:: 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;