holds
f . x is non-empty ) & f [/] c1 = f [/] c2 implies c1 = c2 )
assume that
A1: f <> {} and
A2: f is non-empty and
A3: for x being set st x in dom f holds
f . x is non-empty and
A4: f [/] c1 = f [/] c2 ; ::_thesis: c1 = c2
consider x being set  such that
A5: x in dom f by A1, XBOOLE_0:def_1;
 dom f = dom (f [/] c2) by Def39;
then A6: (f [/] c2) . x = (f . x) (/) c2 by A5, Def39;
 dom f = dom (f [/] c1) by Def39;
then A7: (f [/] c1) . x = (f . x) (/) c1 by A5, Def39;
 f . x in rng f by A5, FUNCT_1:def_3;
then A8: f . x <> {} by A2, RELAT_1:def_9;
 f . x is non-empty by A3, A5;
hence  c1 = c2 by A4, A8, A7, A6, Th33; ::_thesis: verum
end;

definition
let Y be complex-functions-membered set ;
let f be Y -valued Function;
let g be complex-valued Function;
deffunc H1( set ) ->  set = (f . $1) + (g . $1);
funcf <+> g ->  Function means :Def41: :: VALUED_2:def 41
( dom it = (dom f) /\ (dom g) & ( for x being set st x in dom it holds
it . x = (f . x) + (g . x) ) );
existence
 ex b1 being Function st
( dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) + (g . x) ) )
proof
 ex F being Function st
( dom F = (dom f) /\ (dom g) & ( for x being set st x in (dom f) /\ (dom g) holds
F . x = H1(x) ) ) from FUNCT_1:sch_3();
hence  ex b1 being Function st
( dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) + (g . x) ) ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function st dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) + (g . x) ) & dom b2 = (dom f) /\ (dom g) & ( for x being set st x in dom b2 holds
b2 . x = (f . x) + (g . x) ) holds
b1 = b2
proof
let F, G be Function; ::_thesis: ( dom F = (dom f) /\ (dom g) & ( for x being set st x in dom F holds
F . x = (f . x) + (g . x) ) & dom G = (dom f) /\ (dom g) & ( for x being set st x in dom G holds
G . x = (f . x) + (g . x) ) implies F = G )
assume that
A1: dom F = (dom f) /\ (dom g) and
A2: for x being set st x in dom F holds
F . x = H1(x) and
A3: dom G = (dom f) /\ (dom g) and
A4: for x being set st x in dom G holds
G . x = H1(x) ; ::_thesis: F = G
thus  dom F = dom G by A1, A3; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom F or F . b1 = G . b1 )
let x be set ; ::_thesis: ( not x in dom F or F . x = G . x )
assume A5: x in dom F ; ::_thesis: F . x = G . x
hence F . x = H1(x) by A2
.= G . x by A1, A3, A4, A5 ;
::_thesis: verum
end;
end;

:: deftheorem Def41 defines <+> VALUED_2:def_41_:_
 for Y being complex-functions-membered set
for f being b1 -valued Function
for g being complex-valued Function
for b4 being Function holds
( b4 = f <+> g iff ( dom b4 = (dom f) /\ (dom g) & ( for x being set st x in dom b4 holds
b4 . x = (f . x) + (g . x) ) ) );

definition
let X be set ;
let Y be complex-functions-membered set ;
let f be PartFunc of X,Y;
let g be complex-valued Function;
:: original: <+>
redefine funcf <+> g ->  PartFunc of (X /\ (dom g)),(C_PFuncs (DOMS Y));
coherence
 f <+> g is PartFunc of (X /\ (dom g)),(C_PFuncs (DOMS Y))
proof
set h = f <+> g;
A1: dom (f <+> g) = (dom f) /\ (dom g) by Def41;
 rng (f <+> g) c= C_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <+> g) or y in C_PFuncs (DOMS Y) )
assume  y in rng (f <+> g) ; ::_thesis: y in C_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f <+> g) and
A3: (f <+> g) . x = y by FUNCT_1:def_3;
A4: (f <+> g) . x = (f . x) + (g . x) by A2, Def41;
then reconsider y = y as Function by A3;
A5: rng y c= COMPLEX
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in COMPLEX )
thus  ( not b in rng y or b in COMPLEX ) by A3, A4, XCMPLX_0:def_2; ::_thesis: verum
end;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_2;
then  y is PartFunc of (DOMS Y),COMPLEX by A6, A5, RELSET_1:4;
hence  y in C_PFuncs (DOMS Y) by Def8; ::_thesis: verum
end;
hence  f <+> g is PartFunc of (X /\ (dom g)),(C_PFuncs (DOMS Y)) by A1, RELSET_1:4, XBOOLE_1:27; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be real-functions-membered set ;
let f be PartFunc of X,Y;
let g be real-valued Function;
:: original: <+>
redefine funcf <+> g ->  PartFunc of (X /\ (dom g)),(R_PFuncs (DOMS Y));
coherence
 f <+> g is PartFunc of (X /\ (dom g)),(R_PFuncs (DOMS Y))
proof
set h = f <+> g;
A1: dom (f <+> g) = (dom f) /\ (dom g) by Def41;
 rng (f <+> g) c= R_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <+> g) or y in R_PFuncs (DOMS Y) )
assume  y in rng (f <+> g) ; ::_thesis: y in R_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f <+> g) and
A3: (f <+> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <+> g) . x = (f . x) + (g . x) by A2, Def41;
A5: rng y c= REAL
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in REAL )
thus  ( not b in rng y or b in REAL ) by A3, A4, XREAL_0:def_1; ::_thesis: verum
end;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_2;
then  y is PartFunc of (DOMS Y),REAL by A6, A5, RELSET_1:4;
hence  y in R_PFuncs (DOMS Y) by Def12; ::_thesis: verum
end;
hence  f <+> g is PartFunc of (X /\ (dom g)),(R_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be rational-functions-membered set ;
let f be PartFunc of X,Y;
let g be RAT -valued Function;
:: original: <+>
redefine funcf <+> g ->  PartFunc of (X /\ (dom g)),(Q_PFuncs (DOMS Y));
coherence
 f <+> g is PartFunc of (X /\ (dom g)),(Q_PFuncs (DOMS Y))
proof
set h = f <+> g;
A1: dom (f <+> g) = (dom f) /\ (dom g) by Def41;
 rng (f <+> g) c= Q_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <+> g) or y in Q_PFuncs (DOMS Y) )
assume  y in rng (f <+> g) ; ::_thesis: y in Q_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f <+> g) and
A3: (f <+> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <+> g) . x = (f . x) + (g . x) by A2, Def41;
A5: rng y c= RAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in RAT )
thus  ( not b in rng y or b in RAT ) by A3, A4, RAT_1:def_2; ::_thesis: verum
end;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_2;
then  y is PartFunc of (DOMS Y),RAT by A6, A5, RELSET_1:4;
hence  y in Q_PFuncs (DOMS Y) by Def14; ::_thesis: verum
end;
hence  f <+> g is PartFunc of (X /\ (dom g)),(Q_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be integer-functions-membered set ;
let f be PartFunc of X,Y;
let g be INT -valued Function;
:: original: <+>
redefine funcf <+> g ->  PartFunc of (X /\ (dom g)),(I_PFuncs (DOMS Y));
coherence
 f <+> g is PartFunc of (X /\ (dom g)),(I_PFuncs (DOMS Y))
proof
set h = f <+> g;
A1: dom (f <+> g) = (dom f) /\ (dom g) by Def41;
 rng (f <+> g) c= I_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <+> g) or y in I_PFuncs (DOMS Y) )
assume  y in rng (f <+> g) ; ::_thesis: y in I_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f <+> g) and
A3: (f <+> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <+> g) . x = (f . x) + (g . x) by A2, Def41;
A5: rng y c= INT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in INT )
thus  ( not b in rng y or b in INT ) by A3, A4, INT_1:def_2; ::_thesis: verum
end;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_2;
then  y is PartFunc of (DOMS Y),INT by A6, A5, RELSET_1:4;
hence  y in I_PFuncs (DOMS Y) by Def16; ::_thesis: verum
end;
hence  f <+> g is PartFunc of (X /\ (dom g)),(I_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be natural-functions-membered set ;
let f be PartFunc of X,Y;
let g be natural-valued Function;
:: original: <+>
redefine funcf <+> g ->  PartFunc of (X /\ (dom g)),(N_PFuncs (DOMS Y));
coherence
 f <+> g is PartFunc of (X /\ (dom g)),(N_PFuncs (DOMS Y))
proof
set h = f <+> g;
A1: dom (f <+> g) = (dom f) /\ (dom g) by Def41;
 rng (f <+> g) c= N_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <+> g) or y in N_PFuncs (DOMS Y) )
assume  y in rng (f <+> g) ; ::_thesis: y in N_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f <+> g) and
A3: (f <+> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <+> g) . x = (f . x) + (g . x) by A2, Def41;
A5: rng y c= NAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in NAT )
thus  ( not b in rng y or b in NAT ) by A3, A4, ORDINAL1:def_12; ::_thesis: verum
end;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_2;
then  y is PartFunc of (DOMS Y),NAT by A6, A5, RELSET_1:4;
hence  y in N_PFuncs (DOMS Y) by Def18; ::_thesis: verum
end;
hence  f <+> g is PartFunc of (X /\ (dom g)),(N_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

theorem :: VALUED_2:59
for X being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <+> g) <+> h = f <+> (g + h)
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <+> g) <+> h = f <+> (g + h)
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <+> g) <+> h = f <+> (g + h)
let f be PartFunc of X,Y; ::_thesis: for g, h being complex-valued Function holds (f <+> g) <+> h = f <+> (g + h)
let g, h be complex-valued Function; ::_thesis: (f <+> g) <+> h = f <+> (g + h)
set f1 = f <+> g;
A1: dom (g + h) = (dom g) /\ (dom h) by VALUED_1:def_1;
A2: dom ((f <+> g) <+> h) = (dom (f <+> g)) /\ (dom h) by Def41;
 ( dom (f <+> g) = (dom f) /\ (dom g) & dom (f <+> (g + h)) = (dom f) /\ (dom (g + h)) ) by Def41;
hence A3: dom ((f <+> g) <+> h) = dom (f <+> (g + h)) by A2, A1, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f <+> g) <+> h) or ((f <+> g) <+> h) . b1 = (f <+> (g + h)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f <+> g) <+> h) or ((f <+> g) <+> h) . x = (f <+> (g + h)) . x )
assume A4: x in dom ((f <+> g) <+> h) ; ::_thesis: ((f <+> g) <+> h) . x = (f <+> (g + h)) . x
then A5: x in dom (f <+> g) by A2, XBOOLE_0:def_4;
A6: x in dom (g + h) by A3, A4, XBOOLE_0:def_4;
thus ((f <+> g) <+> h) . x = ((f <+> g) . x) + (h . x) by A4, Def41
.= ((f . x) + (g . x)) + (h . x) by A5, Def41
.= (f . x) + ((g . x) + (h . x)) by Th12
.= (f . x) + ((g + h) . x) by A6, VALUED_1:def_1
.= (f <+> (g + h)) . x by A3, A4, Def41 ; ::_thesis: verum
end;

theorem :: VALUED_2:60
for X being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function holds <-> (f <+> g) = (<-> f) <+> (- g)
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function holds <-> (f <+> g) = (<-> f) <+> (- g)
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for g being complex-valued Function holds <-> (f <+> g) = (<-> f) <+> (- g)
let f be PartFunc of X,Y; ::_thesis: for g being complex-valued Function holds <-> (f <+> g) = (<-> f) <+> (- g)
let g be complex-valued Function; ::_thesis: <-> (f <+> g) = (<-> f) <+> (- g)
set f1 = f <+> g;
set f2 = <-> f;
A1: dom (<-> (f <+> g)) = dom (f <+> g) by Def33;
A2: ( dom (f <+> g) = (dom f) /\ (dom g) & dom (<-> f) = dom f ) by Def33, Def41;
 dom (- g) = dom g by VALUED_1:8;
hence A3: dom (<-> (f <+> g)) = dom ((<-> f) <+> (- g)) by A1, A2, Def41; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (<-> (f <+> g)) or (<-> (f <+> g)) . b1 = ((<-> f) <+> (- g)) . b1 )
let x be set ; ::_thesis: ( not x in dom (<-> (f <+> g)) or (<-> (f <+> g)) . x = ((<-> f) <+> (- g)) . x )
assume A4: x in dom (<-> (f <+> g)) ; ::_thesis: (<-> (f <+> g)) . x = ((<-> f) <+> (- g)) . x
then A5: x in dom (<-> f) by A1, A2, XBOOLE_0:def_4;
thus (<-> (f <+> g)) . x = - ((f <+> g) . x) by A4, Def33
.= - ((f . x) + (g . x)) by A1, A4, Def41
.= (- (f . x)) - (g . x) by Th10
.= (- (f . x)) + ((- g) . x) by VALUED_1:8
.= ((<-> f) . x) + ((- g) . x) by A5, Def33
.= ((<-> f) <+> (- g)) . x by A3, A4, Def41 ; ::_thesis: verum
end;

definition
let Y be complex-functions-membered set ;
let f be Y -valued Function;
let g be complex-valued Function;
funcf <-> g ->  Function equals :: VALUED_2:def 42
f <+> (- g);
coherence
 f <+> (- g) is Function ;
end;

:: deftheorem  defines <-> VALUED_2:def_42_:_
 for Y being complex-functions-membered set
for f being b1 -valued Function
for g being complex-valued Function holds f <-> g = f <+> (- g);

theorem Th61: :: VALUED_2:61
for X being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function holds dom (f <-> g) = (dom f) /\ (dom g)
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function holds dom (f <-> g) = (dom f) /\ (dom g)
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for g being complex-valued Function holds dom (f <-> g) = (dom f) /\ (dom g)
let f be PartFunc of X,Y; ::_thesis: for g being complex-valued Function holds dom (f <-> g) = (dom f) /\ (dom g)
let g be complex-valued Function; ::_thesis: dom (f <-> g) = (dom f) /\ (dom g)
thus  dom (f <-> g) = (dom f) /\ (dom (- g)) by Def41
.= (dom f) /\ (dom g) by VALUED_1:8 ; ::_thesis: verum
end;

theorem Th62: :: VALUED_2:62
for X, x being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function st x in dom (f <-> g) holds
(f <-> g) . x = (f . x) - (g . x)
proof
let X, x be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function st x in dom (f <-> g) holds
(f <-> g) . x = (f . x) - (g . x)
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for g being complex-valued Function st x in dom (f <-> g) holds
(f <-> g) . x = (f . x) - (g . x)
let f be PartFunc of X,Y; ::_thesis: for g being complex-valued Function st x in dom (f <-> g) holds
(f <-> g) . x = (f . x) - (g . x)
let g be complex-valued Function; ::_thesis: ( x in dom (f <-> g) implies (f <-> g) . x = (f . x) - (g . x) )
assume  x in dom (f <-> g) ; ::_thesis: (f <-> g) . x = (f . x) - (g . x)
hence (f <-> g) . x = (f . x) + ((- g) . x) by Def41
.= (f . x) - (g . x) by VALUED_1:8 ;
::_thesis: verum
end;

definition
let X be set ;
let Y be complex-functions-membered set ;
let f be PartFunc of X,Y;
let g be complex-valued Function;
:: original: <->
redefine funcf <-> g ->  PartFunc of (X /\ (dom g)),(C_PFuncs (DOMS Y));
coherence
 f <-> g is PartFunc of (X /\ (dom g)),(C_PFuncs (DOMS Y))
proof
set h = f <-> g;
A1: dom (f <-> g) = (dom f) /\ (dom g) by Th61;
 rng (f <-> g) c= C_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <-> g) or y in C_PFuncs (DOMS Y) )
assume  y in rng (f <-> g) ; ::_thesis: y in C_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f <-> g) and
A3: (f <-> g) . x = y by FUNCT_1:def_3;
A4: (f <-> g) . x = (f . x) - (g . x) by A2, Th62;
then reconsider y = y as Function by A3;
A5: rng y c= COMPLEX
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in COMPLEX )
thus  ( not b in rng y or b in COMPLEX ) by A3, A4, XCMPLX_0:def_2; ::_thesis: verum
end;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 (f <-> g) . x = (f . x) - (g . x) by A2, Th62;
then  dom y = dom (f . x) by A3, VALUED_1:def_2;
then  y is PartFunc of (DOMS Y),COMPLEX by A6, A5, RELSET_1:4;
hence  y in C_PFuncs (DOMS Y) by Def8; ::_thesis: verum
end;
hence  f <-> g is PartFunc of (X /\ (dom g)),(C_PFuncs (DOMS Y)) by A1, RELSET_1:4, XBOOLE_1:27; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be real-functions-membered set ;
let f be PartFunc of X,Y;
let g be real-valued Function;
:: original: <->
redefine funcf <-> g ->  PartFunc of (X /\ (dom g)),(R_PFuncs (DOMS Y));
coherence
 f <-> g is PartFunc of (X /\ (dom g)),(R_PFuncs (DOMS Y))
proof
set h = f <-> g;
A1: dom (f <-> g) = (dom f) /\ (dom g) by Th61;
 rng (f <-> g) c= R_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <-> g) or y in R_PFuncs (DOMS Y) )
assume  y in rng (f <-> g) ; ::_thesis: y in R_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f <-> g) and
A3: (f <-> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <-> g) . x = (f . x) - (g . x) by A2, Th62;
A5: rng y c= REAL
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in REAL )
thus  ( not b in rng y or b in REAL ) by A3, A4, XREAL_0:def_1; ::_thesis: verum
end;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_2;
then  y is PartFunc of (DOMS Y),REAL by A6, A5, RELSET_1:4;
hence  y in R_PFuncs (DOMS Y) by Def12; ::_thesis: verum
end;
hence  f <-> g is PartFunc of (X /\ (dom g)),(R_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be rational-functions-membered set ;
let f be PartFunc of X,Y;
let g be RAT -valued Function;
:: original: <->
redefine funcf <-> g ->  PartFunc of (X /\ (dom g)),(Q_PFuncs (DOMS Y));
coherence
 f <-> g is PartFunc of (X /\ (dom g)),(Q_PFuncs (DOMS Y))
proof
set h = f <-> g;
A1: dom (f <-> g) = (dom f) /\ (dom g) by Th61;
 rng (f <-> g) c= Q_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <-> g) or y in Q_PFuncs (DOMS Y) )
assume  y in rng (f <-> g) ; ::_thesis: y in Q_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f <-> g) and
A3: (f <-> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <-> g) . x = (f . x) - (g . x) by A2, Th62;
A5: rng y c= RAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in RAT )
thus  ( not b in rng y or b in RAT ) by A3, A4, RAT_1:def_2; ::_thesis: verum
end;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_2;
then  y is PartFunc of (DOMS Y),RAT by A6, A5, RELSET_1:4;
hence  y in Q_PFuncs (DOMS Y) by Def14; ::_thesis: verum
end;
hence  f <-> g is PartFunc of (X /\ (dom g)),(Q_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be integer-functions-membered set ;
let f be PartFunc of X,Y;
let g be INT -valued Function;
:: original: <->
redefine funcf <-> g ->  PartFunc of (X /\ (dom g)),(I_PFuncs (DOMS Y));
coherence
 f <-> g is PartFunc of (X /\ (dom g)),(I_PFuncs (DOMS Y))
proof
set h = f <-> g;
A1: dom (f <-> g) = (dom f) /\ (dom g) by Th61;
 rng (f <-> g) c= I_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <-> g) or y in I_PFuncs (DOMS Y) )
assume  y in rng (f <-> g) ; ::_thesis: y in I_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f <-> g) and
A3: (f <-> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <-> g) . x = (f . x) - (g . x) by A2, Th62;
A5: rng y c= INT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in INT )
thus  ( not b in rng y or b in INT ) by A3, A4, INT_1:def_2; ::_thesis: verum
end;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_2;
then  y is PartFunc of (DOMS Y),INT by A6, A5, RELSET_1:4;
hence  y in I_PFuncs (DOMS Y) by Def16; ::_thesis: verum
end;
hence  f <-> g is PartFunc of (X /\ (dom g)),(I_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

theorem :: VALUED_2:63
for X being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function holds f <-> (- g) = f <+> g ;

theorem :: VALUED_2:64
for X being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function holds <-> (f <-> g) = (<-> f) <+> g
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function holds <-> (f <-> g) = (<-> f) <+> g
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for g being complex-valued Function holds <-> (f <-> g) = (<-> f) <+> g
let f be PartFunc of X,Y; ::_thesis: for g being complex-valued Function holds <-> (f <-> g) = (<-> f) <+> g
let g be complex-valued Function; ::_thesis: <-> (f <-> g) = (<-> f) <+> g
set f1 = f <-> g;
set f2 = <-> f;
A1: dom (<-> (f <-> g)) = dom (f <-> g) by Def33;
A2: ( dom (f <-> g) = (dom f) /\ (dom g) & dom (<-> f) = dom f ) by Def33, Th61;
hence A3: dom (<-> (f <-> g)) = dom ((<-> f) <+> g) by A1, Def41; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (<-> (f <-> g)) or (<-> (f <-> g)) . b1 = ((<-> f) <+> g) . b1 )
let x be set ; ::_thesis: ( not x in dom (<-> (f <-> g)) or (<-> (f <-> g)) . x = ((<-> f) <+> g) . x )
assume A4: x in dom (<-> (f <-> g)) ; ::_thesis: (<-> (f <-> g)) . x = ((<-> f) <+> g) . x
then A5: x in dom (<-> f) by A1, A2, XBOOLE_0:def_4;
thus (<-> (f <-> g)) . x = - ((f <-> g) . x) by A4, Def33
.= - ((f . x) - (g . x)) by A1, A4, Th62
.= (- (f . x)) + (g . x) by Th11
.= ((<-> f) . x) + (g . x) by A5, Def33
.= ((<-> f) <+> g) . x by A3, A4, Def41 ; ::_thesis: verum
end;

theorem :: VALUED_2:65
for X being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <+> g) <-> h = f <+> (g - h)
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <+> g) <-> h = f <+> (g - h)
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <+> g) <-> h = f <+> (g - h)
let f be PartFunc of X,Y; ::_thesis: for g, h being complex-valued Function holds (f <+> g) <-> h = f <+> (g - h)
let g, h be complex-valued Function; ::_thesis: (f <+> g) <-> h = f <+> (g - h)
set f1 = f <+> g;
A1: dom (g - h) = (dom g) /\ (dom h) by VALUED_1:12;
A2: dom ((f <+> g) <-> h) = (dom (f <+> g)) /\ (dom h) by Th61;
 ( dom (f <+> g) = (dom f) /\ (dom g) & dom (f <+> (g - h)) = (dom f) /\ (dom (g - h)) ) by Def41;
hence A3: dom ((f <+> g) <-> h) = dom (f <+> (g - h)) by A2, A1, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f <+> g) <-> h) or ((f <+> g) <-> h) . b1 = (f <+> (g - h)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f <+> g) <-> h) or ((f <+> g) <-> h) . x = (f <+> (g - h)) . x )
assume A4: x in dom ((f <+> g) <-> h) ; ::_thesis: ((f <+> g) <-> h) . x = (f <+> (g - h)) . x
then A5: x in dom (f <+> g) by A2, XBOOLE_0:def_4;
A6: x in dom (g - h) by A3, A4, XBOOLE_0:def_4;
thus ((f <+> g) <-> h) . x = ((f <+> g) . x) - (h . x) by A4, Th62
.= ((f . x) + (g . x)) - (h . x) by A5, Def41
.= (f . x) + ((g . x) - (h . x)) by Th13
.= (f . x) + ((g - h) . x) by A6, VALUED_1:13
.= (f <+> (g - h)) . x by A3, A4, Def41 ; ::_thesis: verum
end;

theorem :: VALUED_2:66
for X being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <-> g) <+> h = f <-> (g - h)
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <-> g) <+> h = f <-> (g - h)
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <-> g) <+> h = f <-> (g - h)
let f be PartFunc of X,Y; ::_thesis: for g, h being complex-valued Function holds (f <-> g) <+> h = f <-> (g - h)
let g, h be complex-valued Function; ::_thesis: (f <-> g) <+> h = f <-> (g - h)
set f1 = f <-> g;
A1: dom (g - h) = (dom g) /\ (dom h) by VALUED_1:12;
A2: dom ((f <-> g) <+> h) = (dom (f <-> g)) /\ (dom h) by Def41;
 ( dom (f <-> g) = (dom f) /\ (dom g) & dom (f <-> (g - h)) = (dom f) /\ (dom (g - h)) ) by Th61;
hence A3: dom ((f <-> g) <+> h) = dom (f <-> (g - h)) by A2, A1, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f <-> g) <+> h) or ((f <-> g) <+> h) . b1 = (f <-> (g - h)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f <-> g) <+> h) or ((f <-> g) <+> h) . x = (f <-> (g - h)) . x )
assume A4: x in dom ((f <-> g) <+> h) ; ::_thesis: ((f <-> g) <+> h) . x = (f <-> (g - h)) . x
then A5: x in dom (f <-> g) by A2, XBOOLE_0:def_4;
A6: x in dom (g - h) by A3, A4, XBOOLE_0:def_4;
thus ((f <-> g) <+> h) . x = ((f <-> g) . x) + (h . x) by A4, Def41
.= ((f . x) - (g . x)) + (h . x) by A5, Th62
.= (f . x) - ((g . x) - (h . x)) by Th14
.= (f . x) - ((g - h) . x) by A6, VALUED_1:13
.= (f <-> (g - h)) . x by A3, A4, Th62 ; ::_thesis: verum
end;

theorem :: VALUED_2:67
for X being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <-> g) <-> h = f <-> (g + h)
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <-> g) <-> h = f <-> (g + h)
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <-> g) <-> h = f <-> (g + h)
let f be PartFunc of X,Y; ::_thesis: for g, h being complex-valued Function holds (f <-> g) <-> h = f <-> (g + h)
let g, h be complex-valued Function; ::_thesis: (f <-> g) <-> h = f <-> (g + h)
set f1 = f <-> g;
A1: dom (g + h) = (dom g) /\ (dom h) by VALUED_1:def_1;
A2: dom ((f <-> g) <-> h) = (dom (f <-> g)) /\ (dom h) by Th61;
 ( dom (f <-> g) = (dom f) /\ (dom g) & dom (f <-> (g + h)) = (dom f) /\ (dom (g + h)) ) by Th61;
hence A3: dom ((f <-> g) <-> h) = dom (f <-> (g + h)) by A2, A1, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f <-> g) <-> h) or ((f <-> g) <-> h) . b1 = (f <-> (g + h)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f <-> g) <-> h) or ((f <-> g) <-> h) . x = (f <-> (g + h)) . x )
assume A4: x in dom ((f <-> g) <-> h) ; ::_thesis: ((f <-> g) <-> h) . x = (f <-> (g + h)) . x
then A5: x in dom (f <-> g) by A2, XBOOLE_0:def_4;
A6: x in dom (g + h) by A3, A4, XBOOLE_0:def_4;
thus ((f <-> g) <-> h) . x = ((f <-> g) . x) - (h . x) by A4, Th62
.= ((f . x) - (g . x)) - (h . x) by A5, Th62
.= (f . x) - ((g . x) + (h . x)) by Th15
.= (f . x) - ((g + h) . x) by A6, VALUED_1:def_1
.= (f <-> (g + h)) . x by A3, A4, Th62 ; ::_thesis: verum
end;

definition
let Y be complex-functions-membered set ;
let f be Y -valued Function;
let g be complex-valued Function;
deffunc H1( set ) ->  set = (f . $1) (#) (g . $1);
funcf <#> g ->  Function means :Def43: :: VALUED_2:def 43
( dom it = (dom f) /\ (dom g) & ( for x being set st x in dom it holds
it . x = (f . x) (#) (g . x) ) );
existence
 ex b1 being Function st
( dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) (#) (g . x) ) )
proof
 ex F being Function st
( dom F = (dom f) /\ (dom g) & ( for x being set st x in (dom f) /\ (dom g) holds
F . x = H1(x) ) ) from FUNCT_1:sch_3();
hence  ex b1 being Function st
( dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) (#) (g . x) ) ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function st dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) (#) (g . x) ) & dom b2 = (dom f) /\ (dom g) & ( for x being set st x in dom b2 holds
b2 . x = (f . x) (#) (g . x) ) holds
b1 = b2
proof
let F, G be Function; ::_thesis: ( dom F = (dom f) /\ (dom g) & ( for x being set st x in dom F holds
F . x = (f . x) (#) (g . x) ) & dom G = (dom f) /\ (dom g) & ( for x being set st x in dom G holds
G . x = (f . x) (#) (g . x) ) implies F = G )
assume that
A1: dom F = (dom f) /\ (dom g) and
A2: for x being set st x in dom F holds
F . x = H1(x) and
A3: dom G = (dom f) /\ (dom g) and
A4: for x being set st x in dom G holds
G . x = H1(x) ; ::_thesis: F = G
thus  dom F = dom G by A1, A3; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom F or F . b1 = G . b1 )
let x be set ; ::_thesis: ( not x in dom F or F . x = G . x )
assume A5: x in dom F ; ::_thesis: F . x = G . x
hence F . x = H1(x) by A2
.= G . x by A1, A3, A4, A5 ;
::_thesis: verum
end;
end;

:: deftheorem Def43 defines <#> VALUED_2:def_43_:_
 for Y being complex-functions-membered set
for f being b1 -valued Function
for g being complex-valued Function
for b4 being Function holds
( b4 = f <#> g iff ( dom b4 = (dom f) /\ (dom g) & ( for x being set st x in dom b4 holds
b4 . x = (f . x) (#) (g . x) ) ) );

definition
let X be set ;
let Y be complex-functions-membered set ;
let f be PartFunc of X,Y;
let g be complex-valued Function;
:: original: <#>
redefine funcf <#> g ->  PartFunc of (X /\ (dom g)),(C_PFuncs (DOMS Y));
coherence
 f <#> g is PartFunc of (X /\ (dom g)),(C_PFuncs (DOMS Y))
proof
set h = f <#> g;
A1: dom (f <#> g) = (dom f) /\ (dom g) by Def43;
 rng (f <#> g) c= C_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <#> g) or y in C_PFuncs (DOMS Y) )
assume  y in rng (f <#> g) ; ::_thesis: y in C_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f <#> g) and
A3: (f <#> g) . x = y by FUNCT_1:def_3;
A4: (f <#> g) . x = (f . x) (#) (g . x) by A2, Def43;
then reconsider y = y as Function by A3;
A5: rng y c= COMPLEX
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in COMPLEX )
thus  ( not b in rng y or b in COMPLEX ) by A3, A4, XCMPLX_0:def_2; ::_thesis: verum
end;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_5;
then  y is PartFunc of (DOMS Y),COMPLEX by A6, A5, RELSET_1:4;
hence  y in C_PFuncs (DOMS Y) by Def8; ::_thesis: verum
end;
hence  f <#> g is PartFunc of (X /\ (dom g)),(C_PFuncs (DOMS Y)) by A1, RELSET_1:4, XBOOLE_1:27; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be real-functions-membered set ;
let f be PartFunc of X,Y;
let g be real-valued Function;
:: original: <#>
redefine funcf <#> g ->  PartFunc of (X /\ (dom g)),(R_PFuncs (DOMS Y));
coherence
 f <#> g is PartFunc of (X /\ (dom g)),(R_PFuncs (DOMS Y))
proof
set h = f <#> g;
A1: dom (f <#> g) = (dom f) /\ (dom g) by Def43;
 rng (f <#> g) c= R_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <#> g) or y in R_PFuncs (DOMS Y) )
assume  y in rng (f <#> g) ; ::_thesis: y in R_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f <#> g) and
A3: (f <#> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <#> g) . x = (f . x) (#) (g . x) by A2, Def43;
A5: rng y c= REAL
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in REAL )
thus  ( not b in rng y or b in REAL ) by A3, A4, XREAL_0:def_1; ::_thesis: verum
end;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_5;
then  y is PartFunc of (DOMS Y),REAL by A6, A5, RELSET_1:4;
hence  y in R_PFuncs (DOMS Y) by Def12; ::_thesis: verum
end;
hence  f <#> g is PartFunc of (X /\ (dom g)),(R_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be rational-functions-membered set ;
let f be PartFunc of X,Y;
let g be RAT -valued Function;
:: original: <#>
redefine funcf <#> g ->  PartFunc of (X /\ (dom g)),(Q_PFuncs (DOMS Y));
coherence
 f <#> g is PartFunc of (X /\ (dom g)),(Q_PFuncs (DOMS Y))
proof
set h = f <#> g;
A1: dom (f <#> g) = (dom f) /\ (dom g) by Def43;
 rng (f <#> g) c= Q_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <#> g) or y in Q_PFuncs (DOMS Y) )
assume  y in rng (f <#> g) ; ::_thesis: y in Q_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f <#> g) and
A3: (f <#> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <#> g) . x = (f . x) (#) (g . x) by A2, Def43;
A5: rng y c= RAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in RAT )
thus  ( not b in rng y or b in RAT ) by A3, A4, RAT_1:def_2; ::_thesis: verum
end;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_5;
then  y is PartFunc of (DOMS Y),RAT by A6, A5, RELSET_1:4;
hence  y in Q_PFuncs (DOMS Y) by Def14; ::_thesis: verum
end;
hence  f <#> g is PartFunc of (X /\ (dom g)),(Q_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be integer-functions-membered set ;
let f be PartFunc of X,Y;
let g be INT -valued Function;
:: original: <#>
redefine funcf <#> g ->  PartFunc of (X /\ (dom g)),(I_PFuncs (DOMS Y));
coherence
 f <#> g is PartFunc of (X /\ (dom g)),(I_PFuncs (DOMS Y))
proof
set h = f <#> g;
A1: dom (f <#> g) = (dom f) /\ (dom g) by Def43;
 rng (f <#> g) c= I_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <#> g) or y in I_PFuncs (DOMS Y) )
assume  y in rng (f <#> g) ; ::_thesis: y in I_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f <#> g) and
A3: (f <#> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <#> g) . x = (f . x) (#) (g . x) by A2, Def43;
A5: rng y c= INT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in INT )
thus  ( not b in rng y or b in INT ) by A3, A4, INT_1:def_2; ::_thesis: verum
end;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_5;
then  y is PartFunc of (DOMS Y),INT by A6, A5, RELSET_1:4;
hence  y in I_PFuncs (DOMS Y) by Def16; ::_thesis: verum
end;
hence  f <#> g is PartFunc of (X /\ (dom g)),(I_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be natural-functions-membered set ;
let f be PartFunc of X,Y;
let g be natural-valued Function;
:: original: <#>
redefine funcf <#> g ->  PartFunc of (X /\ (dom g)),(N_PFuncs (DOMS Y));
coherence
 f <#> g is PartFunc of (X /\ (dom g)),(N_PFuncs (DOMS Y))
proof
set h = f <#> g;
A1: dom (f <#> g) = (dom f) /\ (dom g) by Def43;
 rng (f <#> g) c= N_PFuncs (DOMS Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <#> g) or y in N_PFuncs (DOMS Y) )
assume  y in rng (f <#> g) ; ::_thesis: y in N_PFuncs (DOMS Y)
then consider x being set  such that
A2: x in dom (f <#> g) and
A3: (f <#> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <#> g) . x = (f . x) (#) (g . x) by A2, Def43;
A5: rng y c= NAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in NAT )
thus  ( not b in rng y or b in NAT ) by A3, A4, ORDINAL1:def_12; ::_thesis: verum
end;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y : verum  }  ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
 dom y = dom (f . x) by A3, A4, VALUED_1:def_5;
then  y is PartFunc of (DOMS Y),NAT by A6, A5, RELSET_1:4;
hence  y in N_PFuncs (DOMS Y) by Def18; ::_thesis: verum
end;
hence  f <#> g is PartFunc of (X /\ (dom g)),(N_PFuncs (DOMS Y)) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

theorem :: VALUED_2:68
for X being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function holds f <#> (- g) = (<-> f) <#> g
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function holds f <#> (- g) = (<-> f) <#> g
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for g being complex-valued Function holds f <#> (- g) = (<-> f) <#> g
let f be PartFunc of X,Y; ::_thesis: for g being complex-valued Function holds f <#> (- g) = (<-> f) <#> g
let g be complex-valued Function; ::_thesis: f <#> (- g) = (<-> f) <#> g
set f1 = <-> f;
A1: ( dom (<-> f) = dom f & dom (f <#> (- g)) = (dom f) /\ (dom (- g)) ) by Def33, Def43;
 dom ((<-> f) <#> g) = (dom (<-> f)) /\ (dom g) by Def43;
hence A2: dom (f <#> (- g)) = dom ((<-> f) <#> g) by A1, VALUED_1:8; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (f <#> (- g)) or (f <#> (- g)) . b1 = ((<-> f) <#> g) . b1 )
let x be set ; ::_thesis: ( not x in dom (f <#> (- g)) or (f <#> (- g)) . x = ((<-> f) <#> g) . x )
assume A3: x in dom (f <#> (- g)) ; ::_thesis: (f <#> (- g)) . x = ((<-> f) <#> g) . x
then A4: x in dom (<-> f) by A1, XBOOLE_0:def_4;
thus (f <#> (- g)) . x = (f . x) (#) ((- g) . x) by A3, Def43
.= (f . x) (#) (- (g . x)) by VALUED_1:8
.= (- (f . x)) (#) (g . x) by Th22
.= ((<-> f) . x) (#) (g . x) by A4, Def33
.= ((<-> f) <#> g) . x by A2, A3, Def43 ; ::_thesis: verum
end;

theorem :: VALUED_2:69
for X being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function holds f <#> (- g) = <-> (f <#> g)
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function holds f <#> (- g) = <-> (f <#> g)
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for g being complex-valued Function holds f <#> (- g) = <-> (f <#> g)
let f be PartFunc of X,Y; ::_thesis: for g being complex-valued Function holds f <#> (- g) = <-> (f <#> g)
let g be complex-valued Function; ::_thesis: f <#> (- g) = <-> (f <#> g)
set f1 = f <#> g;
A1: dom (<-> (f <#> g)) = dom (f <#> g) by Def33;
 ( dom (f <#> g) = (dom f) /\ (dom g) & dom (f <#> (- g)) = (dom f) /\ (dom (- g)) ) by Def43;
hence A2: dom (f <#> (- g)) = dom (<-> (f <#> g)) by A1, VALUED_1:8; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (f <#> (- g)) or (f <#> (- g)) . b1 = (<-> (f <#> g)) . b1 )
let x be set ; ::_thesis: ( not x in dom (f <#> (- g)) or (f <#> (- g)) . x = (<-> (f <#> g)) . x )
assume A3: x in dom (f <#> (- g)) ; ::_thesis: (f <#> (- g)) . x = (<-> (f <#> g)) . x
hence (f <#> (- g)) . x = (f . x) (#) ((- g) . x) by Def43
.= (f . x) (#) (- (g . x)) by VALUED_1:8
.= - ((f . x) (#) (g . x)) by Th24
.= - ((f <#> g) . x) by A1, A2, A3, Def43
.= (<-> (f <#> g)) . x by A2, A3, Def33 ;
::_thesis: verum
end;

theorem :: VALUED_2:70
for X being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <#> g) <#> h = f <#> (g (#) h)
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <#> g) <#> h = f <#> (g (#) h)
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <#> g) <#> h = f <#> (g (#) h)
let f be PartFunc of X,Y; ::_thesis: for g, h being complex-valued Function holds (f <#> g) <#> h = f <#> (g (#) h)
let g, h be complex-valued Function; ::_thesis: (f <#> g) <#> h = f <#> (g (#) h)
set f1 = f <#> g;
A1: dom (g (#) h) = (dom g) /\ (dom h) by VALUED_1:def_4;
A2: dom ((f <#> g) <#> h) = (dom (f <#> g)) /\ (dom h) by Def43;
 ( dom (f <#> g) = (dom f) /\ (dom g) & dom (f <#> (g (#) h)) = (dom f) /\ (dom (g (#) h)) ) by Def43;
hence A3: dom ((f <#> g) <#> h) = dom (f <#> (g (#) h)) by A2, A1, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f <#> g) <#> h) or ((f <#> g) <#> h) . b1 = (f <#> (g (#) h)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f <#> g) <#> h) or ((f <#> g) <#> h) . x = (f <#> (g (#) h)) . x )
assume A4: x in dom ((f <#> g) <#> h) ; ::_thesis: ((f <#> g) <#> h) . x = (f <#> (g (#) h)) . x
then A5: x in dom (f <#> g) by A2, XBOOLE_0:def_4;
A6: x in dom (g (#) h) by A3, A4, XBOOLE_0:def_4;
thus ((f <#> g) <#> h) . x = ((f <#> g) . x) (#) (h . x) by A4, Def43
.= ((f . x) (#) (g . x)) (#) (h . x) by A5, Def43
.= (f . x) (#) ((g . x) * (h . x)) by Th16
.= (f . x) (#) ((g (#) h) . x) by A6, VALUED_1:def_4
.= (f <#> (g (#) h)) . x by A3, A4, Def43 ; ::_thesis: verum
end;

definition
let Y be complex-functions-membered set ;
let f be Y -valued Function;
let g be complex-valued Function;
funcf </> g ->  Function equals :: VALUED_2:def 44
f <#> (g ");
coherence
 f <#> (g ") is Function ;
end;

:: deftheorem  defines </> VALUED_2:def_44_:_
 for Y being complex-functions-membered set
for f being b1 -valued Function
for g being complex-valued Function holds f </> g = f <#> (g ");

theorem Th71: :: VALUED_2:71
for X being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function holds dom (f </> g) = (dom f) /\ (dom g)
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function holds dom (f </> g) = (dom f) /\ (dom g)
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for g being complex-valued Function holds dom (f </> g) = (dom f) /\ (dom g)
let f be PartFunc of X,Y; ::_thesis: for g being complex-valued Function holds dom (f </> g) = (dom f) /\ (dom g)
let g be complex-valued Function; ::_thesis: dom (f </> g) = (dom f) /\ (dom g)
thus  dom (f </> g) = (dom f) /\ (dom (g ")) by Def43
.= (dom f) /\ (dom g) by VALUED_1:def_7 ; ::_thesis: verum
end;

theorem Th72: :: VALUED_2:72
for X, x being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function st x in dom (f </> g) holds
(f </> g) . x = (f . x) (/) (g . x)
proof
let X, x be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g being complex-valued Function st x in dom (f </> g) holds
(f </> g) . x = (f . x) (/) (g . x)
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for g being complex-valued Function st x in dom (f </> g) holds
(f </> g) . x = (f . x) (/) (g . x)
let f be PartFunc of X,Y; ::_thesis: for g being complex-valued Function st x in dom (f </> g) holds
(f </> g) . x = (f . x) (/) (g . x)
let g be complex-valued Function; ::_thesis: ( x in dom (f </> g) implies (f </> g) . x = (f . x) (/) (g . x) )
assume  x in dom (f </> g) ; ::_thesis: (f </> g) . x = (f . x) (/) (g . x)
hence (f </> g) . x = (f . x) (#) ((g ") . x) by Def43
.= (f . x) (/) (g . x) by VALUED_1:10 ;
::_thesis: verum
end;

definition
let X be set ;
let Y be complex-functions-membered set ;
let f be PartFunc of X,Y;
let g be complex-valued Function;
:: original: </>
redefine funcf </> g ->  PartFunc of (X /\ (dom g)),(C_PFuncs (DOMS Y));
coherence
 f </> g is PartFunc of (X /\ (dom g)),(C_PFuncs (DOMS Y))
proof
 f </> g = f <#> (g ") ;
hence  f </> g is PartFunc of (X /\ (dom g)),(C_PFuncs (DOMS Y)) by VALUED_1:def_7; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be real-functions-membered set ;
let f be PartFunc of X,Y;
let g be real-valued Function;
:: original: </>
redefine funcf </> g ->  PartFunc of (X /\ (dom g)),(R_PFuncs (DOMS Y));
coherence
 f </> g is PartFunc of (X /\ (dom g)),(R_PFuncs (DOMS Y))
proof
 f </> g = f <#> (g ") ;
hence  f </> g is PartFunc of (X /\ (dom g)),(R_PFuncs (DOMS Y)) by VALUED_1:def_7; ::_thesis: verum
end;
end;

definition
let X be set ;
let Y be rational-functions-membered set ;
let f be PartFunc of X,Y;
let g be RAT -valued Function;
:: original: </>
redefine funcf </> g ->  PartFunc of (X /\ (dom g)),(Q_PFuncs (DOMS Y));
coherence
 f </> g is PartFunc of (X /\ (dom g)),(Q_PFuncs (DOMS Y))
proof
 f </> g = f <#> (g ") ;
hence  f </> g is PartFunc of (X /\ (dom g)),(Q_PFuncs (DOMS Y)) by VALUED_1:def_7; ::_thesis: verum
end;
end;

theorem :: VALUED_2:73
for X being set
for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <#> g) </> h = f <#> (g /" h)
proof
let X be set ; ::_thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <#> g) </> h = f <#> (g /" h)
let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <#> g) </> h = f <#> (g /" h)
let f be PartFunc of X,Y; ::_thesis: for g, h being complex-valued Function holds (f <#> g) </> h = f <#> (g /" h)
let g, h be complex-valued Function; ::_thesis: (f <#> g) </> h = f <#> (g /" h)
set f1 = f <#> g;
A1: dom (g /" h) = (dom g) /\ (dom h) by VALUED_1:16;
A2: dom ((f <#> g) </> h) = (dom (f <#> g)) /\ (dom h) by Th71;
 ( dom (f <#> g) = (dom f) /\ (dom g) & dom (f <#> (g /" h)) = (dom f) /\ (dom (g /" h)) ) by Def43;
hence A3: dom ((f <#> g) </> h) = dom (f <#> (g /" h)) by A2, A1, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f <#> g) </> h) or ((f <#> g) </> h) . b1 = (f <#> (g /" h)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f <#> g) </> h) or ((f <#> g) </> h) . x = (f <#> (g /" h)) . x )
assume A4: x in dom ((f <#> g) </> h) ; ::_thesis: ((f <#> g) </> h) . x = (f <#> (g /" h)) . x
then A5: x in dom (f <#> g) by A2, XBOOLE_0:def_4;
thus ((f <#> g) </> h) . x = ((f <#> g) . x) (/) (h . x) by A4, Th72
.= ((f . x) (#) (g . x)) (/) (h . x) by A5, Def43
.= (f . x) (#) ((g . x) / (h . x)) by Th16
.= (f . x) (#) ((g /" h) . x) by VALUED_1:17
.= (f <#> (g /" h)) . x by A3, A4, Def43 ; ::_thesis: verum
end;

definition
let Y1, Y2 be complex-functions-membered set ;
let f be Y1 -valued Function;
let g be Y2 -valued Function;
deffunc H1( set ) ->  set = (f . $1) + (g . $1);
funcf <++> g ->  Function means :Def45: :: VALUED_2:def 45
( dom it = (dom f) /\ (dom g) & ( for x being set st x in dom it holds
it . x = (f . x) + (g . x) ) );
existence
 ex b1 being Function st
( dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) + (g . x) ) )
proof
 ex F being Function st
( dom F = (dom f) /\ (dom g) & ( for x being set st x in (dom f) /\ (dom g) holds
F . x = H1(x) ) ) from FUNCT_1:sch_3();
hence  ex b1 being Function st
( dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) + (g . x) ) ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function st dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) + (g . x) ) & dom b2 = (dom f) /\ (dom g) & ( for x being set st x in dom b2 holds
b2 . x = (f . x) + (g . x) ) holds
b1 = b2
proof
let F, G be Function; ::_thesis: ( dom F = (dom f) /\ (dom g) & ( for x being set st x in dom F holds
F . x = (f . x) + (g . x) ) & dom G = (dom f) /\ (dom g) & ( for x being set st x in dom G holds
G . x = (f . x) + (g . x) ) implies F = G )
assume that
A1: dom F = (dom f) /\ (dom g) and
A2: for x being set st x in dom F holds
F . x = H1(x) and
A3: dom G = (dom f) /\ (dom g) and
A4: for x being set st x in dom G holds
G . x = H1(x) ; ::_thesis: F = G
thus  dom F = dom G by A1, A3; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom F or F . b1 = G . b1 )
let x be set ; ::_thesis: ( not x in dom F or F . x = G . x )
assume A5: x in dom F ; ::_thesis: F . x = G . x
hence F . x = H1(x) by A2
.= G . x by A1, A3, A4, A5 ;
::_thesis: verum
end;
end;

:: deftheorem Def45 defines <++> VALUED_2:def_45_:_
 for Y1, Y2 being complex-functions-membered set
for f being b1 -valued Function
for g being b2 -valued Function
for b5 being Function holds
( b5 = f <++> g iff ( dom b5 = (dom f) /\ (dom g) & ( for x being set st x in dom b5 holds
b5 . x = (f . x) + (g . x) ) ) );

definition
let X1, X2 be set ;
let Y1, Y2 be complex-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <++>
redefine funcf <++> g ->  PartFunc of (X1 /\ X2),(C_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <++> g is PartFunc of (X1 /\ X2),(C_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <++> g;
A1: dom (f <++> g) = (dom f) /\ (dom g) by Def45;
 rng (f <++> g) c= C_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <++> g) or y in C_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <++> g) ; ::_thesis: y in C_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <++> g) and
A3: (f <++> g) . x = y by FUNCT_1:def_3;
A4: (f <++> g) . x = (f . x) + (g . x) by A2, Def45;
then reconsider y = y as Function by A3;
A5: rng y c= COMPLEX
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in COMPLEX )
thus  ( not b in rng y or b in COMPLEX ) by A3, A4, XCMPLX_0:def_2; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:def_1;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),COMPLEX by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in C_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def8; ::_thesis: verum
end;
hence  f <++> g is PartFunc of (X1 /\ X2),(C_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4, XBOOLE_1:27; ::_thesis: verum
end;
end;

definition
let X1, X2 be set ;
let Y1, Y2 be real-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <++>
redefine funcf <++> g ->  PartFunc of (X1 /\ X2),(R_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <++> g is PartFunc of (X1 /\ X2),(R_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <++> g;
A1: dom (f <++> g) = (dom f) /\ (dom g) by Def45;
 rng (f <++> g) c= R_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <++> g) or y in R_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <++> g) ; ::_thesis: y in R_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <++> g) and
A3: (f <++> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <++> g) . x = (f . x) + (g . x) by A2, Def45;
A5: rng y c= REAL
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in REAL )
thus  ( not b in rng y or b in REAL ) by A3, A4, XREAL_0:def_1; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:def_1;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),REAL by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in R_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def12; ::_thesis: verum
end;
hence  f <++> g is PartFunc of (X1 /\ X2),(R_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X1, X2 be set ;
let Y1, Y2 be rational-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <++>
redefine funcf <++> g ->  PartFunc of (X1 /\ X2),(Q_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <++> g is PartFunc of (X1 /\ X2),(Q_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <++> g;
A1: dom (f <++> g) = (dom f) /\ (dom g) by Def45;
 rng (f <++> g) c= Q_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <++> g) or y in Q_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <++> g) ; ::_thesis: y in Q_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <++> g) and
A3: (f <++> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <++> g) . x = (f . x) + (g . x) by A2, Def45;
A5: rng y c= RAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in RAT )
thus  ( not b in rng y or b in RAT ) by A3, A4, RAT_1:def_2; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:def_1;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),RAT by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in Q_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def14; ::_thesis: verum
end;
hence  f <++> g is PartFunc of (X1 /\ X2),(Q_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X1, X2 be set ;
let Y1, Y2 be integer-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <++>
redefine funcf <++> g ->  PartFunc of (X1 /\ X2),(I_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <++> g is PartFunc of (X1 /\ X2),(I_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <++> g;
A1: dom (f <++> g) = (dom f) /\ (dom g) by Def45;
 rng (f <++> g) c= I_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <++> g) or y in I_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <++> g) ; ::_thesis: y in I_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <++> g) and
A3: (f <++> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <++> g) . x = (f . x) + (g . x) by A2, Def45;
A5: rng y c= INT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in INT )
thus  ( not b in rng y or b in INT ) by A3, A4, INT_1:def_2; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:def_1;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),INT by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in I_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def16; ::_thesis: verum
end;
hence  f <++> g is PartFunc of (X1 /\ X2),(I_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X1, X2 be set ;
let Y1, Y2 be natural-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <++>
redefine funcf <++> g ->  PartFunc of (X1 /\ X2),(N_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <++> g is PartFunc of (X1 /\ X2),(N_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <++> g;
A1: dom (f <++> g) = (dom f) /\ (dom g) by Def45;
 rng (f <++> g) c= N_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <++> g) or y in N_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <++> g) ; ::_thesis: y in N_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <++> g) and
A3: (f <++> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <++> g) . x = (f . x) + (g . x) by A2, Def45;
A5: rng y c= NAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in NAT )
thus  ( not b in rng y or b in NAT ) by A3, A4, ORDINAL1:def_12; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:def_1;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),NAT by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in N_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def18; ::_thesis: verum
end;
hence  f <++> g is PartFunc of (X1 /\ X2),(N_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

theorem :: VALUED_2:74
for X1, X2 being set
for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <++> f2 = f2 <++> f1
proof
let X1, X2 be set ; ::_thesis: for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <++> f2 = f2 <++> f1
let Y1, Y2 be complex-functions-membered set ; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <++> f2 = f2 <++> f1
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds f1 <++> f2 = f2 <++> f1
let f2 be PartFunc of X2,Y2; ::_thesis: f1 <++> f2 = f2 <++> f1
 dom (f1 <++> f2) = (dom f1) /\ (dom f2) by Def45;
hence A1: dom (f1 <++> f2) = dom (f2 <++> f1) by Def45; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (f1 <++> f2) or (f1 <++> f2) . b1 = (f2 <++> f1) . b1 )
let x be set ; ::_thesis: ( not x in dom (f1 <++> f2) or (f1 <++> f2) . x = (f2 <++> f1) . x )
assume A2: x in dom (f1 <++> f2) ; ::_thesis: (f1 <++> f2) . x = (f2 <++> f1) . x
hence (f1 <++> f2) . x = (f1 . x) + (f2 . x) by Def45
.= (f2 <++> f1) . x by A1, A2, Def45 ;
::_thesis: verum
end;

theorem :: VALUED_2:75
for X, X1, X2 being set
for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <++> f1) <++> f2 = f <++> (f1 <++> f2)
proof
let X, X1, X2 be set ; ::_thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <++> f1) <++> f2 = f <++> (f1 <++> f2)
let Y, Y1, Y2 be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <++> f1) <++> f2 = f <++> (f1 <++> f2)
let f be PartFunc of X,Y; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <++> f1) <++> f2 = f <++> (f1 <++> f2)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds (f <++> f1) <++> f2 = f <++> (f1 <++> f2)
let f2 be PartFunc of X2,Y2; ::_thesis: (f <++> f1) <++> f2 = f <++> (f1 <++> f2)
set f3 = f <++> f1;
set f4 = f1 <++> f2;
A1: dom ((f <++> f1) <++> f2) = (dom (f <++> f1)) /\ (dom f2) by Def45;
A2: dom (f <++> (f1 <++> f2)) = (dom f) /\ (dom (f1 <++> f2)) by Def45;
 ( dom (f <++> f1) = (dom f) /\ (dom f1) & dom (f1 <++> f2) = (dom f1) /\ (dom f2) ) by Def45;
hence A3: dom ((f <++> f1) <++> f2) = dom (f <++> (f1 <++> f2)) by A1, A2, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f <++> f1) <++> f2) or ((f <++> f1) <++> f2) . b1 = (f <++> (f1 <++> f2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f <++> f1) <++> f2) or ((f <++> f1) <++> f2) . x = (f <++> (f1 <++> f2)) . x )
assume A4: x in dom ((f <++> f1) <++> f2) ; ::_thesis: ((f <++> f1) <++> f2) . x = (f <++> (f1 <++> f2)) . x
then A5: x in dom (f1 <++> f2) by A2, A3, XBOOLE_0:def_4;
A6: x in dom (f <++> f1) by A1, A4, XBOOLE_0:def_4;
thus ((f <++> f1) <++> f2) . x = ((f <++> f1) . x) + (f2 . x) by A4, Def45
.= ((f . x) + (f1 . x)) + (f2 . x) by A6, Def45
.= (f . x) + ((f1 . x) + (f2 . x)) by RFUNCT_1:8
.= (f . x) + ((f1 <++> f2) . x) by A5, Def45
.= (f <++> (f1 <++> f2)) . x by A3, A4, Def45 ; ::_thesis: verum
end;

theorem :: VALUED_2:76
for X1, X2 being set
for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds <-> (f1 <++> f2) = (<-> f1) <++> (<-> f2)
proof
let X1, X2 be set ; ::_thesis: for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds <-> (f1 <++> f2) = (<-> f1) <++> (<-> f2)
let Y1, Y2 be complex-functions-membered set ; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds <-> (f1 <++> f2) = (<-> f1) <++> (<-> f2)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds <-> (f1 <++> f2) = (<-> f1) <++> (<-> f2)
let f2 be PartFunc of X2,Y2; ::_thesis: <-> (f1 <++> f2) = (<-> f1) <++> (<-> f2)
set f3 = f1 <++> f2;
set f4 = <-> f1;
set f5 = <-> f2;
A1: dom (f1 <++> f2) = (dom f1) /\ (dom f2) by Def45;
A2: dom (<-> f2) = dom f2 by Def33;
A3: dom (<-> (f1 <++> f2)) = dom (f1 <++> f2) by Def33;
A4: dom (<-> f1) = dom f1 by Def33;
hence A5: dom (<-> (f1 <++> f2)) = dom ((<-> f1) <++> (<-> f2)) by A1, A2, A3, Def45; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (<-> (f1 <++> f2)) or (<-> (f1 <++> f2)) . b1 = ((<-> f1) <++> (<-> f2)) . b1 )
let x be set ; ::_thesis: ( not x in dom (<-> (f1 <++> f2)) or (<-> (f1 <++> f2)) . x = ((<-> f1) <++> (<-> f2)) . x )
assume A6: x in dom (<-> (f1 <++> f2)) ; ::_thesis: (<-> (f1 <++> f2)) . x = ((<-> f1) <++> (<-> f2)) . x
then A7: x in dom (<-> f1) by A1, A4, A3, XBOOLE_0:def_4;
A8: x in dom (<-> f2) by A1, A2, A3, A6, XBOOLE_0:def_4;
thus (<-> (f1 <++> f2)) . x = - ((f1 <++> f2) . x) by A6, Def33
.= - ((f1 . x) + (f2 . x)) by A3, A6, Def45
.= (- (f1 . x)) - (f2 . x) by Th17
.= ((<-> f1) . x) + (- (f2 . x)) by A7, Def33
.= ((<-> f1) . x) + ((<-> f2) . x) by A8, Def33
.= ((<-> f1) <++> (<-> f2)) . x by A5, A6, Def45 ; ::_thesis: verum
end;

definition
let Y1, Y2 be complex-functions-membered set ;
let f be Y1 -valued Function;
let g be Y2 -valued Function;
deffunc H1( set ) ->  set = (f . $1) - (g . $1);
funcf <--> g ->  Function means :Def46: :: VALUED_2:def 46
( dom it = (dom f) /\ (dom g) & ( for x being set st x in dom it holds
it . x = (f . x) - (g . x) ) );
existence
 ex b1 being Function st
( dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) - (g . x) ) )
proof
 ex F being Function st
( dom F = (dom f) /\ (dom g) & ( for x being set st x in (dom f) /\ (dom g) holds
F . x = H1(x) ) ) from FUNCT_1:sch_3();
hence  ex b1 being Function st
( dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) - (g . x) ) ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function st dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) - (g . x) ) & dom b2 = (dom f) /\ (dom g) & ( for x being set st x in dom b2 holds
b2 . x = (f . x) - (g . x) ) holds
b1 = b2
proof
let F, G be Function; ::_thesis: ( dom F = (dom f) /\ (dom g) & ( for x being set st x in dom F holds
F . x = (f . x) - (g . x) ) & dom G = (dom f) /\ (dom g) & ( for x being set st x in dom G holds
G . x = (f . x) - (g . x) ) implies F = G )
assume that
A1: dom F = (dom f) /\ (dom g) and
A2: for x being set st x in dom F holds
F . x = H1(x) and
A3: dom G = (dom f) /\ (dom g) and
A4: for x being set st x in dom G holds
G . x = H1(x) ; ::_thesis: F = G
thus  dom F = dom G by A1, A3; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom F or F . b1 = G . b1 )
let x be set ; ::_thesis: ( not x in dom F or F . x = G . x )
assume A5: x in dom F ; ::_thesis: F . x = G . x
hence F . x = H1(x) by A2
.= G . x by A1, A3, A4, A5 ;
::_thesis: verum
end;
end;

:: deftheorem Def46 defines <--> VALUED_2:def_46_:_
 for Y1, Y2 being complex-functions-membered set
for f being b1 -valued Function
for g being b2 -valued Function
for b5 being Function holds
( b5 = f <--> g iff ( dom b5 = (dom f) /\ (dom g) & ( for x being set st x in dom b5 holds
b5 . x = (f . x) - (g . x) ) ) );

definition
let X1, X2 be set ;
let Y1, Y2 be complex-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <-->
redefine funcf <--> g ->  PartFunc of (X1 /\ X2),(C_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <--> g is PartFunc of (X1 /\ X2),(C_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <--> g;
A1: dom (f <--> g) = (dom f) /\ (dom g) by Def46;
 rng (f <--> g) c= C_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <--> g) or y in C_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <--> g) ; ::_thesis: y in C_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <--> g) and
A3: (f <--> g) . x = y by FUNCT_1:def_3;
A4: (f <--> g) . x = (f . x) - (g . x) by A2, Def46;
then reconsider y = y as Function by A3;
A5: rng y c= COMPLEX
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in COMPLEX )
thus  ( not b in rng y or b in COMPLEX ) by A3, A4, XCMPLX_0:def_2; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:12;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),COMPLEX by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in C_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def8; ::_thesis: verum
end;
hence  f <--> g is PartFunc of (X1 /\ X2),(C_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4, XBOOLE_1:27; ::_thesis: verum
end;
end;

definition
let X1, X2 be set ;
let Y1, Y2 be real-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <-->
redefine funcf <--> g ->  PartFunc of (X1 /\ X2),(R_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <--> g is PartFunc of (X1 /\ X2),(R_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <--> g;
A1: dom (f <--> g) = (dom f) /\ (dom g) by Def46;
 rng (f <--> g) c= R_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <--> g) or y in R_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <--> g) ; ::_thesis: y in R_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <--> g) and
A3: (f <--> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <--> g) . x = (f . x) - (g . x) by A2, Def46;
A5: rng y c= REAL
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in REAL )
thus  ( not b in rng y or b in REAL ) by A3, A4, XREAL_0:def_1; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:12;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),REAL by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in R_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def12; ::_thesis: verum
end;
hence  f <--> g is PartFunc of (X1 /\ X2),(R_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X1, X2 be set ;
let Y1, Y2 be rational-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <-->
redefine funcf <--> g ->  PartFunc of (X1 /\ X2),(Q_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <--> g is PartFunc of (X1 /\ X2),(Q_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <--> g;
A1: dom (f <--> g) = (dom f) /\ (dom g) by Def46;
 rng (f <--> g) c= Q_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <--> g) or y in Q_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <--> g) ; ::_thesis: y in Q_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <--> g) and
A3: (f <--> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <--> g) . x = (f . x) - (g . x) by A2, Def46;
A5: rng y c= RAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in RAT )
thus  ( not b in rng y or b in RAT ) by A3, A4, RAT_1:def_2; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:12;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),RAT by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in Q_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def14; ::_thesis: verum
end;
hence  f <--> g is PartFunc of (X1 /\ X2),(Q_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X1, X2 be set ;
let Y1, Y2 be integer-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <-->
redefine funcf <--> g ->  PartFunc of (X1 /\ X2),(I_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <--> g is PartFunc of (X1 /\ X2),(I_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <--> g;
A1: dom (f <--> g) = (dom f) /\ (dom g) by Def46;
 rng (f <--> g) c= I_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <--> g) or y in I_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <--> g) ; ::_thesis: y in I_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <--> g) and
A3: (f <--> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <--> g) . x = (f . x) - (g . x) by A2, Def46;
A5: rng y c= INT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in INT )
thus  ( not b in rng y or b in INT ) by A3, A4, INT_1:def_2; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:12;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),INT by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in I_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def16; ::_thesis: verum
end;
hence  f <--> g is PartFunc of (X1 /\ X2),(I_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

theorem :: VALUED_2:77
for X1, X2 being set
for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <--> f2 = <-> (f2 <--> f1)
proof
let X1, X2 be set ; ::_thesis: for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <--> f2 = <-> (f2 <--> f1)
let Y1, Y2 be complex-functions-membered set ; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <--> f2 = <-> (f2 <--> f1)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds f1 <--> f2 = <-> (f2 <--> f1)
let f2 be PartFunc of X2,Y2; ::_thesis: f1 <--> f2 = <-> (f2 <--> f1)
set f = f2 <--> f1;
A1: ( dom (f1 <--> f2) = (dom f1) /\ (dom f2) & dom (f2 <--> f1) = (dom f2) /\ (dom f1) ) by Def46;
hence A2: dom (f1 <--> f2) = dom (<-> (f2 <--> f1)) by Def33; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (f1 <--> f2) or (f1 <--> f2) . b1 = (<-> (f2 <--> f1)) . b1 )
let x be set ; ::_thesis: ( not x in dom (f1 <--> f2) or (f1 <--> f2) . x = (<-> (f2 <--> f1)) . x )
assume A3: x in dom (f1 <--> f2) ; ::_thesis: (f1 <--> f2) . x = (<-> (f2 <--> f1)) . x
hence (f1 <--> f2) . x = (f1 . x) - (f2 . x) by Def46
.= - ((f2 . x) - (f1 . x)) by Th18
.= - ((f2 <--> f1) . x) by A1, A3, Def46
.= (<-> (f2 <--> f1)) . x by A2, A3, Def33 ;
::_thesis: verum
end;

theorem :: VALUED_2:78
for X1, X2 being set
for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds <-> (f1 <--> f2) = (<-> f1) <++> f2
proof
let X1, X2 be set ; ::_thesis: for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds <-> (f1 <--> f2) = (<-> f1) <++> f2
let Y1, Y2 be complex-functions-membered set ; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds <-> (f1 <--> f2) = (<-> f1) <++> f2
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds <-> (f1 <--> f2) = (<-> f1) <++> f2
let f2 be PartFunc of X2,Y2; ::_thesis: <-> (f1 <--> f2) = (<-> f1) <++> f2
set f3 = f1 <--> f2;
set f4 = <-> f1;
A1: dom (<-> (f1 <--> f2)) = dom (f1 <--> f2) by Def33;
A2: ( dom (f1 <--> f2) = (dom f1) /\ (dom f2) & dom (<-> f1) = dom f1 ) by Def33, Def46;
hence A3: dom (<-> (f1 <--> f2)) = dom ((<-> f1) <++> f2) by A1, Def45; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (<-> (f1 <--> f2)) or (<-> (f1 <--> f2)) . b1 = ((<-> f1) <++> f2) . b1 )
let x be set ; ::_thesis: ( not x in dom (<-> (f1 <--> f2)) or (<-> (f1 <--> f2)) . x = ((<-> f1) <++> f2) . x )
assume A4: x in dom (<-> (f1 <--> f2)) ; ::_thesis: (<-> (f1 <--> f2)) . x = ((<-> f1) <++> f2) . x
then A5: x in dom (<-> f1) by A2, A1, XBOOLE_0:def_4;
thus (<-> (f1 <--> f2)) . x = - ((f1 <--> f2) . x) by A4, Def33
.= - ((f1 . x) - (f2 . x)) by A1, A4, Def46
.= (- (f1 . x)) - (- (f2 . x)) by Th17
.= ((<-> f1) . x) + (f2 . x) by A5, Def33
.= ((<-> f1) <++> f2) . x by A3, A4, Def45 ; ::_thesis: verum
end;

theorem :: VALUED_2:79
for X, X1, X2 being set
for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <++> f1) <--> f2 = f <++> (f1 <--> f2)
proof
let X, X1, X2 be set ; ::_thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <++> f1) <--> f2 = f <++> (f1 <--> f2)
let Y, Y1, Y2 be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <++> f1) <--> f2 = f <++> (f1 <--> f2)
let f be PartFunc of X,Y; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <++> f1) <--> f2 = f <++> (f1 <--> f2)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds (f <++> f1) <--> f2 = f <++> (f1 <--> f2)
let f2 be PartFunc of X2,Y2; ::_thesis: (f <++> f1) <--> f2 = f <++> (f1 <--> f2)
set f3 = f <++> f1;
set f4 = f1 <--> f2;
A1: dom ((f <++> f1) <--> f2) = (dom (f <++> f1)) /\ (dom f2) by Def46;
A2: dom (f <++> (f1 <--> f2)) = (dom f) /\ (dom (f1 <--> f2)) by Def45;
 ( dom (f <++> f1) = (dom f) /\ (dom f1) & dom (f1 <--> f2) = (dom f1) /\ (dom f2) ) by Def45, Def46;
hence A3: dom ((f <++> f1) <--> f2) = dom (f <++> (f1 <--> f2)) by A1, A2, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f <++> f1) <--> f2) or ((f <++> f1) <--> f2) . b1 = (f <++> (f1 <--> f2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f <++> f1) <--> f2) or ((f <++> f1) <--> f2) . x = (f <++> (f1 <--> f2)) . x )
assume A4: x in dom ((f <++> f1) <--> f2) ; ::_thesis: ((f <++> f1) <--> f2) . x = (f <++> (f1 <--> f2)) . x
then A5: x in dom (f1 <--> f2) by A2, A3, XBOOLE_0:def_4;
A6: x in dom (f <++> f1) by A1, A4, XBOOLE_0:def_4;
thus ((f <++> f1) <--> f2) . x = ((f <++> f1) . x) - (f2 . x) by A4, Def46
.= ((f . x) + (f1 . x)) - (f2 . x) by A6, Def45
.= (f . x) + ((f1 . x) - (f2 . x)) by RFUNCT_1:8
.= (f . x) + ((f1 <--> f2) . x) by A5, Def46
.= (f <++> (f1 <--> f2)) . x by A3, A4, Def45 ; ::_thesis: verum
end;

theorem :: VALUED_2:80
for X, X1, X2 being set
for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <--> f1) <++> f2 = f <--> (f1 <--> f2)
proof
let X, X1, X2 be set ; ::_thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <--> f1) <++> f2 = f <--> (f1 <--> f2)
let Y, Y1, Y2 be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <--> f1) <++> f2 = f <--> (f1 <--> f2)
let f be PartFunc of X,Y; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <--> f1) <++> f2 = f <--> (f1 <--> f2)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds (f <--> f1) <++> f2 = f <--> (f1 <--> f2)
let f2 be PartFunc of X2,Y2; ::_thesis: (f <--> f1) <++> f2 = f <--> (f1 <--> f2)
set f3 = f <--> f1;
set f4 = f1 <--> f2;
A1: dom ((f <--> f1) <++> f2) = (dom (f <--> f1)) /\ (dom f2) by Def45;
A2: dom (f <--> (f1 <--> f2)) = (dom f) /\ (dom (f1 <--> f2)) by Def46;
 ( dom (f <--> f1) = (dom f) /\ (dom f1) & dom (f1 <--> f2) = (dom f1) /\ (dom f2) ) by Def46;
hence A3: dom ((f <--> f1) <++> f2) = dom (f <--> (f1 <--> f2)) by A1, A2, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f <--> f1) <++> f2) or ((f <--> f1) <++> f2) . b1 = (f <--> (f1 <--> f2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f <--> f1) <++> f2) or ((f <--> f1) <++> f2) . x = (f <--> (f1 <--> f2)) . x )
assume A4: x in dom ((f <--> f1) <++> f2) ; ::_thesis: ((f <--> f1) <++> f2) . x = (f <--> (f1 <--> f2)) . x
then A5: x in dom (f1 <--> f2) by A2, A3, XBOOLE_0:def_4;
A6: x in dom (f <--> f1) by A1, A4, XBOOLE_0:def_4;
thus ((f <--> f1) <++> f2) . x = ((f <--> f1) . x) + (f2 . x) by A4, Def45
.= ((f . x) - (f1 . x)) + (f2 . x) by A6, Def46
.= (f . x) - ((f1 . x) - (f2 . x)) by RFUNCT_1:22
.= (f . x) - ((f1 <--> f2) . x) by A5, Def46
.= (f <--> (f1 <--> f2)) . x by A3, A4, Def46 ; ::_thesis: verum
end;

theorem :: VALUED_2:81
for X, X1, X2 being set
for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <--> f1) <--> f2 = f <--> (f1 <++> f2)
proof
let X, X1, X2 be set ; ::_thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <--> f1) <--> f2 = f <--> (f1 <++> f2)
let Y, Y1, Y2 be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <--> f1) <--> f2 = f <--> (f1 <++> f2)
let f be PartFunc of X,Y; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <--> f1) <--> f2 = f <--> (f1 <++> f2)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds (f <--> f1) <--> f2 = f <--> (f1 <++> f2)
let f2 be PartFunc of X2,Y2; ::_thesis: (f <--> f1) <--> f2 = f <--> (f1 <++> f2)
set f3 = f <--> f1;
set f4 = f1 <++> f2;
A1: dom ((f <--> f1) <--> f2) = (dom (f <--> f1)) /\ (dom f2) by Def46;
A2: dom (f <--> (f1 <++> f2)) = (dom f) /\ (dom (f1 <++> f2)) by Def46;
 ( dom (f <--> f1) = (dom f) /\ (dom f1) & dom (f1 <++> f2) = (dom f1) /\ (dom f2) ) by Def45, Def46;
hence A3: dom ((f <--> f1) <--> f2) = dom (f <--> (f1 <++> f2)) by A1, A2, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f <--> f1) <--> f2) or ((f <--> f1) <--> f2) . b1 = (f <--> (f1 <++> f2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f <--> f1) <--> f2) or ((f <--> f1) <--> f2) . x = (f <--> (f1 <++> f2)) . x )
assume A4: x in dom ((f <--> f1) <--> f2) ; ::_thesis: ((f <--> f1) <--> f2) . x = (f <--> (f1 <++> f2)) . x
then A5: x in dom (f1 <++> f2) by A2, A3, XBOOLE_0:def_4;
A6: x in dom (f <--> f1) by A1, A4, XBOOLE_0:def_4;
thus ((f <--> f1) <--> f2) . x = ((f <--> f1) . x) - (f2 . x) by A4, Def46
.= ((f . x) - (f1 . x)) - (f2 . x) by A6, Def46
.= (f . x) - ((f1 . x) + (f2 . x)) by RFUNCT_1:20
.= (f . x) - ((f1 <++> f2) . x) by A5, Def45
.= (f <--> (f1 <++> f2)) . x by A3, A4, Def46 ; ::_thesis: verum
end;

theorem :: VALUED_2:82
for X, X1, X2 being set
for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <--> f1) <--> f2 = (f <--> f2) <--> f1
proof
let X, X1, X2 be set ; ::_thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <--> f1) <--> f2 = (f <--> f2) <--> f1
let Y, Y1, Y2 be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <--> f1) <--> f2 = (f <--> f2) <--> f1
let f be PartFunc of X,Y; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <--> f1) <--> f2 = (f <--> f2) <--> f1
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds (f <--> f1) <--> f2 = (f <--> f2) <--> f1
let f2 be PartFunc of X2,Y2; ::_thesis: (f <--> f1) <--> f2 = (f <--> f2) <--> f1
set f3 = f <--> f1;
set f4 = f <--> f2;
A1: dom ((f <--> f1) <--> f2) = (dom (f <--> f1)) /\ (dom f2) by Def46;
A2: dom ((f <--> f2) <--> f1) = (dom (f <--> f2)) /\ (dom f1) by Def46;
 ( dom (f <--> f1) = (dom f) /\ (dom f1) & dom (f <--> f2) = (dom f) /\ (dom f2) ) by Def46;
hence A3: dom ((f <--> f1) <--> f2) = dom ((f <--> f2) <--> f1) by A1, A2, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f <--> f1) <--> f2) or ((f <--> f1) <--> f2) . b1 = ((f <--> f2) <--> f1) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f <--> f1) <--> f2) or ((f <--> f1) <--> f2) . x = ((f <--> f2) <--> f1) . x )
assume A4: x in dom ((f <--> f1) <--> f2) ; ::_thesis: ((f <--> f1) <--> f2) . x = ((f <--> f2) <--> f1) . x
then A5: x in dom (f <--> f2) by A2, A3, XBOOLE_0:def_4;
A6: x in dom (f <--> f1) by A1, A4, XBOOLE_0:def_4;
thus ((f <--> f1) <--> f2) . x = ((f <--> f1) . x) - (f2 . x) by A4, Def46
.= ((f . x) - (f1 . x)) - (f2 . x) by A6, Def46
.= ((f . x) - (f2 . x)) - (f1 . x) by RFUNCT_1:23
.= ((f <--> f2) . x) - (f1 . x) by A5, Def46
.= ((f <--> f2) <--> f1) . x by A3, A4, Def46 ; ::_thesis: verum
end;

definition
let Y1, Y2 be complex-functions-membered set ;
let f be Y1 -valued Function;
let g be Y2 -valued Function;
deffunc H1( set ) ->  set = (f . $1) (#) (g . $1);
funcf <##> g ->  Function means :Def47: :: VALUED_2:def 47
( dom it = (dom f) /\ (dom g) & ( for x being set st x in dom it holds
it . x = (f . x) (#) (g . x) ) );
existence
 ex b1 being Function st
( dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) (#) (g . x) ) )
proof
 ex F being Function st
( dom F = (dom f) /\ (dom g) & ( for x being set st x in (dom f) /\ (dom g) holds
F . x = H1(x) ) ) from FUNCT_1:sch_3();
hence  ex b1 being Function st
( dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) (#) (g . x) ) ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function st dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) (#) (g . x) ) & dom b2 = (dom f) /\ (dom g) & ( for x being set st x in dom b2 holds
b2 . x = (f . x) (#) (g . x) ) holds
b1 = b2
proof
let F, G be Function; ::_thesis: ( dom F = (dom f) /\ (dom g) & ( for x being set st x in dom F holds
F . x = (f . x) (#) (g . x) ) & dom G = (dom f) /\ (dom g) & ( for x being set st x in dom G holds
G . x = (f . x) (#) (g . x) ) implies F = G )
assume that
A1: dom F = (dom f) /\ (dom g) and
A2: for x being set st x in dom F holds
F . x = H1(x) and
A3: dom G = (dom f) /\ (dom g) and
A4: for x being set st x in dom G holds
G . x = H1(x) ; ::_thesis: F = G
thus  dom F = dom G by A1, A3; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom F or F . b1 = G . b1 )
let x be set ; ::_thesis: ( not x in dom F or F . x = G . x )
assume A5: x in dom F ; ::_thesis: F . x = G . x
hence F . x = H1(x) by A2
.= G . x by A1, A3, A4, A5 ;
::_thesis: verum
end;
end;

:: deftheorem Def47 defines <##> VALUED_2:def_47_:_
 for Y1, Y2 being complex-functions-membered set
for f being b1 -valued Function
for g being b2 -valued Function
for b5 being Function holds
( b5 = f <##> g iff ( dom b5 = (dom f) /\ (dom g) & ( for x being set st x in dom b5 holds
b5 . x = (f . x) (#) (g . x) ) ) );

definition
let X1, X2 be set ;
let Y1, Y2 be complex-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <##>
redefine funcf <##> g ->  PartFunc of (X1 /\ X2),(C_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <##> g is PartFunc of (X1 /\ X2),(C_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <##> g;
A1: dom (f <##> g) = (dom f) /\ (dom g) by Def47;
 rng (f <##> g) c= C_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <##> g) or y in C_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <##> g) ; ::_thesis: y in C_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <##> g) and
A3: (f <##> g) . x = y by FUNCT_1:def_3;
A4: (f <##> g) . x = (f . x) (#) (g . x) by A2, Def47;
then reconsider y = y as Function by A3;
A5: rng y c= COMPLEX
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in COMPLEX )
thus  ( not b in rng y or b in COMPLEX ) by A3, A4, XCMPLX_0:def_2; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:def_4;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),COMPLEX by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in C_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def8; ::_thesis: verum
end;
hence  f <##> g is PartFunc of (X1 /\ X2),(C_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4, XBOOLE_1:27; ::_thesis: verum
end;
end;

definition
let X1, X2 be set ;
let Y1, Y2 be real-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <##>
redefine funcf <##> g ->  PartFunc of (X1 /\ X2),(R_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <##> g is PartFunc of (X1 /\ X2),(R_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <##> g;
A1: dom (f <##> g) = (dom f) /\ (dom g) by Def47;
 rng (f <##> g) c= R_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <##> g) or y in R_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <##> g) ; ::_thesis: y in R_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <##> g) and
A3: (f <##> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <##> g) . x = (f . x) (#) (g . x) by A2, Def47;
A5: rng y c= REAL
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in REAL )
thus  ( not b in rng y or b in REAL ) by A3, A4, XREAL_0:def_1; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:def_4;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),REAL by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in R_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def12; ::_thesis: verum
end;
hence  f <##> g is PartFunc of (X1 /\ X2),(R_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X1, X2 be set ;
let Y1, Y2 be rational-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <##>
redefine funcf <##> g ->  PartFunc of (X1 /\ X2),(Q_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <##> g is PartFunc of (X1 /\ X2),(Q_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <##> g;
A1: dom (f <##> g) = (dom f) /\ (dom g) by Def47;
 rng (f <##> g) c= Q_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <##> g) or y in Q_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <##> g) ; ::_thesis: y in Q_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <##> g) and
A3: (f <##> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <##> g) . x = (f . x) (#) (g . x) by A2, Def47;
A5: rng y c= RAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in RAT )
thus  ( not b in rng y or b in RAT ) by A3, A4, RAT_1:def_2; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:def_4;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),RAT by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in Q_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def14; ::_thesis: verum
end;
hence  f <##> g is PartFunc of (X1 /\ X2),(Q_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X1, X2 be set ;
let Y1, Y2 be integer-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <##>
redefine funcf <##> g ->  PartFunc of (X1 /\ X2),(I_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <##> g is PartFunc of (X1 /\ X2),(I_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <##> g;
A1: dom (f <##> g) = (dom f) /\ (dom g) by Def47;
 rng (f <##> g) c= I_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <##> g) or y in I_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <##> g) ; ::_thesis: y in I_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <##> g) and
A3: (f <##> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <##> g) . x = (f . x) (#) (g . x) by A2, Def47;
A5: rng y c= INT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in INT )
thus  ( not b in rng y or b in INT ) by A3, A4, INT_1:def_2; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:def_4;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),INT by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in I_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def16; ::_thesis: verum
end;
hence  f <##> g is PartFunc of (X1 /\ X2),(I_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X1, X2 be set ;
let Y1, Y2 be natural-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <##>
redefine funcf <##> g ->  PartFunc of (X1 /\ X2),(N_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <##> g is PartFunc of (X1 /\ X2),(N_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <##> g;
A1: dom (f <##> g) = (dom f) /\ (dom g) by Def47;
 rng (f <##> g) c= N_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <##> g) or y in N_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <##> g) ; ::_thesis: y in N_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <##> g) and
A3: (f <##> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <##> g) . x = (f . x) (#) (g . x) by A2, Def47;
A5: rng y c= NAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in NAT )
thus  ( not b in rng y or b in NAT ) by A3, A4, ORDINAL1:def_12; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:def_4;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),NAT by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in N_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def18; ::_thesis: verum
end;
hence  f <##> g is PartFunc of (X1 /\ X2),(N_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

theorem Th83: :: VALUED_2:83
for X1, X2 being set
for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <##> f2 = f2 <##> f1
proof
let X1, X2 be set ; ::_thesis: for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <##> f2 = f2 <##> f1
let Y1, Y2 be complex-functions-membered set ; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <##> f2 = f2 <##> f1
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds f1 <##> f2 = f2 <##> f1
let f2 be PartFunc of X2,Y2; ::_thesis: f1 <##> f2 = f2 <##> f1
 dom (f1 <##> f2) = (dom f1) /\ (dom f2) by Def47;
hence A1: dom (f1 <##> f2) = dom (f2 <##> f1) by Def47; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (f1 <##> f2) or (f1 <##> f2) . b1 = (f2 <##> f1) . b1 )
let x be set ; ::_thesis: ( not x in dom (f1 <##> f2) or (f1 <##> f2) . x = (f2 <##> f1) . x )
assume A2: x in dom (f1 <##> f2) ; ::_thesis: (f1 <##> f2) . x = (f2 <##> f1) . x
hence (f1 <##> f2) . x = (f1 . x) (#) (f2 . x) by Def47
.= (f2 <##> f1) . x by A1, A2, Def47 ;
::_thesis: verum
end;

theorem :: VALUED_2:84
for X, X1, X2 being set
for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <##> f1) <##> f2 = f <##> (f1 <##> f2)
proof
let X, X1, X2 be set ; ::_thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <##> f1) <##> f2 = f <##> (f1 <##> f2)
let Y, Y1, Y2 be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <##> f1) <##> f2 = f <##> (f1 <##> f2)
let f be PartFunc of X,Y; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <##> f1) <##> f2 = f <##> (f1 <##> f2)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds (f <##> f1) <##> f2 = f <##> (f1 <##> f2)
let f2 be PartFunc of X2,Y2; ::_thesis: (f <##> f1) <##> f2 = f <##> (f1 <##> f2)
set f3 = f <##> f1;
set f4 = f1 <##> f2;
A1: dom ((f <##> f1) <##> f2) = (dom (f <##> f1)) /\ (dom f2) by Def47;
A2: dom (f <##> (f1 <##> f2)) = (dom f) /\ (dom (f1 <##> f2)) by Def47;
 ( dom (f <##> f1) = (dom f) /\ (dom f1) & dom (f1 <##> f2) = (dom f1) /\ (dom f2) ) by Def47;
hence A3: dom ((f <##> f1) <##> f2) = dom (f <##> (f1 <##> f2)) by A1, A2, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f <##> f1) <##> f2) or ((f <##> f1) <##> f2) . b1 = (f <##> (f1 <##> f2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f <##> f1) <##> f2) or ((f <##> f1) <##> f2) . x = (f <##> (f1 <##> f2)) . x )
assume A4: x in dom ((f <##> f1) <##> f2) ; ::_thesis: ((f <##> f1) <##> f2) . x = (f <##> (f1 <##> f2)) . x
then A5: x in dom (f1 <##> f2) by A2, A3, XBOOLE_0:def_4;
A6: x in dom (f <##> f1) by A1, A4, XBOOLE_0:def_4;
thus ((f <##> f1) <##> f2) . x = ((f <##> f1) . x) (#) (f2 . x) by A4, Def47
.= ((f . x) (#) (f1 . x)) (#) (f2 . x) by A6, Def47
.= (f . x) (#) ((f1 . x) (#) (f2 . x)) by RFUNCT_1:9
.= (f . x) (#) ((f1 <##> f2) . x) by A5, Def47
.= (f <##> (f1 <##> f2)) . x by A3, A4, Def47 ; ::_thesis: verum
end;

theorem :: VALUED_2:85
for X1, X2 being set
for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (<-> f1) <##> f2 = <-> (f1 <##> f2)
proof
let X1, X2 be set ; ::_thesis: for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (<-> f1) <##> f2 = <-> (f1 <##> f2)
let Y1, Y2 be complex-functions-membered set ; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (<-> f1) <##> f2 = <-> (f1 <##> f2)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds (<-> f1) <##> f2 = <-> (f1 <##> f2)
let f2 be PartFunc of X2,Y2; ::_thesis: (<-> f1) <##> f2 = <-> (f1 <##> f2)
set f3 = f1 <##> f2;
set f4 = <-> f1;
A1: ( dom (f1 <##> f2) = (dom f1) /\ (dom f2) & dom (<-> f1) = dom f1 ) by Def33, Def47;
 dom ((<-> f1) <##> f2) = (dom (<-> f1)) /\ (dom f2) by Def47;
hence A2: dom ((<-> f1) <##> f2) = dom (<-> (f1 <##> f2)) by A1, Def33; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((<-> f1) <##> f2) or ((<-> f1) <##> f2) . b1 = (<-> (f1 <##> f2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((<-> f1) <##> f2) or ((<-> f1) <##> f2) . x = (<-> (f1 <##> f2)) . x )
assume A3: x in dom ((<-> f1) <##> f2) ; ::_thesis: ((<-> f1) <##> f2) . x = (<-> (f1 <##> f2)) . x
then A4: x in dom (f1 <##> f2) by A1, Def47;
then A5: x in dom (<-> f1) by A1, XBOOLE_0:def_4;
thus ((<-> f1) <##> f2) . x = ((<-> f1) . x) (#) (f2 . x) by A3, Def47
.= (- (f1 . x)) (#) (f2 . x) by A5, Def33
.= - ((f1 . x) (#) (f2 . x)) by Th25
.= - ((f1 <##> f2) . x) by A4, Def47
.= (<-> (f1 <##> f2)) . x by A2, A3, Def33 ; ::_thesis: verum
end;

theorem :: VALUED_2:86
for X1, X2 being set
for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <##> (<-> f2) = <-> (f1 <##> f2)
proof
let X1, X2 be set ; ::_thesis: for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <##> (<-> f2) = <-> (f1 <##> f2)
let Y1, Y2 be complex-functions-membered set ; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <##> (<-> f2) = <-> (f1 <##> f2)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds f1 <##> (<-> f2) = <-> (f1 <##> f2)
let f2 be PartFunc of X2,Y2; ::_thesis: f1 <##> (<-> f2) = <-> (f1 <##> f2)
set f3 = f1 <##> f2;
set f4 = <-> f2;
A1: ( dom (f1 <##> f2) = (dom f1) /\ (dom f2) & dom (<-> f2) = dom f2 ) by Def33, Def47;
 dom (f1 <##> (<-> f2)) = (dom f1) /\ (dom (<-> f2)) by Def47;
hence A2: dom (f1 <##> (<-> f2)) = dom (<-> (f1 <##> f2)) by A1, Def33; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (f1 <##> (<-> f2)) or (f1 <##> (<-> f2)) . b1 = (<-> (f1 <##> f2)) . b1 )
let x be set ; ::_thesis: ( not x in dom (f1 <##> (<-> f2)) or (f1 <##> (<-> f2)) . x = (<-> (f1 <##> f2)) . x )
assume A3: x in dom (f1 <##> (<-> f2)) ; ::_thesis: (f1 <##> (<-> f2)) . x = (<-> (f1 <##> f2)) . x
then A4: x in dom (f1 <##> f2) by A1, Def47;
then A5: x in dom (<-> f2) by A1, XBOOLE_0:def_4;
thus (f1 <##> (<-> f2)) . x = (f1 . x) (#) ((<-> f2) . x) by A3, Def47
.= (f1 . x) (#) (- (f2 . x)) by A5, Def33
.= - ((f1 . x) (#) (f2 . x)) by Th25
.= - ((f1 <##> f2) . x) by A4, Def47
.= (<-> (f1 <##> f2)) . x by A2, A3, Def33 ; ::_thesis: verum
end;

theorem Th87: :: VALUED_2:87
for X, X1, X2 being set
for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f <##> (f1 <++> f2) = (f <##> f1) <++> (f <##> f2)
proof
let X, X1, X2 be set ; ::_thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f <##> (f1 <++> f2) = (f <##> f1) <++> (f <##> f2)
let Y, Y1, Y2 be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f <##> (f1 <++> f2) = (f <##> f1) <++> (f <##> f2)
let f be PartFunc of X,Y; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f <##> (f1 <++> f2) = (f <##> f1) <++> (f <##> f2)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds f <##> (f1 <++> f2) = (f <##> f1) <++> (f <##> f2)
let f2 be PartFunc of X2,Y2; ::_thesis: f <##> (f1 <++> f2) = (f <##> f1) <++> (f <##> f2)
set f3 = f <##> f1;
set f4 = f <##> f2;
set f5 = f1 <++> f2;
A1: dom (f <##> (f1 <++> f2)) = (dom f) /\ (dom (f1 <++> f2)) by Def47;
A2: dom (f1 <++> f2) = (dom f1) /\ (dom f2) by Def45;
A3: dom ((f <##> f1) <++> (f <##> f2)) = (dom (f <##> f1)) /\ (dom (f <##> f2)) by Def45;
 ( dom (f <##> f1) = (dom f) /\ (dom f1) & dom (f <##> f2) = (dom f) /\ (dom f2) ) by Def47;
hence A4: dom (f <##> (f1 <++> f2)) = dom ((f <##> f1) <++> (f <##> f2)) by A1, A3, A2, Lm1; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (f <##> (f1 <++> f2)) or (f <##> (f1 <++> f2)) . b1 = ((f <##> f1) <++> (f <##> f2)) . b1 )
let x be set ; ::_thesis: ( not x in dom (f <##> (f1 <++> f2)) or (f <##> (f1 <++> f2)) . x = ((f <##> f1) <++> (f <##> f2)) . x )
assume A5: x in dom (f <##> (f1 <++> f2)) ; ::_thesis: (f <##> (f1 <++> f2)) . x = ((f <##> f1) <++> (f <##> f2)) . x
then A6: x in dom (f <##> f1) by A3, A4, XBOOLE_0:def_4;
A7: x in dom (f1 <++> f2) by A1, A5, XBOOLE_0:def_4;
A8: x in dom (f <##> f2) by A3, A4, A5, XBOOLE_0:def_4;
thus (f <##> (f1 <++> f2)) . x = (f . x) (#) ((f1 <++> f2) . x) by A5, Def47
.= (f . x) (#) ((f1 . x) + (f2 . x)) by A7, Def45
.= ((f . x) (#) (f1 . x)) + ((f . x) (#) (f2 . x)) by RFUNCT_1:10
.= ((f <##> f1) . x) + ((f . x) (#) (f2 . x)) by A6, Def47
.= ((f <##> f1) . x) + ((f <##> f2) . x) by A8, Def47
.= ((f <##> f1) <++> (f <##> f2)) . x by A4, A5, Def45 ; ::_thesis: verum
end;

theorem :: VALUED_2:88
for X, X1, X2 being set
for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <++> f2) <##> f = (f1 <##> f) <++> (f2 <##> f)
proof
let X, X1, X2 be set ; ::_thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <++> f2) <##> f = (f1 <##> f) <++> (f2 <##> f)
let Y, Y1, Y2 be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <++> f2) <##> f = (f1 <##> f) <++> (f2 <##> f)
let f be PartFunc of X,Y; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <++> f2) <##> f = (f1 <##> f) <++> (f2 <##> f)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds (f1 <++> f2) <##> f = (f1 <##> f) <++> (f2 <##> f)
let f2 be PartFunc of X2,Y2; ::_thesis: (f1 <++> f2) <##> f = (f1 <##> f) <++> (f2 <##> f)
set f3 = f1 <##> f;
set f4 = f2 <##> f;
set f5 = f1 <++> f2;
A1: ( f1 <##> f = f <##> f1 & f2 <##> f = f <##> f2 ) by Th83;
thus (f1 <++> f2) <##> f = f <##> (f1 <++> f2) by Th83
.= (f1 <##> f) <++> (f2 <##> f) by A1, Th87 ; ::_thesis: verum
end;

theorem Th89: :: VALUED_2:89
for X, X1, X2 being set
for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f <##> (f1 <--> f2) = (f <##> f1) <--> (f <##> f2)
proof
let X, X1, X2 be set ; ::_thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f <##> (f1 <--> f2) = (f <##> f1) <--> (f <##> f2)
let Y, Y1, Y2 be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f <##> (f1 <--> f2) = (f <##> f1) <--> (f <##> f2)
let f be PartFunc of X,Y; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f <##> (f1 <--> f2) = (f <##> f1) <--> (f <##> f2)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds f <##> (f1 <--> f2) = (f <##> f1) <--> (f <##> f2)
let f2 be PartFunc of X2,Y2; ::_thesis: f <##> (f1 <--> f2) = (f <##> f1) <--> (f <##> f2)
set f3 = f <##> f1;
set f4 = f <##> f2;
set f5 = f1 <--> f2;
A1: dom (f <##> (f1 <--> f2)) = (dom f) /\ (dom (f1 <--> f2)) by Def47;
A2: dom (f1 <--> f2) = (dom f1) /\ (dom f2) by Def46;
A3: dom ((f <##> f1) <--> (f <##> f2)) = (dom (f <##> f1)) /\ (dom (f <##> f2)) by Def46;
 ( dom (f <##> f1) = (dom f) /\ (dom f1) & dom (f <##> f2) = (dom f) /\ (dom f2) ) by Def47;
hence A4: dom (f <##> (f1 <--> f2)) = dom ((f <##> f1) <--> (f <##> f2)) by A1, A3, A2, Lm1; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (f <##> (f1 <--> f2)) or (f <##> (f1 <--> f2)) . b1 = ((f <##> f1) <--> (f <##> f2)) . b1 )
let x be set ; ::_thesis: ( not x in dom (f <##> (f1 <--> f2)) or (f <##> (f1 <--> f2)) . x = ((f <##> f1) <--> (f <##> f2)) . x )
assume A5: x in dom (f <##> (f1 <--> f2)) ; ::_thesis: (f <##> (f1 <--> f2)) . x = ((f <##> f1) <--> (f <##> f2)) . x
then A6: x in dom (f <##> f1) by A3, A4, XBOOLE_0:def_4;
A7: x in dom (f1 <--> f2) by A1, A5, XBOOLE_0:def_4;
A8: x in dom (f <##> f2) by A3, A4, A5, XBOOLE_0:def_4;
thus (f <##> (f1 <--> f2)) . x = (f . x) (#) ((f1 <--> f2) . x) by A5, Def47
.= (f . x) (#) ((f1 . x) - (f2 . x)) by A7, Def46
.= ((f . x) (#) (f1 . x)) - ((f . x) (#) (f2 . x)) by RFUNCT_1:15
.= ((f <##> f1) . x) - ((f . x) (#) (f2 . x)) by A6, Def47
.= ((f <##> f1) . x) - ((f <##> f2) . x) by A8, Def47
.= ((f <##> f1) <--> (f <##> f2)) . x by A4, A5, Def46 ; ::_thesis: verum
end;

theorem :: VALUED_2:90
for X, X1, X2 being set
for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <--> f2) <##> f = (f1 <##> f) <--> (f2 <##> f)
proof
let X, X1, X2 be set ; ::_thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <--> f2) <##> f = (f1 <##> f) <--> (f2 <##> f)
let Y, Y1, Y2 be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <--> f2) <##> f = (f1 <##> f) <--> (f2 <##> f)
let f be PartFunc of X,Y; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <--> f2) <##> f = (f1 <##> f) <--> (f2 <##> f)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds (f1 <--> f2) <##> f = (f1 <##> f) <--> (f2 <##> f)
let f2 be PartFunc of X2,Y2; ::_thesis: (f1 <--> f2) <##> f = (f1 <##> f) <--> (f2 <##> f)
set f3 = f1 <##> f;
set f4 = f2 <##> f;
set f5 = f1 <--> f2;
A1: ( f1 <##> f = f <##> f1 & f2 <##> f = f <##> f2 ) by Th83;
thus (f1 <--> f2) <##> f = f <##> (f1 <--> f2) by Th83
.= (f1 <##> f) <--> (f2 <##> f) by A1, Th89 ; ::_thesis: verum
end;

definition
let Y1, Y2 be complex-functions-membered set ;
let f be Y1 -valued Function;
let g be Y2 -valued Function;
deffunc H1( set ) ->  set = (f . $1) /" (g . $1);
funcf <//> g ->  Function means :Def48: :: VALUED_2:def 48
( dom it = (dom f) /\ (dom g) & ( for x being set st x in dom it holds
it . x = (f . x) /" (g . x) ) );
existence
 ex b1 being Function st
( dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) /" (g . x) ) )
proof
 ex F being Function st
( dom F = (dom f) /\ (dom g) & ( for x being set st x in (dom f) /\ (dom g) holds
F . x = H1(x) ) ) from FUNCT_1:sch_3();
hence  ex b1 being Function st
( dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) /" (g . x) ) ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function st dom b1 = (dom f) /\ (dom g) & ( for x being set st x in dom b1 holds
b1 . x = (f . x) /" (g . x) ) & dom b2 = (dom f) /\ (dom g) & ( for x being set st x in dom b2 holds
b2 . x = (f . x) /" (g . x) ) holds
b1 = b2
proof
let F, G be Function; ::_thesis: ( dom F = (dom f) /\ (dom g) & ( for x being set st x in dom F holds
F . x = (f . x) /" (g . x) ) & dom G = (dom f) /\ (dom g) & ( for x being set st x in dom G holds
G . x = (f . x) /" (g . x) ) implies F = G )
assume that
A1: dom F = (dom f) /\ (dom g) and
A2: for x being set st x in dom F holds
F . x = H1(x) and
A3: dom G = (dom f) /\ (dom g) and
A4: for x being set st x in dom G holds
G . x = H1(x) ; ::_thesis: F = G
thus  dom F = dom G by A1, A3; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom F or F . b1 = G . b1 )
let x be set ; ::_thesis: ( not x in dom F or F . x = G . x )
assume A5: x in dom F ; ::_thesis: F . x = G . x
hence F . x = H1(x) by A2
.= G . x by A1, A3, A4, A5 ;
::_thesis: verum
end;
end;

:: deftheorem Def48 defines <//> VALUED_2:def_48_:_
 for Y1, Y2 being complex-functions-membered set
for f being b1 -valued Function
for g being b2 -valued Function
for b5 being Function holds
( b5 = f <//> g iff ( dom b5 = (dom f) /\ (dom g) & ( for x being set st x in dom b5 holds
b5 . x = (f . x) /" (g . x) ) ) );

definition
let X1, X2 be set ;
let Y1, Y2 be complex-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <//>
redefine funcf <//> g ->  PartFunc of (X1 /\ X2),(C_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <//> g is PartFunc of (X1 /\ X2),(C_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <//> g;
A1: dom (f <//> g) = (dom f) /\ (dom g) by Def48;
 rng (f <//> g) c= C_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <//> g) or y in C_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <//> g) ; ::_thesis: y in C_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <//> g) and
A3: (f <//> g) . x = y by FUNCT_1:def_3;
A4: (f <//> g) . x = (f . x) /" (g . x) by A2, Def48;
then reconsider y = y as Function by A3;
A5: rng y c= COMPLEX
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in COMPLEX )
thus  ( not b in rng y or b in COMPLEX ) by A3, A4, XCMPLX_0:def_2; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:16;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),COMPLEX by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in C_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def8; ::_thesis: verum
end;
hence  f <//> g is PartFunc of (X1 /\ X2),(C_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4, XBOOLE_1:27; ::_thesis: verum
end;
end;

definition
let X1, X2 be set ;
let Y1, Y2 be real-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <//>
redefine funcf <//> g ->  PartFunc of (X1 /\ X2),(R_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <//> g is PartFunc of (X1 /\ X2),(R_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <//> g;
A1: dom (f <//> g) = (dom f) /\ (dom g) by Def48;
 rng (f <//> g) c= R_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <//> g) or y in R_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <//> g) ; ::_thesis: y in R_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <//> g) and
A3: (f <//> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <//> g) . x = (f . x) /" (g . x) by A2, Def48;
A5: rng y c= REAL
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in REAL )
thus  ( not b in rng y or b in REAL ) by A3, A4, XREAL_0:def_1; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:16;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),REAL by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in R_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def12; ::_thesis: verum
end;
hence  f <//> g is PartFunc of (X1 /\ X2),(R_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let X1, X2 be set ;
let Y1, Y2 be rational-functions-membered set ;
let f be PartFunc of X1,Y1;
let g be PartFunc of X2,Y2;
:: original: <//>
redefine funcf <//> g ->  PartFunc of (X1 /\ X2),(Q_PFuncs ((DOMS Y1) /\ (DOMS Y2)));
coherence
 f <//> g is PartFunc of (X1 /\ X2),(Q_PFuncs ((DOMS Y1) /\ (DOMS Y2)))
proof
set h = f <//> g;
A1: dom (f <//> g) = (dom f) /\ (dom g) by Def48;
 rng (f <//> g) c= Q_PFuncs ((DOMS Y1) /\ (DOMS Y2))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f <//> g) or y in Q_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume  y in rng (f <//> g) ; ::_thesis: y in Q_PFuncs ((DOMS Y1) /\ (DOMS Y2))
then consider x being set  such that
A2: x in dom (f <//> g) and
A3: (f <//> g) . x = y by FUNCT_1:def_3;
reconsider y = y as Function by A3;
A4: (f <//> g) . x = (f . x) /" (g . x) by A2, Def48;
A5: rng y c= RAT
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng y or b in RAT )
thus  ( not b in rng y or b in RAT ) by A3, A4, RAT_1:def_2; ::_thesis: verum
end;
 x in dom g by A1, A2, XBOOLE_0:def_4;
then  g . x in Y2 by PARTFUN1:4;
then  dom (g . x) in  {  (dom m) where m is Element of Y2 : verum  }  ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
 x in dom f by A1, A2, XBOOLE_0:def_4;
then  f . x in Y1 by PARTFUN1:4;
then  dom (f . x) in  {  (dom m) where m is Element of Y1 : verum  }  ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
 dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:16;
then  y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),RAT by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence  y in Q_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def14; ::_thesis: verum
end;
hence  f <//> g is PartFunc of (X1 /\ X2),(Q_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4; ::_thesis: verum
end;
end;

theorem :: VALUED_2:91
for X1, X2 being set
for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (<-> f1) <//> f2 = <-> (f1 <//> f2)
proof
let X1, X2 be set ; ::_thesis: for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (<-> f1) <//> f2 = <-> (f1 <//> f2)
let Y1, Y2 be complex-functions-membered set ; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (<-> f1) <//> f2 = <-> (f1 <//> f2)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds (<-> f1) <//> f2 = <-> (f1 <//> f2)
let f2 be PartFunc of X2,Y2; ::_thesis: (<-> f1) <//> f2 = <-> (f1 <//> f2)
set f3 = f1 <//> f2;
set f4 = <-> f1;
A1: ( dom (f1 <//> f2) = (dom f1) /\ (dom f2) & dom (<-> f1) = dom f1 ) by Def33, Def48;
 dom ((<-> f1) <//> f2) = (dom (<-> f1)) /\ (dom f2) by Def48;
hence A2: dom ((<-> f1) <//> f2) = dom (<-> (f1 <//> f2)) by A1, Def33; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((<-> f1) <//> f2) or ((<-> f1) <//> f2) . b1 = (<-> (f1 <//> f2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((<-> f1) <//> f2) or ((<-> f1) <//> f2) . x = (<-> (f1 <//> f2)) . x )
assume A3: x in dom ((<-> f1) <//> f2) ; ::_thesis: ((<-> f1) <//> f2) . x = (<-> (f1 <//> f2)) . x
then A4: x in dom (f1 <//> f2) by A1, Def48;
then A5: x in dom (<-> f1) by A1, XBOOLE_0:def_4;
thus ((<-> f1) <//> f2) . x = ((<-> f1) . x) /" (f2 . x) by A3, Def48
.= (- (f1 . x)) /" (f2 . x) by A5, Def33
.= - ((f1 . x) /" (f2 . x)) by Th25
.= - ((f1 <//> f2) . x) by A4, Def48
.= (<-> (f1 <//> f2)) . x by A2, A3, Def33 ; ::_thesis: verum
end;

theorem :: VALUED_2:92
for X1, X2 being set
for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <//> (<-> f2) = <-> (f1 <//> f2)
proof
let X1, X2 be set ; ::_thesis: for Y1, Y2 being complex-functions-membered set
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <//> (<-> f2) = <-> (f1 <//> f2)
let Y1, Y2 be complex-functions-membered set ; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds f1 <//> (<-> f2) = <-> (f1 <//> f2)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds f1 <//> (<-> f2) = <-> (f1 <//> f2)
let f2 be PartFunc of X2,Y2; ::_thesis: f1 <//> (<-> f2) = <-> (f1 <//> f2)
set f3 = f1 <//> f2;
set f4 = <-> f2;
A1: ( dom (f1 <//> f2) = (dom f1) /\ (dom f2) & dom (<-> f2) = dom f2 ) by Def33, Def48;
 dom (f1 <//> (<-> f2)) = (dom f1) /\ (dom (<-> f2)) by Def48;
hence A2: dom (f1 <//> (<-> f2)) = dom (<-> (f1 <//> f2)) by A1, Def33; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom (f1 <//> (<-> f2)) or (f1 <//> (<-> f2)) . b1 = (<-> (f1 <//> f2)) . b1 )
let x be set ; ::_thesis: ( not x in dom (f1 <//> (<-> f2)) or (f1 <//> (<-> f2)) . x = (<-> (f1 <//> f2)) . x )
assume A3: x in dom (f1 <//> (<-> f2)) ; ::_thesis: (f1 <//> (<-> f2)) . x = (<-> (f1 <//> f2)) . x
then A4: x in dom (f1 <//> f2) by A1, Def48;
then A5: x in dom (<-> f2) by A1, XBOOLE_0:def_4;
thus (f1 <//> (<-> f2)) . x = (f1 . x) /" ((<-> f2) . x) by A3, Def48
.= (f1 . x) /" (- (f2 . x)) by A5, Def33
.= - ((f1 . x) /" (f2 . x)) by Th27
.= - ((f1 <//> f2) . x) by A4, Def48
.= (<-> (f1 <//> f2)) . x by A2, A3, Def33 ; ::_thesis: verum
end;

theorem :: VALUED_2:93
for X, X1, X2 being set
for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <##> f1) <//> f2 = f <##> (f1 <//> f2)
proof
let X, X1, X2 be set ; ::_thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <##> f1) <//> f2 = f <##> (f1 <//> f2)
let Y, Y1, Y2 be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <##> f1) <//> f2 = f <##> (f1 <//> f2)
let f be PartFunc of X,Y; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <##> f1) <//> f2 = f <##> (f1 <//> f2)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds (f <##> f1) <//> f2 = f <##> (f1 <//> f2)
let f2 be PartFunc of X2,Y2; ::_thesis: (f <##> f1) <//> f2 = f <##> (f1 <//> f2)
set f3 = f <##> f1;
set f4 = f1 <//> f2;
A1: dom ((f <##> f1) <//> f2) = (dom (f <##> f1)) /\ (dom f2) by Def48;
A2: dom (f <##> (f1 <//> f2)) = (dom f) /\ (dom (f1 <//> f2)) by Def47;
 ( dom (f <##> f1) = (dom f) /\ (dom f1) & dom (f1 <//> f2) = (dom f1) /\ (dom f2) ) by Def47, Def48;
hence A3: dom ((f <##> f1) <//> f2) = dom (f <##> (f1 <//> f2)) by A1, A2, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f <##> f1) <//> f2) or ((f <##> f1) <//> f2) . b1 = (f <##> (f1 <//> f2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f <##> f1) <//> f2) or ((f <##> f1) <//> f2) . x = (f <##> (f1 <//> f2)) . x )
assume A4: x in dom ((f <##> f1) <//> f2) ; ::_thesis: ((f <##> f1) <//> f2) . x = (f <##> (f1 <//> f2)) . x
then A5: x in dom (f1 <//> f2) by A2, A3, XBOOLE_0:def_4;
A6: x in dom (f <##> f1) by A1, A4, XBOOLE_0:def_4;
thus ((f <##> f1) <//> f2) . x = ((f <##> f1) . x) /" (f2 . x) by A4, Def48
.= ((f . x) (#) (f1 . x)) /" (f2 . x) by A6, Def47
.= (f . x) (#) ((f1 . x) /" (f2 . x)) by Th19
.= (f . x) (#) ((f1 <//> f2) . x) by A5, Def48
.= (f <##> (f1 <//> f2)) . x by A3, A4, Def47 ; ::_thesis: verum
end;

theorem :: VALUED_2:94
for X, X1, X2 being set
for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <//> f1) <##> f2 = (f <##> f2) <//> f1
proof
let X, X1, X2 be set ; ::_thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <//> f1) <##> f2 = (f <##> f2) <//> f1
let Y, Y1, Y2 be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <//> f1) <##> f2 = (f <##> f2) <//> f1
let f be PartFunc of X,Y; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <//> f1) <##> f2 = (f <##> f2) <//> f1
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds (f <//> f1) <##> f2 = (f <##> f2) <//> f1
let f2 be PartFunc of X2,Y2; ::_thesis: (f <//> f1) <##> f2 = (f <##> f2) <//> f1
set f3 = f <//> f1;
set f4 = f <##> f2;
A1: dom ((f <//> f1) <##> f2) = (dom (f <//> f1)) /\ (dom f2) by Def47;
A2: dom ((f <##> f2) <//> f1) = (dom (f <##> f2)) /\ (dom f1) by Def48;
 ( dom (f <//> f1) = (dom f) /\ (dom f1) & dom (f <##> f2) = (dom f) /\ (dom f2) ) by Def47, Def48;
hence A3: dom ((f <//> f1) <##> f2) = dom ((f <##> f2) <//> f1) by A1, A2, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f <//> f1) <##> f2) or ((f <//> f1) <##> f2) . b1 = ((f <##> f2) <//> f1) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f <//> f1) <##> f2) or ((f <//> f1) <##> f2) . x = ((f <##> f2) <//> f1) . x )
assume A4: x in dom ((f <//> f1) <##> f2) ; ::_thesis: ((f <//> f1) <##> f2) . x = ((f <##> f2) <//> f1) . x
then A5: x in dom (f <##> f2) by A2, A3, XBOOLE_0:def_4;
A6: x in dom (f <//> f1) by A1, A4, XBOOLE_0:def_4;
thus ((f <//> f1) <##> f2) . x = ((f <//> f1) . x) (#) (f2 . x) by A4, Def47
.= ((f . x) /" (f1 . x)) (#) (f2 . x) by A6, Def48
.= ((f . x) (#) (f2 . x)) /" (f1 . x) by Th20
.= ((f <##> f2) . x) /" (f1 . x) by A5, Def47
.= ((f <##> f2) <//> f1) . x by A3, A4, Def48 ; ::_thesis: verum
end;

theorem :: VALUED_2:95
for X, X1, X2 being set
for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <//> f1) <//> f2 = f <//> (f1 <##> f2)
proof
let X, X1, X2 be set ; ::_thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <//> f1) <//> f2 = f <//> (f1 <##> f2)
let Y, Y1, Y2 be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <//> f1) <//> f2 = f <//> (f1 <##> f2)
let f be PartFunc of X,Y; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f <//> f1) <//> f2 = f <//> (f1 <##> f2)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds (f <//> f1) <//> f2 = f <//> (f1 <##> f2)
let f2 be PartFunc of X2,Y2; ::_thesis: (f <//> f1) <//> f2 = f <//> (f1 <##> f2)
set f3 = f <//> f1;
set f4 = f1 <##> f2;
A1: dom ((f <//> f1) <//> f2) = (dom (f <//> f1)) /\ (dom f2) by Def48;
A2: dom (f <//> (f1 <##> f2)) = (dom f) /\ (dom (f1 <##> f2)) by Def48;
 ( dom (f <//> f1) = (dom f) /\ (dom f1) & dom (f1 <##> f2) = (dom f1) /\ (dom f2) ) by Def47, Def48;
hence A3: dom ((f <//> f1) <//> f2) = dom (f <//> (f1 <##> f2)) by A1, A2, XBOOLE_1:16; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f <//> f1) <//> f2) or ((f <//> f1) <//> f2) . b1 = (f <//> (f1 <##> f2)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f <//> f1) <//> f2) or ((f <//> f1) <//> f2) . x = (f <//> (f1 <##> f2)) . x )
assume A4: x in dom ((f <//> f1) <//> f2) ; ::_thesis: ((f <//> f1) <//> f2) . x = (f <//> (f1 <##> f2)) . x
then A5: x in dom (f1 <##> f2) by A2, A3, XBOOLE_0:def_4;
A6: x in dom (f <//> f1) by A1, A4, XBOOLE_0:def_4;
thus ((f <//> f1) <//> f2) . x = ((f <//> f1) . x) /" (f2 . x) by A4, Def48
.= ((f . x) /" (f1 . x)) /" (f2 . x) by A6, Def48
.= (f . x) /" ((f1 . x) (#) (f2 . x)) by Th21
.= (f . x) /" ((f1 <##> f2) . x) by A5, Def47
.= (f <//> (f1 <##> f2)) . x by A3, A4, Def48 ; ::_thesis: verum
end;

theorem :: VALUED_2:96
for X, X1, X2 being set
for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <++> f2) <//> f = (f1 <//> f) <++> (f2 <//> f)
proof
let X, X1, X2 be set ; ::_thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <++> f2) <//> f = (f1 <//> f) <++> (f2 <//> f)
let Y, Y1, Y2 be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <++> f2) <//> f = (f1 <//> f) <++> (f2 <//> f)
let f be PartFunc of X,Y; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <++> f2) <//> f = (f1 <//> f) <++> (f2 <//> f)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds (f1 <++> f2) <//> f = (f1 <//> f) <++> (f2 <//> f)
let f2 be PartFunc of X2,Y2; ::_thesis: (f1 <++> f2) <//> f = (f1 <//> f) <++> (f2 <//> f)
set f3 = f1 <//> f;
set f4 = f2 <//> f;
set f5 = f1 <++> f2;
A1: dom ((f1 <++> f2) <//> f) = (dom f) /\ (dom (f1 <++> f2)) by Def48;
A2: dom (f1 <++> f2) = (dom f1) /\ (dom f2) by Def45;
A3: dom ((f1 <//> f) <++> (f2 <//> f)) = (dom (f1 <//> f)) /\ (dom (f2 <//> f)) by Def45;
 ( dom (f1 <//> f) = (dom f1) /\ (dom f) & dom (f2 <//> f) = (dom f2) /\ (dom f) ) by Def48;
hence A4: dom ((f1 <++> f2) <//> f) = dom ((f1 <//> f) <++> (f2 <//> f)) by A1, A3, A2, Lm1; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f1 <++> f2) <//> f) or ((f1 <++> f2) <//> f) . b1 = ((f1 <//> f) <++> (f2 <//> f)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f1 <++> f2) <//> f) or ((f1 <++> f2) <//> f) . x = ((f1 <//> f) <++> (f2 <//> f)) . x )
assume A5: x in dom ((f1 <++> f2) <//> f) ; ::_thesis: ((f1 <++> f2) <//> f) . x = ((f1 <//> f) <++> (f2 <//> f)) . x
then A6: x in dom (f1 <//> f) by A3, A4, XBOOLE_0:def_4;
A7: x in dom (f1 <++> f2) by A1, A5, XBOOLE_0:def_4;
A8: x in dom (f2 <//> f) by A3, A4, A5, XBOOLE_0:def_4;
thus ((f1 <++> f2) <//> f) . x = ((f1 <++> f2) . x) /" (f . x) by A5, Def48
.= ((f1 . x) + (f2 . x)) /" (f . x) by A7, Def45
.= ((f1 . x) /" (f . x)) + ((f2 . x) /" (f . x)) by RFUNCT_1:10
.= ((f1 <//> f) . x) + ((f2 . x) /" (f . x)) by A6, Def48
.= ((f1 <//> f) . x) + ((f2 <//> f) . x) by A8, Def48
.= ((f1 <//> f) <++> (f2 <//> f)) . x by A4, A5, Def45 ; ::_thesis: verum
end;

theorem :: VALUED_2:97
for X, X1, X2 being set
for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <--> f2) <//> f = (f1 <//> f) <--> (f2 <//> f)
proof
let X, X1, X2 be set ; ::_thesis: for Y, Y1, Y2 being complex-functions-membered set
for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <--> f2) <//> f = (f1 <//> f) <--> (f2 <//> f)
let Y, Y1, Y2 be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y
for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <--> f2) <//> f = (f1 <//> f) <--> (f2 <//> f)
let f be PartFunc of X,Y; ::_thesis: for f1 being PartFunc of X1,Y1
for f2 being PartFunc of X2,Y2 holds (f1 <--> f2) <//> f = (f1 <//> f) <--> (f2 <//> f)
let f1 be PartFunc of X1,Y1; ::_thesis: for f2 being PartFunc of X2,Y2 holds (f1 <--> f2) <//> f = (f1 <//> f) <--> (f2 <//> f)
let f2 be PartFunc of X2,Y2; ::_thesis: (f1 <--> f2) <//> f = (f1 <//> f) <--> (f2 <//> f)
set f3 = f1 <//> f;
set f4 = f2 <//> f;
set f5 = f1 <--> f2;
A1: dom ((f1 <--> f2) <//> f) = (dom f) /\ (dom (f1 <--> f2)) by Def48;
A2: dom (f1 <--> f2) = (dom f1) /\ (dom f2) by Def46;
A3: dom ((f1 <//> f) <--> (f2 <//> f)) = (dom (f1 <//> f)) /\ (dom (f2 <//> f)) by Def46;
 ( dom (f1 <//> f) = (dom f1) /\ (dom f) & dom (f2 <//> f) = (dom f2) /\ (dom f) ) by Def48;
hence A4: dom ((f1 <--> f2) <//> f) = dom ((f1 <//> f) <--> (f2 <//> f)) by A1, A3, A2, Lm1; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in dom ((f1 <--> f2) <//> f) or ((f1 <--> f2) <//> f) . b1 = ((f1 <//> f) <--> (f2 <//> f)) . b1 )
let x be set ; ::_thesis: ( not x in dom ((f1 <--> f2) <//> f) or ((f1 <--> f2) <//> f) . x = ((f1 <//> f) <--> (f2 <//> f)) . x )
assume A5: x in dom ((f1 <--> f2) <//> f) ; ::_thesis: ((f1 <--> f2) <//> f) . x = ((f1 <//> f) <--> (f2 <//> f)) . x
then A6: x in dom (f1 <//> f) by A3, A4, XBOOLE_0:def_4;
A7: x in dom (f1 <--> f2) by A1, A5, XBOOLE_0:def_4;
A8: x in dom (f2 <//> f) by A3, A4, A5, XBOOLE_0:def_4;
thus ((f1 <--> f2) <//> f) . x = ((f1 <--> f2) . x) /" (f . x) by A5, Def48
.= ((f1 . x) - (f2 . x)) /" (f . x) by A7, Def46
.= ((f1 . x) /" (f . x)) - ((f2 . x) /" (f . x)) by RFUNCT_1:14
.= ((f1 <//> f) . x) - ((f2 . x) /" (f . x)) by A6, Def48
.= ((f1 <//> f) . x) - ((f2 <//> f) . x) by A8, Def48
.= ((f1 <//> f) <--> (f2 <//> f)) . x by A4, A5, Def46 ; ::_thesis: verum
end;