:: RFUNCT_3 semantic presentation begin definition let r be real number ; func max+ r -> Real equals :: RFUNCT_3:def 1 max (r,0); correctness coherence max (r,0) is Real; by XREAL_0:def_1; func max- r -> Real equals :: RFUNCT_3:def 2 max ((- r),0); correctness coherence max ((- r),0) is Real; by XREAL_0:def_1; end; :: deftheorem defines max+ RFUNCT_3:def_1_:_ for r being real number holds max+ r = max (r,0); :: deftheorem defines max- RFUNCT_3:def_2_:_ for r being real number holds max- r = max ((- r),0); theorem Th1: :: RFUNCT_3:1 for r being real number holds r = (max+ r) - (max- r) proof let r be real number ; ::_thesis: r = (max+ r) - (max- r) now__::_thesis:_(_(_0_<=_r_&_r_=_(max+_r)_-_(max-_r)_)_or_(_r_<_0_&_r_=_(max+_r)_-_(max-_r)_)_) percases ( 0 <= r or r < 0 ) ; caseA1: 0 <= r ; ::_thesis: r = (max+ r) - (max- r) then max- r = 0 by XXREAL_0:def_10; hence r = (max+ r) - (max- r) by A1, XXREAL_0:def_10; ::_thesis: verum end; case r < 0 ; ::_thesis: r = (max+ r) - (max- r) then ( max+ r = 0 & max- r = - r ) by XXREAL_0:def_10; hence r = (max+ r) - (max- r) ; ::_thesis: verum end; end; end; hence r = (max+ r) - (max- r) ; ::_thesis: verum end; theorem Th2: :: RFUNCT_3:2 for r being real number holds abs r = (max+ r) + (max- r) proof let r be real number ; ::_thesis: abs r = (max+ r) + (max- r) now__::_thesis:_(_(_0_<=_r_&_abs_r_=_(max+_r)_+_(max-_r)_)_or_(_r_<_0_&_abs_r_=_(max+_r)_+_(max-_r)_)_) percases ( 0 <= r or r < 0 ) ; caseA1: 0 <= r ; ::_thesis: abs r = (max+ r) + (max- r) then ( max+ r = r & max- r = 0 ) by XXREAL_0:def_10; hence abs r = (max+ r) + (max- r) by A1, ABSVALUE:def_1; ::_thesis: verum end; caseA2: r < 0 ; ::_thesis: abs r = (max+ r) + (max- r) then ( max+ r = 0 & max- r = - r ) by XXREAL_0:def_10; hence abs r = (max+ r) + (max- r) by A2, ABSVALUE:def_1; ::_thesis: verum end; end; end; hence abs r = (max+ r) + (max- r) ; ::_thesis: verum end; theorem Th3: :: RFUNCT_3:3 for r being real number holds 2 * (max+ r) = r + (abs r) proof let r be real number ; ::_thesis: 2 * (max+ r) = r + (abs r) thus r + (abs r) = ((max+ r) - (max- r)) + (abs r) by Th1 .= ((max+ r) - (max- r)) + ((max+ r) + (max- r)) by Th2 .= 2 * (max+ r) ; ::_thesis: verum end; theorem Th4: :: RFUNCT_3:4 for r, s being real number st 0 <= r holds max+ (r * s) = r * (max+ s) proof let r, s be real number ; ::_thesis: ( 0 <= r implies max+ (r * s) = r * (max+ s) ) assume A1: 0 <= r ; ::_thesis: max+ (r * s) = r * (max+ s) now__::_thesis:_(_(_0_<=_s_&_max+_(r_*_s)_=_r_*_(max+_s)_)_or_(_s_<_0_&_max+_(r_*_s)_=_r_*_(max+_s)_)_) percases ( 0 <= s or s < 0 ) ; caseA2: 0 <= s ; ::_thesis: max+ (r * s) = r * (max+ s) then max+ (r * s) = r * s by A1, XXREAL_0:def_10; hence max+ (r * s) = r * (max+ s) by A2, XXREAL_0:def_10; ::_thesis: verum end; caseA3: s < 0 ; ::_thesis: max+ (r * s) = r * (max+ s) then max+ s = 0 by XXREAL_0:def_10; hence max+ (r * s) = r * (max+ s) by A1, A3, XXREAL_0:def_10; ::_thesis: verum end; end; end; hence max+ (r * s) = r * (max+ s) ; ::_thesis: verum end; theorem Th5: :: RFUNCT_3:5 for r, s being real number holds max+ (r + s) <= (max+ r) + (max+ s) proof let r, s be real number ; ::_thesis: max+ (r + s) <= (max+ r) + (max+ s) A1: ( 0 <= max (r,0) & 0 <= max (s,0) ) by XXREAL_0:25; A2: ( r <= max (r,0) & s <= max (s,0) ) by XXREAL_0:25; now__::_thesis:_(_(_0_<=_r_+_s_&_max+_(r_+_s)_<=_(max+_r)_+_(max+_s)_)_or_(_r_+_s_<_0_&_max+_(r_+_s)_<=_(max+_r)_+_(max+_s)_)_) percases ( 0 <= r + s or r + s < 0 ) ; case 0 <= r + s ; ::_thesis: max+ (r + s) <= (max+ r) + (max+ s) then max+ (r + s) = r + s by XXREAL_0:def_10; hence max+ (r + s) <= (max+ r) + (max+ s) by A2, XREAL_1:7; ::_thesis: verum end; case r + s < 0 ; ::_thesis: max+ (r + s) <= (max+ r) + (max+ s) then max+ (r + s) = 0 + 0 by XXREAL_0:def_10; hence max+ (r + s) <= (max+ r) + (max+ s) by A1; ::_thesis: verum end; end; end; hence max+ (r + s) <= (max+ r) + (max+ s) ; ::_thesis: verum end; Lm1: for n being Element of NAT for D being non empty set for f being FinSequence of D st len f <= n holds f | n = f proof let n be Element of NAT ; ::_thesis: for D being non empty set for f being FinSequence of D st len f <= n holds f | n = f let D be non empty set ; ::_thesis: for f being FinSequence of D st len f <= n holds f | n = f let f be FinSequence of D; ::_thesis: ( len f <= n implies f | n = f ) A1: dom f = Seg (len f) by FINSEQ_1:def_3; assume len f <= n ; ::_thesis: f | n = f then ( f | n = f | (Seg n) & dom f c= Seg n ) by A1, FINSEQ_1:5, FINSEQ_1:def_15; hence f | n = f by RELAT_1:68; ::_thesis: verum end; Lm2: for f being Function for x being set st not x in rng f holds f " {x} = {} proof let f be Function; ::_thesis: for x being set st not x in rng f holds f " {x} = {} let x be set ; ::_thesis: ( not x in rng f implies f " {x} = {} ) assume A1: not x in rng f ; ::_thesis: f " {x} = {} rng f misses {x} proof set y = the Element of (rng f) /\ {x}; assume (rng f) /\ {x} <> {} ; :: according to XBOOLE_0:def_7 ::_thesis: contradiction then ( the Element of (rng f) /\ {x} in rng f & the Element of (rng f) /\ {x} in {x} ) by XBOOLE_0:def_4; hence contradiction by A1, TARSKI:def_1; ::_thesis: verum end; hence f " {x} = {} by RELAT_1:138; ::_thesis: verum end; begin theorem Th6: :: RFUNCT_3:6 for D being non empty set for F being PartFunc of D,REAL for r, s being real number st r <> 0 holds F " {(s / r)} = (r (#) F) " {s} proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for r, s being real number st r <> 0 holds F " {(s / r)} = (r (#) F) " {s} let F be PartFunc of D,REAL; ::_thesis: for r, s being real number st r <> 0 holds F " {(s / r)} = (r (#) F) " {s} let r, s be real number ; ::_thesis: ( r <> 0 implies F " {(s / r)} = (r (#) F) " {s} ) assume A1: r <> 0 ; ::_thesis: F " {(s / r)} = (r (#) F) " {s} thus F " {(s / r)} c= (r (#) F) " {s} :: according to XBOOLE_0:def_10 ::_thesis: (r (#) F) " {s} c= F " {(s / r)} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " {(s / r)} or x in (r (#) F) " {s} ) assume A2: x in F " {(s / r)} ; ::_thesis: x in (r (#) F) " {s} then reconsider d = x as Element of D ; d in dom F by A2, FUNCT_1:def_7; then A3: d in dom (r (#) F) by VALUED_1:def_5; F . d in {(s / r)} by A2, FUNCT_1:def_7; then F . d = s / r by TARSKI:def_1; then r * (F . d) = s by A1, XCMPLX_1:87; then (r (#) F) . d = s by A3, VALUED_1:def_5; then (r (#) F) . d in {s} by TARSKI:def_1; hence x in (r (#) F) " {s} by A3, FUNCT_1:def_7; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (r (#) F) " {s} or x in F " {(s / r)} ) assume A4: x in (r (#) F) " {s} ; ::_thesis: x in F " {(s / r)} then reconsider d = x as Element of D ; A5: d in dom (r (#) F) by A4, FUNCT_1:def_7; (r (#) F) . d in {s} by A4, FUNCT_1:def_7; then (r (#) F) . d = s by TARSKI:def_1; then r * (F . d) = s by A5, VALUED_1:def_5; then F . d = s / r by A1, XCMPLX_1:89; then A6: F . d in {(s / r)} by TARSKI:def_1; d in dom F by A5, VALUED_1:def_5; hence x in F " {(s / r)} by A6, FUNCT_1:def_7; ::_thesis: verum end; theorem Th7: :: RFUNCT_3:7 for D being non empty set for F being PartFunc of D,REAL holds (0 (#) F) " {0} = dom F proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL holds (0 (#) F) " {0} = dom F let F be PartFunc of D,REAL; ::_thesis: (0 (#) F) " {0} = dom F thus (0 (#) F) " {0} c= dom F :: according to XBOOLE_0:def_10 ::_thesis: dom F c= (0 (#) F) " {0} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (0 (#) F) " {0} or x in dom F ) assume A1: x in (0 (#) F) " {0} ; ::_thesis: x in dom F then reconsider d = x as Element of D ; d in dom (0 (#) F) by A1, FUNCT_1:def_7; hence x in dom F by VALUED_1:def_5; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom F or x in (0 (#) F) " {0} ) assume A2: x in dom F ; ::_thesis: x in (0 (#) F) " {0} then reconsider d = x as Element of D ; A3: d in dom (0 (#) F) by A2, VALUED_1:def_5; then (0 (#) F) . d = 0 * (F . d) by VALUED_1:def_5 .= 0 ; then (0 (#) F) . d in {0} by TARSKI:def_1; hence x in (0 (#) F) " {0} by A3, FUNCT_1:def_7; ::_thesis: verum end; theorem Th8: :: RFUNCT_3:8 for D being non empty set for F being PartFunc of D,REAL for r being Real st 0 < r holds (abs F) " {r} = F " {(- r),r} proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for r being Real st 0 < r holds (abs F) " {r} = F " {(- r),r} let F be PartFunc of D,REAL; ::_thesis: for r being Real st 0 < r holds (abs F) " {r} = F " {(- r),r} let r be Real; ::_thesis: ( 0 < r implies (abs F) " {r} = F " {(- r),r} ) assume A1: 0 < r ; ::_thesis: (abs F) " {r} = F " {(- r),r} A2: dom (abs F) = dom F by VALUED_1:def_11; thus (abs F) " {r} c= F " {(- r),r} :: according to XBOOLE_0:def_10 ::_thesis: F " {(- r),r} c= (abs F) " {r} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (abs F) " {r} or x in F " {(- r),r} ) assume A3: x in (abs F) " {r} ; ::_thesis: x in F " {(- r),r} then reconsider rr = x as Element of D ; (abs F) . rr in {r} by A3, FUNCT_1:def_7; then abs (F . rr) in {r} by VALUED_1:18; then A4: abs (F . rr) = r by TARSKI:def_1; A5: rr in dom (abs F) by A3, FUNCT_1:def_7; now__::_thesis:_(_(_0_<=_F_._rr_&_x_in_F_"_{(-_r),r}_)_or_(_F_._rr_<_0_&_x_in_F_"_{(-_r),r}_)_) percases ( 0 <= F . rr or F . rr < 0 ) ; case 0 <= F . rr ; ::_thesis: x in F " {(- r),r} then F . rr = r by A4, ABSVALUE:def_1; then F . rr in {(- r),r} by TARSKI:def_2; hence x in F " {(- r),r} by A2, A5, FUNCT_1:def_7; ::_thesis: verum end; case F . rr < 0 ; ::_thesis: x in F " {(- r),r} then - (F . rr) = r by A4, ABSVALUE:def_1; then F . rr in {(- r),r} by TARSKI:def_2; hence x in F " {(- r),r} by A2, A5, FUNCT_1:def_7; ::_thesis: verum end; end; end; hence x in F " {(- r),r} ; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " {(- r),r} or x in (abs F) " {r} ) assume A6: x in F " {(- r),r} ; ::_thesis: x in (abs F) " {r} then reconsider rr = x as Element of D ; A7: rr in dom F by A6, FUNCT_1:def_7; A8: F . rr in {(- r),r} by A6, FUNCT_1:def_7; now__::_thesis:_(_(_F_._rr_=_-_r_&_x_in_(abs_F)_"_{r}_)_or_(_F_._rr_=_r_&_x_in_(abs_F)_"_{r}_)_) percases ( F . rr = - r or F . rr = r ) by A8, TARSKI:def_2; case F . rr = - r ; ::_thesis: x in (abs F) " {r} then r = abs (- (F . rr)) by A1, ABSVALUE:def_1 .= abs (F . rr) by COMPLEX1:52 .= (abs F) . rr by VALUED_1:18 ; then (abs F) . rr in {r} by TARSKI:def_1; hence x in (abs F) " {r} by A2, A7, FUNCT_1:def_7; ::_thesis: verum end; case F . rr = r ; ::_thesis: x in (abs F) " {r} then r = abs (F . rr) by A1, ABSVALUE:def_1 .= (abs F) . rr by VALUED_1:18 ; then (abs F) . rr in {r} by TARSKI:def_1; hence x in (abs F) " {r} by A2, A7, FUNCT_1:def_7; ::_thesis: verum end; end; end; hence x in (abs F) " {r} ; ::_thesis: verum end; theorem Th9: :: RFUNCT_3:9 for D being non empty set for F being PartFunc of D,REAL holds (abs F) " {0} = F " {0} proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL holds (abs F) " {0} = F " {0} let F be PartFunc of D,REAL; ::_thesis: (abs F) " {0} = F " {0} A1: dom (abs F) = dom F by VALUED_1:def_11; thus (abs F) " {0} c= F " {0} :: according to XBOOLE_0:def_10 ::_thesis: F " {0} c= (abs F) " {0} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (abs F) " {0} or x in F " {0} ) assume A2: x in (abs F) " {0} ; ::_thesis: x in F " {0} then reconsider r = x as Element of D ; (abs F) . r in {0} by A2, FUNCT_1:def_7; then abs (F . r) in {0} by VALUED_1:18; then abs (F . r) = 0 by TARSKI:def_1; then F . r = 0 by ABSVALUE:2; then A3: F . r in {0} by TARSKI:def_1; r in dom (abs F) by A2, FUNCT_1:def_7; hence x in F " {0} by A1, A3, FUNCT_1:def_7; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " {0} or x in (abs F) " {0} ) assume A4: x in F " {0} ; ::_thesis: x in (abs F) " {0} then reconsider r = x as Element of D ; F . r in {0} by A4, FUNCT_1:def_7; then F . r = 0 by TARSKI:def_1; then abs (F . r) = 0 by ABSVALUE:2; then (abs F) . r = 0 by VALUED_1:18; then A5: (abs F) . r in {0} by TARSKI:def_1; r in dom F by A4, FUNCT_1:def_7; hence x in (abs F) " {0} by A1, A5, FUNCT_1:def_7; ::_thesis: verum end; theorem Th10: :: RFUNCT_3:10 for D being non empty set for F being PartFunc of D,REAL for r being Real st r < 0 holds (abs F) " {r} = {} proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for r being Real st r < 0 holds (abs F) " {r} = {} let F be PartFunc of D,REAL; ::_thesis: for r being Real st r < 0 holds (abs F) " {r} = {} let r be Real; ::_thesis: ( r < 0 implies (abs F) " {r} = {} ) assume A1: r < 0 ; ::_thesis: (abs F) " {r} = {} set x = the Element of (abs F) " {r}; assume A2: (abs F) " {r} <> {} ; ::_thesis: contradiction then reconsider x = the Element of (abs F) " {r} as Element of D by TARSKI:def_3; (abs F) . x in {r} by A2, FUNCT_1:def_7; then abs (F . x) in {r} by VALUED_1:18; then abs (F . x) = r by TARSKI:def_1; hence contradiction by A1, COMPLEX1:46; ::_thesis: verum end; theorem Th11: :: RFUNCT_3:11 for D, C being non empty set for F being PartFunc of D,REAL for G being PartFunc of C,REAL for r being Real st r <> 0 holds ( F,G are_fiberwise_equipotent iff r (#) F,r (#) G are_fiberwise_equipotent ) proof let D, C be non empty set ; ::_thesis: for F being PartFunc of D,REAL for G being PartFunc of C,REAL for r being Real st r <> 0 holds ( F,G are_fiberwise_equipotent iff r (#) F,r (#) G are_fiberwise_equipotent ) let F be PartFunc of D,REAL; ::_thesis: for G being PartFunc of C,REAL for r being Real st r <> 0 holds ( F,G are_fiberwise_equipotent iff r (#) F,r (#) G are_fiberwise_equipotent ) let G be PartFunc of C,REAL; ::_thesis: for r being Real st r <> 0 holds ( F,G are_fiberwise_equipotent iff r (#) F,r (#) G are_fiberwise_equipotent ) let r be Real; ::_thesis: ( r <> 0 implies ( F,G are_fiberwise_equipotent iff r (#) F,r (#) G are_fiberwise_equipotent ) ) assume A1: r <> 0 ; ::_thesis: ( F,G are_fiberwise_equipotent iff r (#) F,r (#) G are_fiberwise_equipotent ) A2: ( rng (r (#) F) c= REAL & rng (r (#) G) c= REAL ) ; thus ( F,G are_fiberwise_equipotent implies r (#) F,r (#) G are_fiberwise_equipotent ) ::_thesis: ( r (#) F,r (#) G are_fiberwise_equipotent implies F,G are_fiberwise_equipotent ) proof assume A3: F,G are_fiberwise_equipotent ; ::_thesis: r (#) F,r (#) G are_fiberwise_equipotent now__::_thesis:_for_x_being_Real_holds_card_(Coim_((r_(#)_F),x))_=_card_(Coim_((r_(#)_G),x)) let x be Real; ::_thesis: card (Coim ((r (#) F),x)) = card (Coim ((r (#) G),x)) ( Coim (F,(x / r)) = Coim ((r (#) F),x) & Coim (G,(x / r)) = Coim ((r (#) G),x) ) by A1, Th6; hence card (Coim ((r (#) F),x)) = card (Coim ((r (#) G),x)) by A3, CLASSES1:def_9; ::_thesis: verum end; hence r (#) F,r (#) G are_fiberwise_equipotent by A2, CLASSES1:79; ::_thesis: verum end; assume A4: r (#) F,r (#) G are_fiberwise_equipotent ; ::_thesis: F,G are_fiberwise_equipotent A5: now__::_thesis:_for_x_being_Real_holds_card_(Coim_(F,x))_=_card_(Coim_(G,x)) let x be Real; ::_thesis: card (Coim (F,x)) = card (Coim (G,x)) A6: G " {((r * x) / r)} = Coim ((r (#) G),(r * x)) by A1, Th6; ( (r * x) / r = x & F " {((r * x) / r)} = Coim ((r (#) F),(r * x)) ) by A1, Th6, XCMPLX_1:89; hence card (Coim (F,x)) = card (Coim (G,x)) by A4, A6, CLASSES1:def_9; ::_thesis: verum end; ( rng F c= REAL & rng G c= REAL ) ; hence F,G are_fiberwise_equipotent by A5, CLASSES1:79; ::_thesis: verum end; theorem :: RFUNCT_3:12 for D, C being non empty set for F being PartFunc of D,REAL for G being PartFunc of C,REAL holds ( F,G are_fiberwise_equipotent iff - F, - G are_fiberwise_equipotent ) proof let D, C be non empty set ; ::_thesis: for F being PartFunc of D,REAL for G being PartFunc of C,REAL holds ( F,G are_fiberwise_equipotent iff - F, - G are_fiberwise_equipotent ) let F be PartFunc of D,REAL; ::_thesis: for G being PartFunc of C,REAL holds ( F,G are_fiberwise_equipotent iff - F, - G are_fiberwise_equipotent ) let G be PartFunc of C,REAL; ::_thesis: ( F,G are_fiberwise_equipotent iff - F, - G are_fiberwise_equipotent ) - F = (- 1) (#) F ; hence ( F,G are_fiberwise_equipotent iff - F, - G are_fiberwise_equipotent ) by Th11; ::_thesis: verum end; theorem :: RFUNCT_3:13 for D, C being non empty set for F being PartFunc of D,REAL for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds abs F, abs G are_fiberwise_equipotent proof let D, C be non empty set ; ::_thesis: for F being PartFunc of D,REAL for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds abs F, abs G are_fiberwise_equipotent let F be PartFunc of D,REAL; ::_thesis: for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds abs F, abs G are_fiberwise_equipotent let G be PartFunc of C,REAL; ::_thesis: ( F,G are_fiberwise_equipotent implies abs F, abs G are_fiberwise_equipotent ) assume A1: F,G are_fiberwise_equipotent ; ::_thesis: abs F, abs G are_fiberwise_equipotent A2: now__::_thesis:_for_r_being_Real_holds_card_(Coim_((abs_F),r))_=_card_(Coim_((abs_G),r)) let r be Real; ::_thesis: card (Coim ((abs F),r)) = card (Coim ((abs G),r)) now__::_thesis:_(_(_0_<_r_&_card_((abs_F)_"_{r})_=_card_((abs_G)_"_{r})_)_or_(_0_=_r_&_card_((abs_F)_"_{r})_=_card_((abs_G)_"_{r})_)_or_(_r_<_0_&_card_((abs_F)_"_{r})_=_card_((abs_G)_"_{r})_)_) percases ( 0 < r or 0 = r or r < 0 ) ; case 0 < r ; ::_thesis: card ((abs F) " {r}) = card ((abs G) " {r}) then ( (abs F) " {r} = F " {(- r),r} & (abs G) " {r} = G " {(- r),r} ) by Th8; hence card ((abs F) " {r}) = card ((abs G) " {r}) by A1, CLASSES1:78; ::_thesis: verum end; case 0 = r ; ::_thesis: card ((abs F) " {r}) = card ((abs G) " {r}) then ( (abs F) " {r} = F " {r} & (abs G) " {r} = G " {r} ) by Th9; hence card ((abs F) " {r}) = card ((abs G) " {r}) by A1, CLASSES1:78; ::_thesis: verum end; caseA3: r < 0 ; ::_thesis: card ((abs F) " {r}) = card ((abs G) " {r}) then (abs F) " {r} = {} by Th10; hence card ((abs F) " {r}) = card ((abs G) " {r}) by A3, Th10; ::_thesis: verum end; end; end; hence card (Coim ((abs F),r)) = card (Coim ((abs G),r)) ; ::_thesis: verum end; ( rng (abs F) c= REAL & rng (abs G) c= REAL ) ; hence abs F, abs G are_fiberwise_equipotent by A2, CLASSES1:79; ::_thesis: verum end; definition let X, Y be set ; mode PartFunc-set of X,Y -> set means :Def3: :: RFUNCT_3:def 3 for x being Element of it holds x is PartFunc of X,Y; existence ex b1 being set st for x being Element of b1 holds x is PartFunc of X,Y proof reconsider h = {} as PartFunc of X,Y by RELSET_1:12; take {h} ; ::_thesis: for x being Element of {h} holds x is PartFunc of X,Y thus for x being Element of {h} holds x is PartFunc of X,Y by TARSKI:def_1; ::_thesis: verum end; end; :: deftheorem Def3 defines PartFunc-set RFUNCT_3:def_3_:_ for X, Y being set for b3 being set holds ( b3 is PartFunc-set of X,Y iff for x being Element of b3 holds x is PartFunc of X,Y ); registration let X, Y be set ; cluster non empty for PartFunc-set of X,Y; existence not for b1 being PartFunc-set of X,Y holds b1 is empty proof reconsider h = {} as PartFunc of X,Y by RELSET_1:12; {h} is PartFunc-set of X,Y proof let x be Element of {h}; :: according to RFUNCT_3:def_3 ::_thesis: x is PartFunc of X,Y thus x is PartFunc of X,Y by TARSKI:def_1; ::_thesis: verum end; hence not for b1 being PartFunc-set of X,Y holds b1 is empty ; ::_thesis: verum end; end; definition let X, Y be set ; mode PFUNC_DOMAIN of X,Y is non empty PartFunc-set of X,Y; end; definition let X, Y be set ; :: original: PFuncs redefine func PFuncs (X,Y) -> PartFunc-set of X,Y; coherence PFuncs (X,Y) is PartFunc-set of X,Y proof for x being Element of PFuncs (X,Y) holds x is PartFunc of X,Y by PARTFUN1:47; hence PFuncs (X,Y) is PartFunc-set of X,Y by Def3; ::_thesis: verum end; let P be non empty PartFunc-set of X,Y; :: original: Element redefine mode Element of P -> PartFunc of X,Y; coherence for b1 being Element of P holds b1 is PartFunc of X,Y by Def3; end; definition let D, C be non empty set ; let X be Subset of D; let c be Element of C; :: original: --> redefine funcX --> c -> Element of PFuncs (D,C); coherence X --> c is Element of PFuncs (D,C) proof X --> c is PartFunc of D,C ; hence X --> c is Element of PFuncs (D,C) by PARTFUN1:45; ::_thesis: verum end; end; registration let D be non empty set ; let E be real-membered set ; cluster -> real-valued for Element of PFuncs (D,E); coherence for b1 being Element of PFuncs (D,E) holds b1 is real-valued ; end; definition let D be non empty set ; let E be real-membered set ; let F1, F2 be Element of PFuncs (D,E); :: original: + redefine funcF1 + F2 -> Element of PFuncs (D,REAL); coherence F1 + F2 is Element of PFuncs (D,REAL) proof reconsider F1 = F1, F2 = F2 as PartFunc of D,E ; F1 + F2 is PartFunc of D,REAL ; hence F1 + F2 is Element of PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum end; :: original: - redefine funcF1 - F2 -> Element of PFuncs (D,REAL); coherence F1 - F2 is Element of PFuncs (D,REAL) proof reconsider F1 = F1, F2 = F2 as PartFunc of D,E ; F1 - F2 is PartFunc of D,REAL ; hence F1 - F2 is Element of PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum end; :: original: (#) redefine funcF1 (#) F2 -> Element of PFuncs (D,REAL); coherence F1 (#) F2 is Element of PFuncs (D,REAL) proof reconsider F1 = F1, F2 = F2 as PartFunc of D,E ; F1 (#) F2 is PartFunc of D,REAL ; hence F1 (#) F2 is Element of PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum end; :: original: / redefine funcF1 / F2 -> Element of PFuncs (D,REAL); coherence F1 / F2 is Element of PFuncs (D,REAL) proof reconsider F1 = F1, F2 = F2 as PartFunc of D,E ; F1 / F2 is PartFunc of D,REAL ; hence F1 / F2 is Element of PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum end; end; definition let D be non empty set ; let E be real-membered set ; let F be Element of PFuncs (D,E); :: original: |. redefine func abs F -> Element of PFuncs (D,REAL); coherence |.F.| is Element of PFuncs (D,REAL) proof reconsider F = F as PartFunc of D,E ; abs F is PartFunc of D,REAL ; hence |.F.| is Element of PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum end; :: original: - redefine func - F -> Element of PFuncs (D,REAL); coherence - F is Element of PFuncs (D,REAL) proof reconsider F = F as PartFunc of D,E ; - F is PartFunc of D,REAL ; hence - F is Element of PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum end; :: original: ^ redefine funcF ^ -> Element of PFuncs (D,REAL); coherence F ^ is Element of PFuncs (D,REAL) proof reconsider F = F as PartFunc of D,E ; F ^ is PartFunc of D,REAL ; hence F ^ is Element of PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum end; end; definition let D be non empty set ; let E be real-membered set ; let F be Element of PFuncs (D,E); let r be real number ; :: original: (#) redefine funcr (#) F -> Element of PFuncs (D,REAL); coherence r (#) F is Element of PFuncs (D,REAL) proof reconsider F = F as PartFunc of D,E ; r (#) F is PartFunc of D,REAL ; hence r (#) F is Element of PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum end; end; definition let D be non empty set ; func addpfunc D -> BinOp of (PFuncs (D,REAL)) means :Def4: :: RFUNCT_3:def 4 for F1, F2 being Element of PFuncs (D,REAL) holds it . (F1,F2) = F1 + F2; existence ex b1 being BinOp of (PFuncs (D,REAL)) st for F1, F2 being Element of PFuncs (D,REAL) holds b1 . (F1,F2) = F1 + F2 proof deffunc H1( Element of PFuncs (D,REAL), Element of PFuncs (D,REAL)) -> Element of PFuncs (D,REAL) = $1 + $2; ex o being BinOp of (PFuncs (D,REAL)) st for a, b being Element of PFuncs (D,REAL) holds o . (a,b) = H1(a,b) from BINOP_1:sch_4(); hence ex b1 being BinOp of (PFuncs (D,REAL)) st for F1, F2 being Element of PFuncs (D,REAL) holds b1 . (F1,F2) = F1 + F2 ; ::_thesis: verum end; uniqueness for b1, b2 being BinOp of (PFuncs (D,REAL)) st ( for F1, F2 being Element of PFuncs (D,REAL) holds b1 . (F1,F2) = F1 + F2 ) & ( for F1, F2 being Element of PFuncs (D,REAL) holds b2 . (F1,F2) = F1 + F2 ) holds b1 = b2 proof let o1, o2 be BinOp of (PFuncs (D,REAL)); ::_thesis: ( ( for F1, F2 being Element of PFuncs (D,REAL) holds o1 . (F1,F2) = F1 + F2 ) & ( for F1, F2 being Element of PFuncs (D,REAL) holds o2 . (F1,F2) = F1 + F2 ) implies o1 = o2 ) assume that A1: for f1, f2 being Element of PFuncs (D,REAL) holds o1 . (f1,f2) = f1 + f2 and A2: for f1, f2 being Element of PFuncs (D,REAL) holds o2 . (f1,f2) = f1 + f2 ; ::_thesis: o1 = o2 now__::_thesis:_for_f1,_f2_being_Element_of_PFuncs_(D,REAL)_holds_o1_._(f1,f2)_=_o2_._(f1,f2) let f1, f2 be Element of PFuncs (D,REAL); ::_thesis: o1 . (f1,f2) = o2 . (f1,f2) o1 . (f1,f2) = f1 + f2 by A1; hence o1 . (f1,f2) = o2 . (f1,f2) by A2; ::_thesis: verum end; hence o1 = o2 by BINOP_1:2; ::_thesis: verum end; end; :: deftheorem Def4 defines addpfunc RFUNCT_3:def_4_:_ for D being non empty set for b2 being BinOp of (PFuncs (D,REAL)) holds ( b2 = addpfunc D iff for F1, F2 being Element of PFuncs (D,REAL) holds b2 . (F1,F2) = F1 + F2 ); theorem Th14: :: RFUNCT_3:14 for D being non empty set holds addpfunc D is commutative proof let D be non empty set ; ::_thesis: addpfunc D is commutative let F1, F2 be Element of PFuncs (D,REAL); :: according to BINOP_1:def_2 ::_thesis: (addpfunc D) . (F1,F2) = (addpfunc D) . (F2,F1) set o = addpfunc D; thus (addpfunc D) . (F1,F2) = F2 + F1 by Def4 .= (addpfunc D) . (F2,F1) by Def4 ; ::_thesis: verum end; theorem Th15: :: RFUNCT_3:15 for D being non empty set holds addpfunc D is associative proof let D be non empty set ; ::_thesis: addpfunc D is associative let F1, F2, F3 be Element of PFuncs (D,REAL); :: according to BINOP_1:def_3 ::_thesis: (addpfunc D) . (F1,((addpfunc D) . (F2,F3))) = (addpfunc D) . (((addpfunc D) . (F1,F2)),F3) set o = addpfunc D; thus (addpfunc D) . (F1,((addpfunc D) . (F2,F3))) = (addpfunc D) . (F1,(F2 + F3)) by Def4 .= F1 + (F2 + F3) by Def4 .= (F1 + F2) + F3 by RFUNCT_1:8 .= ((addpfunc D) . (F1,F2)) + F3 by Def4 .= (addpfunc D) . (((addpfunc D) . (F1,F2)),F3) by Def4 ; ::_thesis: verum end; theorem Th16: :: RFUNCT_3:16 for D being non empty set holds ([#] D) --> 0 is_a_unity_wrt addpfunc D proof let D be non empty set ; ::_thesis: ([#] D) --> 0 is_a_unity_wrt addpfunc D set F = ([#] D) --> 0; A1: dom (([#] D) --> 0) = D by FUNCOP_1:13; A2: now__::_thesis:_for_G_being_Element_of_PFuncs_(D,REAL)_holds_(addpfunc_D)_._(G,(([#]_D)_-->_0))_=_G let G be Element of PFuncs (D,REAL); ::_thesis: (addpfunc D) . (G,(([#] D) --> 0)) = G A3: now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_(G_+_(([#]_D)_-->_0))_holds_ (G_+_(([#]_D)_-->_0))_._d_=_G_._d let d be Element of D; ::_thesis: ( d in dom (G + (([#] D) --> 0)) implies (G + (([#] D) --> 0)) . d = G . d ) assume d in dom (G + (([#] D) --> 0)) ; ::_thesis: (G + (([#] D) --> 0)) . d = G . d hence (G + (([#] D) --> 0)) . d = (G . d) + ((([#] D) --> 0) . d) by VALUED_1:def_1 .= (G . d) + 0 by FUNCOP_1:7 .= G . d ; ::_thesis: verum end; (dom G) /\ D = dom G by XBOOLE_1:28; then dom (G + (([#] D) --> 0)) = dom G by A1, VALUED_1:def_1; then G + (([#] D) --> 0) = G by A3, PARTFUN1:5; hence (addpfunc D) . (G,(([#] D) --> 0)) = G by Def4; ::_thesis: verum end; addpfunc D is commutative by Th14; hence ([#] D) --> 0 is_a_unity_wrt addpfunc D by A2, BINOP_1:5; ::_thesis: verum end; theorem Th17: :: RFUNCT_3:17 for D being non empty set holds the_unity_wrt (addpfunc D) = ([#] D) --> 0 proof let D be non empty set ; ::_thesis: the_unity_wrt (addpfunc D) = ([#] D) --> 0 ([#] D) --> 0 is_a_unity_wrt addpfunc D by Th16; hence the_unity_wrt (addpfunc D) = ([#] D) --> 0 by BINOP_1:def_8; ::_thesis: verum end; theorem Th18: :: RFUNCT_3:18 for D being non empty set holds addpfunc D is having_a_unity proof let D be non empty set ; ::_thesis: addpfunc D is having_a_unity take ([#] D) --> 0 ; :: according to SETWISEO:def_2 ::_thesis: ([#] D) --> 0 is_a_unity_wrt addpfunc D thus ([#] D) --> 0 is_a_unity_wrt addpfunc D by Th16; ::_thesis: verum end; definition let D be non empty set ; let f be FinSequence of PFuncs (D,REAL); func Sum f -> Element of PFuncs (D,REAL) equals :: RFUNCT_3:def 5 (addpfunc D) $$ f; correctness coherence (addpfunc D) $$ f is Element of PFuncs (D,REAL); ; end; :: deftheorem defines Sum RFUNCT_3:def_5_:_ for D being non empty set for f being FinSequence of PFuncs (D,REAL) holds Sum f = (addpfunc D) $$ f; theorem Th19: :: RFUNCT_3:19 for D being non empty set holds Sum (<*> (PFuncs (D,REAL))) = ([#] D) --> 0 proof let D be non empty set ; ::_thesis: Sum (<*> (PFuncs (D,REAL))) = ([#] D) --> 0 set o = addpfunc D; set o0 = ([#] D) --> 0; the_unity_wrt (addpfunc D) = ([#] D) --> 0 by Th17; hence Sum (<*> (PFuncs (D,REAL))) = ([#] D) --> 0 by Th18, FINSOP_1:10; ::_thesis: verum end; theorem Th20: :: RFUNCT_3:20 for D being non empty set for f being FinSequence of PFuncs (D,REAL) for G being Element of PFuncs (D,REAL) holds Sum (f ^ <*G*>) = (Sum f) + G proof let D be non empty set ; ::_thesis: for f being FinSequence of PFuncs (D,REAL) for G being Element of PFuncs (D,REAL) holds Sum (f ^ <*G*>) = (Sum f) + G let f be FinSequence of PFuncs (D,REAL); ::_thesis: for G being Element of PFuncs (D,REAL) holds Sum (f ^ <*G*>) = (Sum f) + G let G be Element of PFuncs (D,REAL); ::_thesis: Sum (f ^ <*G*>) = (Sum f) + G set o = addpfunc D; thus Sum (f ^ <*G*>) = (addpfunc D) . (((addpfunc D) $$ f),G) by Th18, FINSOP_1:4 .= (Sum f) + G by Def4 ; ::_thesis: verum end; theorem Th21: :: RFUNCT_3:21 for D being non empty set for f1, f2 being FinSequence of PFuncs (D,REAL) holds Sum (f1 ^ f2) = (Sum f1) + (Sum f2) proof let D be non empty set ; ::_thesis: for f1, f2 being FinSequence of PFuncs (D,REAL) holds Sum (f1 ^ f2) = (Sum f1) + (Sum f2) let f1, f2 be FinSequence of PFuncs (D,REAL); ::_thesis: Sum (f1 ^ f2) = (Sum f1) + (Sum f2) set o = addpfunc D; addpfunc D is associative by Th15; hence Sum (f1 ^ f2) = (addpfunc D) . ((Sum f1),(Sum f2)) by Th18, FINSOP_1:5 .= (Sum f1) + (Sum f2) by Def4 ; ::_thesis: verum end; theorem :: RFUNCT_3:22 for D being non empty set for f being FinSequence of PFuncs (D,REAL) for G being Element of PFuncs (D,REAL) holds Sum (<*G*> ^ f) = G + (Sum f) proof let D be non empty set ; ::_thesis: for f being FinSequence of PFuncs (D,REAL) for G being Element of PFuncs (D,REAL) holds Sum (<*G*> ^ f) = G + (Sum f) let f be FinSequence of PFuncs (D,REAL); ::_thesis: for G being Element of PFuncs (D,REAL) holds Sum (<*G*> ^ f) = G + (Sum f) let G be Element of PFuncs (D,REAL); ::_thesis: Sum (<*G*> ^ f) = G + (Sum f) thus Sum (<*G*> ^ f) = (Sum <*G*>) + (Sum f) by Th21 .= G + (Sum f) by FINSOP_1:11 ; ::_thesis: verum end; theorem Th23: :: RFUNCT_3:23 for D being non empty set for G1, G2 being Element of PFuncs (D,REAL) holds Sum <*G1,G2*> = G1 + G2 proof let D be non empty set ; ::_thesis: for G1, G2 being Element of PFuncs (D,REAL) holds Sum <*G1,G2*> = G1 + G2 let G1, G2 be Element of PFuncs (D,REAL); ::_thesis: Sum <*G1,G2*> = G1 + G2 thus Sum <*G1,G2*> = Sum (<*G1*> ^ <*G2*>) by FINSEQ_1:def_9 .= (Sum <*G1*>) + G2 by Th20 .= G1 + G2 by FINSOP_1:11 ; ::_thesis: verum end; theorem :: RFUNCT_3:24 for D being non empty set for G1, G2, G3 being Element of PFuncs (D,REAL) holds Sum <*G1,G2,G3*> = (G1 + G2) + G3 proof let D be non empty set ; ::_thesis: for G1, G2, G3 being Element of PFuncs (D,REAL) holds Sum <*G1,G2,G3*> = (G1 + G2) + G3 let G1, G2, G3 be Element of PFuncs (D,REAL); ::_thesis: Sum <*G1,G2,G3*> = (G1 + G2) + G3 thus Sum <*G1,G2,G3*> = Sum (<*G1,G2*> ^ <*G3*>) by FINSEQ_1:43 .= (Sum <*G1,G2*>) + G3 by Th20 .= (G1 + G2) + G3 by Th23 ; ::_thesis: verum end; theorem :: RFUNCT_3:25 for D being non empty set for f, g being FinSequence of PFuncs (D,REAL) st f,g are_fiberwise_equipotent holds Sum f = Sum g proof let D be non empty set ; ::_thesis: for f, g being FinSequence of PFuncs (D,REAL) st f,g are_fiberwise_equipotent holds Sum f = Sum g defpred S1[ Element of NAT ] means for f, g being FinSequence of PFuncs (D,REAL) st f,g are_fiberwise_equipotent & len f = $1 holds Sum f = Sum g; let f, g be FinSequence of PFuncs (D,REAL); ::_thesis: ( f,g are_fiberwise_equipotent implies Sum f = Sum g ) assume A1: f,g are_fiberwise_equipotent ; ::_thesis: Sum f = Sum g A2: len f = len f ; A3: for n being Element of NAT st S1[n] holds S1[n + 1] proof let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A4: S1[n] ; ::_thesis: S1[n + 1] let f, g be FinSequence of PFuncs (D,REAL); ::_thesis: ( f,g are_fiberwise_equipotent & len f = n + 1 implies Sum f = Sum g ) assume that A5: f,g are_fiberwise_equipotent and A6: len f = n + 1 ; ::_thesis: Sum f = Sum g 0 + 1 <= n + 1 by NAT_1:13; then A7: n + 1 in dom f by A6, FINSEQ_3:25; then reconsider a = f . (n + 1) as Element of PFuncs (D,REAL) by FINSEQ_2:11; rng f = rng g by A5, CLASSES1:75; then a in rng g by A7, FUNCT_1:def_3; then consider m being Nat such that A8: m in dom g and A9: g . m = a by FINSEQ_2:10; A10: g = (g | m) ^ (g /^ m) by RFINSEQ:8; set gg = g /^ m; set gm = g | m; m <= len g by A8, FINSEQ_3:25; then A11: len (g | m) = m by FINSEQ_1:59; set fn = f | n; A12: f = (f | n) ^ <*a*> by A6, RFINSEQ:7; A13: 1 <= m by A8, FINSEQ_3:25; then max (0,(m - 1)) = m - 1 by FINSEQ_2:4; then reconsider m1 = m - 1 as Element of NAT by FINSEQ_2:5; A14: m = m1 + 1 ; then m1 <= m by NAT_1:11; then A15: Seg m1 c= Seg m by FINSEQ_1:5; m in Seg m by A13, FINSEQ_1:1; then (g | m) . m = a by A8, A9, RFINSEQ:6; then A16: g | m = ((g | m) | m1) ^ <*a*> by A11, A14, RFINSEQ:7; A17: (g | m) | m1 = (g | m) | (Seg m1) by FINSEQ_1:def_15 .= (g | (Seg m)) | (Seg m1) by FINSEQ_1:def_15 .= g | ((Seg m) /\ (Seg m1)) by RELAT_1:71 .= g | (Seg m1) by A15, XBOOLE_1:28 .= g | m1 by FINSEQ_1:def_15 ; now__::_thesis:_for_x_being_set_holds_card_(Coim_((f_|_n),x))_=_card_(Coim_(((g_|_m1)_^_(g_/^_m)),x)) let x be set ; ::_thesis: card (Coim ((f | n),x)) = card (Coim (((g | m1) ^ (g /^ m)),x)) card (Coim (f,x)) = card (Coim (g,x)) by A5, CLASSES1:def_9; then (card ((f | n) " {x})) + (card (<*a*> " {x})) = card ((((g | m1) ^ <*a*>) ^ (g /^ m)) " {x}) by A10, A16, A17, A12, FINSEQ_3:57 .= (card (((g | m1) ^ <*a*>) " {x})) + (card ((g /^ m) " {x})) by FINSEQ_3:57 .= ((card ((g | m1) " {x})) + (card (<*a*> " {x}))) + (card ((g /^ m) " {x})) by FINSEQ_3:57 .= ((card ((g | m1) " {x})) + (card ((g /^ m) " {x}))) + (card (<*a*> " {x})) .= (card (((g | m1) ^ (g /^ m)) " {x})) + (card (<*a*> " {x})) by FINSEQ_3:57 ; hence card (Coim ((f | n),x)) = card (Coim (((g | m1) ^ (g /^ m)),x)) ; ::_thesis: verum end; then A18: f | n,(g | m1) ^ (g /^ m) are_fiberwise_equipotent by CLASSES1:def_9; len (f | n) = n by A6, FINSEQ_1:59, NAT_1:11; then Sum (f | n) = Sum ((g | m1) ^ (g /^ m)) by A4, A18; hence Sum f = (Sum ((g | m1) ^ (g /^ m))) + (Sum <*a*>) by A12, Th21 .= ((Sum (g | m1)) + (Sum (g /^ m))) + (Sum <*a*>) by Th21 .= ((Sum (g | m1)) + (Sum <*a*>)) + (Sum (g /^ m)) by RFUNCT_1:8 .= (Sum (g | m)) + (Sum (g /^ m)) by A16, A17, Th21 .= Sum g by A10, Th21 ; ::_thesis: verum end; A19: S1[ 0 ] proof let f, g be FinSequence of PFuncs (D,REAL); ::_thesis: ( f,g are_fiberwise_equipotent & len f = 0 implies Sum f = Sum g ) assume ( f,g are_fiberwise_equipotent & len f = 0 ) ; ::_thesis: Sum f = Sum g then A20: ( len g = 0 & f = <*> (PFuncs (D,REAL)) ) by RFINSEQ:3; then g = <*> (PFuncs (D,REAL)) ; hence Sum f = Sum g by A20; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A19, A3); hence Sum f = Sum g by A1, A2; ::_thesis: verum end; definition let D be non empty set ; let f be FinSequence; func CHI (f,D) -> FinSequence of PFuncs (D,REAL) means :Def6: :: RFUNCT_3:def 6 ( len it = len f & ( for n being Element of NAT st n in dom it holds it . n = chi ((f . n),D) ) ); existence ex b1 being FinSequence of PFuncs (D,REAL) st ( len b1 = len f & ( for n being Element of NAT st n in dom b1 holds b1 . n = chi ((f . n),D) ) ) proof deffunc H1( Nat) -> Element of K19(K20(D,REAL)) = chi ((f . $1),D); consider p being FinSequence such that A1: len p = len f and A2: for n being Nat st n in dom p holds p . n = H1(n) from FINSEQ_1:sch_2(); rng p c= PFuncs (D,REAL) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng p or x in PFuncs (D,REAL) ) assume x in rng p ; ::_thesis: x in PFuncs (D,REAL) then consider n being Nat such that A3: ( n in dom p & p . n = x ) by FINSEQ_2:10; x = chi ((f . n),D) by A2, A3; hence x in PFuncs (D,REAL) by PARTFUN1:45; ::_thesis: verum end; then reconsider p = p as FinSequence of PFuncs (D,REAL) by FINSEQ_1:def_4; take p ; ::_thesis: ( len p = len f & ( for n being Element of NAT st n in dom p holds p . n = chi ((f . n),D) ) ) thus len p = len f by A1; ::_thesis: for n being Element of NAT st n in dom p holds p . n = chi ((f . n),D) let n be Element of NAT ; ::_thesis: ( n in dom p implies p . n = chi ((f . n),D) ) assume n in dom p ; ::_thesis: p . n = chi ((f . n),D) hence p . n = chi ((f . n),D) by A2; ::_thesis: verum end; uniqueness for b1, b2 being FinSequence of PFuncs (D,REAL) st len b1 = len f & ( for n being Element of NAT st n in dom b1 holds b1 . n = chi ((f . n),D) ) & len b2 = len f & ( for n being Element of NAT st n in dom b2 holds b2 . n = chi ((f . n),D) ) holds b1 = b2 proof let p1, p2 be FinSequence of PFuncs (D,REAL); ::_thesis: ( len p1 = len f & ( for n being Element of NAT st n in dom p1 holds p1 . n = chi ((f . n),D) ) & len p2 = len f & ( for n being Element of NAT st n in dom p2 holds p2 . n = chi ((f . n),D) ) implies p1 = p2 ) assume that A4: len p1 = len f and A5: for n being Element of NAT st n in dom p1 holds p1 . n = chi ((f . n),D) and A6: len p2 = len f and A7: for n being Element of NAT st n in dom p2 holds p2 . n = chi ((f . n),D) ; ::_thesis: p1 = p2 A8: ( dom p1 = Seg (len p1) & dom p2 = Seg (len p2) ) by FINSEQ_1:def_3; now__::_thesis:_for_n_being_Nat_st_n_in_dom_p1_holds_ p1_._n_=_p2_._n let n be Nat; ::_thesis: ( n in dom p1 implies p1 . n = p2 . n ) assume A9: n in dom p1 ; ::_thesis: p1 . n = p2 . n then p1 . n = chi ((f . n),D) by A5; hence p1 . n = p2 . n by A4, A6, A7, A8, A9; ::_thesis: verum end; hence p1 = p2 by A4, A6, FINSEQ_2:9; ::_thesis: verum end; end; :: deftheorem Def6 defines CHI RFUNCT_3:def_6_:_ for D being non empty set for f being FinSequence for b3 being FinSequence of PFuncs (D,REAL) holds ( b3 = CHI (f,D) iff ( len b3 = len f & ( for n being Element of NAT st n in dom b3 holds b3 . n = chi ((f . n),D) ) ) ); definition let D be non empty set ; let f be FinSequence of PFuncs (D,REAL); let R be FinSequence of REAL ; funcR (#) f -> FinSequence of PFuncs (D,REAL) means :Def7: :: RFUNCT_3:def 7 ( len it = min ((len R),(len f)) & ( for n being Element of NAT st n in dom it holds for F being PartFunc of D,REAL for r being Real st r = R . n & F = f . n holds it . n = r (#) F ) ); existence ex b1 being FinSequence of PFuncs (D,REAL) st ( len b1 = min ((len R),(len f)) & ( for n being Element of NAT st n in dom b1 holds for F being PartFunc of D,REAL for r being Real st r = R . n & F = f . n holds b1 . n = r (#) F ) ) proof defpred S1[ Nat, set ] means for F being PartFunc of D,REAL for r being Real st r = R . $1 & F = f . $1 holds $2 = r (#) F; set m = min ((len R),(len f)); A1: min ((len R),(len f)) <= len f by XXREAL_0:17; A2: for n being Nat st n in Seg (min ((len R),(len f))) holds ex x being Element of PFuncs (D,REAL) st S1[n,x] proof let n be Nat; ::_thesis: ( n in Seg (min ((len R),(len f))) implies ex x being Element of PFuncs (D,REAL) st S1[n,x] ) reconsider r = R . n as Real ; assume A3: n in Seg (min ((len R),(len f))) ; ::_thesis: ex x being Element of PFuncs (D,REAL) st S1[n,x] then n <= min ((len R),(len f)) by FINSEQ_1:1; then A4: n <= len f by A1, XXREAL_0:2; 1 <= n by A3, FINSEQ_1:1; then n in dom f by A4, FINSEQ_3:25; then reconsider F = f . n as Element of PFuncs (D,REAL) by FINSEQ_2:11; reconsider a = r (#) F as Element of PFuncs (D,REAL) ; take a ; ::_thesis: S1[n,a] thus S1[n,a] ; ::_thesis: verum end; consider p being FinSequence of PFuncs (D,REAL) such that A5: ( dom p = Seg (min ((len R),(len f))) & ( for n being Nat st n in Seg (min ((len R),(len f))) holds S1[n,p . n] ) ) from FINSEQ_1:sch_5(A2); take p ; ::_thesis: ( len p = min ((len R),(len f)) & ( for n being Element of NAT st n in dom p holds for F being PartFunc of D,REAL for r being Real st r = R . n & F = f . n holds p . n = r (#) F ) ) thus len p = min ((len R),(len f)) by A5, FINSEQ_1:def_3; ::_thesis: for n being Element of NAT st n in dom p holds for F being PartFunc of D,REAL for r being Real st r = R . n & F = f . n holds p . n = r (#) F let n be Element of NAT ; ::_thesis: ( n in dom p implies for F being PartFunc of D,REAL for r being Real st r = R . n & F = f . n holds p . n = r (#) F ) assume n in dom p ; ::_thesis: for F being PartFunc of D,REAL for r being Real st r = R . n & F = f . n holds p . n = r (#) F hence for F being PartFunc of D,REAL for r being Real st r = R . n & F = f . n holds p . n = r (#) F by A5; ::_thesis: verum end; uniqueness for b1, b2 being FinSequence of PFuncs (D,REAL) st len b1 = min ((len R),(len f)) & ( for n being Element of NAT st n in dom b1 holds for F being PartFunc of D,REAL for r being Real st r = R . n & F = f . n holds b1 . n = r (#) F ) & len b2 = min ((len R),(len f)) & ( for n being Element of NAT st n in dom b2 holds for F being PartFunc of D,REAL for r being Real st r = R . n & F = f . n holds b2 . n = r (#) F ) holds b1 = b2 proof set m = min ((len R),(len f)); let p1, p2 be FinSequence of PFuncs (D,REAL); ::_thesis: ( len p1 = min ((len R),(len f)) & ( for n being Element of NAT st n in dom p1 holds for F being PartFunc of D,REAL for r being Real st r = R . n & F = f . n holds p1 . n = r (#) F ) & len p2 = min ((len R),(len f)) & ( for n being Element of NAT st n in dom p2 holds for F being PartFunc of D,REAL for r being Real st r = R . n & F = f . n holds p2 . n = r (#) F ) implies p1 = p2 ) assume that A6: len p1 = min ((len R),(len f)) and A7: for n being Element of NAT st n in dom p1 holds for F being PartFunc of D,REAL for r being Real st r = R . n & F = f . n holds p1 . n = r (#) F and A8: len p2 = min ((len R),(len f)) and A9: for n being Element of NAT st n in dom p2 holds for F being PartFunc of D,REAL for r being Real st r = R . n & F = f . n holds p2 . n = r (#) F ; ::_thesis: p1 = p2 A10: dom p1 = Seg (min ((len R),(len f))) by A6, FINSEQ_1:def_3; A11: ( dom p1 = Seg (len p1) & dom p2 = Seg (len p2) ) by FINSEQ_1:def_3; A12: min ((len R),(len f)) <= len f by XXREAL_0:17; now__::_thesis:_for_n_being_Nat_st_n_in_dom_p1_holds_ p1_._n_=_p2_._n let n be Nat; ::_thesis: ( n in dom p1 implies p1 . n = p2 . n ) reconsider r = R . n as Real ; assume A13: n in dom p1 ; ::_thesis: p1 . n = p2 . n then n <= min ((len R),(len f)) by A10, FINSEQ_1:1; then A14: n <= len f by A12, XXREAL_0:2; 1 <= n by A10, A13, FINSEQ_1:1; then n in dom f by A14, FINSEQ_3:25; then reconsider F = f . n as Element of PFuncs (D,REAL) by FINSEQ_2:11; p1 . n = r (#) F by A7, A13; hence p1 . n = p2 . n by A6, A8, A9, A11, A13; ::_thesis: verum end; hence p1 = p2 by A6, A8, FINSEQ_2:9; ::_thesis: verum end; end; :: deftheorem Def7 defines (#) RFUNCT_3:def_7_:_ for D being non empty set for f being FinSequence of PFuncs (D,REAL) for R being FinSequence of REAL for b4 being FinSequence of PFuncs (D,REAL) holds ( b4 = R (#) f iff ( len b4 = min ((len R),(len f)) & ( for n being Element of NAT st n in dom b4 holds for F being PartFunc of D,REAL for r being Real st r = R . n & F = f . n holds b4 . n = r (#) F ) ) ); definition let D be non empty set ; let f be FinSequence of PFuncs (D,REAL); let d be Element of D; funcf # d -> FinSequence of REAL means :Def8: :: RFUNCT_3:def 8 ( len it = len f & ( for n being Element of NAT st n in dom it holds it . n = (f . n) . d ) ); existence ex b1 being FinSequence of REAL st ( len b1 = len f & ( for n being Element of NAT st n in dom b1 holds b1 . n = (f . n) . d ) ) proof defpred S1[ Nat, set ] means $2 = (f . $1) . d; A1: for n being Nat st n in Seg (len f) holds ex x being Element of REAL st S1[n,x] proof let n be Nat; ::_thesis: ( n in Seg (len f) implies ex x being Element of REAL st S1[n,x] ) assume n in Seg (len f) ; ::_thesis: ex x being Element of REAL st S1[n,x] then n in dom f by FINSEQ_1:def_3; then reconsider G = f . n as Element of PFuncs (D,REAL) by FINSEQ_2:11; take G . d ; ::_thesis: S1[n,G . d] thus S1[n,G . d] ; ::_thesis: verum end; consider p being FinSequence of REAL such that A2: dom p = Seg (len f) and A3: for n being Nat st n in Seg (len f) holds S1[n,p . n] from FINSEQ_1:sch_5(A1); take p ; ::_thesis: ( len p = len f & ( for n being Element of NAT st n in dom p holds p . n = (f . n) . d ) ) thus len p = len f by A2, FINSEQ_1:def_3; ::_thesis: for n being Element of NAT st n in dom p holds p . n = (f . n) . d thus for n being Element of NAT st n in dom p holds p . n = (f . n) . d by A2, A3; ::_thesis: verum end; uniqueness for b1, b2 being FinSequence of REAL st len b1 = len f & ( for n being Element of NAT st n in dom b1 holds b1 . n = (f . n) . d ) & len b2 = len f & ( for n being Element of NAT st n in dom b2 holds b2 . n = (f . n) . d ) holds b1 = b2 proof let p1, p2 be FinSequence of REAL ; ::_thesis: ( len p1 = len f & ( for n being Element of NAT st n in dom p1 holds p1 . n = (f . n) . d ) & len p2 = len f & ( for n being Element of NAT st n in dom p2 holds p2 . n = (f . n) . d ) implies p1 = p2 ) assume that A4: len p1 = len f and A5: for n being Element of NAT st n in dom p1 holds p1 . n = (f . n) . d and A6: len p2 = len f and A7: for n being Element of NAT st n in dom p2 holds p2 . n = (f . n) . d ; ::_thesis: p1 = p2 A8: dom p1 = Seg (len p1) by FINSEQ_1:def_3; A9: dom p2 = Seg (len p2) by FINSEQ_1:def_3; now__::_thesis:_for_n_being_Nat_st_n_in_dom_p1_holds_ p1_._n_=_p2_._n let n be Nat; ::_thesis: ( n in dom p1 implies p1 . n = p2 . n ) assume A10: n in dom p1 ; ::_thesis: p1 . n = p2 . n then p1 . n = (f . n) . d by A5; hence p1 . n = p2 . n by A4, A6, A7, A8, A9, A10; ::_thesis: verum end; hence p1 = p2 by A4, A6, FINSEQ_2:9; ::_thesis: verum end; end; :: deftheorem Def8 defines # RFUNCT_3:def_8_:_ for D being non empty set for f being FinSequence of PFuncs (D,REAL) for d being Element of D for b4 being FinSequence of REAL holds ( b4 = f # d iff ( len b4 = len f & ( for n being Element of NAT st n in dom b4 holds b4 . n = (f . n) . d ) ) ); definition let D, C be non empty set ; let f be FinSequence of PFuncs (D,C); let d be Element of D; predd is_common_for_dom f means :Def9: :: RFUNCT_3:def 9 for n being Element of NAT st n in dom f holds d in dom (f . n); end; :: deftheorem Def9 defines is_common_for_dom RFUNCT_3:def_9_:_ for D, C being non empty set for f being FinSequence of PFuncs (D,C) for d being Element of D holds ( d is_common_for_dom f iff for n being Element of NAT st n in dom f holds d in dom (f . n) ); theorem Th26: :: RFUNCT_3:26 for D, C being non empty set for f being FinSequence of PFuncs (D,C) for d being Element of D for n being Element of NAT st d is_common_for_dom f & n <> 0 holds d is_common_for_dom f | n proof let D1, D2 be non empty set ; ::_thesis: for f being FinSequence of PFuncs (D1,D2) for d being Element of D1 for n being Element of NAT st d is_common_for_dom f & n <> 0 holds d is_common_for_dom f | n let f be FinSequence of PFuncs (D1,D2); ::_thesis: for d being Element of D1 for n being Element of NAT st d is_common_for_dom f & n <> 0 holds d is_common_for_dom f | n let d1 be Element of D1; ::_thesis: for n being Element of NAT st d1 is_common_for_dom f & n <> 0 holds d1 is_common_for_dom f | n let n be Element of NAT ; ::_thesis: ( d1 is_common_for_dom f & n <> 0 implies d1 is_common_for_dom f | n ) assume that A1: d1 is_common_for_dom f and A2: n <> 0 ; ::_thesis: d1 is_common_for_dom f | n let m be Element of NAT ; :: according to RFUNCT_3:def_9 ::_thesis: ( m in dom (f | n) implies d1 in dom ((f | n) . m) ) assume A3: m in dom (f | n) ; ::_thesis: d1 in dom ((f | n) . m) set G = (f | n) . m; now__::_thesis:_(_(_n_>=_len_f_&_d1_in_dom_((f_|_n)_._m)_)_or_(_n_<_len_f_&_d1_in_dom_((f_|_n)_._m)_)_) percases ( n >= len f or n < len f ) ; case n >= len f ; ::_thesis: d1 in dom ((f | n) . m) then f | n = f by Lm1; hence d1 in dom ((f | n) . m) by A1, A3, Def9; ::_thesis: verum end; caseA4: n < len f ; ::_thesis: d1 in dom ((f | n) . m) 0 + 1 <= n by A2, NAT_1:13; then A5: n in dom f by A4, FINSEQ_3:25; ( dom (f | n) = Seg (len (f | n)) & len (f | n) = n ) by A4, FINSEQ_1:59, FINSEQ_1:def_3; then ( (f | n) . m = f . m & m in dom f ) by A3, A5, RFINSEQ:6; hence d1 in dom ((f | n) . m) by A1, Def9; ::_thesis: verum end; end; end; hence d1 in dom ((f | n) . m) ; ::_thesis: verum end; theorem :: RFUNCT_3:27 for D, C being non empty set for f being FinSequence of PFuncs (D,C) for d being Element of D for n being Element of NAT st d is_common_for_dom f holds d is_common_for_dom f /^ n proof let D1, D2 be non empty set ; ::_thesis: for f being FinSequence of PFuncs (D1,D2) for d being Element of D1 for n being Element of NAT st d is_common_for_dom f holds d is_common_for_dom f /^ n let f be FinSequence of PFuncs (D1,D2); ::_thesis: for d being Element of D1 for n being Element of NAT st d is_common_for_dom f holds d is_common_for_dom f /^ n let d1 be Element of D1; ::_thesis: for n being Element of NAT st d1 is_common_for_dom f holds d1 is_common_for_dom f /^ n let n be Element of NAT ; ::_thesis: ( d1 is_common_for_dom f implies d1 is_common_for_dom f /^ n ) assume A1: d1 is_common_for_dom f ; ::_thesis: d1 is_common_for_dom f /^ n let m be Element of NAT ; :: according to RFUNCT_3:def_9 ::_thesis: ( m in dom (f /^ n) implies d1 in dom ((f /^ n) . m) ) set fn = f /^ n; assume A2: m in dom (f /^ n) ; ::_thesis: d1 in dom ((f /^ n) . m) set G = (f /^ n) . m; now__::_thesis:_(_(_len_f_<_n_&_d1_in_dom_((f_/^_n)_._m)_)_or_(_n_<=_len_f_&_d1_in_dom_((f_/^_n)_._m)_)_) percases ( len f < n or n <= len f ) ; case len f < n ; ::_thesis: d1 in dom ((f /^ n) . m) hence d1 in dom ((f /^ n) . m) by A2, RELAT_1:38, RFINSEQ:def_1; ::_thesis: verum end; caseA3: n <= len f ; ::_thesis: d1 in dom ((f /^ n) . m) ( 1 <= m & m <= m + n ) by A2, FINSEQ_3:25, NAT_1:11; then A4: 1 <= m + n by XXREAL_0:2; A5: m <= len (f /^ n) by A2, FINSEQ_3:25; len (f /^ n) = (len f) - n by A3, RFINSEQ:def_1; then m + n <= len f by A5, XREAL_1:19; then A6: m + n in dom f by A4, FINSEQ_3:25; (f /^ n) . m = f . (m + n) by A2, A3, RFINSEQ:def_1; hence d1 in dom ((f /^ n) . m) by A1, A6, Def9; ::_thesis: verum end; end; end; hence d1 in dom ((f /^ n) . m) ; ::_thesis: verum end; theorem Th28: :: RFUNCT_3:28 for D being non empty set for d being Element of D for f being FinSequence of PFuncs (D,REAL) st len f <> 0 holds ( d is_common_for_dom f iff d in dom (Sum f) ) proof let D be non empty set ; ::_thesis: for d being Element of D for f being FinSequence of PFuncs (D,REAL) st len f <> 0 holds ( d is_common_for_dom f iff d in dom (Sum f) ) let d be Element of D; ::_thesis: for f being FinSequence of PFuncs (D,REAL) st len f <> 0 holds ( d is_common_for_dom f iff d in dom (Sum f) ) defpred S1[ Element of NAT ] means for f being FinSequence of PFuncs (D,REAL) st len f = $1 & len f <> 0 holds ( d is_common_for_dom f iff d in dom (Sum f) ); let f be FinSequence of PFuncs (D,REAL); ::_thesis: ( len f <> 0 implies ( d is_common_for_dom f iff d in dom (Sum f) ) ) assume A1: len f <> 0 ; ::_thesis: ( d is_common_for_dom f iff d in dom (Sum f) ) A2: for n being Element of NAT st S1[n] holds S1[n + 1] proof let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A3: S1[n] ; ::_thesis: S1[n + 1] let f be FinSequence of PFuncs (D,REAL); ::_thesis: ( len f = n + 1 & len f <> 0 implies ( d is_common_for_dom f iff d in dom (Sum f) ) ) assume that A4: len f = n + 1 and len f <> 0 ; ::_thesis: ( d is_common_for_dom f iff d in dom (Sum f) ) A5: dom f = Seg (len f) by FINSEQ_1:def_3; now__::_thesis:_(_(_n_=_0_&_(_d_is_common_for_dom_f_implies_d_in_dom_(Sum_f)_)_&_(_d_in_dom_(Sum_f)_implies_d_is_common_for_dom_f_)_)_or_(_n_<>_0_&_(_d_is_common_for_dom_f_implies_d_in_dom_(Sum_f)_)_&_(_d_in_dom_(Sum_f)_implies_d_is_common_for_dom_f_)_)_) percases ( n = 0 or n <> 0 ) ; caseA6: n = 0 ; ::_thesis: ( ( d is_common_for_dom f implies d in dom (Sum f) ) & ( d in dom (Sum f) implies d is_common_for_dom f ) ) then A7: 1 in dom f by A4, FINSEQ_3:25; then reconsider G = f . 1 as Element of PFuncs (D,REAL) by FINSEQ_2:11; f = <*G*> by A4, A6, FINSEQ_1:40; then A8: Sum f = G by FINSOP_1:11; hence ( d is_common_for_dom f implies d in dom (Sum f) ) by A7, Def9; ::_thesis: ( d in dom (Sum f) implies d is_common_for_dom f ) assume d in dom (Sum f) ; ::_thesis: d is_common_for_dom f then for m being Element of NAT st m in dom f holds d in dom (f . m) by A4, A5, A6, A8, FINSEQ_1:2, TARSKI:def_1; hence d is_common_for_dom f by Def9; ::_thesis: verum end; caseA9: n <> 0 ; ::_thesis: ( ( d is_common_for_dom f implies d in dom (Sum f) ) & ( d in dom (Sum f) implies d is_common_for_dom f ) ) A10: n <= n + 1 by NAT_1:11; 0 + 1 <= n by A9, NAT_1:13; then A11: n in dom f by A4, A10, FINSEQ_3:25; 0 + 1 <= n + 1 by NAT_1:13; then A12: n + 1 in dom f by A4, FINSEQ_3:25; then reconsider G = f . (n + 1) as Element of PFuncs (D,REAL) by FINSEQ_2:11; set fn = f | n; A13: len (f | n) = n by A4, FINSEQ_1:59, NAT_1:11; f = (f | n) ^ <*G*> by A4, RFINSEQ:7; then A14: Sum f = (Sum (f | n)) + G by Th20; thus ( d is_common_for_dom f implies d in dom (Sum f) ) ::_thesis: ( d in dom (Sum f) implies d is_common_for_dom f ) proof assume A15: d is_common_for_dom f ; ::_thesis: d in dom (Sum f) then d is_common_for_dom f | n by A9, Th26; then A16: d in dom (Sum (f | n)) by A3, A9, A13; d in dom G by A12, A15, Def9; then d in (dom (Sum (f | n))) /\ (dom G) by A16, XBOOLE_0:def_4; hence d in dom (Sum f) by A14, VALUED_1:def_1; ::_thesis: verum end; assume d in dom (Sum f) ; ::_thesis: d is_common_for_dom f then A17: d in (dom (Sum (f | n))) /\ (dom G) by A14, VALUED_1:def_1; then d in dom (Sum (f | n)) by XBOOLE_0:def_4; then A18: d is_common_for_dom f | n by A3, A9, A13; now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_f_holds_ d_in_dom_(f_._m) let m be Element of NAT ; ::_thesis: ( m in dom f implies d in dom (f . m) ) assume A19: m in dom f ; ::_thesis: d in dom (f . m) set F = f . m; A20: m <= len f by A19, FINSEQ_3:25; A21: 1 <= m by A19, FINSEQ_3:25; now__::_thesis:_(_(_m_=_len_f_&_d_in_dom_(f_._m)_)_or_(_m_<>_len_f_&_d_in_dom_(f_._m)_)_) percases ( m = len f or m <> len f ) ; case m = len f ; ::_thesis: d in dom (f . m) hence d in dom (f . m) by A4, A17, XBOOLE_0:def_4; ::_thesis: verum end; case m <> len f ; ::_thesis: d in dom (f . m) then m < len f by A20, XXREAL_0:1; then m <= n by A4, NAT_1:13; then A22: m in Seg n by A21, FINSEQ_1:1; then ( dom (f | n) = Seg (len (f | n)) & f . m = (f | n) . m ) by A11, FINSEQ_1:def_3, RFINSEQ:6; hence d in dom (f . m) by A13, A18, A22, Def9; ::_thesis: verum end; end; end; hence d in dom (f . m) ; ::_thesis: verum end; hence d is_common_for_dom f by Def9; ::_thesis: verum end; end; end; hence ( d is_common_for_dom f iff d in dom (Sum f) ) ; ::_thesis: verum end; A23: S1[ 0 ] ; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A23, A2); hence ( d is_common_for_dom f iff d in dom (Sum f) ) by A1; ::_thesis: verum end; theorem Th29: :: RFUNCT_3:29 for D being non empty set for f being FinSequence of PFuncs (D,REAL) for d being Element of D for n being Element of NAT holds (f | n) # d = (f # d) | n proof let D1 be non empty set ; ::_thesis: for f being FinSequence of PFuncs (D1,REAL) for d being Element of D1 for n being Element of NAT holds (f | n) # d = (f # d) | n let f be FinSequence of PFuncs (D1,REAL); ::_thesis: for d being Element of D1 for n being Element of NAT holds (f | n) # d = (f # d) | n let d1 be Element of D1; ::_thesis: for n being Element of NAT holds (f | n) # d1 = (f # d1) | n let n be Element of NAT ; ::_thesis: (f | n) # d1 = (f # d1) | n A1: len (f # d1) = len f by Def8; A2: len ((f | n) # d1) = len (f | n) by Def8; now__::_thesis:_(_(_len_f_<=_n_&_(f_|_n)_#_d1_=_(f_#_d1)_|_n_)_or_(_n_<_len_f_&_(f_|_n)_#_d1_=_(f_#_d1)_|_n_)_) percases ( len f <= n or n < len f ) ; caseA3: len f <= n ; ::_thesis: (f | n) # d1 = (f # d1) | n then f | n = f by Lm1; hence (f | n) # d1 = (f # d1) | n by A1, A3, Lm1; ::_thesis: verum end; caseA4: n < len f ; ::_thesis: (f | n) # d1 = (f # d1) | n then A5: len (f | n) = n by FINSEQ_1:59; A6: len ((f # d1) | n) = n by A1, A4, FINSEQ_1:59; A7: ( dom f = Seg (len f) & dom (f # d1) = Seg (len (f # d1)) ) by FINSEQ_1:def_3; A8: dom ((f | n) # d1) = Seg (len ((f | n) # d1)) by FINSEQ_1:def_3; now__::_thesis:_(_(_n_=_0_&_(f_|_n)_#_d1_=_(f_#_d1)_|_n_)_or_(_n_<>_0_&_(f_|_n)_#_d1_=_(f_#_d1)_|_n_)_) percases ( n = 0 or n <> 0 ) ; caseA9: n = 0 ; ::_thesis: (f | n) # d1 = (f # d1) | n then (f # d1) | n = <*> REAL ; hence (f | n) # d1 = (f # d1) | n by A2, A9; ::_thesis: verum end; caseA10: n <> 0 ; ::_thesis: (f | n) # d1 = (f # d1) | n A11: dom ((f # d1) | n) = Seg (len (f | n)) by A5, A6, FINSEQ_1:def_3; 0 + 1 <= n by A10, NAT_1:13; then A12: n in dom f by A4, FINSEQ_3:25; now__::_thesis:_for_m_being_Nat_st_m_in_dom_((f_#_d1)_|_n)_holds_ ((f_#_d1)_|_n)_._m_=_((f_|_n)_#_d1)_._m let m be Nat; ::_thesis: ( m in dom ((f # d1) | n) implies ((f # d1) | n) . m = ((f | n) # d1) . m ) assume A13: m in dom ((f # d1) | n) ; ::_thesis: ((f # d1) | n) . m = ((f | n) # d1) . m then A14: m in dom (f # d1) by A1, A5, A7, A12, A11, RFINSEQ:6; then reconsider G = f . m as Element of PFuncs (D1,REAL) by A1, A7, FINSEQ_2:11; ((f # d1) | n) . m = (f # d1) . m by A1, A5, A7, A12, A11, A13, RFINSEQ:6; then A15: ((f # d1) | n) . m = G . d1 by A14, Def8; (f | n) . m = G by A5, A12, A11, A13, RFINSEQ:6; hence ((f # d1) | n) . m = ((f | n) # d1) . m by A2, A8, A11, A13, A15, Def8; ::_thesis: verum end; hence (f | n) # d1 = (f # d1) | n by A2, A5, A6, FINSEQ_2:9; ::_thesis: verum end; end; end; hence (f | n) # d1 = (f # d1) | n ; ::_thesis: verum end; end; end; hence (f | n) # d1 = (f # d1) | n ; ::_thesis: verum end; theorem Th30: :: RFUNCT_3:30 for D being non empty set for f being FinSequence for d being Element of D holds d is_common_for_dom CHI (f,D) proof let D be non empty set ; ::_thesis: for f being FinSequence for d being Element of D holds d is_common_for_dom CHI (f,D) let f be FinSequence; ::_thesis: for d being Element of D holds d is_common_for_dom CHI (f,D) let d be Element of D; ::_thesis: d is_common_for_dom CHI (f,D) let n be Element of NAT ; :: according to RFUNCT_3:def_9 ::_thesis: ( n in dom (CHI (f,D)) implies d in dom ((CHI (f,D)) . n) ) assume n in dom (CHI (f,D)) ; ::_thesis: d in dom ((CHI (f,D)) . n) then (CHI (f,D)) . n = chi ((f . n),D) by Def6; then dom ((CHI (f,D)) . n) = D by RFUNCT_1:61; hence d in dom ((CHI (f,D)) . n) ; ::_thesis: verum end; theorem Th31: :: RFUNCT_3:31 for D being non empty set for d being Element of D for f being FinSequence of PFuncs (D,REAL) for R being FinSequence of REAL st d is_common_for_dom f holds d is_common_for_dom R (#) f proof let D be non empty set ; ::_thesis: for d being Element of D for f being FinSequence of PFuncs (D,REAL) for R being FinSequence of REAL st d is_common_for_dom f holds d is_common_for_dom R (#) f let d be Element of D; ::_thesis: for f being FinSequence of PFuncs (D,REAL) for R being FinSequence of REAL st d is_common_for_dom f holds d is_common_for_dom R (#) f let f be FinSequence of PFuncs (D,REAL); ::_thesis: for R being FinSequence of REAL st d is_common_for_dom f holds d is_common_for_dom R (#) f let R be FinSequence of REAL ; ::_thesis: ( d is_common_for_dom f implies d is_common_for_dom R (#) f ) assume A1: d is_common_for_dom f ; ::_thesis: d is_common_for_dom R (#) f set m = min ((len R),(len f)); let n be Element of NAT ; :: according to RFUNCT_3:def_9 ::_thesis: ( n in dom (R (#) f) implies d in dom ((R (#) f) . n) ) assume A2: n in dom (R (#) f) ; ::_thesis: d in dom ((R (#) f) . n) set G = (R (#) f) . n; len (R (#) f) = min ((len R),(len f)) by Def7; then ( min ((len R),(len f)) <= len f & n <= min ((len R),(len f)) ) by A2, FINSEQ_3:25, XXREAL_0:17; then A3: n <= len f by XXREAL_0:2; 1 <= n by A2, FINSEQ_3:25; then A4: n in dom f by A3, FINSEQ_3:25; then reconsider F = f . n as Element of PFuncs (D,REAL) by FINSEQ_2:11; A5: d in dom F by A1, A4, Def9; reconsider r = R . n as Real ; (R (#) f) . n = r (#) F by A2, Def7; hence d in dom ((R (#) f) . n) by A5, VALUED_1:def_5; ::_thesis: verum end; theorem :: RFUNCT_3:32 for D being non empty set for f being FinSequence for R being FinSequence of REAL for d being Element of D holds d is_common_for_dom R (#) (CHI (f,D)) by Th30, Th31; theorem :: RFUNCT_3:33 for D being non empty set for d being Element of D for f being FinSequence of PFuncs (D,REAL) st d is_common_for_dom f holds (Sum f) . d = Sum (f # d) proof let D be non empty set ; ::_thesis: for d being Element of D for f being FinSequence of PFuncs (D,REAL) st d is_common_for_dom f holds (Sum f) . d = Sum (f # d) let d be Element of D; ::_thesis: for f being FinSequence of PFuncs (D,REAL) st d is_common_for_dom f holds (Sum f) . d = Sum (f # d) defpred S1[ Element of NAT ] means for f being FinSequence of PFuncs (D,REAL) st len f = $1 & d is_common_for_dom f holds (Sum f) . d = Sum (f # d); let f be FinSequence of PFuncs (D,REAL); ::_thesis: ( d is_common_for_dom f implies (Sum f) . d = Sum (f # d) ) assume A1: d is_common_for_dom f ; ::_thesis: (Sum f) . d = Sum (f # d) A2: len f = len f ; A3: for n being Element of NAT st S1[n] holds S1[n + 1] proof let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A4: S1[n] ; ::_thesis: S1[n + 1] let f be FinSequence of PFuncs (D,REAL); ::_thesis: ( len f = n + 1 & d is_common_for_dom f implies (Sum f) . d = Sum (f # d) ) assume that A5: len f = n + 1 and A6: d is_common_for_dom f ; ::_thesis: (Sum f) . d = Sum (f # d) set fn = f | n; A7: len (f | n) = n by A5, FINSEQ_1:59, NAT_1:11; 0 + 1 <= n + 1 by NAT_1:13; then A8: n + 1 in dom f by A5, FINSEQ_3:25; then reconsider G = f . (n + 1) as Element of PFuncs (D,REAL) by FINSEQ_2:11; A9: ( dom f = Seg (len f) & dom (f # d) = Seg (len (f # d)) ) by FINSEQ_1:def_3; f = (f | n) ^ <*G*> by A5, RFINSEQ:7; then A10: Sum f = (Sum (f | n)) + G by Th20; A11: len (f # d) = len f by Def8; A12: d in dom G by A6, A8, Def9; now__::_thesis:_(_(_n_=_0_&_(Sum_f)_._d_=_Sum_(f_#_d)_)_or_(_n_<>_0_&_(Sum_f)_._d_=_Sum_(f_#_d)_)_) percases ( n = 0 or n <> 0 ) ; caseA13: n = 0 ; ::_thesis: (Sum f) . d = Sum (f # d) then A14: len (f # d) = 1 by A5, Def8; then A15: 1 in dom (f # d) by FINSEQ_3:25; A16: now__::_thesis:_for_m_being_Nat_st_m_in_Seg_1_holds_ (f_#_d)_._m_=_<*(G_._d)*>_._m let m be Nat; ::_thesis: ( m in Seg 1 implies (f # d) . m = <*(G . d)*> . m ) assume m in Seg 1 ; ::_thesis: (f # d) . m = <*(G . d)*> . m then A17: m = 1 by FINSEQ_1:2, TARSKI:def_1; hence (f # d) . m = G . d by A13, A15, Def8 .= <*(G . d)*> . m by A17, FINSEQ_1:40 ; ::_thesis: verum end; ( len <*(G . d)*> = 1 & dom (f # d) = Seg 1 ) by A14, FINSEQ_1:40, FINSEQ_1:def_3; then A18: f # d = <*(G . d)*> by A14, A16, FINSEQ_2:9; f = <*G*> by A5, A13, FINSEQ_1:40; hence (Sum f) . d = G . d by FINSOP_1:11 .= Sum (f # d) by A18, FINSOP_1:11 ; ::_thesis: verum end; caseA19: n <> 0 ; ::_thesis: (Sum f) . d = Sum (f # d) A20: (f # d) . (n + 1) = G . d by A9, A11, A8, Def8; d is_common_for_dom f | n by A6, A19, Th26; then d in dom (Sum (f | n)) by A7, A19, Th28; then d in (dom (Sum (f | n))) /\ (dom G) by A12, XBOOLE_0:def_4; then d in dom ((Sum (f | n)) + G) by VALUED_1:def_1; hence (Sum f) . d = ((Sum (f | n)) . d) + (G . d) by A10, VALUED_1:def_1 .= (Sum ((f | n) # d)) + (G . d) by A4, A6, A7, A19, Th26 .= (Sum ((f # d) | n)) + (G . d) by Th29 .= Sum (((f # d) | n) ^ <*(G . d)*>) by RVSUM_1:74 .= Sum (f # d) by A5, A11, A20, RFINSEQ:7 ; ::_thesis: verum end; end; end; hence (Sum f) . d = Sum (f # d) ; ::_thesis: verum end; A21: S1[ 0 ] proof let f be FinSequence of PFuncs (D,REAL); ::_thesis: ( len f = 0 & d is_common_for_dom f implies (Sum f) . d = Sum (f # d) ) assume that A22: len f = 0 and d is_common_for_dom f ; ::_thesis: (Sum f) . d = Sum (f # d) f = <*> (PFuncs (D,REAL)) by A22; then A23: (Sum f) . d = (([#] D) --> 0) . d by Th19 .= 0 by FUNCOP_1:7 ; len (f # d) = 0 by A22, Def8; then f # d = <*> (PFuncs (D,REAL)) ; hence (Sum f) . d = Sum (f # d) by A23, RVSUM_1:72; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A21, A3); hence (Sum f) . d = Sum (f # d) by A1, A2; ::_thesis: verum end; definition let D be non empty set ; let F be PartFunc of D,REAL; func max+ F -> PartFunc of D,REAL means :Def10: :: RFUNCT_3:def 10 ( dom it = dom F & ( for d being Element of D st d in dom it holds it . d = max+ (F . d) ) ); existence ex b1 being PartFunc of D,REAL st ( dom b1 = dom F & ( for d being Element of D st d in dom b1 holds b1 . d = max+ (F . d) ) ) proof deffunc H1( set ) -> Real = max+ (F . $1); defpred S1[ set ] means $1 in dom F; consider f being PartFunc of D,REAL such that A1: for d being Element of D holds ( d in dom f iff S1[d] ) and A2: for d being Element of D st d in dom f holds f . d = H1(d) from SEQ_1:sch_3(); take f ; ::_thesis: ( dom f = dom F & ( for d being Element of D st d in dom f holds f . d = max+ (F . d) ) ) thus dom f = dom F ::_thesis: for d being Element of D st d in dom f holds f . d = max+ (F . d) proof thus dom f c= dom F :: according to XBOOLE_0:def_10 ::_thesis: dom F c= dom f proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom f or x in dom F ) assume x in dom f ; ::_thesis: x in dom F hence x in dom F by A1; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom F or x in dom f ) assume x in dom F ; ::_thesis: x in dom f hence x in dom f by A1; ::_thesis: verum end; thus for d being Element of D st d in dom f holds f . d = max+ (F . d) by A2; ::_thesis: verum end; uniqueness for b1, b2 being PartFunc of D,REAL st dom b1 = dom F & ( for d being Element of D st d in dom b1 holds b1 . d = max+ (F . d) ) & dom b2 = dom F & ( for d being Element of D st d in dom b2 holds b2 . d = max+ (F . d) ) holds b1 = b2 proof deffunc H1( set ) -> Real = max+ (F . $1); for f, g being PartFunc of D,REAL st dom f = dom F & ( for c being Element of D st c in dom f holds f . c = H1(c) ) & dom g = dom F & ( for c being Element of D st c in dom g holds g . c = H1(c) ) holds f = g from SEQ_1:sch_4(); hence for b1, b2 being PartFunc of D,REAL st dom b1 = dom F & ( for d being Element of D st d in dom b1 holds b1 . d = max+ (F . d) ) & dom b2 = dom F & ( for d being Element of D st d in dom b2 holds b2 . d = max+ (F . d) ) holds b1 = b2 ; ::_thesis: verum end; func max- F -> PartFunc of D,REAL means :Def11: :: RFUNCT_3:def 11 ( dom it = dom F & ( for d being Element of D st d in dom it holds it . d = max- (F . d) ) ); existence ex b1 being PartFunc of D,REAL st ( dom b1 = dom F & ( for d being Element of D st d in dom b1 holds b1 . d = max- (F . d) ) ) proof deffunc H1( set ) -> Real = max- (F . $1); defpred S1[ set ] means $1 in dom F; consider f being PartFunc of D,REAL such that A3: for d being Element of D holds ( d in dom f iff S1[d] ) and A4: for d being Element of D st d in dom f holds f . d = H1(d) from SEQ_1:sch_3(); take f ; ::_thesis: ( dom f = dom F & ( for d being Element of D st d in dom f holds f . d = max- (F . d) ) ) thus dom f = dom F ::_thesis: for d being Element of D st d in dom f holds f . d = max- (F . d) proof thus dom f c= dom F :: according to XBOOLE_0:def_10 ::_thesis: dom F c= dom f proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom f or x in dom F ) assume x in dom f ; ::_thesis: x in dom F hence x in dom F by A3; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom F or x in dom f ) assume x in dom F ; ::_thesis: x in dom f hence x in dom f by A3; ::_thesis: verum end; thus for d being Element of D st d in dom f holds f . d = max- (F . d) by A4; ::_thesis: verum end; uniqueness for b1, b2 being PartFunc of D,REAL st dom b1 = dom F & ( for d being Element of D st d in dom b1 holds b1 . d = max- (F . d) ) & dom b2 = dom F & ( for d being Element of D st d in dom b2 holds b2 . d = max- (F . d) ) holds b1 = b2 proof deffunc H1( set ) -> Real = max- (F . $1); for f, g being PartFunc of D,REAL st dom f = dom F & ( for c being Element of D st c in dom f holds f . c = H1(c) ) & dom g = dom F & ( for c being Element of D st c in dom g holds g . c = H1(c) ) holds f = g from SEQ_1:sch_4(); hence for b1, b2 being PartFunc of D,REAL st dom b1 = dom F & ( for d being Element of D st d in dom b1 holds b1 . d = max- (F . d) ) & dom b2 = dom F & ( for d being Element of D st d in dom b2 holds b2 . d = max- (F . d) ) holds b1 = b2 ; ::_thesis: verum end; end; :: deftheorem Def10 defines max+ RFUNCT_3:def_10_:_ for D being non empty set for F, b3 being PartFunc of D,REAL holds ( b3 = max+ F iff ( dom b3 = dom F & ( for d being Element of D st d in dom b3 holds b3 . d = max+ (F . d) ) ) ); :: deftheorem Def11 defines max- RFUNCT_3:def_11_:_ for D being non empty set for F, b3 being PartFunc of D,REAL holds ( b3 = max- F iff ( dom b3 = dom F & ( for d being Element of D st d in dom b3 holds b3 . d = max- (F . d) ) ) ); theorem :: RFUNCT_3:34 for D being non empty set for F being PartFunc of D,REAL holds ( F = (max+ F) - (max- F) & abs F = (max+ F) + (max- F) & 2 (#) (max+ F) = F + (abs F) ) proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL holds ( F = (max+ F) - (max- F) & abs F = (max+ F) + (max- F) & 2 (#) (max+ F) = F + (abs F) ) let F be PartFunc of D,REAL; ::_thesis: ( F = (max+ F) - (max- F) & abs F = (max+ F) + (max- F) & 2 (#) (max+ F) = F + (abs F) ) A1: dom F = (dom F) /\ (dom F) ; A2: dom (max+ F) = dom F by Def10; A3: dom (max- F) = dom F by Def11; dom (- (max- F)) = dom (max- F) by VALUED_1:def_5; then A4: dom F = dom ((max+ F) + (- (max- F))) by A2, A3, A1, VALUED_1:def_1; now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_F_holds_ ((max+_F)_-_(max-_F))_._d_=_F_._d let d be Element of D; ::_thesis: ( d in dom F implies ((max+ F) - (max- F)) . d = F . d ) assume A5: d in dom F ; ::_thesis: ((max+ F) - (max- F)) . d = F . d hence ((max+ F) - (max- F)) . d = ((max+ F) . d) - ((max- F) . d) by A4, VALUED_1:13 .= (max+ (F . d)) - ((max- F) . d) by A2, A5, Def10 .= (max+ (F . d)) - (max- (F . d)) by A3, A5, Def11 .= F . d by Th1 ; ::_thesis: verum end; hence F = (max+ F) - (max- F) by A4, PARTFUN1:5; ::_thesis: ( abs F = (max+ F) + (max- F) & 2 (#) (max+ F) = F + (abs F) ) A6: dom (abs F) = dom F by VALUED_1:def_11; then A7: dom (abs F) = dom ((max+ F) + (max- F)) by A2, A3, A1, VALUED_1:def_1; now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_(abs_F)_holds_ ((max+_F)_+_(max-_F))_._d_=_(abs_F)_._d let d be Element of D; ::_thesis: ( d in dom (abs F) implies ((max+ F) + (max- F)) . d = (abs F) . d ) assume A8: d in dom (abs F) ; ::_thesis: ((max+ F) + (max- F)) . d = (abs F) . d hence ((max+ F) + (max- F)) . d = ((max+ F) . d) + ((max- F) . d) by A7, VALUED_1:def_1 .= (max+ (F . d)) + ((max- F) . d) by A2, A6, A8, Def10 .= (max+ (F . d)) + (max- (F . d)) by A3, A6, A8, Def11 .= abs (F . d) by Th2 .= (abs F) . d by VALUED_1:18 ; ::_thesis: verum end; hence abs F = (max+ F) + (max- F) by A7, PARTFUN1:5; ::_thesis: 2 (#) (max+ F) = F + (abs F) A9: dom (2 (#) (max+ F)) = dom (max+ F) by VALUED_1:def_5; then A10: dom (2 (#) (max+ F)) = dom (F + (abs F)) by A2, A6, A1, VALUED_1:def_1; now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_F_holds_ (2_(#)_(max+_F))_._d_=_(F_+_(abs_F))_._d let d be Element of D; ::_thesis: ( d in dom F implies (2 (#) (max+ F)) . d = (F + (abs F)) . d ) assume A11: d in dom F ; ::_thesis: (2 (#) (max+ F)) . d = (F + (abs F)) . d hence (2 (#) (max+ F)) . d = 2 * ((max+ F) . d) by A2, A9, VALUED_1:def_5 .= 2 * (max+ (F . d)) by A2, A11, Def10 .= (F . d) + (abs (F . d)) by Th3 .= (F . d) + ((abs F) . d) by VALUED_1:18 .= (F + (abs F)) . d by A2, A9, A10, A11, VALUED_1:def_1 ; ::_thesis: verum end; hence 2 (#) (max+ F) = F + (abs F) by A2, A9, A10, PARTFUN1:5; ::_thesis: verum end; theorem Th35: :: RFUNCT_3:35 for D being non empty set for F being PartFunc of D,REAL for r being Real st 0 < r holds F " {r} = (max+ F) " {r} proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for r being Real st 0 < r holds F " {r} = (max+ F) " {r} let F be PartFunc of D,REAL; ::_thesis: for r being Real st 0 < r holds F " {r} = (max+ F) " {r} let r be Real; ::_thesis: ( 0 < r implies F " {r} = (max+ F) " {r} ) A1: dom (max+ F) = dom F by Def10; assume A2: 0 < r ; ::_thesis: F " {r} = (max+ F) " {r} thus F " {r} c= (max+ F) " {r} :: according to XBOOLE_0:def_10 ::_thesis: (max+ F) " {r} c= F " {r} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " {r} or x in (max+ F) " {r} ) assume A3: x in F " {r} ; ::_thesis: x in (max+ F) " {r} then reconsider d = x as Element of D ; F . d in {r} by A3, FUNCT_1:def_7; then A4: F . d = r by TARSKI:def_1; A5: d in dom F by A3, FUNCT_1:def_7; then (max+ F) . d = max+ (F . d) by A1, Def10 .= r by A2, A4, XXREAL_0:def_10 ; then (max+ F) . d in {r} by TARSKI:def_1; hence x in (max+ F) " {r} by A1, A5, FUNCT_1:def_7; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (max+ F) " {r} or x in F " {r} ) assume A6: x in (max+ F) " {r} ; ::_thesis: x in F " {r} then reconsider d = x as Element of D ; (max+ F) . d in {r} by A6, FUNCT_1:def_7; then A7: (max+ F) . d = r by TARSKI:def_1; A8: d in dom F by A1, A6, FUNCT_1:def_7; then (max+ F) . d = max+ (F . d) by A1, Def10 .= max ((F . d),0) ; then F . d = r by A2, A7, XXREAL_0:16; then F . d in {r} by TARSKI:def_1; hence x in F " {r} by A8, FUNCT_1:def_7; ::_thesis: verum end; theorem Th36: :: RFUNCT_3:36 for D being non empty set for F being PartFunc of D,REAL holds F " (left_closed_halfline 0) = (max+ F) " {0} proof set li = left_closed_halfline 0; let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL holds F " (left_closed_halfline 0) = (max+ F) " {0} let F be PartFunc of D,REAL; ::_thesis: F " (left_closed_halfline 0) = (max+ F) " {0} A1: dom (max+ F) = dom F by Def10; A2: left_closed_halfline 0 = { s where s is Real : s <= 0 } by XXREAL_1:231; thus F " (left_closed_halfline 0) c= (max+ F) " {0} :: according to XBOOLE_0:def_10 ::_thesis: (max+ F) " {0} c= F " (left_closed_halfline 0) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " (left_closed_halfline 0) or x in (max+ F) " {0} ) assume A3: x in F " (left_closed_halfline 0) ; ::_thesis: x in (max+ F) " {0} then reconsider d = x as Element of D ; F . d in left_closed_halfline 0 by A3, FUNCT_1:def_7; then ex s being Real st ( s = F . d & s <= 0 ) by A2; then A4: max ((F . d),0) = 0 by XXREAL_0:def_10; A5: d in dom F by A3, FUNCT_1:def_7; then (max+ F) . d = max+ (F . d) by A1, Def10 .= max ((F . d),0) ; then (max+ F) . d in {0} by A4, TARSKI:def_1; hence x in (max+ F) " {0} by A1, A5, FUNCT_1:def_7; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (max+ F) " {0} or x in F " (left_closed_halfline 0) ) assume A6: x in (max+ F) " {0} ; ::_thesis: x in F " (left_closed_halfline 0) then reconsider d = x as Element of D ; (max+ F) . d in {0} by A6, FUNCT_1:def_7; then A7: (max+ F) . d = 0 by TARSKI:def_1; A8: d in dom F by A1, A6, FUNCT_1:def_7; then (max+ F) . d = max+ (F . d) by A1, Def10 .= max ((F . d),0) ; then F . d <= 0 by A7, XXREAL_0:def_10; then F . d in left_closed_halfline 0 by A2; hence x in F " (left_closed_halfline 0) by A8, FUNCT_1:def_7; ::_thesis: verum end; theorem Th37: :: RFUNCT_3:37 for D being non empty set for F being PartFunc of D,REAL for d being Element of D holds 0 <= (max+ F) . d proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for d being Element of D holds 0 <= (max+ F) . d let F be PartFunc of D,REAL; ::_thesis: for d being Element of D holds 0 <= (max+ F) . d let d be Element of D; ::_thesis: 0 <= (max+ F) . d A1: dom F = dom (max+ F) by Def10; percases ( d in dom F or not d in dom F ) ; suppose d in dom F ; ::_thesis: 0 <= (max+ F) . d then (max+ F) . d = max+ (F . d) by A1, Def10 .= max ((F . d),0) ; hence 0 <= (max+ F) . d by XXREAL_0:25; ::_thesis: verum end; suppose not d in dom F ; ::_thesis: 0 <= (max+ F) . d hence 0 <= (max+ F) . d by A1, FUNCT_1:def_2; ::_thesis: verum end; end; end; theorem Th38: :: RFUNCT_3:38 for D being non empty set for F being PartFunc of D,REAL for r being Real st 0 < r holds F " {(- r)} = (max- F) " {r} proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for r being Real st 0 < r holds F " {(- r)} = (max- F) " {r} let F be PartFunc of D,REAL; ::_thesis: for r being Real st 0 < r holds F " {(- r)} = (max- F) " {r} let r be Real; ::_thesis: ( 0 < r implies F " {(- r)} = (max- F) " {r} ) A1: dom (max- F) = dom F by Def11; assume A2: 0 < r ; ::_thesis: F " {(- r)} = (max- F) " {r} thus F " {(- r)} c= (max- F) " {r} :: according to XBOOLE_0:def_10 ::_thesis: (max- F) " {r} c= F " {(- r)} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " {(- r)} or x in (max- F) " {r} ) assume A3: x in F " {(- r)} ; ::_thesis: x in (max- F) " {r} then reconsider d = x as Element of D ; F . d in {(- r)} by A3, FUNCT_1:def_7; then A4: F . d = - r by TARSKI:def_1; A5: d in dom F by A3, FUNCT_1:def_7; then (max- F) . d = max- (F . d) by A1, Def11 .= r by A2, A4, XXREAL_0:def_10 ; then (max- F) . d in {r} by TARSKI:def_1; hence x in (max- F) " {r} by A1, A5, FUNCT_1:def_7; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (max- F) " {r} or x in F " {(- r)} ) assume A6: x in (max- F) " {r} ; ::_thesis: x in F " {(- r)} then reconsider d = x as Element of D ; (max- F) . d in {r} by A6, FUNCT_1:def_7; then A7: (max- F) . d = r by TARSKI:def_1; A8: d in dom F by A1, A6, FUNCT_1:def_7; then (max- F) . d = max- (F . d) by A1, Def11 .= max ((- (F . d)),0) ; then - (F . d) = r by A2, A7, XXREAL_0:16; then F . d in {(- r)} by TARSKI:def_1; hence x in F " {(- r)} by A8, FUNCT_1:def_7; ::_thesis: verum end; theorem Th39: :: RFUNCT_3:39 for D being non empty set for F being PartFunc of D,REAL holds F " (right_closed_halfline 0) = (max- F) " {0} proof set li = right_closed_halfline 0; let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL holds F " (right_closed_halfline 0) = (max- F) " {0} let F be PartFunc of D,REAL; ::_thesis: F " (right_closed_halfline 0) = (max- F) " {0} A1: dom (max- F) = dom F by Def11; A2: right_closed_halfline 0 = { s where s is Real : 0 <= s } by XXREAL_1:232; thus F " (right_closed_halfline 0) c= (max- F) " {0} :: according to XBOOLE_0:def_10 ::_thesis: (max- F) " {0} c= F " (right_closed_halfline 0) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " (right_closed_halfline 0) or x in (max- F) " {0} ) assume A3: x in F " (right_closed_halfline 0) ; ::_thesis: x in (max- F) " {0} then reconsider d = x as Element of D ; F . d in right_closed_halfline 0 by A3, FUNCT_1:def_7; then ex s being Real st ( s = F . d & 0 <= s ) by A2; then A4: max ((- (F . d)),0) = 0 by XXREAL_0:def_10; A5: d in dom F by A3, FUNCT_1:def_7; then (max- F) . d = max- (F . d) by A1, Def11 .= max ((- (F . d)),0) ; then (max- F) . d in {0} by A4, TARSKI:def_1; hence x in (max- F) " {0} by A1, A5, FUNCT_1:def_7; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (max- F) " {0} or x in F " (right_closed_halfline 0) ) assume A6: x in (max- F) " {0} ; ::_thesis: x in F " (right_closed_halfline 0) then reconsider d = x as Element of D ; (max- F) . d in {0} by A6, FUNCT_1:def_7; then A7: (max- F) . d = 0 by TARSKI:def_1; A8: d in dom F by A1, A6, FUNCT_1:def_7; then (max- F) . d = max- (F . d) by A1, Def11 .= max ((- (F . d)),0) ; then - (F . d) <= - 0 by A7, XXREAL_0:def_10; then 0 <= F . d by XREAL_1:24; then F . d in right_closed_halfline 0 by A2; hence x in F " (right_closed_halfline 0) by A8, FUNCT_1:def_7; ::_thesis: verum end; theorem Th40: :: RFUNCT_3:40 for D being non empty set for F being PartFunc of D,REAL for d being Element of D holds 0 <= (max- F) . d proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for d being Element of D holds 0 <= (max- F) . d let F be PartFunc of D,REAL; ::_thesis: for d being Element of D holds 0 <= (max- F) . d let d be Element of D; ::_thesis: 0 <= (max- F) . d A1: dom F = dom (max- F) by Def11; percases ( d in dom F or not d in dom F ) ; suppose d in dom F ; ::_thesis: 0 <= (max- F) . d then (max- F) . d = max- (F . d) by A1, Def11 .= max ((- (F . d)),0) ; hence 0 <= (max- F) . d by XXREAL_0:25; ::_thesis: verum end; suppose not d in dom F ; ::_thesis: 0 <= (max- F) . d hence 0 <= (max- F) . d by A1, FUNCT_1:def_2; ::_thesis: verum end; end; end; theorem :: RFUNCT_3:41 for D, C being non empty set for F being PartFunc of D,REAL for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds max+ F, max+ G are_fiberwise_equipotent proof set li = left_closed_halfline 0; let D, C be non empty set ; ::_thesis: for F being PartFunc of D,REAL for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds max+ F, max+ G are_fiberwise_equipotent let F be PartFunc of D,REAL; ::_thesis: for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds max+ F, max+ G are_fiberwise_equipotent let G be PartFunc of C,REAL; ::_thesis: ( F,G are_fiberwise_equipotent implies max+ F, max+ G are_fiberwise_equipotent ) assume A1: F,G are_fiberwise_equipotent ; ::_thesis: max+ F, max+ G are_fiberwise_equipotent A2: now__::_thesis:_for_r_being_Real_holds_card_(Coim_((max+_F),r))_=_card_(Coim_((max+_G),r)) let r be Real; ::_thesis: card (Coim ((max+ F),r)) = card (Coim ((max+ G),r)) now__::_thesis:_(_(_0_<_r_&_card_(Coim_((max+_F),r))_=_card_(Coim_((max+_G),r))_)_or_(_r_=_0_&_card_((max+_F)_"_{r})_=_card_((max+_G)_"_{r})_)_or_(_r_<_0_&_card_((max+_F)_"_{r})_=_card_((max+_G)_"_{r})_)_) percases ( 0 < r or r = 0 or r < 0 ) ; case 0 < r ; ::_thesis: card (Coim ((max+ F),r)) = card (Coim ((max+ G),r)) then ( Coim (F,r) = Coim ((max+ F),r) & Coim (G,r) = Coim ((max+ G),r) ) by Th35; hence card (Coim ((max+ F),r)) = card (Coim ((max+ G),r)) by A1, CLASSES1:def_9; ::_thesis: verum end; caseA3: r = 0 ; ::_thesis: card ((max+ F) " {r}) = card ((max+ G) " {r}) ( F " (left_closed_halfline 0) = (max+ F) " {0} & G " (left_closed_halfline 0) = (max+ G) " {0} ) by Th36; hence card ((max+ F) " {r}) = card ((max+ G) " {r}) by A1, A3, CLASSES1:78; ::_thesis: verum end; caseA4: r < 0 ; ::_thesis: card ((max+ F) " {r}) = card ((max+ G) " {r}) now__::_thesis:_not_r_in_rng_(max+_F) assume r in rng (max+ F) ; ::_thesis: contradiction then ex d being Element of D st ( d in dom (max+ F) & (max+ F) . d = r ) by PARTFUN1:3; hence contradiction by A4, Th37; ::_thesis: verum end; then A5: (max+ F) " {r} = {} by Lm2; now__::_thesis:_not_r_in_rng_(max+_G) assume r in rng (max+ G) ; ::_thesis: contradiction then ex c being Element of C st ( c in dom (max+ G) & (max+ G) . c = r ) by PARTFUN1:3; hence contradiction by A4, Th37; ::_thesis: verum end; hence card ((max+ F) " {r}) = card ((max+ G) " {r}) by A5, Lm2; ::_thesis: verum end; end; end; hence card (Coim ((max+ F),r)) = card (Coim ((max+ G),r)) ; ::_thesis: verum end; ( rng (max+ F) c= REAL & rng (max+ G) c= REAL ) ; hence max+ F, max+ G are_fiberwise_equipotent by A2, CLASSES1:79; ::_thesis: verum end; theorem :: RFUNCT_3:42 for D, C being non empty set for F being PartFunc of D,REAL for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds max- F, max- G are_fiberwise_equipotent proof set li = right_closed_halfline 0; let D, C be non empty set ; ::_thesis: for F being PartFunc of D,REAL for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds max- F, max- G are_fiberwise_equipotent let F be PartFunc of D,REAL; ::_thesis: for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds max- F, max- G are_fiberwise_equipotent let G be PartFunc of C,REAL; ::_thesis: ( F,G are_fiberwise_equipotent implies max- F, max- G are_fiberwise_equipotent ) assume A1: F,G are_fiberwise_equipotent ; ::_thesis: max- F, max- G are_fiberwise_equipotent A2: now__::_thesis:_for_r_being_Real_holds_card_(Coim_((max-_F),r))_=_card_(Coim_((max-_G),r)) let r be Real; ::_thesis: card (Coim ((max- F),r)) = card (Coim ((max- G),r)) now__::_thesis:_(_(_0_<_r_&_card_(Coim_((max-_F),r))_=_card_(Coim_((max-_G),r))_)_or_(_r_=_0_&_card_((max-_F)_"_{r})_=_card_((max-_G)_"_{r})_)_or_(_r_<_0_&_card_((max-_F)_"_{r})_=_card_((max-_G)_"_{r})_)_) percases ( 0 < r or r = 0 or r < 0 ) ; case 0 < r ; ::_thesis: card (Coim ((max- F),r)) = card (Coim ((max- G),r)) then ( Coim (F,(- r)) = (max- F) " {r} & Coim (G,(- r)) = (max- G) " {r} ) by Th38; hence card (Coim ((max- F),r)) = card (Coim ((max- G),r)) by A1, CLASSES1:def_9; ::_thesis: verum end; caseA3: r = 0 ; ::_thesis: card ((max- F) " {r}) = card ((max- G) " {r}) ( F " (right_closed_halfline 0) = (max- F) " {0} & G " (right_closed_halfline 0) = (max- G) " {0} ) by Th39; hence card ((max- F) " {r}) = card ((max- G) " {r}) by A1, A3, CLASSES1:78; ::_thesis: verum end; caseA4: r < 0 ; ::_thesis: card ((max- F) " {r}) = card ((max- G) " {r}) now__::_thesis:_not_r_in_rng_(max-_F) assume r in rng (max- F) ; ::_thesis: contradiction then ex d being Element of D st ( d in dom (max- F) & (max- F) . d = r ) by PARTFUN1:3; hence contradiction by A4, Th40; ::_thesis: verum end; then A5: (max- F) " {r} = {} by Lm2; now__::_thesis:_not_r_in_rng_(max-_G) assume r in rng (max- G) ; ::_thesis: contradiction then ex c being Element of C st ( c in dom (max- G) & (max- G) . c = r ) by PARTFUN1:3; hence contradiction by A4, Th40; ::_thesis: verum end; hence card ((max- F) " {r}) = card ((max- G) " {r}) by A5, Lm2; ::_thesis: verum end; end; end; hence card (Coim ((max- F),r)) = card (Coim ((max- G),r)) ; ::_thesis: verum end; ( rng (max- F) c= REAL & rng (max- G) c= REAL ) ; hence max- F, max- G are_fiberwise_equipotent by A2, CLASSES1:79; ::_thesis: verum end; registration let D be non empty set ; let F be finite PartFunc of D,REAL; cluster max+ F -> finite ; coherence max+ F is finite proof dom F is finite ; then dom (max+ F) is finite by Def10; hence max+ F is finite by FINSET_1:10; ::_thesis: verum end; cluster max- F -> finite ; coherence max- F is finite proof dom F is finite ; then dom (max- F) is finite by Def11; hence max- F is finite by FINSET_1:10; ::_thesis: verum end; end; theorem :: RFUNCT_3:43 for D, C being non empty set for F being finite PartFunc of D,REAL for G being finite PartFunc of C,REAL st max+ F, max+ G are_fiberwise_equipotent & max- F, max- G are_fiberwise_equipotent holds F,G are_fiberwise_equipotent proof let D, C be non empty set ; ::_thesis: for F being finite PartFunc of D,REAL for G being finite PartFunc of C,REAL st max+ F, max+ G are_fiberwise_equipotent & max- F, max- G are_fiberwise_equipotent holds F,G are_fiberwise_equipotent let F be finite PartFunc of D,REAL; ::_thesis: for G being finite PartFunc of C,REAL st max+ F, max+ G are_fiberwise_equipotent & max- F, max- G are_fiberwise_equipotent holds F,G are_fiberwise_equipotent let G be finite PartFunc of C,REAL; ::_thesis: ( max+ F, max+ G are_fiberwise_equipotent & max- F, max- G are_fiberwise_equipotent implies F,G are_fiberwise_equipotent ) assume that A1: max+ F, max+ G are_fiberwise_equipotent and A2: max- F, max- G are_fiberwise_equipotent ; ::_thesis: F,G are_fiberwise_equipotent set lh = left_closed_halfline 0; set rh = right_closed_halfline 0; set fp0 = (max+ F) " {0}; set fm0 = (max- F) " {0}; set gp0 = (max+ G) " {0}; set gm0 = (max- G) " {0}; A3: (left_closed_halfline 0) /\ (right_closed_halfline 0) = [.0,0.] by XXREAL_1:272 .= {0} by XXREAL_1:17 ; F " (rng F) c= F " REAL by RELAT_1:143; then A4: ( F " REAL c= dom F & dom F c= F " REAL ) by RELAT_1:132, RELAT_1:134; A5: ( F " (left_closed_halfline 0) = (max+ F) " {0} & F " (right_closed_halfline 0) = (max- F) " {0} ) by Th36, Th39; G " (rng G) c= G " REAL by RELAT_1:143; then A6: ( G " REAL c= dom G & dom G c= G " REAL ) by RELAT_1:132, RELAT_1:134; A7: ( G " (left_closed_halfline 0) = (max+ G) " {0} & G " (right_closed_halfline 0) = (max- G) " {0} ) by Th36, Th39; reconsider fp0 = (max+ F) " {0}, fm0 = (max- F) " {0}, gp0 = (max+ G) " {0}, gm0 = (max- G) " {0} as finite set ; A8: (left_closed_halfline 0) \/ (right_closed_halfline 0) = REAL \ ].0,0.[ by XXREAL_1:398 .= REAL \ {} by XXREAL_1:28 .= REAL ; then fp0 \/ fm0 = F " REAL by A5, RELAT_1:140; then A9: fp0 \/ fm0 = dom F by A4, XBOOLE_0:def_10; gp0 \/ gm0 = G " ((left_closed_halfline 0) \/ (right_closed_halfline 0)) by A7, RELAT_1:140; then A10: gp0 \/ gm0 = dom G by A8, A6, XBOOLE_0:def_10; card (fp0 \/ fm0) = ((card fp0) + (card fm0)) - (card (fp0 /\ fm0)) by CARD_2:45; then A11: card (fp0 /\ fm0) = ((card fp0) + (card fm0)) - (card (fp0 \/ fm0)) ; card (gp0 \/ gm0) = ((card gp0) + (card gm0)) - (card (gp0 /\ gm0)) by CARD_2:45; then A12: card (gp0 /\ gm0) = ((card gp0) + (card gm0)) - (card (gp0 \/ gm0)) ; A13: ( dom F = dom (max+ F) & dom G = dom (max+ G) ) by Def10; A14: now__::_thesis:_for_r_being_Real_holds_card_(Coim_(F,r))_=_card_(Coim_(G,r)) let r be Real; ::_thesis: card (Coim (F,r)) = card (Coim (G,r)) A15: ( card fp0 = card gp0 & card fm0 = card gm0 ) by A1, A2, CLASSES1:78; now__::_thesis:_(_(_0_<_r_&_card_(Coim_(F,r))_=_card_(Coim_(G,r))_)_or_(_0_=_r_&_card_(F_"_{r})_=_card_(G_"_{r})_)_or_(_r_<_0_&_card_(Coim_(F,r))_=_card_(Coim_(G,r))_)_) percases ( 0 < r or 0 = r or r < 0 ) ; case 0 < r ; ::_thesis: card (Coim (F,r)) = card (Coim (G,r)) then ( Coim (F,r) = Coim ((max+ F),r) & Coim (G,r) = Coim ((max+ G),r) ) by Th35; hence card (Coim (F,r)) = card (Coim (G,r)) by A1, CLASSES1:def_9; ::_thesis: verum end; case 0 = r ; ::_thesis: card (F " {r}) = card (G " {r}) then ( F " {r} = (F " (left_closed_halfline 0)) /\ (F " (right_closed_halfline 0)) & G " {r} = (G " (left_closed_halfline 0)) /\ (G " (right_closed_halfline 0)) ) by A3, FUNCT_1:68; hence card (F " {r}) = card (G " {r}) by A1, A13, A5, A7, A11, A12, A9, A10, A15, CLASSES1:81; ::_thesis: verum end; caseA16: r < 0 ; ::_thesis: card (Coim (F,r)) = card (Coim (G,r)) A17: - (- r) = r ; 0 < - r by A16, XREAL_1:58; then ( Coim (F,r) = Coim ((max- F),(- r)) & Coim (G,r) = Coim ((max- G),(- r)) ) by A17, Th38; hence card (Coim (F,r)) = card (Coim (G,r)) by A2, CLASSES1:def_9; ::_thesis: verum end; end; end; hence card (Coim (F,r)) = card (Coim (G,r)) ; ::_thesis: verum end; ( rng F c= REAL & rng G c= REAL ) ; hence F,G are_fiberwise_equipotent by A14, CLASSES1:79; ::_thesis: verum end; theorem Th44: :: RFUNCT_3:44 for D being non empty set for F being PartFunc of D,REAL for X being set holds (max+ F) | X = max+ (F | X) proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for X being set holds (max+ F) | X = max+ (F | X) let F be PartFunc of D,REAL; ::_thesis: for X being set holds (max+ F) | X = max+ (F | X) let X be set ; ::_thesis: (max+ F) | X = max+ (F | X) A1: dom ((max+ F) | X) = (dom (max+ F)) /\ X by RELAT_1:61; A2: (dom (max+ F)) /\ X = (dom F) /\ X by Def10 .= dom (F | X) by RELAT_1:61 ; A3: dom (F | X) = dom (max+ (F | X)) by Def10; now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_((max+_F)_|_X)_holds_ ((max+_F)_|_X)_._d_=_(max+_(F_|_X))_._d let d be Element of D; ::_thesis: ( d in dom ((max+ F) | X) implies ((max+ F) | X) . d = (max+ (F | X)) . d ) assume A4: d in dom ((max+ F) | X) ; ::_thesis: ((max+ F) | X) . d = (max+ (F | X)) . d then A5: d in dom (max+ F) by A1, XBOOLE_0:def_4; thus ((max+ F) | X) . d = (max+ F) . d by A4, FUNCT_1:47 .= max+ (F . d) by A5, Def10 .= max+ ((F | X) . d) by A1, A2, A4, FUNCT_1:47 .= (max+ (F | X)) . d by A1, A2, A3, A4, Def10 ; ::_thesis: verum end; hence (max+ F) | X = max+ (F | X) by A2, A3, PARTFUN1:5, RELAT_1:61; ::_thesis: verum end; theorem :: RFUNCT_3:45 for D being non empty set for F being PartFunc of D,REAL for X being set holds (max- F) | X = max- (F | X) proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for X being set holds (max- F) | X = max- (F | X) let F be PartFunc of D,REAL; ::_thesis: for X being set holds (max- F) | X = max- (F | X) let X be set ; ::_thesis: (max- F) | X = max- (F | X) A1: dom ((max- F) | X) = (dom (max- F)) /\ X by RELAT_1:61; A2: (dom (max- F)) /\ X = (dom F) /\ X by Def11 .= dom (F | X) by RELAT_1:61 ; A3: dom (F | X) = dom (max- (F | X)) by Def11; now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_((max-_F)_|_X)_holds_ ((max-_F)_|_X)_._d_=_(max-_(F_|_X))_._d let d be Element of D; ::_thesis: ( d in dom ((max- F) | X) implies ((max- F) | X) . d = (max- (F | X)) . d ) assume A4: d in dom ((max- F) | X) ; ::_thesis: ((max- F) | X) . d = (max- (F | X)) . d then A5: d in dom (max- F) by A1, XBOOLE_0:def_4; thus ((max- F) | X) . d = (max- F) . d by A4, FUNCT_1:47 .= max- (F . d) by A5, Def11 .= max- ((F | X) . d) by A1, A2, A4, FUNCT_1:47 .= (max- (F | X)) . d by A1, A2, A3, A4, Def11 ; ::_thesis: verum end; hence (max- F) | X = max- (F | X) by A2, A3, PARTFUN1:5, RELAT_1:61; ::_thesis: verum end; theorem Th46: :: RFUNCT_3:46 for D being non empty set for F being PartFunc of D,REAL st ( for d being Element of D st d in dom F holds F . d >= 0 ) holds max+ F = F proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL st ( for d being Element of D st d in dom F holds F . d >= 0 ) holds max+ F = F let F be PartFunc of D,REAL; ::_thesis: ( ( for d being Element of D st d in dom F holds F . d >= 0 ) implies max+ F = F ) A1: dom (max+ F) = dom F by Def10; assume A2: for d being Element of D st d in dom F holds F . d >= 0 ; ::_thesis: max+ F = F now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_F_holds_ (max+_F)_._d_=_F_._d let d be Element of D; ::_thesis: ( d in dom F implies (max+ F) . d = F . d ) assume A3: d in dom F ; ::_thesis: (max+ F) . d = F . d then A4: F . d >= 0 by A2; thus (max+ F) . d = max+ (F . d) by A1, A3, Def10 .= F . d by A4, XXREAL_0:def_10 ; ::_thesis: verum end; hence max+ F = F by A1, PARTFUN1:5; ::_thesis: verum end; theorem :: RFUNCT_3:47 for D being non empty set for F being PartFunc of D,REAL st ( for d being Element of D st d in dom F holds F . d <= 0 ) holds max- F = - F proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL st ( for d being Element of D st d in dom F holds F . d <= 0 ) holds max- F = - F let F be PartFunc of D,REAL; ::_thesis: ( ( for d being Element of D st d in dom F holds F . d <= 0 ) implies max- F = - F ) A1: dom (max- F) = dom F by Def11; assume A2: for d being Element of D st d in dom F holds F . d <= 0 ; ::_thesis: max- F = - F A3: now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_F_holds_ (max-_F)_._d_=_(-_F)_._d let d be Element of D; ::_thesis: ( d in dom F implies (max- F) . d = (- F) . d ) assume A4: d in dom F ; ::_thesis: (max- F) . d = (- F) . d then A5: F . d <= 0 by A2; thus (max- F) . d = max- (F . d) by A1, A4, Def11 .= - (F . d) by A5, XXREAL_0:def_10 .= (- F) . d by VALUED_1:8 ; ::_thesis: verum end; dom F = dom (- F) by VALUED_1:8; hence max- F = - F by A1, A3, PARTFUN1:5; ::_thesis: verum end; theorem Th48: :: RFUNCT_3:48 for D being non empty set for F being PartFunc of D,REAL holds F - 0 = F proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL holds F - 0 = F let F be PartFunc of D,REAL; ::_thesis: F - 0 = F A1: now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_F_holds_ (F_-_0)_._d_=_F_._d let d be Element of D; ::_thesis: ( d in dom F implies (F - 0) . d = F . d ) assume d in dom F ; ::_thesis: (F - 0) . d = F . d hence (F - 0) . d = (F . d) - 0 by VALUED_1:3 .= F . d ; ::_thesis: verum end; dom (F - 0) = dom F by VALUED_1:3; hence F - 0 = F by A1, PARTFUN1:5; ::_thesis: verum end; theorem :: RFUNCT_3:49 for D being non empty set for F being PartFunc of D,REAL for r being Real for X being set holds (F | X) - r = (F - r) | X proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for r being Real for X being set holds (F | X) - r = (F - r) | X let F be PartFunc of D,REAL; ::_thesis: for r being Real for X being set holds (F | X) - r = (F - r) | X let r be Real; ::_thesis: for X being set holds (F | X) - r = (F - r) | X let X be set ; ::_thesis: (F | X) - r = (F - r) | X A1: dom ((F | X) - r) = dom (F | X) by VALUED_1:3; A2: dom (F | X) = (dom F) /\ X by RELAT_1:61; A3: (dom F) /\ X = (dom (F - r)) /\ X by VALUED_1:3 .= dom ((F - r) | X) by RELAT_1:61 ; now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_((F_|_X)_-_r)_holds_ ((F_|_X)_-_r)_._d_=_((F_-_r)_|_X)_._d let d be Element of D; ::_thesis: ( d in dom ((F | X) - r) implies ((F | X) - r) . d = ((F - r) | X) . d ) assume A4: d in dom ((F | X) - r) ; ::_thesis: ((F | X) - r) . d = ((F - r) | X) . d then A5: d in dom F by A1, A2, XBOOLE_0:def_4; thus ((F | X) - r) . d = ((F | X) . d) - r by A1, A4, VALUED_1:3 .= (F . d) - r by A1, A4, FUNCT_1:47 .= (F - r) . d by A5, VALUED_1:3 .= ((F - r) | X) . d by A1, A2, A3, A4, FUNCT_1:47 ; ::_thesis: verum end; hence (F | X) - r = (F - r) | X by A2, A3, PARTFUN1:5, VALUED_1:3; ::_thesis: verum end; theorem Th50: :: RFUNCT_3:50 for D being non empty set for F being PartFunc of D,REAL for r, s being Real holds F " {(s + r)} = (F - r) " {s} proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for r, s being Real holds F " {(s + r)} = (F - r) " {s} let F be PartFunc of D,REAL; ::_thesis: for r, s being Real holds F " {(s + r)} = (F - r) " {s} let r, s be Real; ::_thesis: F " {(s + r)} = (F - r) " {s} thus F " {(s + r)} c= (F - r) " {s} :: according to XBOOLE_0:def_10 ::_thesis: (F - r) " {s} c= F " {(s + r)} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " {(s + r)} or x in (F - r) " {s} ) assume A1: x in F " {(s + r)} ; ::_thesis: x in (F - r) " {s} then reconsider d = x as Element of D ; A2: d in dom F by A1, FUNCT_1:def_7; F . d in {(s + r)} by A1, FUNCT_1:def_7; then F . d = s + r by TARSKI:def_1; then (F . d) - r = s ; then (F - r) . d = s by A2, VALUED_1:3; then A3: (F - r) . d in {s} by TARSKI:def_1; d in dom (F - r) by A2, VALUED_1:3; hence x in (F - r) " {s} by A3, FUNCT_1:def_7; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (F - r) " {s} or x in F " {(s + r)} ) assume A4: x in (F - r) " {s} ; ::_thesis: x in F " {(s + r)} then reconsider d = x as Element of D ; d in dom (F - r) by A4, FUNCT_1:def_7; then A5: d in dom F by VALUED_1:3; (F - r) . d in {s} by A4, FUNCT_1:def_7; then (F - r) . d = s by TARSKI:def_1; then (F . d) - r = s by A5, VALUED_1:3; then F . d in {(s + r)} by TARSKI:def_1; hence x in F " {(s + r)} by A5, FUNCT_1:def_7; ::_thesis: verum end; theorem :: RFUNCT_3:51 for D, C being non empty set for F being PartFunc of D,REAL for G being PartFunc of C,REAL for r being Real holds ( F,G are_fiberwise_equipotent iff F - r,G - r are_fiberwise_equipotent ) proof let D, C be non empty set ; ::_thesis: for F being PartFunc of D,REAL for G being PartFunc of C,REAL for r being Real holds ( F,G are_fiberwise_equipotent iff F - r,G - r are_fiberwise_equipotent ) let F be PartFunc of D,REAL; ::_thesis: for G being PartFunc of C,REAL for r being Real holds ( F,G are_fiberwise_equipotent iff F - r,G - r are_fiberwise_equipotent ) let G be PartFunc of C,REAL; ::_thesis: for r being Real holds ( F,G are_fiberwise_equipotent iff F - r,G - r are_fiberwise_equipotent ) let r be Real; ::_thesis: ( F,G are_fiberwise_equipotent iff F - r,G - r are_fiberwise_equipotent ) A1: ( rng (F - r) c= REAL & rng (G - r) c= REAL ) ; thus ( F,G are_fiberwise_equipotent implies F - r,G - r are_fiberwise_equipotent ) ::_thesis: ( F - r,G - r are_fiberwise_equipotent implies F,G are_fiberwise_equipotent ) proof assume A2: F,G are_fiberwise_equipotent ; ::_thesis: F - r,G - r are_fiberwise_equipotent now__::_thesis:_for_s_being_Real_holds_card_(Coim_((F_-_r),s))_=_card_(Coim_((G_-_r),s)) let s be Real; ::_thesis: card (Coim ((F - r),s)) = card (Coim ((G - r),s)) thus card (Coim ((F - r),s)) = card (Coim (F,(s + r))) by Th50 .= card (Coim (G,(s + r))) by A2, CLASSES1:def_9 .= card (Coim ((G - r),s)) by Th50 ; ::_thesis: verum end; hence F - r,G - r are_fiberwise_equipotent by A1, CLASSES1:79; ::_thesis: verum end; assume A3: F - r,G - r are_fiberwise_equipotent ; ::_thesis: F,G are_fiberwise_equipotent A4: now__::_thesis:_for_s_being_Real_holds_card_(Coim_(F,s))_=_card_(Coim_(G,s)) let s be Real; ::_thesis: card (Coim (F,s)) = card (Coim (G,s)) A5: s = (s - r) + r ; hence card (Coim (F,s)) = card (Coim ((F - r),(s - r))) by Th50 .= card (Coim ((G - r),(s - r))) by A3, CLASSES1:def_9 .= card (Coim (G,s)) by A5, Th50 ; ::_thesis: verum end; ( rng F c= REAL & rng G c= REAL ) ; hence F,G are_fiberwise_equipotent by A4, CLASSES1:79; ::_thesis: verum end; definition let F be PartFunc of REAL,REAL; let X be set ; predF is_convex_on X means :Def12: :: RFUNCT_3:def 12 ( X c= dom F & ( for p being Real st 0 <= p & p <= 1 holds for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) ) ); end; :: deftheorem Def12 defines is_convex_on RFUNCT_3:def_12_:_ for F being PartFunc of REAL,REAL for X being set holds ( F is_convex_on X iff ( X c= dom F & ( for p being Real st 0 <= p & p <= 1 holds for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) ) ) ); theorem Th52: :: RFUNCT_3:52 for a, b being Real for F being PartFunc of REAL,REAL holds ( F is_convex_on [.a,b.] iff ( [.a,b.] c= dom F & ( for p being Real st 0 <= p & p <= 1 holds for r, s being Real st r in [.a,b.] & s in [.a,b.] holds F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) ) ) ) proof let a, b be Real; ::_thesis: for F being PartFunc of REAL,REAL holds ( F is_convex_on [.a,b.] iff ( [.a,b.] c= dom F & ( for p being Real st 0 <= p & p <= 1 holds for r, s being Real st r in [.a,b.] & s in [.a,b.] holds F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) ) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_convex_on [.a,b.] iff ( [.a,b.] c= dom f & ( for p being Real st 0 <= p & p <= 1 holds for r, s being Real st r in [.a,b.] & s in [.a,b.] holds f . ((p * r) + ((1 - p) * s)) <= (p * (f . r)) + ((1 - p) * (f . s)) ) ) ) set ab = { r where r is Real : ( a <= r & r <= b ) } ; A1: [.a,b.] = { r where r is Real : ( a <= r & r <= b ) } by RCOMP_1:def_1; thus ( f is_convex_on [.a,b.] implies ( [.a,b.] c= dom f & ( for p being Real st 0 <= p & p <= 1 holds for x, y being Real st x in [.a,b.] & y in [.a,b.] holds f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ) ) ) ::_thesis: ( [.a,b.] c= dom f & ( for p being Real st 0 <= p & p <= 1 holds for r, s being Real st r in [.a,b.] & s in [.a,b.] holds f . ((p * r) + ((1 - p) * s)) <= (p * (f . r)) + ((1 - p) * (f . s)) ) implies f is_convex_on [.a,b.] ) proof assume A2: f is_convex_on [.a,b.] ; ::_thesis: ( [.a,b.] c= dom f & ( for p being Real st 0 <= p & p <= 1 holds for x, y being Real st x in [.a,b.] & y in [.a,b.] holds f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ) ) hence [.a,b.] c= dom f by Def12; ::_thesis: for p being Real st 0 <= p & p <= 1 holds for x, y being Real st x in [.a,b.] & y in [.a,b.] holds f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for x, y being Real st x in [.a,b.] & y in [.a,b.] holds f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ) assume that A3: 0 <= p and A4: p <= 1 ; ::_thesis: for x, y being Real st x in [.a,b.] & y in [.a,b.] holds f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) A5: 0 <= 1 - p by A4, XREAL_1:48; A6: (p * b) + ((1 - p) * b) = b ; A7: (p * a) + ((1 - p) * a) = a ; let x, y be Real; ::_thesis: ( x in [.a,b.] & y in [.a,b.] implies f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ) assume that A8: x in [.a,b.] and A9: y in [.a,b.] ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) A10: ex r2 being Real st ( r2 = y & a <= r2 & r2 <= b ) by A1, A9; then A11: (1 - p) * y <= (1 - p) * b by A5, XREAL_1:64; A12: ex r1 being Real st ( r1 = x & a <= r1 & r1 <= b ) by A1, A8; then p * x <= p * b by A3, XREAL_1:64; then A13: (p * x) + ((1 - p) * y) <= b by A11, A6, XREAL_1:7; A14: (1 - p) * a <= (1 - p) * y by A5, A10, XREAL_1:64; p * a <= p * x by A3, A12, XREAL_1:64; then a <= (p * x) + ((1 - p) * y) by A14, A7, XREAL_1:7; then (p * x) + ((1 - p) * y) in { r where r is Real : ( a <= r & r <= b ) } by A13; hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) by A1, A2, A3, A4, A8, A9, Def12; ::_thesis: verum end; assume that A15: [.a,b.] c= dom f and A16: for p being Real st 0 <= p & p <= 1 holds for x, y being Real st x in [.a,b.] & y in [.a,b.] holds f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ; ::_thesis: f is_convex_on [.a,b.] thus [.a,b.] c= dom f by A15; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds for r, s being Real st r in [.a,b.] & s in [.a,b.] & (p * r) + ((1 - p) * s) in [.a,b.] holds f . ((p * r) + ((1 - p) * s)) <= (p * (f . r)) + ((1 - p) * (f . s)) let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in [.a,b.] & s in [.a,b.] & (p * r) + ((1 - p) * s) in [.a,b.] holds f . ((p * r) + ((1 - p) * s)) <= (p * (f . r)) + ((1 - p) * (f . s)) ) assume A17: ( 0 <= p & p <= 1 ) ; ::_thesis: for r, s being Real st r in [.a,b.] & s in [.a,b.] & (p * r) + ((1 - p) * s) in [.a,b.] holds f . ((p * r) + ((1 - p) * s)) <= (p * (f . r)) + ((1 - p) * (f . s)) let x, y be Real; ::_thesis: ( x in [.a,b.] & y in [.a,b.] & (p * x) + ((1 - p) * y) in [.a,b.] implies f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ) assume that A18: ( x in [.a,b.] & y in [.a,b.] ) and (p * x) + ((1 - p) * y) in [.a,b.] ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) thus f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) by A16, A17, A18; ::_thesis: verum end; theorem :: RFUNCT_3:53 for a, b being Real for F being PartFunc of REAL,REAL holds ( F is_convex_on [.a,b.] iff ( [.a,b.] c= dom F & ( for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds ((F . x1) - (F . x2)) / (x1 - x2) <= ((F . x2) - (F . x3)) / (x2 - x3) ) ) ) proof let a, b be Real; ::_thesis: for F being PartFunc of REAL,REAL holds ( F is_convex_on [.a,b.] iff ( [.a,b.] c= dom F & ( for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds ((F . x1) - (F . x2)) / (x1 - x2) <= ((F . x2) - (F . x3)) / (x2 - x3) ) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_convex_on [.a,b.] iff ( [.a,b.] c= dom f & ( for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds ((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3) ) ) ) thus ( f is_convex_on [.a,b.] implies ( [.a,b.] c= dom f & ( for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds ((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3) ) ) ) ::_thesis: ( [.a,b.] c= dom f & ( for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds ((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3) ) implies f is_convex_on [.a,b.] ) proof assume A1: f is_convex_on [.a,b.] ; ::_thesis: ( [.a,b.] c= dom f & ( for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds ((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3) ) ) hence [.a,b.] c= dom f by Def12; ::_thesis: for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds ((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3) let x1, x2, x3 be Real; ::_thesis: ( x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 implies ((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3) ) assume that A2: ( x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] ) and A3: x1 < x2 and A4: x2 < x3 ; ::_thesis: ((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3) A5: x2 - x3 < 0 by A4, XREAL_1:49; set r = (x2 - x3) / (x1 - x3); A6: ( ((x2 - x3) / (x1 - x3)) * ((f . x2) - (f . x1)) = (((x2 - x3) / (x1 - x3)) * (f . x2)) - (((x2 - x3) / (x1 - x3)) * (f . x1)) & (1 - ((x2 - x3) / (x1 - x3))) * ((f . x3) - (f . x2)) = ((1 - ((x2 - x3) / (x1 - x3))) * (f . x3)) - ((1 - ((x2 - x3) / (x1 - x3))) * (f . x2)) ) ; A7: (x1 - x2) / (x1 - x3) = ((x1 - x3) ") * (x1 - x2) by XCMPLX_0:def_9; x1 < x3 by A3, A4, XXREAL_0:2; then A8: x1 - x3 < 0 by XREAL_1:49; A9: ((x2 - x3) / (x1 - x3)) + ((x1 - x2) / (x1 - x3)) = ((x1 - x2) + (x2 - x3)) / (x1 - x3) by XCMPLX_1:62 .= 1 by A8, XCMPLX_1:60 ; then A10: (((x2 - x3) / (x1 - x3)) * x1) + ((1 - ((x2 - x3) / (x1 - x3))) * x3) = ((x1 * (x2 - x3)) / (x1 - x3)) + (x3 * ((x1 - x2) / (x1 - x3))) by XCMPLX_1:74 .= ((x1 * (x2 - x3)) / (x1 - x3)) + ((x3 * (x1 - x2)) / (x1 - x3)) by XCMPLX_1:74 .= (((x2 * x1) - (x3 * x1)) + ((x1 - x2) * x3)) / (x1 - x3) by XCMPLX_1:62 .= (x2 * (x1 - x3)) / (x1 - x3) .= x2 by A8, XCMPLX_1:89 ; A11: x1 - x2 < 0 by A3, XREAL_1:49; then (x2 - x3) / (x1 - x3) <= 1 by A8, A9, XREAL_1:29, XREAL_1:140; then (((x2 - x3) / (x1 - x3)) * (f . x2)) + ((1 - ((x2 - x3) / (x1 - x3))) * (f . x2)) <= (((x2 - x3) / (x1 - x3)) * (f . x1)) + ((1 - ((x2 - x3) / (x1 - x3))) * (f . x3)) by A1, A2, A5, A8, A10, Def12; then ((x2 - x3) / (x1 - x3)) * ((f . x2) - (f . x1)) <= (1 - ((x2 - x3) / (x1 - x3))) * ((f . x3) - (f . x2)) by A6, XREAL_1:21; then - ((1 - ((x2 - x3) / (x1 - x3))) * ((f . x3) - (f . x2))) <= - (((x2 - x3) / (x1 - x3)) * ((f . x2) - (f . x1))) by XREAL_1:24; then (1 - ((x2 - x3) / (x1 - x3))) * (- ((f . x3) - (f . x2))) <= ((x2 - x3) / (x1 - x3)) * (- ((f . x2) - (f . x1))) ; then (((x1 - x3) ") * (x1 - x2)) * ((f . x2) - (f . x3)) <= (((x1 - x3) ") * (x2 - x3)) * ((f . x1) - (f . x2)) by A9, A7, XCMPLX_0:def_9; then A12: (x1 - x3) * ((((x1 - x3) ") * (x2 - x3)) * ((f . x1) - (f . x2))) <= (x1 - x3) * ((((x1 - x3) ") * (x1 - x2)) * ((f . x2) - (f . x3))) by A8, XREAL_1:65; set v = (x1 - x2) * ((f . x2) - (f . x3)); set u = (x2 - x3) * ((f . x1) - (f . x2)); A13: (x1 - x3) * ((((x1 - x3) ") * (x1 - x2)) * ((f . x2) - (f . x3))) = ((x1 - x3) * ((x1 - x3) ")) * ((x1 - x2) * ((f . x2) - (f . x3))) .= 1 * ((x1 - x2) * ((f . x2) - (f . x3))) by A8, XCMPLX_0:def_7 .= (x1 - x2) * ((f . x2) - (f . x3)) ; (x1 - x3) * ((((x1 - x3) ") * (x2 - x3)) * ((f . x1) - (f . x2))) = ((x1 - x3) * ((x1 - x3) ")) * ((x2 - x3) * ((f . x1) - (f . x2))) .= 1 * ((x2 - x3) * ((f . x1) - (f . x2))) by A8, XCMPLX_0:def_7 .= (x2 - x3) * ((f . x1) - (f . x2)) ; hence ((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3) by A5, A11, A12, A13, XREAL_1:103; ::_thesis: verum end; assume that A14: [.a,b.] c= dom f and A15: for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds ((f . x1) - (f . x2)) / (x1 - x2) <= ((f . x2) - (f . x3)) / (x2 - x3) ; ::_thesis: f is_convex_on [.a,b.] now__::_thesis:_for_p_being_Real_st_0_<=_p_&_p_<=_1_holds_ for_x,_y_being_Real_st_x_in_[.a,b.]_&_y_in_[.a,b.]_holds_ f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y)) let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for x, y being Real st x in [.a,b.] & y in [.a,b.] holds f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ) assume that A16: 0 <= p and A17: p <= 1 ; ::_thesis: for x, y being Real st x in [.a,b.] & y in [.a,b.] holds f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) A18: 0 <= 1 - p by A17, XREAL_1:48; let x, y be Real; ::_thesis: ( x in [.a,b.] & y in [.a,b.] implies f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ) set r = (p * x) + ((1 - p) * y); assume that A19: x in [.a,b.] and A20: y in [.a,b.] ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) A21: (p * y) + ((1 - p) * y) = y ; A22: { s where s is Real : ( a <= s & s <= b ) } = [.a,b.] by RCOMP_1:def_1; then A23: ex t being Real st ( t = y & a <= t & t <= b ) by A20; then A24: (1 - p) * y <= (1 - p) * b by A18, XREAL_1:64; A25: ex t being Real st ( t = x & a <= t & t <= b ) by A22, A19; then ( (p * b) + ((1 - p) * b) = b & p * x <= p * b ) by A16, XREAL_1:64; then A26: (p * x) + ((1 - p) * y) <= b by A24, XREAL_1:7; A27: (1 - p) * a <= (1 - p) * y by A18, A23, XREAL_1:64; ( (p * a) + ((1 - p) * a) = a & p * a <= p * x ) by A16, A25, XREAL_1:64; then a <= (p * x) + ((1 - p) * y) by A27, XREAL_1:7; then A28: (p * x) + ((1 - p) * y) in [.a,b.] by A22, A26; A29: (p * x) + ((1 - p) * x) = x ; now__::_thesis:_(_(_x_=_y_&_f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))_)_or_(_x_<>_y_&_f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))_)_) percases ( x = y or x <> y ) ; case x = y ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ; ::_thesis: verum end; caseA30: x <> y ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) now__::_thesis:_(_(_p_=_0_&_f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))_)_or_(_p_<>_0_&_f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))_)_) percases ( p = 0 or p <> 0 ) ; case p = 0 ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ; ::_thesis: verum end; caseA31: p <> 0 ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) now__::_thesis:_(_(_p_=_1_&_f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))_)_or_(_p_<>_1_&_f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))_)_) percases ( p = 1 or p <> 1 ) ; case p = 1 ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ; ::_thesis: verum end; case p <> 1 ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) then p < 1 by A17, XXREAL_0:1; then A32: 0 < 1 - p by XREAL_1:50; now__::_thesis:_(_(_x_<_y_&_f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))_)_or_(_y_<_x_&_f_._((p_*_x)_+_((1_-_p)_*_y))_<=_(p_*_(f_._x))_+_((1_-_p)_*_(f_._y))_)_) percases ( x < y or y < x ) by A30, XXREAL_0:1; caseA33: x < y ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) then (1 - p) * x < (1 - p) * y by A32, XREAL_1:68; then A34: x < (p * x) + ((1 - p) * y) by A29, XREAL_1:8; then A35: x - ((p * x) + ((1 - p) * y)) < 0 by XREAL_1:49; p * x < p * y by A16, A31, A33, XREAL_1:68; then A36: (p * x) + ((1 - p) * y) < y by A21, XREAL_1:8; then A37: ((p * x) + ((1 - p) * y)) - y < 0 by XREAL_1:49; A38: x - y < 0 by A33, XREAL_1:49; ((f . x) - (f . ((p * x) + ((1 - p) * y)))) / (x - ((p * x) + ((1 - p) * y))) <= ((f . ((p * x) + ((1 - p) * y))) - (f . y)) / (((p * x) + ((1 - p) * y)) - y) by A15, A19, A20, A28, A36, A34; then ((f . x) - (f . ((p * x) + ((1 - p) * y)))) * (p * (x - y)) <= ((f . ((p * x) + ((1 - p) * y))) - (f . y)) * ((1 - p) * (x - y)) by A37, A35, XREAL_1:107; then ((((f . x) - (f . ((p * x) + ((1 - p) * y)))) * p) * (x - y)) / (x - y) >= ((((f . ((p * x) + ((1 - p) * y))) - (f . y)) * (1 - p)) * (x - y)) / (x - y) by A38, XREAL_1:73; then ((((f . ((p * x) + ((1 - p) * y))) - (f . y)) * (1 - p)) * (x - y)) / (x - y) <= ((f . x) - (f . ((p * x) + ((1 - p) * y)))) * p by A38, XCMPLX_1:89; then ((f . ((p * x) + ((1 - p) * y))) * (1 - p)) - ((f . y) * (1 - p)) <= ((f . x) * p) - ((f . ((p * x) + ((1 - p) * y))) * p) by A38, XCMPLX_1:89; then ((f . ((p * x) + ((1 - p) * y))) * p) + (((f . ((p * x) + ((1 - p) * y))) * (1 - p)) - ((f . y) * (1 - p))) <= (f . x) * p by XREAL_1:19; then (((f . ((p * x) + ((1 - p) * y))) * p) + ((f . ((p * x) + ((1 - p) * y))) * (1 - p))) - ((f . y) * (1 - p)) <= (f . x) * p ; hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) by XREAL_1:20; ::_thesis: verum end; caseA39: y < x ; ::_thesis: f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) then (1 - p) * y < (1 - p) * x by A32, XREAL_1:68; then A40: (p * x) + ((1 - p) * y) < x by A29, XREAL_1:8; then A41: ((p * x) + ((1 - p) * y)) - x < 0 by XREAL_1:49; p * y < p * x by A16, A31, A39, XREAL_1:68; then A42: y < (p * x) + ((1 - p) * y) by A21, XREAL_1:8; then A43: y - ((p * x) + ((1 - p) * y)) < 0 by XREAL_1:49; A44: y - x < 0 by A39, XREAL_1:49; ((f . y) - (f . ((p * x) + ((1 - p) * y)))) / (y - ((p * x) + ((1 - p) * y))) <= ((f . ((p * x) + ((1 - p) * y))) - (f . x)) / (((p * x) + ((1 - p) * y)) - x) by A15, A19, A20, A28, A42, A40; then ((f . y) - (f . ((p * x) + ((1 - p) * y)))) * ((1 - p) * (y - x)) <= ((f . ((p * x) + ((1 - p) * y))) - (f . x)) * (p * (y - x)) by A43, A41, XREAL_1:107; then ((((f . y) - (f . ((p * x) + ((1 - p) * y)))) * (1 - p)) * (y - x)) / (y - x) >= ((((f . ((p * x) + ((1 - p) * y))) - (f . x)) * p) * (y - x)) / (y - x) by A44, XREAL_1:73; then ((((f . ((p * x) + ((1 - p) * y))) - (f . x)) * p) * (y - x)) / (y - x) <= ((f . y) - (f . ((p * x) + ((1 - p) * y)))) * (1 - p) by A44, XCMPLX_1:89; then ((f . ((p * x) + ((1 - p) * y))) * p) - ((f . x) * p) <= ((f . y) * (1 - p)) - ((f . ((p * x) + ((1 - p) * y))) * (1 - p)) by A44, XCMPLX_1:89; then (((f . ((p * x) + ((1 - p) * y))) * p) - ((f . x) * p)) + ((f . ((p * x) + ((1 - p) * y))) * (1 - p)) <= (f . y) * (1 - p) by XREAL_1:19; then (((f . ((p * x) + ((1 - p) * y))) * p) + ((f . ((p * x) + ((1 - p) * y))) * (1 - p))) - ((f . x) * p) <= (f . y) * (1 - p) ; hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) by XREAL_1:20; ::_thesis: verum end; end; end; hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ; ::_thesis: verum end; end; end; hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ; ::_thesis: verum end; end; end; hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ; ::_thesis: verum end; end; end; hence f . ((p * x) + ((1 - p) * y)) <= (p * (f . x)) + ((1 - p) * (f . y)) ; ::_thesis: verum end; hence f is_convex_on [.a,b.] by A14, Th52; ::_thesis: verum end; theorem :: RFUNCT_3:54 for F being PartFunc of REAL,REAL for X, Y being set st F is_convex_on X & Y c= X holds F is_convex_on Y proof let F be PartFunc of REAL,REAL; ::_thesis: for X, Y being set st F is_convex_on X & Y c= X holds F is_convex_on Y let X, Y be set ; ::_thesis: ( F is_convex_on X & Y c= X implies F is_convex_on Y ) assume that A1: F is_convex_on X and A2: Y c= X ; ::_thesis: F is_convex_on Y X c= dom F by A1, Def12; hence Y c= dom F by A2, XBOOLE_1:1; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds for r, s being Real st r in Y & s in Y & (p * r) + ((1 - p) * s) in Y holds F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in Y & s in Y & (p * r) + ((1 - p) * s) in Y holds F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) ) assume A3: ( 0 <= p & p <= 1 ) ; ::_thesis: for r, s being Real st r in Y & s in Y & (p * r) + ((1 - p) * s) in Y holds F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) let x, y be Real; ::_thesis: ( x in Y & y in Y & (p * x) + ((1 - p) * y) in Y implies F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) ) assume ( x in Y & y in Y & (p * x) + ((1 - p) * y) in Y ) ; ::_thesis: F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) hence F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) by A1, A2, A3, Def12; ::_thesis: verum end; theorem :: RFUNCT_3:55 for F being PartFunc of REAL,REAL for X being set for r being Real holds ( F is_convex_on X iff F - r is_convex_on X ) proof let F be PartFunc of REAL,REAL; ::_thesis: for X being set for r being Real holds ( F is_convex_on X iff F - r is_convex_on X ) let X be set ; ::_thesis: for r being Real holds ( F is_convex_on X iff F - r is_convex_on X ) let r be Real; ::_thesis: ( F is_convex_on X iff F - r is_convex_on X ) A1: dom F = dom (F - r) by VALUED_1:3; thus ( F is_convex_on X implies F - r is_convex_on X ) ::_thesis: ( F - r is_convex_on X implies F is_convex_on X ) proof assume A2: F is_convex_on X ; ::_thesis: F - r is_convex_on X hence A3: X c= dom (F - r) by A1, Def12; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds (F - r) . ((p * r) + ((1 - p) * s)) <= (p * ((F - r) . r)) + ((1 - p) * ((F - r) . s)) let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds (F - r) . ((p * r) + ((1 - p) * s)) <= (p * ((F - r) . r)) + ((1 - p) * ((F - r) . s)) ) assume A4: ( 0 <= p & p <= 1 ) ; ::_thesis: for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds (F - r) . ((p * r) + ((1 - p) * s)) <= (p * ((F - r) . r)) + ((1 - p) * ((F - r) . s)) let x, y be Real; ::_thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies (F - r) . ((p * x) + ((1 - p) * y)) <= (p * ((F - r) . x)) + ((1 - p) * ((F - r) . y)) ) assume that A5: x in X and A6: y in X and A7: (p * x) + ((1 - p) * y) in X ; ::_thesis: (F - r) . ((p * x) + ((1 - p) * y)) <= (p * ((F - r) . x)) + ((1 - p) * ((F - r) . y)) F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) by A2, A4, A5, A6, A7, Def12; then A8: (F . ((p * x) + ((1 - p) * y))) - r <= ((p * (F . x)) + ((1 - p) * (F . y))) - r by XREAL_1:9; ((p * (F . x)) + ((1 - p) * (F . y))) - r = (p * ((F . x) - r)) + ((1 - p) * ((F . y) - r)) .= (p * ((F - r) . x)) + ((1 - p) * ((F . y) - r)) by A1, A3, A5, VALUED_1:3 .= (p * ((F - r) . x)) + ((1 - p) * ((F - r) . y)) by A1, A3, A6, VALUED_1:3 ; hence (F - r) . ((p * x) + ((1 - p) * y)) <= (p * ((F - r) . x)) + ((1 - p) * ((F - r) . y)) by A1, A3, A7, A8, VALUED_1:3; ::_thesis: verum end; assume A9: F - r is_convex_on X ; ::_thesis: F is_convex_on X hence A10: X c= dom F by A1, Def12; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) ) assume A11: ( 0 <= p & p <= 1 ) ; ::_thesis: for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) let x, y be Real; ::_thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) ) assume that A12: x in X and A13: y in X and A14: (p * x) + ((1 - p) * y) in X ; ::_thesis: F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) (F - r) . ((p * x) + ((1 - p) * y)) <= (p * ((F - r) . x)) + ((1 - p) * ((F - r) . y)) by A9, A11, A12, A13, A14, Def12; then A15: (F . ((p * x) + ((1 - p) * y))) - r <= (p * ((F - r) . x)) + ((1 - p) * ((F - r) . y)) by A10, A14, VALUED_1:3; (p * ((F - r) . x)) + ((1 - p) * ((F - r) . y)) = (p * ((F - r) . x)) + ((1 - p) * ((F . y) - r)) by A10, A13, VALUED_1:3 .= (p * ((F . x) - r)) + (((1 - p) * (F . y)) - ((1 - p) * r)) by A10, A12, VALUED_1:3 .= ((p * (F . x)) + ((1 - p) * (F . y))) - r ; hence F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) by A15, XREAL_1:9; ::_thesis: verum end; theorem :: RFUNCT_3:56 for F being PartFunc of REAL,REAL for X being set for r being Real st 0 < r holds ( F is_convex_on X iff r (#) F is_convex_on X ) proof let F be PartFunc of REAL,REAL; ::_thesis: for X being set for r being Real st 0 < r holds ( F is_convex_on X iff r (#) F is_convex_on X ) let X be set ; ::_thesis: for r being Real st 0 < r holds ( F is_convex_on X iff r (#) F is_convex_on X ) let r be Real; ::_thesis: ( 0 < r implies ( F is_convex_on X iff r (#) F is_convex_on X ) ) assume A1: 0 < r ; ::_thesis: ( F is_convex_on X iff r (#) F is_convex_on X ) A2: dom F = dom (r (#) F) by VALUED_1:def_5; thus ( F is_convex_on X implies r (#) F is_convex_on X ) ::_thesis: ( r (#) F is_convex_on X implies F is_convex_on X ) proof assume A3: F is_convex_on X ; ::_thesis: r (#) F is_convex_on X then A4: X c= dom F by Def12; thus X c= dom (r (#) F) by A2, A3, Def12; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds (r (#) F) . ((p * r) + ((1 - p) * s)) <= (p * ((r (#) F) . r)) + ((1 - p) * ((r (#) F) . s)) let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds (r (#) F) . ((p * r) + ((1 - p) * s)) <= (p * ((r (#) F) . r)) + ((1 - p) * ((r (#) F) . s)) ) assume A5: ( 0 <= p & p <= 1 ) ; ::_thesis: for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds (r (#) F) . ((p * r) + ((1 - p) * s)) <= (p * ((r (#) F) . r)) + ((1 - p) * ((r (#) F) . s)) let x, y be Real; ::_thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies (r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((r (#) F) . x)) + ((1 - p) * ((r (#) F) . y)) ) assume that A6: x in X and A7: y in X and A8: (p * x) + ((1 - p) * y) in X ; ::_thesis: (r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((r (#) F) . x)) + ((1 - p) * ((r (#) F) . y)) F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) by A3, A5, A6, A7, A8, Def12; then r * (F . ((p * x) + ((1 - p) * y))) <= r * ((p * (F . x)) + ((1 - p) * (F . y))) by A1, XREAL_1:64; then (r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * (r * (F . x))) + (((1 - p) * r) * (F . y)) by A2, A4, A8, VALUED_1:def_5; then (r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((r (#) F) . x)) + ((1 - p) * (r * (F . y))) by A2, A4, A6, VALUED_1:def_5; hence (r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((r (#) F) . x)) + ((1 - p) * ((r (#) F) . y)) by A2, A4, A7, VALUED_1:def_5; ::_thesis: verum end; assume A9: r (#) F is_convex_on X ; ::_thesis: F is_convex_on X then A10: X c= dom (r (#) F) by Def12; hence X c= dom F by VALUED_1:def_5; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) ) assume A11: ( 0 <= p & p <= 1 ) ; ::_thesis: for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) let x, y be Real; ::_thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) ) assume that A12: x in X and A13: y in X and A14: (p * x) + ((1 - p) * y) in X ; ::_thesis: F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) (r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((r (#) F) . x)) + ((1 - p) * ((r (#) F) . y)) by A9, A11, A12, A13, A14, Def12; then r * (F . ((p * x) + ((1 - p) * y))) <= (p * ((r (#) F) . x)) + ((1 - p) * ((r (#) F) . y)) by A10, A14, VALUED_1:def_5; then r * (F . ((p * x) + ((1 - p) * y))) <= (p * (r * (F . x))) + ((1 - p) * ((r (#) F) . y)) by A10, A12, VALUED_1:def_5; then r * (F . ((p * x) + ((1 - p) * y))) <= (p * (r * (F . x))) + ((1 - p) * (r * (F . y))) by A10, A13, VALUED_1:def_5; then (r * (F . ((p * x) + ((1 - p) * y)))) / r <= (r * ((p * (F . x)) + ((1 - p) * (F . y)))) / r by A1, XREAL_1:72; then F . ((p * x) + ((1 - p) * y)) <= (r * ((p * (F . x)) + ((1 - p) * (F . y)))) / r by A1, XCMPLX_1:89; hence F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) by A1, XCMPLX_1:89; ::_thesis: verum end; theorem :: RFUNCT_3:57 for F being PartFunc of REAL,REAL for X being set st X c= dom F holds 0 (#) F is_convex_on X proof let F be PartFunc of REAL,REAL; ::_thesis: for X being set st X c= dom F holds 0 (#) F is_convex_on X let X be set ; ::_thesis: ( X c= dom F implies 0 (#) F is_convex_on X ) assume A1: X c= dom F ; ::_thesis: 0 (#) F is_convex_on X hence X c= dom (0 (#) F) by VALUED_1:def_5; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds (0 (#) F) . ((p * r) + ((1 - p) * s)) <= (p * ((0 (#) F) . r)) + ((1 - p) * ((0 (#) F) . s)) let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds (0 (#) F) . ((p * r) + ((1 - p) * s)) <= (p * ((0 (#) F) . r)) + ((1 - p) * ((0 (#) F) . s)) ) assume that 0 <= p and p <= 1 ; ::_thesis: for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds (0 (#) F) . ((p * r) + ((1 - p) * s)) <= (p * ((0 (#) F) . r)) + ((1 - p) * ((0 (#) F) . s)) let x, y be Real; ::_thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies (0 (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((0 (#) F) . x)) + ((1 - p) * ((0 (#) F) . y)) ) assume that A2: x in X and A3: y in X and A4: (p * x) + ((1 - p) * y) in X ; ::_thesis: (0 (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((0 (#) F) . x)) + ((1 - p) * ((0 (#) F) . y)) A5: dom F = dom (0 (#) F) by VALUED_1:def_5; then A6: (0 (#) F) . ((p * x) + ((1 - p) * y)) = 0 * (F . ((p * x) + ((1 - p) * y))) by A1, A4, VALUED_1:def_5 .= 0 ; (p * ((0 (#) F) . x)) + ((1 - p) * ((0 (#) F) . y)) = (p * (0 * (F . x))) + ((1 - p) * ((0 (#) F) . y)) by A1, A5, A2, VALUED_1:def_5 .= (p * 0) + ((1 - p) * (0 * (F . y))) by A1, A5, A3, VALUED_1:def_5 .= 0 + 0 ; hence (0 (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((0 (#) F) . x)) + ((1 - p) * ((0 (#) F) . y)) by A6; ::_thesis: verum end; theorem :: RFUNCT_3:58 for F, G being PartFunc of REAL,REAL for X being set st F is_convex_on X & G is_convex_on X holds F + G is_convex_on X proof let F, G be PartFunc of REAL,REAL; ::_thesis: for X being set st F is_convex_on X & G is_convex_on X holds F + G is_convex_on X let X be set ; ::_thesis: ( F is_convex_on X & G is_convex_on X implies F + G is_convex_on X ) A1: dom (F + G) = (dom F) /\ (dom G) by VALUED_1:def_1; assume A2: ( F is_convex_on X & G is_convex_on X ) ; ::_thesis: F + G is_convex_on X then ( X c= dom F & X c= dom G ) by Def12; hence A3: X c= dom (F + G) by A1, XBOOLE_1:19; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds (F + G) . ((p * r) + ((1 - p) * s)) <= (p * ((F + G) . r)) + ((1 - p) * ((F + G) . s)) let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds (F + G) . ((p * r) + ((1 - p) * s)) <= (p * ((F + G) . r)) + ((1 - p) * ((F + G) . s)) ) assume A4: ( 0 <= p & p <= 1 ) ; ::_thesis: for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds (F + G) . ((p * r) + ((1 - p) * s)) <= (p * ((F + G) . r)) + ((1 - p) * ((F + G) . s)) let x, y be Real; ::_thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies (F + G) . ((p * x) + ((1 - p) * y)) <= (p * ((F + G) . x)) + ((1 - p) * ((F + G) . y)) ) assume that A5: x in X and A6: y in X and A7: (p * x) + ((1 - p) * y) in X ; ::_thesis: (F + G) . ((p * x) + ((1 - p) * y)) <= (p * ((F + G) . x)) + ((1 - p) * ((F + G) . y)) ( F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) & G . ((p * x) + ((1 - p) * y)) <= (p * (G . x)) + ((1 - p) * (G . y)) ) by A2, A4, A5, A6, A7, Def12; then (F . ((p * x) + ((1 - p) * y))) + (G . ((p * x) + ((1 - p) * y))) <= ((p * (F . x)) + ((1 - p) * (F . y))) + ((p * (G . x)) + ((1 - p) * (G . y))) by XREAL_1:7; then A8: (F + G) . ((p * x) + ((1 - p) * y)) <= ((p * (F . x)) + ((1 - p) * (F . y))) + ((p * (G . x)) + ((1 - p) * (G . y))) by A3, A7, VALUED_1:def_1; ((p * (F . x)) + ((1 - p) * (F . y))) + ((p * (G . x)) + ((1 - p) * (G . y))) = ((p * ((F . x) + (G . x))) + ((1 - p) * (F . y))) + ((1 - p) * (G . y)) .= ((p * ((F + G) . x)) + ((1 - p) * (F . y))) + ((1 - p) * (G . y)) by A3, A5, VALUED_1:def_1 .= (p * ((F + G) . x)) + ((1 - p) * ((F . y) + (G . y))) ; hence (F + G) . ((p * x) + ((1 - p) * y)) <= (p * ((F + G) . x)) + ((1 - p) * ((F + G) . y)) by A3, A6, A8, VALUED_1:def_1; ::_thesis: verum end; theorem Th59: :: RFUNCT_3:59 for F being PartFunc of REAL,REAL for X being set for r being Real st F is_convex_on X holds max+ (F - r) is_convex_on X proof let F be PartFunc of REAL,REAL; ::_thesis: for X being set for r being Real st F is_convex_on X holds max+ (F - r) is_convex_on X let X be set ; ::_thesis: for r being Real st F is_convex_on X holds max+ (F - r) is_convex_on X let r be Real; ::_thesis: ( F is_convex_on X implies max+ (F - r) is_convex_on X ) assume A1: F is_convex_on X ; ::_thesis: max+ (F - r) is_convex_on X then A2: X c= dom F by Def12; A3: ( dom F = dom (F - r) & dom (max+ (F - r)) = dom (F - r) ) by Def10, VALUED_1:3; hence X c= dom (max+ (F - r)) by A1, Def12; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds (max+ (F - r)) . ((p * r) + ((1 - p) * s)) <= (p * ((max+ (F - r)) . r)) + ((1 - p) * ((max+ (F - r)) . s)) let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds (max+ (F - r)) . ((p * r) + ((1 - p) * s)) <= (p * ((max+ (F - r)) . r)) + ((1 - p) * ((max+ (F - r)) . s)) ) assume that A4: 0 <= p and A5: p <= 1 ; ::_thesis: for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds (max+ (F - r)) . ((p * r) + ((1 - p) * s)) <= (p * ((max+ (F - r)) . r)) + ((1 - p) * ((max+ (F - r)) . s)) let x, y be Real; ::_thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies (max+ (F - r)) . ((p * x) + ((1 - p) * y)) <= (p * ((max+ (F - r)) . x)) + ((1 - p) * ((max+ (F - r)) . y)) ) assume that A6: x in X and A7: y in X and A8: (p * x) + ((1 - p) * y) in X ; ::_thesis: (max+ (F - r)) . ((p * x) + ((1 - p) * y)) <= (p * ((max+ (F - r)) . x)) + ((1 - p) * ((max+ (F - r)) . y)) F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) by A1, A4, A5, A6, A7, A8, Def12; then (F . ((p * x) + ((1 - p) * y))) - r <= ((p * (F . x)) + ((1 - p) * (F . y))) - r by XREAL_1:9; then A9: max+ ((F . ((p * x) + ((1 - p) * y))) - r) <= max ((((p * (F . x)) + ((1 - p) * (F . y))) - r),0) by XXREAL_0:26; 0 + p <= 1 by A5; then 0 <= 1 - p by XREAL_1:19; then A10: max+ ((1 - p) * ((F - r) . y)) = (1 - p) * (max+ ((F - r) . y)) by Th4; A11: max+ ((p * ((F - r) . x)) + ((1 - p) * ((F - r) . y))) <= (max+ (p * ((F - r) . x))) + (max+ ((1 - p) * ((F - r) . y))) by Th5; A12: max+ (p * ((F - r) . x)) = p * (max+ ((F - r) . x)) by A4, Th4; ((p * (F . x)) + ((1 - p) * (F . y))) - r = (p * ((F . x) - r)) + ((1 - p) * ((F . y) - r)) .= (p * ((F - r) . x)) + ((1 - p) * ((F . y) - r)) by A6, A2, VALUED_1:3 .= (p * ((F - r) . x)) + ((1 - p) * ((F - r) . y)) by A7, A2, VALUED_1:3 ; then max+ ((F . ((p * x) + ((1 - p) * y))) - r) <= (p * (max+ ((F - r) . x))) + ((1 - p) * (max+ ((F - r) . y))) by A9, A11, A12, A10, XXREAL_0:2; then max+ ((F - r) . ((p * x) + ((1 - p) * y))) <= (p * (max+ ((F - r) . x))) + ((1 - p) * (max+ ((F - r) . y))) by A8, A2, VALUED_1:3; then (max+ (F - r)) . ((p * x) + ((1 - p) * y)) <= (p * (max+ ((F - r) . x))) + ((1 - p) * (max+ ((F - r) . y))) by A3, A8, A2, Def10; then (max+ (F - r)) . ((p * x) + ((1 - p) * y)) <= (p * ((max+ (F - r)) . x)) + ((1 - p) * (max+ ((F - r) . y))) by A3, A6, A2, Def10; hence (max+ (F - r)) . ((p * x) + ((1 - p) * y)) <= (p * ((max+ (F - r)) . x)) + ((1 - p) * ((max+ (F - r)) . y)) by A3, A7, A2, Def10; ::_thesis: verum end; theorem :: RFUNCT_3:60 for F being PartFunc of REAL,REAL for X being set st F is_convex_on X holds max+ F is_convex_on X proof let F be PartFunc of REAL,REAL; ::_thesis: for X being set st F is_convex_on X holds max+ F is_convex_on X let X be set ; ::_thesis: ( F is_convex_on X implies max+ F is_convex_on X ) assume F is_convex_on X ; ::_thesis: max+ F is_convex_on X then max+ (F - 0) is_convex_on X by Th59; hence max+ F is_convex_on X by Th48; ::_thesis: verum end; theorem Th61: :: RFUNCT_3:61 id ([#] REAL) is_convex_on REAL proof set i = id ([#] REAL); thus REAL c= dom (id ([#] REAL)) by FUNCT_1:17; :: according to RFUNCT_3:def_12 ::_thesis: for p being Real st 0 <= p & p <= 1 holds for r, s being Real st r in REAL & s in REAL & (p * r) + ((1 - p) * s) in REAL holds (id ([#] REAL)) . ((p * r) + ((1 - p) * s)) <= (p * ((id ([#] REAL)) . r)) + ((1 - p) * ((id ([#] REAL)) . s)) let p be Real; ::_thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in REAL & s in REAL & (p * r) + ((1 - p) * s) in REAL holds (id ([#] REAL)) . ((p * r) + ((1 - p) * s)) <= (p * ((id ([#] REAL)) . r)) + ((1 - p) * ((id ([#] REAL)) . s)) ) assume that 0 <= p and p <= 1 ; ::_thesis: for r, s being Real st r in REAL & s in REAL & (p * r) + ((1 - p) * s) in REAL holds (id ([#] REAL)) . ((p * r) + ((1 - p) * s)) <= (p * ((id ([#] REAL)) . r)) + ((1 - p) * ((id ([#] REAL)) . s)) let x, y be Real; ::_thesis: ( x in REAL & y in REAL & (p * x) + ((1 - p) * y) in REAL implies (id ([#] REAL)) . ((p * x) + ((1 - p) * y)) <= (p * ((id ([#] REAL)) . x)) + ((1 - p) * ((id ([#] REAL)) . y)) ) assume that x in REAL and y in REAL and (p * x) + ((1 - p) * y) in REAL ; ::_thesis: (id ([#] REAL)) . ((p * x) + ((1 - p) * y)) <= (p * ((id ([#] REAL)) . x)) + ((1 - p) * ((id ([#] REAL)) . y)) ( (id ([#] REAL)) . x = x & (id ([#] REAL)) . y = y ) by FUNCT_1:17; hence (id ([#] REAL)) . ((p * x) + ((1 - p) * y)) <= (p * ((id ([#] REAL)) . x)) + ((1 - p) * ((id ([#] REAL)) . y)) by FUNCT_1:17; ::_thesis: verum end; theorem :: RFUNCT_3:62 for r being Real holds max+ ((id ([#] REAL)) - r) is_convex_on REAL by Th59, Th61; definition let D be non empty set ; let F be PartFunc of D,REAL; let X be set ; assume A1: dom (F | X) is finite ; func FinS (F,X) -> non-increasing FinSequence of REAL means :Def13: :: RFUNCT_3:def 13 F | X,it are_fiberwise_equipotent ; existence ex b1 being non-increasing FinSequence of REAL st F | X,b1 are_fiberwise_equipotent proof set x = dom (F | X); consider b being FinSequence such that A2: F | (dom (F | X)),b are_fiberwise_equipotent by A1, RFINSEQ:5; rng (F | (dom (F | X))) = rng b by A2, CLASSES1:75; then reconsider b = b as FinSequence of REAL by FINSEQ_1:def_4; consider a being non-increasing FinSequence of REAL such that A3: b,a are_fiberwise_equipotent by RFINSEQ:22; take a ; ::_thesis: F | X,a are_fiberwise_equipotent dom (F | X) = (dom F) /\ X by RELAT_1:61; then F | (dom (F | X)) = (F | (dom F)) | X by RELAT_1:71 .= F | X by RELAT_1:68 ; hence F | X,a are_fiberwise_equipotent by A2, A3, CLASSES1:76; ::_thesis: verum end; uniqueness for b1, b2 being non-increasing FinSequence of REAL st F | X,b1 are_fiberwise_equipotent & F | X,b2 are_fiberwise_equipotent holds b1 = b2 by CLASSES1:76, RFINSEQ:23; end; :: deftheorem Def13 defines FinS RFUNCT_3:def_13_:_ for D being non empty set for F being PartFunc of D,REAL for X being set st dom (F | X) is finite holds for b4 being non-increasing FinSequence of REAL holds ( b4 = FinS (F,X) iff F | X,b4 are_fiberwise_equipotent ); theorem Th63: :: RFUNCT_3:63 for D being non empty set for F being PartFunc of D,REAL for X being set st dom (F | X) is finite holds FinS (F,(dom (F | X))) = FinS (F,X) proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for X being set st dom (F | X) is finite holds FinS (F,(dom (F | X))) = FinS (F,X) let F be PartFunc of D,REAL; ::_thesis: for X being set st dom (F | X) is finite holds FinS (F,(dom (F | X))) = FinS (F,X) let X be set ; ::_thesis: ( dom (F | X) is finite implies FinS (F,(dom (F | X))) = FinS (F,X) ) A1: F | (dom (F | X)) = F | ((dom F) /\ X) by RELAT_1:61 .= (F | (dom F)) | X by RELAT_1:71 .= F | X by RELAT_1:68 ; assume A2: dom (F | X) is finite ; ::_thesis: FinS (F,(dom (F | X))) = FinS (F,X) then FinS (F,X),F | X are_fiberwise_equipotent by Def13; hence FinS (F,(dom (F | X))) = FinS (F,X) by A2, A1, Def13; ::_thesis: verum end; theorem Th64: :: RFUNCT_3:64 for D being non empty set for F being PartFunc of D,REAL for X being set st dom (F | X) is finite holds FinS ((F | X),X) = FinS (F,X) proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for X being set st dom (F | X) is finite holds FinS ((F | X),X) = FinS (F,X) let F be PartFunc of D,REAL; ::_thesis: for X being set st dom (F | X) is finite holds FinS ((F | X),X) = FinS (F,X) let X be set ; ::_thesis: ( dom (F | X) is finite implies FinS ((F | X),X) = FinS (F,X) ) A1: (F | X) | X = F | X by RELAT_1:72; assume A2: dom (F | X) is finite ; ::_thesis: FinS ((F | X),X) = FinS (F,X) then FinS (F,X),F | X are_fiberwise_equipotent by Def13; hence FinS ((F | X),X) = FinS (F,X) by A2, A1, Def13; ::_thesis: verum end; theorem Th65: :: RFUNCT_3:65 for D being non empty set for d being Element of D for X being set for F being PartFunc of D,REAL st X is finite & d in dom (F | X) holds (FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent proof for D being non empty set for X being finite set for F being PartFunc of D,REAL for x being set st x in dom (F | X) holds (FinS (F,(X \ {x}))) ^ <*(F . x)*>,F | X are_fiberwise_equipotent proof let D be non empty set ; ::_thesis: for X being finite set for F being PartFunc of D,REAL for x being set st x in dom (F | X) holds (FinS (F,(X \ {x}))) ^ <*(F . x)*>,F | X are_fiberwise_equipotent let X be finite set ; ::_thesis: for F being PartFunc of D,REAL for x being set st x in dom (F | X) holds (FinS (F,(X \ {x}))) ^ <*(F . x)*>,F | X are_fiberwise_equipotent let F be PartFunc of D,REAL; ::_thesis: for x being set st x in dom (F | X) holds (FinS (F,(X \ {x}))) ^ <*(F . x)*>,F | X are_fiberwise_equipotent let x be set ; ::_thesis: ( x in dom (F | X) implies (FinS (F,(X \ {x}))) ^ <*(F . x)*>,F | X are_fiberwise_equipotent ) set Y = X \ {x}; set A = (FinS (F,(X \ {x}))) ^ <*(F . x)*>; assume x in dom (F | X) ; ::_thesis: (FinS (F,(X \ {x}))) ^ <*(F . x)*>,F | X are_fiberwise_equipotent then A1: x in (dom F) /\ X by RELAT_1:61; then x in X by XBOOLE_0:def_4; then A2: {x} c= X by ZFMISC_1:31; x in dom F by A1, XBOOLE_0:def_4; then A3: {x} c= dom F by ZFMISC_1:31; dom (F | (X \ {x})) is finite ; then A4: F | (X \ {x}), FinS (F,(X \ {x})) are_fiberwise_equipotent by Def13; now__::_thesis:_for_y_being_set_holds_card_(Coim_((F_|_X),y))_=_card_(Coim_(((FinS_(F,(X_\_{x})))_^_<*(F_._x)*>),y)) let y be set ; ::_thesis: card (Coim ((F | X),y)) = card (Coim (((FinS (F,(X \ {x}))) ^ <*(F . x)*>),y)) A5: X \ {x} misses {x} by XBOOLE_1:79; A6: card (Coim ((F | (X \ {x})),y)) = card (Coim ((FinS (F,(X \ {x}))),y)) by A4, CLASSES1:def_9; A7: dom (F | {x}) = {x} by A3, RELAT_1:62; A8: dom <*(F . x)*> = {1} by FINSEQ_1:2, FINSEQ_1:38; A9: now__::_thesis:_(_(_y_=_F_._x_&_card_((F_|_{x})_"_{y})_=_card_(<*(F_._x)*>_"_{y})_)_or_(_y_<>_F_._x_&_card_((F_|_{x})_"_{y})_=_card_(<*(F_._x)*>_"_{y})_)_) percases ( y = F . x or y <> F . x ) ; caseA10: y = F . x ; ::_thesis: card ((F | {x}) " {y}) = card (<*(F . x)*> " {y}) A11: {x} c= (F | {x}) " {y} proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in {x} or z in (F | {x}) " {y} ) A12: y in {y} by TARSKI:def_1; assume A13: z in {x} ; ::_thesis: z in (F | {x}) " {y} then z = x by TARSKI:def_1; then y = (F | {x}) . z by A7, A10, A13, FUNCT_1:47; hence z in (F | {x}) " {y} by A7, A13, A12, FUNCT_1:def_7; ::_thesis: verum end; (F | {x}) " {y} c= {x} by A7, RELAT_1:132; then (F | {x}) " {y} = {x} by A11, XBOOLE_0:def_10; then A14: card ((F | {x}) " {y}) = 1 by CARD_1:30; A15: {1} c= <*(F . x)*> " {y} proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in {1} or z in <*(F . x)*> " {y} ) A16: y in {y} by TARSKI:def_1; assume A17: z in {1} ; ::_thesis: z in <*(F . x)*> " {y} then z = 1 by TARSKI:def_1; then y = <*(F . x)*> . z by A10, FINSEQ_1:40; hence z in <*(F . x)*> " {y} by A8, A17, A16, FUNCT_1:def_7; ::_thesis: verum end; <*(F . x)*> " {y} c= {1} by A8, RELAT_1:132; then <*(F . x)*> " {y} = {1} by A15, XBOOLE_0:def_10; hence card ((F | {x}) " {y}) = card (<*(F . x)*> " {y}) by A14, CARD_1:30; ::_thesis: verum end; caseA18: y <> F . x ; ::_thesis: card ((F | {x}) " {y}) = card (<*(F . x)*> " {y}) A19: now__::_thesis:_not_<*(F_._x)*>_"_{y}_<>_{} set z = the Element of <*(F . x)*> " {y}; assume A20: <*(F . x)*> " {y} <> {} ; ::_thesis: contradiction then <*(F . x)*> . the Element of <*(F . x)*> " {y} in {y} by FUNCT_1:def_7; then A21: <*(F . x)*> . the Element of <*(F . x)*> " {y} = y by TARSKI:def_1; the Element of <*(F . x)*> " {y} in {1} by A8, A20, FUNCT_1:def_7; then the Element of <*(F . x)*> " {y} = 1 by TARSKI:def_1; hence contradiction by A18, A21, FINSEQ_1:40; ::_thesis: verum end; now__::_thesis:_not_(F_|_{x})_"_{y}_<>_{} set z = the Element of (F | {x}) " {y}; assume A22: (F | {x}) " {y} <> {} ; ::_thesis: contradiction then (F | {x}) . the Element of (F | {x}) " {y} in {y} by FUNCT_1:def_7; then A23: (F | {x}) . the Element of (F | {x}) " {y} = y by TARSKI:def_1; A24: the Element of (F | {x}) " {y} in {x} by A7, A22, FUNCT_1:def_7; then the Element of (F | {x}) " {y} = x by TARSKI:def_1; hence contradiction by A7, A18, A24, A23, FUNCT_1:47; ::_thesis: verum end; hence card ((F | {x}) " {y}) = card (<*(F . x)*> " {y}) by A19; ::_thesis: verum end; end; end; A25: ((F | (X \ {x})) " {y}) \/ ((F | {x}) " {y}) = ((X \ {x}) /\ (F " {y})) \/ ((F | {x}) " {y}) by FUNCT_1:70 .= ((X \ {x}) /\ (F " {y})) \/ ({x} /\ (F " {y})) by FUNCT_1:70 .= ((X \ {x}) \/ {x}) /\ (F " {y}) by XBOOLE_1:23 .= (X \/ {x}) /\ (F " {y}) by XBOOLE_1:39 .= X /\ (F " {y}) by A2, XBOOLE_1:12 .= (F | X) " {y} by FUNCT_1:70 ; ((F | (X \ {x})) " {y}) /\ ((F | {x}) " {y}) = ((X \ {x}) /\ (F " {y})) /\ ((F | {x}) " {y}) by FUNCT_1:70 .= ((X \ {x}) /\ (F " {y})) /\ ({x} /\ (F " {y})) by FUNCT_1:70 .= (((F " {y}) /\ (X \ {x})) /\ {x}) /\ (F " {y}) by XBOOLE_1:16 .= ((F " {y}) /\ ((X \ {x}) /\ {x})) /\ (F " {y}) by XBOOLE_1:16 .= {} /\ (F " {y}) by A5, XBOOLE_0:def_7 .= {} ; hence card (Coim ((F | X),y)) = ((card ((F | (X \ {x})) " {y})) + (card (<*(F . x)*> " {y}))) - (card {}) by A25, A9, CARD_2:45 .= card (Coim (((FinS (F,(X \ {x}))) ^ <*(F . x)*>),y)) by A6, FINSEQ_3:57 ; ::_thesis: verum end; hence (FinS (F,(X \ {x}))) ^ <*(F . x)*>,F | X are_fiberwise_equipotent by CLASSES1:def_9; ::_thesis: verum end; hence for D being non empty set for d being Element of D for X being set for F being PartFunc of D,REAL st X is finite & d in dom (F | X) holds (FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent ; ::_thesis: verum end; theorem Th66: :: RFUNCT_3:66 for D being non empty set for d being Element of D for X being set for F being PartFunc of D,REAL st dom (F | X) is finite & d in dom (F | X) holds (FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent proof let D be non empty set ; ::_thesis: for d being Element of D for X being set for F being PartFunc of D,REAL st dom (F | X) is finite & d in dom (F | X) holds (FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent let d be Element of D; ::_thesis: for X being set for F being PartFunc of D,REAL st dom (F | X) is finite & d in dom (F | X) holds (FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent let X be set ; ::_thesis: for F being PartFunc of D,REAL st dom (F | X) is finite & d in dom (F | X) holds (FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent let F be PartFunc of D,REAL; ::_thesis: ( dom (F | X) is finite & d in dom (F | X) implies (FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent ) set dx = dom (F | X); assume that A1: dom (F | X) is finite and A2: d in dom (F | X) ; ::_thesis: (FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent set Y = X \ {d}; set dy = dom (F | (X \ {d})); A3: dom (F | (X \ {d})) = (dom F) /\ (X \ {d}) by RELAT_1:61; A4: dom (F | X) = (dom F) /\ X by RELAT_1:61; A5: dom (F | (X \ {d})) = (dom (F | X)) \ {d} proof thus dom (F | (X \ {d})) c= (dom (F | X)) \ {d} :: according to XBOOLE_0:def_10 ::_thesis: (dom (F | X)) \ {d} c= dom (F | (X \ {d})) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dom (F | (X \ {d})) or y in (dom (F | X)) \ {d} ) assume A6: y in dom (F | (X \ {d})) ; ::_thesis: y in (dom (F | X)) \ {d} then y in X \ {d} by A3, XBOOLE_0:def_4; then A7: not y in {d} by XBOOLE_0:def_5; y in dom F by A3, A6, XBOOLE_0:def_4; then y in dom (F | X) by A3, A4, A6, XBOOLE_0:def_4; hence y in (dom (F | X)) \ {d} by A7, XBOOLE_0:def_5; ::_thesis: verum end; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in (dom (F | X)) \ {d} or y in dom (F | (X \ {d})) ) assume A8: y in (dom (F | X)) \ {d} ; ::_thesis: y in dom (F | (X \ {d})) then A9: not y in {d} by XBOOLE_0:def_5; A10: y in dom (F | X) by A8, XBOOLE_0:def_5; then y in X by A4, XBOOLE_0:def_4; then A11: y in X \ {d} by A9, XBOOLE_0:def_5; y in dom F by A4, A10, XBOOLE_0:def_4; hence y in dom (F | (X \ {d})) by A3, A11, XBOOLE_0:def_4; ::_thesis: verum end; F | (dom (F | X)) = F | ((dom F) /\ X) by RELAT_1:61 .= (F | (dom F)) | X by RELAT_1:71 .= F | X by RELAT_1:68 ; then (FinS (F,((dom (F | X)) \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent by A1, A2, Th65; hence (FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent by A1, A5, Th63; ::_thesis: verum end; theorem Th67: :: RFUNCT_3:67 for D being non empty set for F being PartFunc of D,REAL for X being set for Y being finite set st Y = dom (F | X) holds len (FinS (F,X)) = card Y proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for X being set for Y being finite set st Y = dom (F | X) holds len (FinS (F,X)) = card Y let F be PartFunc of D,REAL; ::_thesis: for X being set for Y being finite set st Y = dom (F | X) holds len (FinS (F,X)) = card Y let X be set ; ::_thesis: for Y being finite set st Y = dom (F | X) holds len (FinS (F,X)) = card Y let Y be finite set ; ::_thesis: ( Y = dom (F | X) implies len (FinS (F,X)) = card Y ) reconsider fs = dom (FinS (F,X)) as finite set ; A1: dom (FinS (F,X)) = Seg (len (FinS (F,X))) by FINSEQ_1:def_3; assume A2: Y = dom (F | X) ; ::_thesis: len (FinS (F,X)) = card Y FinS (F,X),F | X are_fiberwise_equipotent by A2, Def13; hence card Y = card fs by A2, CLASSES1:81 .= len (FinS (F,X)) by A1, FINSEQ_1:57 ; ::_thesis: verum end; theorem Th68: :: RFUNCT_3:68 for D being non empty set for F being PartFunc of D,REAL holds FinS (F,{}) = <*> REAL proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL holds FinS (F,{}) = <*> REAL let F be PartFunc of D,REAL; ::_thesis: FinS (F,{}) = <*> REAL dom (F | {}) = (dom F) /\ {} by RELAT_1:61 .= {} ; then len (FinS (F,{})) = 0 by Th67, CARD_1:27; hence FinS (F,{}) = <*> REAL ; ::_thesis: verum end; theorem Th69: :: RFUNCT_3:69 for D being non empty set for F being PartFunc of D,REAL for d being Element of D st d in dom F holds FinS (F,{d}) = <*(F . d)*> proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for d being Element of D st d in dom F holds FinS (F,{d}) = <*(F . d)*> let F be PartFunc of D,REAL; ::_thesis: for d being Element of D st d in dom F holds FinS (F,{d}) = <*(F . d)*> let d be Element of D; ::_thesis: ( d in dom F implies FinS (F,{d}) = <*(F . d)*> ) assume d in dom F ; ::_thesis: FinS (F,{d}) = <*(F . d)*> then {d} c= dom F by ZFMISC_1:31; then A1: {d} = (dom F) /\ {d} by XBOOLE_1:28 .= dom (F | {d}) by RELAT_1:61 ; then FinS (F,{d}),F | {d} are_fiberwise_equipotent by Def13; then A2: rng (FinS (F,{d})) = rng (F | {d}) by CLASSES1:75; A3: rng (F | {d}) = {(F . d)} proof thus rng (F | {d}) c= {(F . d)} :: according to XBOOLE_0:def_10 ::_thesis: {(F . d)} c= rng (F | {d}) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (F | {d}) or x in {(F . d)} ) assume x in rng (F | {d}) ; ::_thesis: x in {(F . d)} then consider e being Element of D such that A4: e in dom (F | {d}) and A5: (F | {d}) . e = x by PARTFUN1:3; e = d by A1, A4, TARSKI:def_1; then x = F . d by A4, A5, FUNCT_1:47; hence x in {(F . d)} by TARSKI:def_1; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(F . d)} or x in rng (F | {d}) ) A6: d in dom (F | {d}) by A1, TARSKI:def_1; assume x in {(F . d)} ; ::_thesis: x in rng (F | {d}) then x = F . d by TARSKI:def_1; then x = (F | {d}) . d by A6, FUNCT_1:47; hence x in rng (F | {d}) by A6, FUNCT_1:def_3; ::_thesis: verum end; len (FinS (F,{d})) = card {d} by A1, Th67 .= 1 by CARD_1:30 ; hence FinS (F,{d}) = <*(F . d)*> by A2, A3, FINSEQ_1:39; ::_thesis: verum end; theorem Th70: :: RFUNCT_3:70 for D being non empty set for F being PartFunc of D,REAL for X being set for d being Element of D st dom (F | X) is finite & d in dom (F | X) & (FinS (F,X)) . (len (FinS (F,X))) = F . d holds FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*> proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for X being set for d being Element of D st dom (F | X) is finite & d in dom (F | X) & (FinS (F,X)) . (len (FinS (F,X))) = F . d holds FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*> let F be PartFunc of D,REAL; ::_thesis: for X being set for d being Element of D st dom (F | X) is finite & d in dom (F | X) & (FinS (F,X)) . (len (FinS (F,X))) = F . d holds FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*> let X be set ; ::_thesis: for d being Element of D st dom (F | X) is finite & d in dom (F | X) & (FinS (F,X)) . (len (FinS (F,X))) = F . d holds FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*> let d be Element of D; ::_thesis: ( dom (F | X) is finite & d in dom (F | X) & (FinS (F,X)) . (len (FinS (F,X))) = F . d implies FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*> ) set dx = dom (F | X); set fx = FinS (F,X); set fy = FinS (F,(X \ {d})); assume that A1: dom (F | X) is finite and A2: d in dom (F | X) and A3: (FinS (F,X)) . (len (FinS (F,X))) = F . d ; ::_thesis: FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*> A4: FinS (F,X),F | X are_fiberwise_equipotent by A1, Def13; then rng (FinS (F,X)) = rng (F | X) by CLASSES1:75; then FinS (F,X) <> {} by A2, FUNCT_1:3, RELAT_1:38; then 0 + 1 <= len (FinS (F,X)) by NAT_1:13; then max (0,((len (FinS (F,X))) - 1)) = (len (FinS (F,X))) - 1 by FINSEQ_2:4; then reconsider n = (len (FinS (F,X))) - 1 as Element of NAT by FINSEQ_2:5; len (FinS (F,X)) = n + 1 ; then A5: FinS (F,X) = ((FinS (F,X)) | n) ^ <*(F . d)*> by A3, RFINSEQ:7; A6: (FinS (F,X)) | n is non-increasing by RFINSEQ:20; (FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent by A1, A2, Th66; then FinS (F,X),(FinS (F,(X \ {d}))) ^ <*(F . d)*> are_fiberwise_equipotent by A4, CLASSES1:76; then FinS (F,(X \ {d})),(FinS (F,X)) | n are_fiberwise_equipotent by A5, RFINSEQ:1; hence FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*> by A5, A6, RFINSEQ:23; ::_thesis: verum end; defpred S1[ Element of NAT ] means for D being non empty set for F being PartFunc of D,REAL for X, Y being set for Z being finite set st Z = dom (F | Y) & dom (F | X) is finite & Y c= X & $1 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds F . d1 >= F . d2 ) holds FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))); Lm3: S1[ 0 ] proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for X, Y being set for Z being finite set st Z = dom (F | Y) & dom (F | X) is finite & Y c= X & 0 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds F . d1 >= F . d2 ) holds FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) let F be PartFunc of D,REAL; ::_thesis: for X, Y being set for Z being finite set st Z = dom (F | Y) & dom (F | X) is finite & Y c= X & 0 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds F . d1 >= F . d2 ) holds FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) let X, Y be set ; ::_thesis: for Z being finite set st Z = dom (F | Y) & dom (F | X) is finite & Y c= X & 0 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds F . d1 >= F . d2 ) holds FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) let Z be finite set ; ::_thesis: ( Z = dom (F | Y) & dom (F | X) is finite & Y c= X & 0 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds F . d1 >= F . d2 ) implies FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) ) assume A1: Z = dom (F | Y) ; ::_thesis: ( not dom (F | X) is finite or not Y c= X or not 0 = card Z or ex d1, d2 being Element of D st ( d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) & not F . d1 >= F . d2 ) or FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) ) assume that A2: dom (F | X) is finite and Y c= X and A3: 0 = card Z and for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds F . d1 >= F . d2 ; ::_thesis: FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) A4: dom (F | Y) = {} by A1, A3; A5: dom (F | (X \ Y)) = (dom F) /\ (X \ Y) by RELAT_1:61; ( dom (F | X) = (dom F) /\ X & dom (F | Y) = (dom F) /\ Y ) by RELAT_1:61; then dom (F | (X \ Y)) = (dom (F | X)) \ {} by A5, A4, XBOOLE_1:50 .= dom (F | X) ; then A6: FinS (F,(X \ Y)) = FinS (F,(dom (F | X))) by A2, Th63 .= FinS (F,X) by A2, Th63 ; FinS (F,Y) = FinS (F,{}) by A4, Th63 .= {} by Th68 ; hence FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) by A6, FINSEQ_1:34; ::_thesis: verum end; Lm4: for n being Element of NAT st S1[n] holds S1[n + 1] proof let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A1: S1[n] ; ::_thesis: S1[n + 1] let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for X, Y being set for Z being finite set st Z = dom (F | Y) & dom (F | X) is finite & Y c= X & n + 1 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds F . d1 >= F . d2 ) holds FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) let F be PartFunc of D,REAL; ::_thesis: for X, Y being set for Z being finite set st Z = dom (F | Y) & dom (F | X) is finite & Y c= X & n + 1 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds F . d1 >= F . d2 ) holds FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) let X, Y be set ; ::_thesis: for Z being finite set st Z = dom (F | Y) & dom (F | X) is finite & Y c= X & n + 1 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds F . d1 >= F . d2 ) holds FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) set dx = dom (F | X); set dxy = dom (F | (X \ Y)); set fy = FinS (F,Y); set fxy = FinS (F,(X \ Y)); let dy be finite set ; ::_thesis: ( dy = dom (F | Y) & dom (F | X) is finite & Y c= X & n + 1 = card dy & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds F . d1 >= F . d2 ) implies FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) ) assume that A2: dy = dom (F | Y) and A3: dom (F | X) is finite and A4: Y c= X and A5: n + 1 = card dy and A6: for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds F . d1 >= F . d2 ; ::_thesis: FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) A7: len (FinS (F,Y)) = n + 1 by A2, A5, Th67; A8: F | Y, FinS (F,Y) are_fiberwise_equipotent by A2, Def13; then A9: rng (FinS (F,Y)) = rng (F | Y) by CLASSES1:75; 0 + 1 <= n + 1 by NAT_1:13; then A10: len (FinS (F,Y)) in dom (FinS (F,Y)) by A7, FINSEQ_3:25; then (FinS (F,Y)) . (len (FinS (F,Y))) in rng (FinS (F,Y)) by FUNCT_1:def_3; then consider d being Element of D such that A11: d in dy and A12: (F | Y) . d = (FinS (F,Y)) . (len (FinS (F,Y))) by A2, A9, PARTFUN1:3; A13: dom (F | (X \ Y)) = (dom F) /\ (X \ Y) by RELAT_1:61; A14: dy = (dom F) /\ Y by A2, RELAT_1:61; then A15: d in Y by A11, XBOOLE_0:def_4; then A16: {d} c= X by A4, ZFMISC_1:31; A17: d in dom F by A14, A11, XBOOLE_0:def_4; then A18: {d} c= dom F by ZFMISC_1:31; A19: {d} c= Y by A15, ZFMISC_1:31; A20: (FinS (F,(X \ Y))) ^ <*(F . d)*>,<*(F . d)*> ^ (FinS (F,(X \ Y))) are_fiberwise_equipotent by RFINSEQ:2; set Yd = Y \ {d}; set dyd = dom (F | (Y \ {d})); set xyd = dom (F | (X \ (Y \ {d}))); A21: dom (F | (X \ (Y \ {d}))) = (dom F) /\ (X \ (Y \ {d})) by RELAT_1:61; A22: dom (F | (Y \ {d})) = (dom F) /\ (Y \ {d}) by RELAT_1:61; A23: dom (F | (Y \ {d})) = dy \ {d} proof thus dom (F | (Y \ {d})) c= dy \ {d} :: according to XBOOLE_0:def_10 ::_thesis: dy \ {d} c= dom (F | (Y \ {d})) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dom (F | (Y \ {d})) or y in dy \ {d} ) assume A24: y in dom (F | (Y \ {d})) ; ::_thesis: y in dy \ {d} then y in Y \ {d} by A22, XBOOLE_0:def_4; then A25: not y in {d} by XBOOLE_0:def_5; y in dom F by A22, A24, XBOOLE_0:def_4; then y in dy by A14, A22, A24, XBOOLE_0:def_4; hence y in dy \ {d} by A25, XBOOLE_0:def_5; ::_thesis: verum end; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dy \ {d} or y in dom (F | (Y \ {d})) ) assume A26: y in dy \ {d} ; ::_thesis: y in dom (F | (Y \ {d})) then ( not y in {d} & y in Y ) by A14, XBOOLE_0:def_4, XBOOLE_0:def_5; then A27: y in Y \ {d} by XBOOLE_0:def_5; y in dom F by A14, A26, XBOOLE_0:def_4; hence y in dom (F | (Y \ {d})) by A22, A27, XBOOLE_0:def_4; ::_thesis: verum end; A28: F . d = (FinS (F,Y)) . (len (FinS (F,Y))) by A2, A11, A12, FUNCT_1:47; then A29: FinS (F,Y) = ((FinS (F,Y)) | n) ^ <*(F . d)*> by A7, RFINSEQ:7; reconsider dyd = dom (F | (Y \ {d})) as finite set by A23; A30: X \ (Y \ {d}) = (X \ Y) \/ (X /\ {d}) by XBOOLE_1:52 .= (X \ Y) \/ {d} by A16, XBOOLE_1:28 ; then A31: dom (F | (X \ (Y \ {d}))) = (dom (F | (X \ Y))) \/ ((dom F) /\ {d}) by A13, A21, XBOOLE_1:23 .= (dom (F | (X \ Y))) \/ {d} by A18, XBOOLE_1:28 ; A32: now__::_thesis:_for_d1,_d2_being_Element_of_D_st_d1_in_dyd_&_d2_in_dom_(F_|_(X_\_(Y_\_{d})))_holds_ F_._d1_>=_F_._d2 let d1, d2 be Element of D; ::_thesis: ( d1 in dyd & d2 in dom (F | (X \ (Y \ {d}))) implies F . d1 >= F . d2 ) assume that A33: d1 in dyd and A34: d2 in dom (F | (X \ (Y \ {d}))) ; ::_thesis: F . d1 >= F . d2 now__::_thesis:_(_(_d2_in_dom_(F_|_(X_\_Y))_&_F_._d1_>=_F_._d2_)_or_(_d2_in_{d}_&_F_._d1_>=_F_._d2_)_) percases ( d2 in dom (F | (X \ Y)) or d2 in {d} ) by A31, A34, XBOOLE_0:def_3; case d2 in dom (F | (X \ Y)) ; ::_thesis: F . d1 >= F . d2 hence F . d1 >= F . d2 by A2, A6, A23, A33; ::_thesis: verum end; case d2 in {d} ; ::_thesis: F . d1 >= F . d2 then A35: d2 = d by TARSKI:def_1; (F | Y) . d1 in rng (F | Y) by A2, A23, A33, FUNCT_1:def_3; then F . d1 in rng (F | Y) by A2, A23, A33, FUNCT_1:47; then consider m being Nat such that A36: m in dom (FinS (F,Y)) and A37: (FinS (F,Y)) . m = F . d1 by A9, FINSEQ_2:10; A38: m <= len (FinS (F,Y)) by A36, FINSEQ_3:25; now__::_thesis:_(_(_m_=_len_(FinS_(F,Y))_&_F_._d1_>=_F_._d2_)_or_(_m_<>_len_(FinS_(F,Y))_&_F_._d1_>=_F_._d2_)_) percases ( m = len (FinS (F,Y)) or m <> len (FinS (F,Y)) ) ; case m = len (FinS (F,Y)) ; ::_thesis: F . d1 >= F . d2 hence F . d1 >= F . d2 by A2, A11, A12, A35, A37, FUNCT_1:47; ::_thesis: verum end; case m <> len (FinS (F,Y)) ; ::_thesis: F . d1 >= F . d2 then m < len (FinS (F,Y)) by A38, XXREAL_0:1; hence F . d1 >= F . d2 by A10, A28, A35, A36, A37, RFINSEQ:19; ::_thesis: verum end; end; end; hence F . d1 >= F . d2 ; ::_thesis: verum end; end; end; hence F . d1 >= F . d2 ; ::_thesis: verum end; dom (F | X) = (dom F) /\ X by RELAT_1:61; then A39: dom (F | (X \ Y)) is finite by A3, A13, FINSET_1:1, XBOOLE_1:26; then F | (X \ Y), FinS (F,(X \ Y)) are_fiberwise_equipotent by Def13; then A40: rng (FinS (F,(X \ Y))) = rng (F | (X \ Y)) by CLASSES1:75; A41: <*(F . d)*> ^ (FinS (F,(X \ Y))) is non-increasing proof set xfy = <*(F . d)*> ^ (FinS (F,(X \ Y))); let n be Element of NAT ; :: according to RFINSEQ:def_3 ::_thesis: ( not n in dom (<*(F . d)*> ^ (FinS (F,(X \ Y)))) or not n + 1 in dom (<*(F . d)*> ^ (FinS (F,(X \ Y)))) or (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1) <= (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n ) assume that A42: n in dom (<*(F . d)*> ^ (FinS (F,(X \ Y)))) and A43: n + 1 in dom (<*(F . d)*> ^ (FinS (F,(X \ Y)))) ; ::_thesis: (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1) <= (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n A44: 1 <= n by A42, FINSEQ_3:25; then max (0,(n - 1)) = n - 1 by FINSEQ_2:4; then reconsider n1 = n - 1 as Element of NAT by FINSEQ_2:5; set r = (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n; set s = (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1); A45: len <*(F . d)*> = 1 by FINSEQ_1:40; then len (<*(F . d)*> ^ (FinS (F,(X \ Y)))) = 1 + (len (FinS (F,(X \ Y)))) by FINSEQ_1:22; then A46: len (FinS (F,(X \ Y))) = (len (<*(F . d)*> ^ (FinS (F,(X \ Y))))) - 1 ; A47: n + 1 <= len (<*(F . d)*> ^ (FinS (F,(X \ Y)))) by A43, FINSEQ_3:25; then n1 + 1 <= len (FinS (F,(X \ Y))) by A46, XREAL_1:19; then A48: n1 + 1 in dom (FinS (F,(X \ Y))) by A44, FINSEQ_3:25; then (FinS (F,(X \ Y))) . (n1 + 1) in rng (FinS (F,(X \ Y))) by FUNCT_1:def_3; then consider d1 being Element of D such that A49: ( d1 in dom (F | (X \ Y)) & (F | (X \ Y)) . d1 = (FinS (F,(X \ Y))) . (n1 + 1) ) by A40, PARTFUN1:3; 1 < n + 1 by A44, NAT_1:13; then A50: (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1) = (FinS (F,(X \ Y))) . ((n + 1) - 1) by A45, A47, FINSEQ_1:24 .= (FinS (F,(X \ Y))) . (n1 + 1) ; A51: n <= len (<*(F . d)*> ^ (FinS (F,(X \ Y)))) by A42, FINSEQ_3:25; then A52: n1 <= len (FinS (F,(X \ Y))) by A46, XREAL_1:9; A53: ( F . d1 = (FinS (F,(X \ Y))) . (n1 + 1) & F . d >= F . d1 ) by A2, A6, A11, A49, FUNCT_1:47; now__::_thesis:_(<*(F_._d)*>_^_(FinS_(F,(X_\_Y))))_._n_>=_(<*(F_._d)*>_^_(FinS_(F,(X_\_Y))))_._(n_+_1) percases ( n = 1 or n <> 1 ) ; suppose n = 1 ; ::_thesis: (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n >= (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1) hence (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n >= (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1) by A50, A53, FINSEQ_1:41; ::_thesis: verum end; suppose n <> 1 ; ::_thesis: (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n >= (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1) then A54: 1 < n by A44, XXREAL_0:1; then 1 + 1 <= n by NAT_1:13; then 1 <= n1 by XREAL_1:19; then A55: n1 in dom (FinS (F,(X \ Y))) by A52, FINSEQ_3:25; (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n = (FinS (F,(X \ Y))) . n1 by A45, A51, A54, FINSEQ_1:24; hence (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n >= (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1) by A48, A50, A55, RFINSEQ:def_3; ::_thesis: verum end; end; end; hence (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . n >= (<*(F . d)*> ^ (FinS (F,(X \ Y)))) . (n + 1) ; ::_thesis: verum end; d in {d} by TARSKI:def_1; then d in X \ (Y \ {d}) by A30, XBOOLE_0:def_3; then A56: d in dom (F | (X \ (Y \ {d}))) by A21, A17, XBOOLE_0:def_4; (X \ (Y \ {d})) \ {d} = X \ ((Y \ {d}) \/ {d}) by XBOOLE_1:41 .= X \ (Y \/ {d}) by XBOOLE_1:39 .= X \ Y by A19, XBOOLE_1:12 ; then (FinS (F,(X \ Y))) ^ <*(F . d)*>,F | (X \ (Y \ {d})) are_fiberwise_equipotent by A39, A31, A56, Th66; then <*(F . d)*> ^ (FinS (F,(X \ Y))),F | (X \ (Y \ {d})) are_fiberwise_equipotent by A20, CLASSES1:76; then A57: <*(F . d)*> ^ (FinS (F,(X \ Y))) = FinS (F,(X \ (Y \ {d}))) by A39, A31, A41, Def13; {d} c= dy by A11, ZFMISC_1:31; then card dyd = (card dy) - (card {d}) by A23, CARD_2:44 .= (n + 1) - 1 by A5, CARD_1:30 .= n ; then FinS (F,X) = (FinS (F,(Y \ {d}))) ^ (FinS (F,(X \ (Y \ {d})))) by A1, A3, A4, A32, XBOOLE_1:1; then A58: FinS (F,X) = ((FinS (F,(Y \ {d}))) ^ <*(F . d)*>) ^ (FinS (F,(X \ Y))) by A57, FINSEQ_1:32; A59: (FinS (F,Y)) | n is non-increasing by RFINSEQ:20; F | Y,(FinS (F,(Y \ {d}))) ^ <*(F . d)*> are_fiberwise_equipotent by A2, A11, Th66; then (FinS (F,(Y \ {d}))) ^ <*(F . d)*>, FinS (F,Y) are_fiberwise_equipotent by A8, CLASSES1:76; then FinS (F,(Y \ {d})),(FinS (F,Y)) | n are_fiberwise_equipotent by A29, RFINSEQ:1; hence FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) by A58, A29, A59, RFINSEQ:23; ::_thesis: verum end; theorem :: RFUNCT_3:71 for D being non empty set for F being PartFunc of D,REAL for X, Y being set st dom (F | X) is finite & Y c= X & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds F . d1 >= F . d2 ) holds FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) proof A1: for n being Element of NAT holds S1[n] from NAT_1:sch_1(Lm3, Lm4); let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for X, Y being set st dom (F | X) is finite & Y c= X & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds F . d1 >= F . d2 ) holds FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) let F be PartFunc of D,REAL; ::_thesis: for X, Y being set st dom (F | X) is finite & Y c= X & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds F . d1 >= F . d2 ) holds FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) let X, Y be set ; ::_thesis: ( dom (F | X) is finite & Y c= X & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds F . d1 >= F . d2 ) implies FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) ) assume that A2: dom (F | X) is finite and A3: Y c= X and A4: for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds F . d1 >= F . d2 ; ::_thesis: FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) F | Y c= F | X by A3, RELAT_1:75; then reconsider dFY = dom (F | Y) as finite set by A2, FINSET_1:1, RELAT_1:11; card dFY = card dFY ; hence FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y))) by A1, A2, A3, A4; ::_thesis: verum end; theorem Th72: :: RFUNCT_3:72 for D being non empty set for F being PartFunc of D,REAL for r being Real for X being set for d being Element of D st dom (F | X) is finite & d in dom (F | X) holds ( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d iff (FinS (F,X)) . (len (FinS (F,X))) = F . d ) proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for r being Real for X being set for d being Element of D st dom (F | X) is finite & d in dom (F | X) holds ( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d iff (FinS (F,X)) . (len (FinS (F,X))) = F . d ) let F be PartFunc of D,REAL; ::_thesis: for r being Real for X being set for d being Element of D st dom (F | X) is finite & d in dom (F | X) holds ( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d iff (FinS (F,X)) . (len (FinS (F,X))) = F . d ) let r be Real; ::_thesis: for X being set for d being Element of D st dom (F | X) is finite & d in dom (F | X) holds ( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d iff (FinS (F,X)) . (len (FinS (F,X))) = F . d ) let X be set ; ::_thesis: for d being Element of D st dom (F | X) is finite & d in dom (F | X) holds ( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d iff (FinS (F,X)) . (len (FinS (F,X))) = F . d ) let d be Element of D; ::_thesis: ( dom (F | X) is finite & d in dom (F | X) implies ( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d iff (FinS (F,X)) . (len (FinS (F,X))) = F . d ) ) set dx = dom (F | X); set drx = dom ((F - r) | X); set frx = FinS ((F - r),X); set fx = FinS (F,X); assume that A1: dom (F | X) is finite and A2: d in dom (F | X) ; ::_thesis: ( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d iff (FinS (F,X)) . (len (FinS (F,X))) = F . d ) reconsider dx = dom (F | X) as finite set by A1; A3: dom ((F - r) | X) = (dom (F - r)) /\ X by RELAT_1:61 .= (dom F) /\ X by VALUED_1:3 .= dx by RELAT_1:61 ; then FinS (F,X),F | X are_fiberwise_equipotent by Def13; then A4: rng (FinS (F,X)) = rng (F | X) by CLASSES1:75; then FinS (F,X) <> {} by A2, FUNCT_1:3, RELAT_1:38; then 0 + 1 <= len (FinS (F,X)) by NAT_1:13; then A5: len (FinS (F,X)) in dom (FinS (F,X)) by FINSEQ_3:25; (F | X) . d in rng (F | X) by A2, FUNCT_1:def_3; then F . d in rng (F | X) by A2, FUNCT_1:47; then consider n being Nat such that A6: n in dom (FinS (F,X)) and A7: (FinS (F,X)) . n = F . d by A4, FINSEQ_2:10; A8: dom (FinS (F,X)) = Seg (len (FinS (F,X))) by FINSEQ_1:def_3; FinS ((F - r),X),(F - r) | X are_fiberwise_equipotent by A3, Def13; then A9: rng (FinS ((F - r),X)) = rng ((F - r) | X) by CLASSES1:75; A10: ( len (FinS (F,X)) = card dx & dom (FinS ((F - r),X)) = Seg (len (FinS ((F - r),X))) ) by Th67, FINSEQ_1:def_3; A11: len (FinS ((F - r),X)) = card dx by A3, Th67; then (FinS ((F - r),X)) . (len (FinS ((F - r),X))) in rng (FinS ((F - r),X)) by A10, A8, A5, FUNCT_1:def_3; then consider d1 being Element of D such that A12: d1 in dom ((F - r) | X) and A13: ((F - r) | X) . d1 = (FinS ((F - r),X)) . (len (FinS ((F - r),X))) by A9, PARTFUN1:3; (F | X) . d1 = F . d1 by A3, A12, FUNCT_1:47; then F . d1 in rng (F | X) by A3, A12, FUNCT_1:def_3; then consider m being Nat such that A14: m in dom (FinS (F,X)) and A15: (FinS (F,X)) . m = F . d1 by A4, FINSEQ_2:10; A16: dom (F - r) = dom F by VALUED_1:3; A17: dom ((F - r) | X) = (dom (F - r)) /\ X by RELAT_1:61; then A18: d1 in dom (F - r) by A12, XBOOLE_0:def_4; A19: (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d1 by A12, A13, FUNCT_1:47 .= (F . d1) - r by A16, A18, VALUED_1:3 ; A20: d in dom (F - r) by A2, A3, A17, XBOOLE_0:def_4; then A21: (F - r) . d = (F . d) - r by A16, VALUED_1:3; A22: n <= len (FinS (F,X)) by A6, FINSEQ_3:25; thus ( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d implies (FinS (F,X)) . (len (FinS (F,X))) = F . d ) ::_thesis: ( (FinS (F,X)) . (len (FinS (F,X))) = F . d implies (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d ) proof (FinS (F,X)) . (len (FinS (F,X))) in rng (FinS (F,X)) by A5, FUNCT_1:def_3; then consider d1 being Element of D such that A23: d1 in dx and A24: (F | X) . d1 = (FinS (F,X)) . (len (FinS (F,X))) by A4, PARTFUN1:3; A25: d1 in dom (F - r) by A3, A17, A23, XBOOLE_0:def_4; A26: F . d1 = (FinS (F,X)) . (len (FinS (F,X))) by A23, A24, FUNCT_1:47; ((F - r) | X) . d1 = (F - r) . d1 by A3, A23, FUNCT_1:47 .= (F . d1) - r by A16, A25, VALUED_1:3 ; then (F . d1) - r in rng ((F - r) | X) by A3, A23, FUNCT_1:def_3; then consider m being Nat such that A27: m in dom (FinS ((F - r),X)) and A28: (FinS ((F - r),X)) . m = (F . d1) - r by A9, FINSEQ_2:10; A29: m <= len (FinS ((F - r),X)) by A27, FINSEQ_3:25; assume that A30: (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d and A31: (FinS (F,X)) . (len (FinS (F,X))) <> F . d ; ::_thesis: contradiction n < len (FinS (F,X)) by A7, A22, A31, XXREAL_0:1; then A32: F . d >= F . d1 by A5, A6, A7, A26, RFINSEQ:19; now__::_thesis:_(_(_len_(FinS_((F_-_r),X))_=_m_&_contradiction_)_or_(_len_(FinS_((F_-_r),X))_<>_m_&_contradiction_)_) percases ( len (FinS ((F - r),X)) = m or len (FinS ((F - r),X)) <> m ) ; case len (FinS ((F - r),X)) = m ; ::_thesis: contradiction then (F . d1) + (- r) = (F . d) - r by A16, A20, A30, A28, VALUED_1:3; hence contradiction by A31, A23, A24, FUNCT_1:47; ::_thesis: verum end; case len (FinS ((F - r),X)) <> m ; ::_thesis: contradiction then m < len (FinS ((F - r),X)) by A29, XXREAL_0:1; then (F . d1) - r >= (F . d) - r by A11, A10, A8, A21, A5, A30, A27, A28, RFINSEQ:19; then F . d1 >= F . d by XREAL_1:9; hence contradiction by A31, A26, A32, XXREAL_0:1; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; assume that A33: (FinS (F,X)) . (len (FinS (F,X))) = F . d and A34: (FinS ((F - r),X)) . (len (FinS ((F - r),X))) <> (F - r) . d ; ::_thesis: contradiction ((F - r) | X) . d in rng ((F - r) | X) by A2, A3, FUNCT_1:def_3; then (F - r) . d in rng ((F - r) | X) by A2, A3, FUNCT_1:47; then consider n1 being Nat such that A35: n1 in dom (FinS ((F - r),X)) and A36: (FinS ((F - r),X)) . n1 = (F . d) - r by A9, A21, FINSEQ_2:10; n1 <= len (FinS ((F - r),X)) by A35, FINSEQ_3:25; then n1 < len (FinS ((F - r),X)) by A21, A34, A36, XXREAL_0:1; then (F . d) - r >= (F . d1) - r by A11, A10, A8, A5, A19, A35, A36, RFINSEQ:19; then A37: F . d >= F . d1 by XREAL_1:9; A38: m <= len (FinS (F,X)) by A14, FINSEQ_3:25; now__::_thesis:_(_(_len_(FinS_(F,X))_=_m_&_contradiction_)_or_(_len_(FinS_(F,X))_<>_m_&_contradiction_)_) percases ( len (FinS (F,X)) = m or len (FinS (F,X)) <> m ) ; case len (FinS (F,X)) = m ; ::_thesis: contradiction hence contradiction by A16, A20, A33, A34, A19, A15, VALUED_1:3; ::_thesis: verum end; case len (FinS (F,X)) <> m ; ::_thesis: contradiction then m < len (FinS (F,X)) by A38, XXREAL_0:1; then F . d1 >= F . d by A5, A33, A14, A15, RFINSEQ:19; hence contradiction by A21, A34, A19, A37, XXREAL_0:1; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; theorem Th73: :: RFUNCT_3:73 for D being non empty set for F being PartFunc of D,REAL for r being Real for X being set for Z being finite set st Z = dom (F | X) holds FinS ((F - r),X) = (FinS (F,X)) - ((card Z) |-> r) proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for r being Real for X being set for Z being finite set st Z = dom (F | X) holds FinS ((F - r),X) = (FinS (F,X)) - ((card Z) |-> r) let F be PartFunc of D,REAL; ::_thesis: for r being Real for X being set for Z being finite set st Z = dom (F | X) holds FinS ((F - r),X) = (FinS (F,X)) - ((card Z) |-> r) let r be Real; ::_thesis: for X being set for Z being finite set st Z = dom (F | X) holds FinS ((F - r),X) = (FinS (F,X)) - ((card Z) |-> r) let X be set ; ::_thesis: for Z being finite set st Z = dom (F | X) holds FinS ((F - r),X) = (FinS (F,X)) - ((card Z) |-> r) defpred S2[ Element of NAT ] means for X being set for G being finite set st G = dom (F | X) & $1 = card G holds FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r); A1: for n being Element of NAT st S2[n] holds S2[n + 1] proof let n be Element of NAT ; ::_thesis: ( S2[n] implies S2[n + 1] ) assume A2: S2[n] ; ::_thesis: S2[n + 1] let X be set ; ::_thesis: for G being finite set st G = dom (F | X) & n + 1 = card G holds FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) let G be finite set ; ::_thesis: ( G = dom (F | X) & n + 1 = card G implies FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) ) assume A3: G = dom (F | X) ; ::_thesis: ( not n + 1 = card G or FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) ) set frx = FinS ((F - r),X); set fx = FinS (F,X); A4: dom ((F - r) | X) = (dom (F - r)) /\ X by RELAT_1:61 .= (dom F) /\ X by VALUED_1:3 .= dom (F | X) by RELAT_1:61 ; then A5: len (FinS ((F - r),X)) = card G by A3, Th67; A6: FinS ((F - r),X),(F - r) | X are_fiberwise_equipotent by A3, A4, Def13; then A7: rng (FinS ((F - r),X)) = rng ((F - r) | X) by CLASSES1:75; assume A8: n + 1 = card G ; ::_thesis: FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) then A9: len (FinS (F,X)) = n + 1 by A3, Th67; 0 + 1 <= n + 1 by NAT_1:13; then len (FinS ((F - r),X)) in dom (FinS ((F - r),X)) by A8, A5, FINSEQ_3:25; then (FinS ((F - r),X)) . (len (FinS ((F - r),X))) in rng (FinS ((F - r),X)) by FUNCT_1:def_3; then consider d being Element of D such that A10: d in dom ((F - r) | X) and A11: ((F - r) | X) . d = (FinS ((F - r),X)) . (len (FinS ((F - r),X))) by A7, PARTFUN1:3; set Y = X \ {d}; set dx = dom (F | X); set dy = dom (F | (X \ {d})); set fry = FinS ((F - r),(X \ {d})); set fy = FinS (F,(X \ {d})); set n1r = (n + 1) |-> r; set nr = n |-> r; A12: {d} c= dom (F | X) by A4, A10, ZFMISC_1:31; (F - r) . d = (FinS ((F - r),X)) . (len (FinS ((F - r),X))) by A10, A11, FUNCT_1:47; then A13: (FinS (F,X)) . (len (FinS (F,X))) = F . d by A3, A4, A10, Th72; len (FinS (F,X)) = card G by A3, Th67; then A14: FinS (F,X) = ((FinS (F,X)) | n) ^ <*(F . d)*> by A8, A13, RFINSEQ:7; FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*> by A3, A4, A10, A13, Th70; then A15: FinS (F,(X \ {d})) = (FinS (F,X)) | n by A14, FINSEQ_1:33; A16: dom ((FinS (F,(X \ {d}))) - (n |-> r)) = Seg (len ((FinS (F,(X \ {d}))) - (n |-> r))) by FINSEQ_1:def_3; A17: dom (F | (X \ {d})) = (dom F) /\ (X \ {d}) by RELAT_1:61; A18: dom (F | X) = (dom F) /\ X by RELAT_1:61; A19: dom (F | (X \ {d})) = (dom (F | X)) \ {d} proof thus dom (F | (X \ {d})) c= (dom (F | X)) \ {d} :: according to XBOOLE_0:def_10 ::_thesis: (dom (F | X)) \ {d} c= dom (F | (X \ {d})) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dom (F | (X \ {d})) or y in (dom (F | X)) \ {d} ) assume A20: y in dom (F | (X \ {d})) ; ::_thesis: y in (dom (F | X)) \ {d} then y in X \ {d} by A17, XBOOLE_0:def_4; then A21: not y in {d} by XBOOLE_0:def_5; y in dom F by A17, A20, XBOOLE_0:def_4; then y in dom (F | X) by A17, A18, A20, XBOOLE_0:def_4; hence y in (dom (F | X)) \ {d} by A21, XBOOLE_0:def_5; ::_thesis: verum end; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in (dom (F | X)) \ {d} or y in dom (F | (X \ {d})) ) assume A22: y in (dom (F | X)) \ {d} ; ::_thesis: y in dom (F | (X \ {d})) then A23: not y in {d} by XBOOLE_0:def_5; A24: y in dom (F | X) by A22, XBOOLE_0:def_5; then y in X by A18, XBOOLE_0:def_4; then A25: y in X \ {d} by A23, XBOOLE_0:def_5; y in dom F by A18, A24, XBOOLE_0:def_4; hence y in dom (F | (X \ {d})) by A17, A25, XBOOLE_0:def_4; ::_thesis: verum end; then reconsider dx = dom (F | X), dy = dom (F | (X \ {d})) as finite set by A3; A26: card dy = (card dx) - (card {d}) by A12, A19, CARD_2:44 .= (n + 1) - 1 by A3, A8, CARD_1:30 .= n ; then ( len (n |-> r) = n & len (FinS (F,(X \ {d}))) = n ) by Th67, CARD_1:def_7; then A27: len ((FinS (F,(X \ {d}))) - (n |-> r)) = n by FINSEQ_2:72; (F - r) . d = (FinS ((F - r),X)) . (len (FinS ((F - r),X))) by A10, A11, FUNCT_1:47; then A28: FinS ((F - r),X) = ((FinS ((F - r),X)) | n) ^ <*((F - r) . d)*> by A8, A5, RFINSEQ:7; (FinS ((F - r),(X \ {d}))) ^ <*((F - r) . d)*>,(F - r) | X are_fiberwise_equipotent by A3, A4, A10, Th66; then (FinS ((F - r),(X \ {d}))) ^ <*((F - r) . d)*>, FinS ((F - r),X) are_fiberwise_equipotent by A6, CLASSES1:76; then ( (FinS ((F - r),X)) | n is non-increasing & FinS ((F - r),(X \ {d})),(FinS ((F - r),X)) | n are_fiberwise_equipotent ) by A28, RFINSEQ:1, RFINSEQ:20; then A29: FinS ((F - r),(X \ {d})) = (FinS ((F - r),X)) | n by RFINSEQ:23; len ((n + 1) |-> r) = n + 1 by CARD_1:def_7; then A30: len ((FinS (F,X)) - ((n + 1) |-> r)) = n + 1 by A9, FINSEQ_2:72; then A31: dom ((FinS (F,X)) - ((n + 1) |-> r)) = Seg (n + 1) by FINSEQ_1:def_3; dom ((F - r) | X) = (dom (F - r)) /\ X by RELAT_1:61; then d in dom (F - r) by A10, XBOOLE_0:def_4; then d in dom F by VALUED_1:3; then (F - r) . d = (F . d) - r by VALUED_1:3; then A32: <*((F - r) . d)*> = <*(F . d)*> - <*r*> by RVSUM_1:29; A33: n < n + 1 by NAT_1:13; A34: dom (FinS (F,X)) = Seg (len (FinS (F,X))) by FINSEQ_1:def_3; ( len <*(F . d)*> = 1 & len <*r*> = 1 ) by FINSEQ_1:40; then A35: len (<*(F . d)*> - <*r*>) = 1 by FINSEQ_2:72; then A36: len (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) = n + 1 by A27, FINSEQ_1:22; 1 in Seg 1 by FINSEQ_1:1; then A37: 1 in dom (<*(F . d)*> - <*r*>) by A35, FINSEQ_1:def_3; A38: ( <*(F . d)*> . 1 = F . d & <*r*> . 1 = r ) by FINSEQ_1:40; A39: now__::_thesis:_for_m_being_Nat_st_m_in_dom_((FinS_(F,X))_-_((n_+_1)_|->_r))_holds_ ((FinS_(F,X))_-_((n_+_1)_|->_r))_._m_=_(((FinS_(F,(X_\_{d})))_-_(n_|->_r))_^_(<*(F_._d)*>_-_<*r*>))_._m let m be Nat; ::_thesis: ( m in dom ((FinS (F,X)) - ((n + 1) |-> r)) implies ((FinS (F,X)) - ((n + 1) |-> r)) . b1 = (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) . b1 ) assume A40: m in dom ((FinS (F,X)) - ((n + 1) |-> r)) ; ::_thesis: ((FinS (F,X)) - ((n + 1) |-> r)) . b1 = (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) . b1 percases ( m = n + 1 or m <> n + 1 ) ; supposeA41: m = n + 1 ; ::_thesis: ((FinS (F,X)) - ((n + 1) |-> r)) . b1 = (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) . b1 then A42: ((n + 1) |-> r) . m = r by FINSEQ_1:4, FUNCOP_1:7; (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) . m = (<*(F . d)*> - <*r*>) . ((n + 1) - n) by A27, A36, A33, A41, FINSEQ_1:24 .= (F . d) - r by A38, A37, VALUED_1:13 ; hence ((FinS (F,X)) - ((n + 1) |-> r)) . m = (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) . m by A13, A9, A40, A41, A42, VALUED_1:13; ::_thesis: verum end; supposeA43: m <> n + 1 ; ::_thesis: (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) . b1 = ((FinS (F,X)) - ((n + 1) |-> r)) . b1 m <= n + 1 by A31, A40, FINSEQ_1:1; then m < n + 1 by A43, XXREAL_0:1; then A44: m <= n by NAT_1:13; reconsider s = (FinS (F,X)) . m as Real ; A45: n <= n + 1 by NAT_1:11; A46: ((n + 1) |-> r) . m = r by A31, A40, FUNCOP_1:7; A47: 1 <= m by A31, A40, FINSEQ_1:1; then A48: m in Seg n by A44, FINSEQ_1:1; then A49: m in dom ((FinS (F,(X \ {d}))) - (n |-> r)) by A27, FINSEQ_1:def_3; 1 <= n by A47, A44, XXREAL_0:2; then n in Seg (n + 1) by A45, FINSEQ_1:1; then A50: ((FinS (F,X)) | n) . m = (FinS (F,X)) . m by A9, A34, A48, RFINSEQ:6; ( (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) . m = ((FinS (F,(X \ {d}))) - (n |-> r)) . m & (n |-> r) . m = r ) by A27, A16, A48, FINSEQ_1:def_7, FUNCOP_1:7; hence (((FinS (F,(X \ {d}))) - (n |-> r)) ^ (<*(F . d)*> - <*r*>)) . m = s - r by A15, A50, A49, VALUED_1:13 .= ((FinS (F,X)) - ((n + 1) |-> r)) . m by A40, A46, VALUED_1:13 ; ::_thesis: verum end; end; end; FinS ((F - r),(X \ {d})) = (FinS (F,(X \ {d}))) - (n |-> r) by A2, A26; hence FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) by A8, A28, A29, A32, A30, A36, A39, FINSEQ_2:9; ::_thesis: verum end; A51: S2[ 0 ] proof let X be set ; ::_thesis: for G being finite set st G = dom (F | X) & 0 = card G holds FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) let G be finite set ; ::_thesis: ( G = dom (F | X) & 0 = card G implies FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) ) assume A52: G = dom (F | X) ; ::_thesis: ( not 0 = card G or FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) ) assume 0 = card G ; ::_thesis: FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) then A53: dom (F | X) = {} by A52; then FinS (F,X) = FinS (F,{}) by Th63 .= <*> REAL by Th68 ; then A54: (FinS (F,X)) - ((card G) |-> r) = <*> REAL by FINSEQ_2:73; dom ((F - r) | X) = (dom (F - r)) /\ X by RELAT_1:61 .= (dom F) /\ X by VALUED_1:3 .= dom (F | X) by RELAT_1:61 ; hence FinS ((F - r),X) = FinS ((F - r),{}) by A53, Th63 .= (FinS (F,X)) - ((card G) |-> r) by A54, Th68 ; ::_thesis: verum end; A55: for n being Element of NAT holds S2[n] from NAT_1:sch_1(A51, A1); let G be finite set ; ::_thesis: ( G = dom (F | X) implies FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) ) assume G = dom (F | X) ; ::_thesis: FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) hence FinS ((F - r),X) = (FinS (F,X)) - ((card G) |-> r) by A55; ::_thesis: verum end; theorem :: RFUNCT_3:74 for D being non empty set for F being PartFunc of D,REAL for X being set st dom (F | X) is finite & ( for d being Element of D st d in dom (F | X) holds F . d >= 0 ) holds FinS ((max+ F),X) = FinS (F,X) proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for X being set st dom (F | X) is finite & ( for d being Element of D st d in dom (F | X) holds F . d >= 0 ) holds FinS ((max+ F),X) = FinS (F,X) let F be PartFunc of D,REAL; ::_thesis: for X being set st dom (F | X) is finite & ( for d being Element of D st d in dom (F | X) holds F . d >= 0 ) holds FinS ((max+ F),X) = FinS (F,X) let X be set ; ::_thesis: ( dom (F | X) is finite & ( for d being Element of D st d in dom (F | X) holds F . d >= 0 ) implies FinS ((max+ F),X) = FinS (F,X) ) assume that A1: dom (F | X) is finite and A2: for d being Element of D st d in dom (F | X) holds F . d >= 0 ; ::_thesis: FinS ((max+ F),X) = FinS (F,X) now__::_thesis:_for_d_being_Element_of_D_st_d_in_dom_(F_|_X)_holds_ (F_|_X)_._d_>=_0 let d be Element of D; ::_thesis: ( d in dom (F | X) implies (F | X) . d >= 0 ) assume A3: d in dom (F | X) ; ::_thesis: (F | X) . d >= 0 then F . d >= 0 by A2; hence (F | X) . d >= 0 by A3, FUNCT_1:47; ::_thesis: verum end; then A4: F | X = max+ (F | X) by Th46 .= (max+ F) | X by Th44 ; hence FinS (F,X) = FinS (((max+ F) | X),X) by A1, Th64 .= FinS ((max+ F),X) by A1, A4, Th64 ; ::_thesis: verum end; theorem :: RFUNCT_3:75 for D being non empty set for F being PartFunc of D,REAL for X being set for r being Real for Z being finite set st Z = dom (F | X) & rng (F | X) = {r} holds FinS (F,X) = (card Z) |-> r proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for X being set for r being Real for Z being finite set st Z = dom (F | X) & rng (F | X) = {r} holds FinS (F,X) = (card Z) |-> r let F be PartFunc of D,REAL; ::_thesis: for X being set for r being Real for Z being finite set st Z = dom (F | X) & rng (F | X) = {r} holds FinS (F,X) = (card Z) |-> r let X be set ; ::_thesis: for r being Real for Z being finite set st Z = dom (F | X) & rng (F | X) = {r} holds FinS (F,X) = (card Z) |-> r let r be Real; ::_thesis: for Z being finite set st Z = dom (F | X) & rng (F | X) = {r} holds FinS (F,X) = (card Z) |-> r let dx be finite set ; ::_thesis: ( dx = dom (F | X) & rng (F | X) = {r} implies FinS (F,X) = (card dx) |-> r ) assume A1: dx = dom (F | X) ; ::_thesis: ( not rng (F | X) = {r} or FinS (F,X) = (card dx) |-> r ) set fx = FinS (F,X); assume A2: rng (F | X) = {r} ; ::_thesis: FinS (F,X) = (card dx) |-> r F | X, FinS (F,X) are_fiberwise_equipotent by A1, Def13; then A3: rng (FinS (F,X)) = {r} by A2, CLASSES1:75; A4: dom (FinS (F,X)) = Seg (len (FinS (F,X))) by FINSEQ_1:def_3; len (FinS (F,X)) = card dx by A1, Th67; hence FinS (F,X) = (card dx) |-> r by A3, A4, FUNCOP_1:9; ::_thesis: verum end; theorem Th76: :: RFUNCT_3:76 for D being non empty set for F being PartFunc of D,REAL for X, Y being set st dom (F | (X \/ Y)) is finite & X misses Y holds FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for X, Y being set st dom (F | (X \/ Y)) is finite & X misses Y holds FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent let F be PartFunc of D,REAL; ::_thesis: for X, Y being set st dom (F | (X \/ Y)) is finite & X misses Y holds FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent let X, Y be set ; ::_thesis: ( dom (F | (X \/ Y)) is finite & X misses Y implies FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent ) assume A1: dom (F | (X \/ Y)) is finite ; ::_thesis: ( not X misses Y or FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent ) F | Y c= F | (X \/ Y) by RELAT_1:75, XBOOLE_1:7; then reconsider dfy = dom (F | Y) as finite set by A1, FINSET_1:1, RELAT_1:11; defpred S2[ Element of NAT ] means for Y being set for Z being finite set st Z = dom (F | Y) & dom (F | (X \/ Y)) is finite & X /\ Y = {} & $1 = card Z holds FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent ; A2: card dfy = card dfy ; A3: for n being Element of NAT st S2[n] holds S2[n + 1] proof let n be Element of NAT ; ::_thesis: ( S2[n] implies S2[n + 1] ) assume A4: S2[n] ; ::_thesis: S2[n + 1] let Y be set ; ::_thesis: for Z being finite set st Z = dom (F | Y) & dom (F | (X \/ Y)) is finite & X /\ Y = {} & n + 1 = card Z holds FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent let Z be finite set ; ::_thesis: ( Z = dom (F | Y) & dom (F | (X \/ Y)) is finite & X /\ Y = {} & n + 1 = card Z implies FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent ) assume that A5: Z = dom (F | Y) and A6: dom (F | (X \/ Y)) is finite and A7: X /\ Y = {} and A8: n + 1 = card Z ; ::_thesis: FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent set x = the Element of dom (F | Y); reconsider x = the Element of dom (F | Y) as Element of D by A5, A8, CARD_1:27, TARSKI:def_3; set y1 = Y \ {x}; A9: dom (F | Y) = (dom F) /\ Y by RELAT_1:61; now__::_thesis:_not_x_in_X assume A10: x in X ; ::_thesis: contradiction x in Y by A5, A8, A9, CARD_1:27, XBOOLE_0:def_4; hence contradiction by A7, A10, XBOOLE_0:def_4; ::_thesis: verum end; then X \ {x} = X by ZFMISC_1:57; then A11: (X \/ Y) \ {x} = X \/ (Y \ {x}) by XBOOLE_1:42; A12: dom (F | (Y \ {x})) = (dom F) /\ (Y \ {x}) by RELAT_1:61; A13: dom (F | (Y \ {x})) = (dom (F | Y)) \ {x} proof thus dom (F | (Y \ {x})) c= (dom (F | Y)) \ {x} :: according to XBOOLE_0:def_10 ::_thesis: (dom (F | Y)) \ {x} c= dom (F | (Y \ {x})) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dom (F | (Y \ {x})) or y in (dom (F | Y)) \ {x} ) assume A14: y in dom (F | (Y \ {x})) ; ::_thesis: y in (dom (F | Y)) \ {x} then y in Y \ {x} by A12, XBOOLE_0:def_4; then A15: not y in {x} by XBOOLE_0:def_5; y in dom F by A12, A14, XBOOLE_0:def_4; then y in dom (F | Y) by A12, A9, A14, XBOOLE_0:def_4; hence y in (dom (F | Y)) \ {x} by A15, XBOOLE_0:def_5; ::_thesis: verum end; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in (dom (F | Y)) \ {x} or y in dom (F | (Y \ {x})) ) assume A16: y in (dom (F | Y)) \ {x} ; ::_thesis: y in dom (F | (Y \ {x})) then A17: not y in {x} by XBOOLE_0:def_5; A18: y in dom (F | Y) by A16, XBOOLE_0:def_5; then y in Y by A9, XBOOLE_0:def_4; then A19: y in Y \ {x} by A17, XBOOLE_0:def_5; y in dom F by A9, A18, XBOOLE_0:def_4; hence y in dom (F | (Y \ {x})) by A12, A19, XBOOLE_0:def_4; ::_thesis: verum end; then reconsider dFy = dom (F | (Y \ {x})) as finite set by A5; {x} c= dom (F | Y) by A5, A8, CARD_1:27, ZFMISC_1:31; then A20: card dFy = (n + 1) - (card {x}) by A5, A8, A13, CARD_2:44 .= (n + 1) - 1 by CARD_1:30 .= n ; X \/ (Y \ {x}) c= X \/ Y by XBOOLE_1:13; then (dom F) /\ (X \/ (Y \ {x})) c= (dom F) /\ (X \/ Y) by XBOOLE_1:27; then dom (F | (X \/ (Y \ {x}))) c= (dom F) /\ (X \/ Y) by RELAT_1:61; then A21: dom (F | (X \/ (Y \ {x}))) c= dom (F | (X \/ Y)) by RELAT_1:61; A22: FinS (F,(X \/ Y)),F | (X \/ Y) are_fiberwise_equipotent by A6, Def13; dom (F | (X \/ Y)) = (dom F) /\ (X \/ Y) by RELAT_1:61 .= ((dom F) /\ X) \/ ((dom F) /\ Y) by XBOOLE_1:23 .= (dom (F | X)) \/ ((dom F) /\ Y) by RELAT_1:61 .= (dom (F | X)) \/ (dom (F | Y)) by RELAT_1:61 ; then x in dom (F | (X \/ Y)) by A5, A8, CARD_1:27, XBOOLE_0:def_3; then (FinS (F,(X \/ (Y \ {x})))) ^ <*(F . x)*>,F | (X \/ Y) are_fiberwise_equipotent by A6, A11, Th66; then A23: (FinS (F,(X \/ (Y \ {x})))) ^ <*(F . x)*>, FinS (F,(X \/ Y)) are_fiberwise_equipotent by A22, CLASSES1:76; X /\ (Y \ {x}) c= X /\ Y by XBOOLE_1:27; then FinS (F,(X \/ (Y \ {x}))),(FinS (F,X)) ^ (FinS (F,(Y \ {x}))) are_fiberwise_equipotent by A4, A6, A7, A21, A20, XBOOLE_1:3; then (FinS (F,(X \/ (Y \ {x})))) ^ <*(F . x)*>,((FinS (F,X)) ^ (FinS (F,(Y \ {x})))) ^ <*(F . x)*> are_fiberwise_equipotent by RFINSEQ:1; then A24: (FinS (F,(X \/ (Y \ {x})))) ^ <*(F . x)*>,(FinS (F,X)) ^ ((FinS (F,(Y \ {x}))) ^ <*(F . x)*>) are_fiberwise_equipotent by FINSEQ_1:32; ( (FinS (F,(Y \ {x}))) ^ <*(F . x)*>,F | Y are_fiberwise_equipotent & FinS (F,Y),F | Y are_fiberwise_equipotent ) by A5, A8, Def13, Th66, CARD_1:27; then (FinS (F,(Y \ {x}))) ^ <*(F . x)*>, FinS (F,Y) are_fiberwise_equipotent by CLASSES1:76; then A25: ((FinS (F,(Y \ {x}))) ^ <*(F . x)*>) ^ (FinS (F,X)),(FinS (F,Y)) ^ (FinS (F,X)) are_fiberwise_equipotent by RFINSEQ:1; (FinS (F,X)) ^ ((FinS (F,(Y \ {x}))) ^ <*(F . x)*>),((FinS (F,(Y \ {x}))) ^ <*(F . x)*>) ^ (FinS (F,X)) are_fiberwise_equipotent by RFINSEQ:2; then ( (FinS (F,Y)) ^ (FinS (F,X)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent & (FinS (F,X)) ^ ((FinS (F,(Y \ {x}))) ^ <*(F . x)*>),(FinS (F,Y)) ^ (FinS (F,X)) are_fiberwise_equipotent ) by A25, CLASSES1:76, RFINSEQ:2; then (FinS (F,X)) ^ ((FinS (F,(Y \ {x}))) ^ <*(F . x)*>),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent by CLASSES1:76; then (FinS (F,(X \/ (Y \ {x})))) ^ <*(F . x)*>,(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent by A24, CLASSES1:76; hence FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent by A23, CLASSES1:76; ::_thesis: verum end; A26: S2[ 0 ] proof let Y be set ; ::_thesis: for Z being finite set st Z = dom (F | Y) & dom (F | (X \/ Y)) is finite & X /\ Y = {} & 0 = card Z holds FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent let Z be finite set ; ::_thesis: ( Z = dom (F | Y) & dom (F | (X \/ Y)) is finite & X /\ Y = {} & 0 = card Z implies FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent ) assume that A27: Z = dom (F | Y) and A28: dom (F | (X \/ Y)) is finite and X /\ Y = {} and A29: 0 = card Z ; ::_thesis: FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent A30: dom (F | (X \/ Y)) = (dom F) /\ (X \/ Y) by RELAT_1:61 .= ((dom F) /\ X) \/ ((dom F) /\ Y) by XBOOLE_1:23 .= (dom (F | X)) \/ ((dom F) /\ Y) by RELAT_1:61 .= (dom (F | X)) \/ (dom (F | Y)) by RELAT_1:61 ; then A31: dom (F | X) is finite by A28, FINSET_1:1, XBOOLE_1:7; A32: dom (F | Y) = {} by A27, A29; then FinS (F,(X \/ Y)) = FinS (F,(dom (F | X))) by A28, A30, Th63 .= FinS (F,X) by A31, Th63 .= (FinS (F,X)) ^ (<*> REAL) by FINSEQ_1:34 .= (FinS (F,X)) ^ (FinS (F,(dom (F | Y)))) by A32, Th68 .= (FinS (F,X)) ^ (FinS (F,Y)) by A27, Th63 ; hence FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent ; ::_thesis: verum end; A33: for n being Element of NAT holds S2[n] from NAT_1:sch_1(A26, A3); assume X /\ Y = {} ; :: according to XBOOLE_0:def_7 ::_thesis: FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent hence FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent by A1, A33, A2; ::_thesis: verum end; definition let D be non empty set ; let F be PartFunc of D,REAL; let X be set ; func Sum (F,X) -> Real equals :: RFUNCT_3:def 14 Sum (FinS (F,X)); correctness coherence Sum (FinS (F,X)) is Real; ; end; :: deftheorem defines Sum RFUNCT_3:def_14_:_ for D being non empty set for F being PartFunc of D,REAL for X being set holds Sum (F,X) = Sum (FinS (F,X)); theorem Th77: :: RFUNCT_3:77 for D being non empty set for F being PartFunc of D,REAL for X being set for r being Real st dom (F | X) is finite holds Sum ((r (#) F),X) = r * (Sum (F,X)) proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for X being set for r being Real st dom (F | X) is finite holds Sum ((r (#) F),X) = r * (Sum (F,X)) let F be PartFunc of D,REAL; ::_thesis: for X being set for r being Real st dom (F | X) is finite holds Sum ((r (#) F),X) = r * (Sum (F,X)) let X be set ; ::_thesis: for r being Real st dom (F | X) is finite holds Sum ((r (#) F),X) = r * (Sum (F,X)) let r be Real; ::_thesis: ( dom (F | X) is finite implies Sum ((r (#) F),X) = r * (Sum (F,X)) ) set x = dom (F | X); assume A1: dom (F | X) is finite ; ::_thesis: Sum ((r (#) F),X) = r * (Sum (F,X)) then reconsider FX = F | X as finite Function by FINSET_1:10; dom ((r (#) F) | X) = dom (r (#) (F | X)) by RFUNCT_1:49 .= dom (F | X) by VALUED_1:def_5 ; then reconsider rFX = (r (#) F) | X as finite Function by A1, FINSET_1:10; consider b being FinSequence such that A2: F | (dom (F | X)),b are_fiberwise_equipotent by A1, RFINSEQ:5; rng (F | (dom (F | X))) = rng b by A2, CLASSES1:75; then reconsider b = b as FinSequence of REAL by FINSEQ_1:def_4; dom (F | X) = (dom F) /\ X by RELAT_1:61; then A3: F | (dom (F | X)) = (F | (dom F)) | X by RELAT_1:71 .= F | X by RELAT_1:68 ; then A4: rng b = rng (F | X) by A2, CLASSES1:75; A5: now__::_thesis:_for_x_being_Real_holds_card_(Coim_((r_*_b),x))_=_card_(Coim_(rFX,x)) let x be Real; ::_thesis: card (Coim ((r * b),x)) = card (Coim (rFX,x)) A6: len (r * b) = len b by FINSEQ_2:33; now__::_thesis:_(_(_not_x_in_rng_(r_*_b)_&_card_((r_*_b)_"_{x})_=_card_(rFX_"_{x})_)_or_(_x_in_rng_(r_*_b)_&_card_((r_*_b)_"_{x})_=_card_(rFX_"_{x})_)_) percases ( not x in rng (r * b) or x in rng (r * b) ) ; caseA7: not x in rng (r * b) ; ::_thesis: card ((r * b) " {x}) = card (rFX " {x}) A8: now__::_thesis:_not_x_in_rng_((r_(#)_F)_|_X) assume x in rng ((r (#) F) | X) ; ::_thesis: contradiction then x in rng (r (#) (F | X)) by RFUNCT_1:49; then consider d being Element of D such that A9: d in dom (r (#) (F | X)) and A10: (r (#) (F | X)) . d = x by PARTFUN1:3; d in dom (F | X) by A9, VALUED_1:def_5; then (F | X) . d in rng (F | X) by FUNCT_1:def_3; then consider n being Nat such that A11: n in dom b and A12: b . n = (F | X) . d by A4, FINSEQ_2:10; x = r * ((F | X) . d) by A9, A10, VALUED_1:def_5; then A13: x = (r * b) . n by A12, RVSUM_1:44; n in dom (r * b) by A6, A11, FINSEQ_3:29; hence contradiction by A7, A13, FUNCT_1:def_3; ::_thesis: verum end; (r * b) " {x} = {} by A7, Lm2; hence card ((r * b) " {x}) = card (rFX " {x}) by A8, Lm2; ::_thesis: verum end; case x in rng (r * b) ; ::_thesis: card ((r * b) " {x}) = card (rFX " {x}) then consider n being Nat such that n in dom (r * b) and A14: (r * b) . n = x by FINSEQ_2:10; reconsider p = b . n as Real ; A15: x = r * p by A14, RVSUM_1:44; now__::_thesis:_(_(_r_=_0_&_card_((r_*_b)_"_{x})_=_card_(rFX_"_{x})_)_or_(_r_<>_0_&_card_(Coim_((r_*_b),x))_=_card_(Coim_(rFX,x))_)_) percases ( r = 0 or r <> 0 ) ; caseA16: r = 0 ; ::_thesis: card ((r * b) " {x}) = card (rFX " {x}) then A17: (r * b) " {x} = dom b by A15, RFINSEQ:25; dom FX = (r (#) (F | X)) " {x} by A15, A16, Th7 .= ((r (#) F) | X) " {x} by RFUNCT_1:49 ; hence card ((r * b) " {x}) = card (rFX " {x}) by A2, A3, A17, CLASSES1:81; ::_thesis: verum end; caseA18: r <> 0 ; ::_thesis: card (Coim ((r * b),x)) = card (Coim (rFX,x)) then A19: Coim ((r * b),x) = Coim (b,(x / r)) by RFINSEQ:24; Coim (((r (#) F) | X),x) = (r (#) (F | X)) " {x} by RFUNCT_1:49 .= Coim (FX,(x / r)) by A18, Th6 ; hence card (Coim ((r * b),x)) = card (Coim (rFX,x)) by A2, A3, A19, CLASSES1:def_9; ::_thesis: verum end; end; end; hence card ((r * b) " {x}) = card (rFX " {x}) ; ::_thesis: verum end; end; end; hence card (Coim ((r * b),x)) = card (Coim (rFX,x)) ; ::_thesis: verum end; ( rng (r * b) c= REAL & rng ((r (#) F) | X) c= REAL ) ; then A20: r * b,(r (#) F) | X are_fiberwise_equipotent by A5, CLASSES1:79; F | X, FinS (F,X) are_fiberwise_equipotent by A1, Def13; then A21: Sum b = Sum (F,X) by A2, A3, CLASSES1:76, RFINSEQ:9; dom ((r (#) F) | X) = (dom (r (#) F)) /\ X by RELAT_1:61 .= (dom F) /\ X by VALUED_1:def_5 .= dom (F | X) by RELAT_1:61 ; then (r (#) F) | X, FinS ((r (#) F),X) are_fiberwise_equipotent by A1, Def13; hence Sum ((r (#) F),X) = Sum (r * b) by A20, CLASSES1:76, RFINSEQ:9 .= r * (Sum (F,X)) by A21, RVSUM_1:87 ; ::_thesis: verum end; theorem Th78: :: RFUNCT_3:78 for D being non empty set for F, G being PartFunc of D,REAL for X being set for Y being finite set st Y = dom (F | X) & dom (F | X) = dom (G | X) holds Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) proof let D be non empty set ; ::_thesis: for F, G being PartFunc of D,REAL for X being set for Y being finite set st Y = dom (F | X) & dom (F | X) = dom (G | X) holds Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) let F, G be PartFunc of D,REAL; ::_thesis: for X being set for Y being finite set st Y = dom (F | X) & dom (F | X) = dom (G | X) holds Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) let X be set ; ::_thesis: for Y being finite set st Y = dom (F | X) & dom (F | X) = dom (G | X) holds Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) let Y be finite set ; ::_thesis: ( Y = dom (F | X) & dom (F | X) = dom (G | X) implies Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) ) assume A1: Y = dom (F | X) ; ::_thesis: ( not dom (F | X) = dom (G | X) or Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) ) defpred S2[ Element of NAT ] means for F, G being PartFunc of D,REAL for X being set for Y being finite set st card Y = $1 & Y = dom (F | X) & dom (F | X) = dom (G | X) holds Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)); A2: card Y = card Y ; A3: for n being Element of NAT st S2[n] holds S2[n + 1] proof let n be Element of NAT ; ::_thesis: ( S2[n] implies S2[n + 1] ) assume A4: S2[n] ; ::_thesis: S2[n + 1] let F, G be PartFunc of D,REAL; ::_thesis: for X being set for Y being finite set st card Y = n + 1 & Y = dom (F | X) & dom (F | X) = dom (G | X) holds Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) let X be set ; ::_thesis: for Y being finite set st card Y = n + 1 & Y = dom (F | X) & dom (F | X) = dom (G | X) holds Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) let dx be finite set ; ::_thesis: ( card dx = n + 1 & dx = dom (F | X) & dom (F | X) = dom (G | X) implies Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) ) set gx = dom (G | X); assume that A5: card dx = n + 1 and A6: dx = dom (F | X) and A7: dom (F | X) = dom (G | X) ; ::_thesis: Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) set x = the Element of dx; reconsider x = the Element of dx as Element of D by A5, A6, CARD_1:27, TARSKI:def_3; A8: dx = (dom F) /\ X by A6, RELAT_1:61; then A9: x in dom F by A5, CARD_1:27, XBOOLE_0:def_4; set Y = X \ {x}; set dy = dom (F | (X \ {x})); set gy = dom (G | (X \ {x})); A10: dom (G | X) = (dom G) /\ X by RELAT_1:61; then x in dom G by A5, A6, A7, CARD_1:27, XBOOLE_0:def_4; then x in (dom F) /\ (dom G) by A9, XBOOLE_0:def_4; then A11: x in dom (F + G) by VALUED_1:def_1; A12: dom (F | (X \ {x})) = (dom F) /\ (X \ {x}) by RELAT_1:61; A13: dom (F | (X \ {x})) = dx \ {x} proof thus dom (F | (X \ {x})) c= dx \ {x} :: according to XBOOLE_0:def_10 ::_thesis: dx \ {x} c= dom (F | (X \ {x})) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dom (F | (X \ {x})) or y in dx \ {x} ) assume A14: y in dom (F | (X \ {x})) ; ::_thesis: y in dx \ {x} then y in X \ {x} by A12, XBOOLE_0:def_4; then A15: not y in {x} by XBOOLE_0:def_5; y in dom F by A12, A14, XBOOLE_0:def_4; then y in dx by A12, A8, A14, XBOOLE_0:def_4; hence y in dx \ {x} by A15, XBOOLE_0:def_5; ::_thesis: verum end; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dx \ {x} or y in dom (F | (X \ {x})) ) assume A16: y in dx \ {x} ; ::_thesis: y in dom (F | (X \ {x})) then ( not y in {x} & y in X ) by A8, XBOOLE_0:def_4, XBOOLE_0:def_5; then A17: y in X \ {x} by XBOOLE_0:def_5; y in dom F by A8, A16, XBOOLE_0:def_4; hence y in dom (F | (X \ {x})) by A12, A17, XBOOLE_0:def_4; ::_thesis: verum end; then reconsider dy = dom (F | (X \ {x})) as finite set ; A18: dom (G | (X \ {x})) = (dom G) /\ (X \ {x}) by RELAT_1:61; A19: dy = dom (G | (X \ {x})) proof thus dy c= dom (G | (X \ {x})) :: according to XBOOLE_0:def_10 ::_thesis: dom (G | (X \ {x})) c= dy proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dy or y in dom (G | (X \ {x})) ) assume A20: y in dy ; ::_thesis: y in dom (G | (X \ {x})) then y in dom F by A12, XBOOLE_0:def_4; then y in dom (G | X) by A6, A7, A12, A8, A20, XBOOLE_0:def_4; then A21: y in dom G by A10, XBOOLE_0:def_4; y in X \ {x} by A12, A20, XBOOLE_0:def_4; hence y in dom (G | (X \ {x})) by A18, A21, XBOOLE_0:def_4; ::_thesis: verum end; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dom (G | (X \ {x})) or y in dy ) assume A22: y in dom (G | (X \ {x})) ; ::_thesis: y in dy then y in dom G by A18, XBOOLE_0:def_4; then y in dx by A6, A7, A18, A10, A22, XBOOLE_0:def_4; then A23: y in dom F by A8, XBOOLE_0:def_4; y in X \ {x} by A18, A22, XBOOLE_0:def_4; hence y in dy by A12, A23, XBOOLE_0:def_4; ::_thesis: verum end; {x} c= dx by A5, CARD_1:27, ZFMISC_1:31; then card dy = (card dx) - (card {x}) by A13, CARD_2:44 .= (n + 1) - 1 by A5, CARD_1:30 .= n ; then A24: Sum ((F + G),(X \ {x})) = (Sum (F,(X \ {x}))) + (Sum (G,(X \ {x}))) by A4, A19; A25: dom ((F + G) | X) = dom ((F | X) + (G | X)) by RFUNCT_1:44 .= dx /\ (dom (G | X)) by A6, VALUED_1:def_1 ; then A26: FinS ((F + G),X),(F + G) | X are_fiberwise_equipotent by Def13; x in X by A5, A8, CARD_1:27, XBOOLE_0:def_4; then x in (dom (F + G)) /\ X by A11, XBOOLE_0:def_4; then x in dom ((F + G) | X) by RELAT_1:61; then A27: (FinS ((F + G),(X \ {x}))) ^ <*((F + G) . x)*>,(F + G) | X are_fiberwise_equipotent by A25, Th66; ( (FinS (F,(X \ {x}))) ^ <*(F . x)*>,F | X are_fiberwise_equipotent & FinS (F,X),F | X are_fiberwise_equipotent ) by A5, A6, Def13, Th66, CARD_1:27; then A28: Sum (F,X) = Sum ((FinS (F,(X \ {x}))) ^ <*(F . x)*>) by CLASSES1:76, RFINSEQ:9 .= (Sum (F,(X \ {x}))) + (F . x) by RVSUM_1:74 ; ( (FinS (G,(X \ {x}))) ^ <*(G . x)*>,G | X are_fiberwise_equipotent & FinS (G,X),G | X are_fiberwise_equipotent ) by A5, A6, A7, Def13, Th66, CARD_1:27; then Sum (G,X) = Sum ((FinS (G,(X \ {x}))) ^ <*(G . x)*>) by CLASSES1:76, RFINSEQ:9 .= (Sum (G,(X \ {x}))) + (G . x) by RVSUM_1:74 ; hence (Sum (F,X)) + (Sum (G,X)) = (Sum (FinS ((F + G),(X \ {x})))) + ((F . x) + (G . x)) by A24, A28 .= (Sum (FinS ((F + G),(X \ {x})))) + ((F + G) . x) by A11, VALUED_1:def_1 .= Sum ((FinS ((F + G),(X \ {x}))) ^ <*((F + G) . x)*>) by RVSUM_1:74 .= Sum ((F + G),X) by A27, A26, CLASSES1:76, RFINSEQ:9 ; ::_thesis: verum end; A29: S2[ 0 ] proof let F, G be PartFunc of D,REAL; ::_thesis: for X being set for Y being finite set st card Y = 0 & Y = dom (F | X) & dom (F | X) = dom (G | X) holds Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) let X be set ; ::_thesis: for Y being finite set st card Y = 0 & Y = dom (F | X) & dom (F | X) = dom (G | X) holds Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) let Y be finite set ; ::_thesis: ( card Y = 0 & Y = dom (F | X) & dom (F | X) = dom (G | X) implies Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) ) assume that A30: card Y = 0 and A31: Y = dom (F | X) and A32: dom (F | X) = dom (G | X) ; ::_thesis: Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) dom (F | X) = {} by A30, A31; then A33: rng (F | X) = {} by RELAT_1:42; (F + G) | X = (F | X) + (G | X) by RFUNCT_1:44; then dom ((F + G) | X) = (dom (F | X)) /\ (dom (G | X)) by VALUED_1:def_1 .= {} by A30, A31, A32 ; then ( rng ((F + G) | X) = {} & FinS ((F + G),X),(F + G) | X are_fiberwise_equipotent ) by Def13, RELAT_1:42; then A34: rng (FinS ((F + G),X)) = {} by CLASSES1:75; FinS (F,X),F | X are_fiberwise_equipotent by A31, Def13; then rng (FinS (F,X)) = {} by A33, CLASSES1:75; then A35: Sum (F,X) = 0 by RELAT_1:41, RVSUM_1:72; dom (G | X) = {} by A30, A31, A32; then A36: rng (G | X) = {} by RELAT_1:42; FinS (G,X),G | X are_fiberwise_equipotent by A31, A32, Def13; then rng (FinS (G,X)) = {} by A36, CLASSES1:75; then (Sum (F,X)) + (Sum (G,X)) = 0 + 0 by A35, RELAT_1:41, RVSUM_1:72; hence Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) by A34, RELAT_1:41, RVSUM_1:72; ::_thesis: verum end; A37: for n being Element of NAT holds S2[n] from NAT_1:sch_1(A29, A3); assume dom (F | X) = dom (G | X) ; ::_thesis: Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) hence Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X)) by A1, A37, A2; ::_thesis: verum end; theorem :: RFUNCT_3:79 for D being non empty set for F, G being PartFunc of D,REAL for X being set st dom (F | X) is finite & dom (F | X) = dom (G | X) holds Sum ((F - G),X) = (Sum (F,X)) - (Sum (G,X)) proof let D be non empty set ; ::_thesis: for F, G being PartFunc of D,REAL for X being set st dom (F | X) is finite & dom (F | X) = dom (G | X) holds Sum ((F - G),X) = (Sum (F,X)) - (Sum (G,X)) let F, G be PartFunc of D,REAL; ::_thesis: for X being set st dom (F | X) is finite & dom (F | X) = dom (G | X) holds Sum ((F - G),X) = (Sum (F,X)) - (Sum (G,X)) let X be set ; ::_thesis: ( dom (F | X) is finite & dom (F | X) = dom (G | X) implies Sum ((F - G),X) = (Sum (F,X)) - (Sum (G,X)) ) assume A1: ( dom (F | X) is finite & dom (F | X) = dom (G | X) ) ; ::_thesis: Sum ((F - G),X) = (Sum (F,X)) - (Sum (G,X)) dom (((- 1) (#) G) | X) = (dom ((- 1) (#) G)) /\ X by RELAT_1:61 .= (dom G) /\ X by VALUED_1:def_5 .= dom (G | X) by RELAT_1:61 ; hence Sum ((F - G),X) = (Sum (F,X)) + (Sum (((- 1) (#) G),X)) by A1, Th78 .= (Sum (F,X)) + ((- 1) * (Sum (G,X))) by A1, Th77 .= (Sum (F,X)) - (Sum (G,X)) ; ::_thesis: verum end; theorem :: RFUNCT_3:80 for D being non empty set for F being PartFunc of D,REAL for X being set for r being Real for Y being finite set st Y = dom (F | X) holds Sum ((F - r),X) = (Sum (F,X)) - (r * (card Y)) proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for X being set for r being Real for Y being finite set st Y = dom (F | X) holds Sum ((F - r),X) = (Sum (F,X)) - (r * (card Y)) let F be PartFunc of D,REAL; ::_thesis: for X being set for r being Real for Y being finite set st Y = dom (F | X) holds Sum ((F - r),X) = (Sum (F,X)) - (r * (card Y)) let X be set ; ::_thesis: for r being Real for Y being finite set st Y = dom (F | X) holds Sum ((F - r),X) = (Sum (F,X)) - (r * (card Y)) let r be Real; ::_thesis: for Y being finite set st Y = dom (F | X) holds Sum ((F - r),X) = (Sum (F,X)) - (r * (card Y)) set fx = FinS (F,X); let Y be finite set ; ::_thesis: ( Y = dom (F | X) implies Sum ((F - r),X) = (Sum (F,X)) - (r * (card Y)) ) set dr = (card Y) |-> r; assume A1: Y = dom (F | X) ; ::_thesis: Sum ((F - r),X) = (Sum (F,X)) - (r * (card Y)) then len (FinS (F,X)) = card Y by Th67; then reconsider xf = FinS (F,X), rd = (card Y) |-> r as Element of (card Y) -tuples_on REAL by FINSEQ_2:92; FinS ((F - r),X) = (FinS (F,X)) - ((card Y) |-> r) by A1, Th73; hence Sum ((F - r),X) = (Sum xf) - (Sum rd) by RVSUM_1:90 .= (Sum (F,X)) - (r * (card Y)) by RVSUM_1:80 ; ::_thesis: verum end; theorem :: RFUNCT_3:81 for D being non empty set for F being PartFunc of D,REAL holds Sum (F,{}) = 0 by Th68, RVSUM_1:72; theorem :: RFUNCT_3:82 for D being non empty set for F being PartFunc of D,REAL for d being Element of D st d in dom F holds Sum (F,{d}) = F . d proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for d being Element of D st d in dom F holds Sum (F,{d}) = F . d let F be PartFunc of D,REAL; ::_thesis: for d being Element of D st d in dom F holds Sum (F,{d}) = F . d let d be Element of D; ::_thesis: ( d in dom F implies Sum (F,{d}) = F . d ) assume d in dom F ; ::_thesis: Sum (F,{d}) = F . d hence Sum (F,{d}) = Sum <*(F . d)*> by Th69 .= F . d by FINSOP_1:11 ; ::_thesis: verum end; theorem :: RFUNCT_3:83 for D being non empty set for F being PartFunc of D,REAL for X, Y being set st dom (F | (X \/ Y)) is finite & X misses Y holds Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y)) proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for X, Y being set st dom (F | (X \/ Y)) is finite & X misses Y holds Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y)) let F be PartFunc of D,REAL; ::_thesis: for X, Y being set st dom (F | (X \/ Y)) is finite & X misses Y holds Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y)) let X, Y be set ; ::_thesis: ( dom (F | (X \/ Y)) is finite & X misses Y implies Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y)) ) assume ( dom (F | (X \/ Y)) is finite & X misses Y ) ; ::_thesis: Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y)) hence Sum (F,(X \/ Y)) = Sum ((FinS (F,X)) ^ (FinS (F,Y))) by Th76, RFINSEQ:9 .= (Sum (F,X)) + (Sum (F,Y)) by RVSUM_1:75 ; ::_thesis: verum end; theorem :: RFUNCT_3:84 for D being non empty set for F being PartFunc of D,REAL for X, Y being set st dom (F | (X \/ Y)) is finite & dom (F | X) misses dom (F | Y) holds Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y)) proof let D be non empty set ; ::_thesis: for F being PartFunc of D,REAL for X, Y being set st dom (F | (X \/ Y)) is finite & dom (F | X) misses dom (F | Y) holds Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y)) let F be PartFunc of D,REAL; ::_thesis: for X, Y being set st dom (F | (X \/ Y)) is finite & dom (F | X) misses dom (F | Y) holds Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y)) let X, Y be set ; ::_thesis: ( dom (F | (X \/ Y)) is finite & dom (F | X) misses dom (F | Y) implies Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y)) ) assume that A1: dom (F | (X \/ Y)) is finite and A2: dom (F | X) misses dom (F | Y) ; ::_thesis: Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y)) A3: dom (F | (X \/ Y)) = (dom F) /\ (X \/ Y) by RELAT_1:61 .= ((dom F) /\ X) \/ ((dom F) /\ Y) by XBOOLE_1:23 .= (dom (F | X)) \/ ((dom F) /\ Y) by RELAT_1:61 .= (dom (F | X)) \/ (dom (F | Y)) by RELAT_1:61 ; then dom (F | X) is finite by A1, FINSET_1:1, XBOOLE_1:7; then A4: FinS (F,X) = FinS (F,(dom (F | X))) by Th63; dom (F | Y) is finite by A1, A3, FINSET_1:1, XBOOLE_1:7; then A5: FinS (F,Y) = FinS (F,(dom (F | Y))) by Th63; A6: dom (F | (dom (F | (X \/ Y)))) = (dom F) /\ (dom (F | (X \/ Y))) by RELAT_1:61 .= (dom F) /\ ((dom F) /\ (X \/ Y)) by RELAT_1:61 .= ((dom F) /\ (dom F)) /\ (X \/ Y) by XBOOLE_1:16 .= dom (F | (X \/ Y)) by RELAT_1:61 ; FinS (F,(X \/ Y)) = FinS (F,(dom (F | (X \/ Y)))) by A1, Th63; hence Sum (F,(X \/ Y)) = Sum ((FinS (F,X)) ^ (FinS (F,Y))) by A1, A2, A3, A4, A5, A6, Th76, RFINSEQ:9 .= (Sum (F,X)) + (Sum (F,Y)) by RVSUM_1:75 ; ::_thesis: verum end;