### The Mizar article:

### Functions and Finite Sequences of Real Numbers

**by****Jaroslaw Kotowicz**

- Received March 15, 1993
Copyright (c) 1993 Association of Mizar Users

- MML identifier: RFINSEQ
- [ MML identifier index ]

environ vocabulary RELAT_1, CARD_1, FUNCT_1, BOOLE, TARSKI, FUNCT_2, FINSET_1, FINSEQ_1, ARYTM_1, SQUARE_1, RLVECT_1, PROB_1, ARYTM_3, VECTSP_1, RFINSEQ; notation TARSKI, XBOOLE_0, SUBSET_1, NUMBERS, XREAL_0, RELAT_1, FUNCT_1, FUNCT_2, FINSEQ_1, CARD_1, REAL_1, NAT_1, SQUARE_1, SEQ_1, FINSEQ_4, FINSET_1, RVSUM_1, TOPREAL1; constructors WELLORD2, FINSEQOP, REAL_1, NAT_1, SQUARE_1, TOPREAL1, FINSEQ_4, SEQ_1, MEMBERED, PARTFUN1, XBOOLE_0; clusters FINSET_1, PRELAMB, FUNCT_1, RELSET_1, FINSEQ_1, FUNCT_2, NAT_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; definitions TARSKI, XBOOLE_0; theorems AXIOMS, TARSKI, FINSEQ_1, FINSEQ_2, FINSEQ_3, REAL_1, NAT_1, FUNCT_1, FUNCT_2, FINSET_1, CARD_1, CARD_2, SQUARE_1, TOPREAL1, RVSUM_1, WELLORD2, ZFMISC_1, AMI_1, RELAT_1, RELSET_1, XBOOLE_0, XBOOLE_1, XCMPLX_1; schemes NAT_1, FINSEQ_1, MATRIX_2, ZFREFLE1, COMPTS_1; begin reserve x,y for set, n,m for Nat, r,s for Real; definition let F,G be Relation; pred F,G are_fiberwise_equipotent means :Def1: for x be set holds Card (F"{x}) = Card (G"{x}); reflexivity; symmetry; end; Lm1: for F be Function, x be set st not x in rng F holds F"{x} = {} proof let F be Function,x; assume A1: not x in rng F; now assume rng F meets {x}; then consider y being set such that A2: y in rng F & y in {x} by XBOOLE_0:3; thus contradiction by A1,A2,TARSKI:def 1; end; hence thesis by RELAT_1:173; end; theorem Th1: for F,G be Function st F,G are_fiberwise_equipotent holds rng F = rng G proof let F,G be Function; assume A1: F,G are_fiberwise_equipotent; thus rng F c= rng G proof let x; assume that A2: x in rng F and A3: not x in rng G; Card (F"{x}) = Card (G"{x}) by A1,Def1; then A4: F"{x},G"{x} are_equipotent by CARD_1:21; consider y such that A5: y in dom F & F.y = x by A2,FUNCT_1:def 5; A6: x in {x} by TARSKI:def 1; G"{x} = {} by A3,Lm1; then F"{x} = {} by A4,CARD_1:46; hence contradiction by A5,A6,FUNCT_1:def 13; end; let x; assume that A7: x in rng G and A8: not x in rng F; Card (G"{x}) = Card (F"{x}) by A1,Def1; then A9: G"{x},F"{x} are_equipotent by CARD_1:21; consider y such that A10: y in dom G & G.y = x by A7,FUNCT_1:def 5; A11: x in {x} by TARSKI:def 1; F"{x} = {} by A8,Lm1; then G"{x} = {} by A9,CARD_1:46; hence contradiction by A10,A11,FUNCT_1:def 13; end; theorem for F,G,H be Function st F,G are_fiberwise_equipotent & F,H are_fiberwise_equipotent holds G,H are_fiberwise_equipotent proof let F,G,H be Function; assume A1: F,G are_fiberwise_equipotent & F,H are_fiberwise_equipotent; let x; thus Card(G"{x}) = Card(F"{x}) by A1,Def1 .= Card(H"{x}) by A1,Def1; end; theorem Th3: for F,G be Function holds F,G are_fiberwise_equipotent iff ex H be Function st dom H = dom F & rng H = dom G & H is one-to-one & F = G*H proof let F,G be Function; thus F,G are_fiberwise_equipotent implies ex H be Function st dom H = dom F & rng H = dom G & H is one-to-one & F = G*H proof assume A1: F,G are_fiberwise_equipotent; defpred P[set,set] means $2 is Function & for f be Function st $2 = f holds dom f = F"{$1} & rng f = G"{$1} & f is one-to-one; A2: for x st x in rng F ex y st P[x,y] proof let x; assume x in rng F; Card(F"{x}) = Card(G"{x}) by A1,Def1; then F"{x},G"{x} are_equipotent by CARD_1:21; then consider H be Function such that A3: H is one-to-one & dom H = F"{x} & rng H = G"{x} by WELLORD2:def 4; take H; thus H is Function; let g be Function; assume g = H; hence thesis by A3; end; consider W be Function such that A4: dom W = rng F & for x st x in rng F holds P[x,W.x] from NonUniqFuncEx(A2); defpred Q[set,set] means for H be Function st H = W.(F.$1) holds $2 = H.$1; set df = dom F, dg = dom G; A5: for x st x in df ex y st y in dg & Q[x,y] proof let x; assume A6: x in df; then A7: F.x in rng F by FUNCT_1:def 5; then reconsider f = W.(F.x) as Function by A4; A8: dom f = F"{F.x} & rng f = G"{F.x} & f is one-to-one by A4,A7; take y = f.x; F.x in {F.x} by TARSKI:def 1; then x in F"{F.x} by A6,FUNCT_1:def 13; then f.x in G"{F.x} by A8,FUNCT_1:def 5; hence y in dg by FUNCT_1:def 13; let g be Function; assume g = W.(F.x); hence thesis; end; consider IT be Function such that A9: dom IT = df & rng IT c= dg & for x st x in df holds Q[x,IT.x] from NonUniqBoundFuncEx(A5); take IT; thus dom IT = df by A9; dom G c= rng IT proof let x; assume A10: x in dg; then A11: G.x in rng G & rng F = rng G by A1,Th1,FUNCT_1:def 5; then reconsider f = W.(G.x) as Function by A4; A12: f is one-to-one & dom f = F"{G.x} & rng f = G"{G.x} by A4,A11; G.x in {G.x} by TARSKI:def 1; then x in rng f by A10,A12,FUNCT_1:def 13; then consider z be set such that A13: z in dom f & f.z = x by FUNCT_1:def 5; A14: z in df & F.z in {G.x} by A12,A13,FUNCT_1:def 13; then F.z = G.x by TARSKI:def 1; then IT.z = x by A9,A13,A14; hence thesis by A9,A14,FUNCT_1:def 5; end; hence rng IT = dg by A9,XBOOLE_0:def 10; now let x,y; assume A15: x in dom IT & y in dom IT & IT.x = IT.y; then A16: F.x in rng F & F.y in rng F by A9,FUNCT_1:def 5; then reconsider f1 = W.(F.x), f2 = W.(F.y) as Function by A4; A17: IT.x = f1.x & IT.y = f2.y by A9,A15; A18: dom f1 = F"{F.x} & rng f1 = G"{F.x} & f1 is one-to-one & dom f2 = F"{F.y} & rng f2 = G"{F.y} & f2 is one-to-one by A4,A16; F.x in {F.x} & F.y in {F.y} by TARSKI:def 1; then A19: x in F"{F.x} & y in F"{F.y} by A9,A15,FUNCT_1:def 13; then f1.x in rng f1 & f2.y in rng f2 by A18,FUNCT_1:def 5; then f1.x in G"{F.x} /\ G"{F.y} by A15,A17,A18,XBOOLE_0:def 3; then f1.x in G"({F.x} /\ {F.y}) by FUNCT_1:137; then G.(f1.x) in {F.x} /\ {F.y} by FUNCT_1:def 13; then G.(f1.x) in {F.x} & G.(f1.x) in {F.y} by XBOOLE_0:def 3; then G.(f1.x) = F.x & G.(f1.x) = F.y by TARSKI:def 1; hence x = y by A15,A17,A18,A19,FUNCT_1:def 8; end; hence IT is one-to-one by FUNCT_1:def 8; A20: dom(G*IT) = df by A9,RELAT_1:46; now let x; assume A21: x in df; then A22: F.x in rng F by FUNCT_1:def 5; then reconsider f = W.(F.x) as Function by A4; A23: f is one-to-one & dom f = F"{F.x} & rng f = G"{F.x} by A4,A22; F.x in {F.x} by TARSKI:def 1; then x in F"{F.x} by A21,FUNCT_1:def 13; then f.x in G"{F.x} by A23,FUNCT_1:def 5; then G.(f.x) in {F.x} by FUNCT_1:def 13; then A24: G.(f.x) = F.x by TARSKI:def 1; IT.x = f.x by A9,A21; hence F.x = (G*IT).x by A9,A21,A24,FUNCT_1:23; end; hence F = G*IT by A20,FUNCT_1:9; end; given H be Function such that A25: dom H = dom F & rng H = dom G & H is one-to-one & F = G*H; let x; set t = H|(F"{x}); A26: t is one-to-one by A25,FUNCT_1:84; F"{x} c= dom F by RELAT_1:167; then A27: dom t = F"{x} by A25,RELAT_1:91; rng t = G"{x} proof thus rng t c= G"{x} proof let z be set; assume z in rng t; then consider y such that A28: y in dom t & t.y = z by FUNCT_1:def 5; F.y in {x} by A27,A28,FUNCT_1:def 13; then A29: F.y = x by TARSKI:def 1; A30: z = H.y by A28,FUNCT_1:68; dom t = dom H /\ F"{x} by FUNCT_1:68; then A31: y in dom H by A28,XBOOLE_0:def 3; then A32: z in dom G by A25,A30,FUNCT_1:def 5; x = G.z by A25,A29,A30,A31,FUNCT_1:23; then G.z in {x} by TARSKI:def 1; hence thesis by A32,FUNCT_1:def 13; end; let z be set; assume z in G"{x}; then A33: z in dom G & G.z in {x} by FUNCT_1:def 13; then A34: G.z = x by TARSKI:def 1; consider y such that A35: y in dom H & H.y = z by A25,A33,FUNCT_1:def 5; F.y = x by A25,A34,A35,FUNCT_1:23; then F.y in {x} by TARSKI:def 1; then A36: y in dom t by A25,A27,A35,FUNCT_1:def 13; then t.y in rng t by FUNCT_1:def 5; hence thesis by A35,A36,FUNCT_1:68; end; then F"{x},G"{x} are_equipotent by A26,A27,WELLORD2:def 4; hence thesis by CARD_1:21; end; theorem Th4: for F,G be Function holds F,G are_fiberwise_equipotent iff for X be set holds Card (F"X) = Card (G"X) proof let F,G be Function; thus F,G are_fiberwise_equipotent implies for X be set holds Card(F"X) = Card(G"X) proof assume F,G are_fiberwise_equipotent; then consider H be Function such that A1: dom H = dom F & rng H = dom G & H is one-to-one & F = G*H by Th3; let X be set; set t = H|(F"X); A2: t is one-to-one by A1,FUNCT_1:84; F"X c= dom F by RELAT_1:167; then A3: dom t = F"X by A1,RELAT_1:91; rng t = G"X proof thus rng t c= G"X proof let z be set; assume z in rng t; then consider y such that A4: y in dom t & t.y = z by FUNCT_1:def 5; A5: F.y in X by A3,A4,FUNCT_1:def 13; A6: z = H.y by A4,FUNCT_1:68; dom t = dom H /\ F"X by FUNCT_1:68; then A7: y in dom H by A4,XBOOLE_0:def 3; then A8: z in dom G by A1,A6,FUNCT_1:def 5; G.z in X by A1,A5,A6,A7,FUNCT_1:23; hence thesis by A8,FUNCT_1:def 13; end; let z be set; assume z in G"X; then A9: z in dom G & G.z in X by FUNCT_1:def 13; then consider y such that A10: y in dom H & H.y = z by A1,FUNCT_1:def 5; F.y in X by A1,A9,A10,FUNCT_1:23; then A11: y in dom t by A1,A3,A10,FUNCT_1:def 13; then t.y in rng t by FUNCT_1:def 5; hence thesis by A10,A11,FUNCT_1:68; end; then F"X,G"X are_equipotent by A2,A3,WELLORD2:def 4; hence thesis by CARD_1:21; end; assume for X be set holds Card(F"X) = Card(G"X); hence for x holds Card(F"{x}) = Card(G"{x}); end; theorem Th5: for D be non empty set, F,G be Function st rng F c= D & rng G c= D holds F,G are_fiberwise_equipotent iff for d be Element of D holds Card (F"{d}) = Card (G"{d}) proof let D be non empty set, F,G be Function;assume A1: rng F c= D & rng G c= D; thus F,G are_fiberwise_equipotent implies for d be Element of D holds Card(F"{d}) = Card(G"{d}) by Def1; assume A2: for d be Element of D holds Card(F"{d}) = Card(G"{d}); let x; now per cases; case not x in rng F; then A3: F"{x} = {} by Lm1; now assume A4: x in rng G; then reconsider d = x as Element of D by A1; Card(G"{d}) = Card(F"{d}) by A2; then G"{d},F"{d} are_equipotent by CARD_1:21; then A5: G"{x} = {} by A3,CARD_1:46; consider y such that A6: y in dom G & G.y = x by A4,FUNCT_1:def 5; G.y in {x} by A6,TARSKI:def 1; hence contradiction by A5,A6,FUNCT_1:def 13; end; hence thesis by A3,Lm1; case x in rng F; then reconsider d = x as Element of D by A1; Card(F"{d}) = Card(G"{d}) by A2; hence Card(F"{x}) = Card(G"{x}); end; hence thesis; end; theorem Th6: for F,G be Function st dom F = dom G holds F,G are_fiberwise_equipotent iff ex P be Permutation of dom F st F = G*P proof let F,G be Function; assume A1: dom F = dom G; thus F,G are_fiberwise_equipotent implies ex P be Permutation of dom F st F = G*P proof assume F,G are_fiberwise_equipotent; then consider I be Function such that A2: dom I = dom F & rng I = dom G & I is one-to-one & F = G*I by Th3; reconsider I as Function of dom F,dom F by A1,A2,FUNCT_2:4; reconsider I as Permutation of dom F by A1,A2,FUNCT_2:83; take I; thus thesis by A2; end; given P be Permutation of dom F such that A3: F = G*P; A4: rng P = dom F & P is one-to-one by FUNCT_2:def 3; P is Function of dom F,dom F & dom F = {} implies dom F ={}; then dom P = dom F by FUNCT_2:def 1; hence thesis by A1,A3,A4,Th3; end; theorem for F,G be Function st F,G are_fiberwise_equipotent holds Card (dom F) = Card (dom G) proof let F,G be Function; assume F,G are_fiberwise_equipotent; then Card(F"(rng F)) = Card(G"(rng F)) & rng F = rng G by Th1,Th4; hence Card(dom F) = Card(G"(rng G)) by RELAT_1:169 .= Card(dom G) by RELAT_1:169; end; definition let F be finite Function, A be set; cluster F"A -> finite; coherence proof dom F is finite & F"A c= dom F by RELAT_1:167; hence thesis by FINSET_1:13; end; end; canceled; theorem for F,G be finite Function holds F,G are_fiberwise_equipotent iff for X be set holds card (F"X) = card (G"X) by Th4; theorem for F,G be finite Function st F,G are_fiberwise_equipotent holds card (dom F) = card (dom G) proof let F,G be finite Function; assume F,G are_fiberwise_equipotent; then card(F"(rng F)) = card(G"(rng F)) & rng F = rng G by Th1,Th4; hence card(dom F) = card(G"(rng G)) by RELAT_1:169 .= card(dom G) by RELAT_1:169; end; theorem for D be non empty set, F,G be finite Function st rng F c= D & rng G c= D holds F,G are_fiberwise_equipotent iff for d be Element of D holds card (F"{d}) = card (G"{d}) by Th5; canceled; theorem for f,g be FinSequence holds f,g are_fiberwise_equipotent iff for X be set holds card (f"X) = card (g"X) by Th4; theorem Th14: for f,g,h be FinSequence holds f,g are_fiberwise_equipotent iff f^h, g^h are_fiberwise_equipotent proof let f,g,h be FinSequence; thus f,g are_fiberwise_equipotent implies f^h, g^h are_fiberwise_equipotent proof assume A1: f,g are_fiberwise_equipotent; now let y; card (f"{y}) = card (g"{y}) by A1,Def1; hence card ((f^h)"{y}) = card(g"{y}) + card(h"{y}) by FINSEQ_3:63 .= card((g^h)"{y}) by FINSEQ_3:63; end; hence thesis by Def1; end; assume A2: f^h,g^h are_fiberwise_equipotent; now let x; card((f^h)"{x}) = card(f"{x})+card(h"{x}) & card((g^h)"{x}) = card(g"{x})+card(h"{x}) & card((f^h)"{x}) = card((g^h)"{x}) by A2,Def1,FINSEQ_3:63; hence card(f"{x}) = card(g"{x}) by XCMPLX_1:2; end; hence thesis by Def1; end; theorem for f,g be FinSequence holds f^g, g^f are_fiberwise_equipotent proof let f,g be FinSequence; let y; thus card((f^g)"{y}) = card(g"{y})+ card(f"{y}) by FINSEQ_3:63 .= card((g^f)"{y}) by FINSEQ_3:63; end; theorem Th16: for f,g be FinSequence st f,g are_fiberwise_equipotent holds len f = len g & dom f = dom g proof let f,g be FinSequence; A1: dom f = Seg len f & Seg len g = dom g by FINSEQ_1:def 3; assume f,g are_fiberwise_equipotent; then A2: card(f"(rng f)) = card(g"(rng f)) & rng f = rng g by Th1,Th4; thus len f = card(Seg len f) by FINSEQ_1:78 .= card(g"(rng g)) by A1,A2,RELAT_1:169 .= card(Seg len g) by A1,RELAT_1:169 .= len g by FINSEQ_1:78; hence thesis by A1; end; theorem for f,g be FinSequence holds f,g are_fiberwise_equipotent iff ex P be Permutation of dom g st f = g*P proof let f,g be FinSequence; thus f,g are_fiberwise_equipotent implies ex P be Permutation of dom g st f = g*P proof assume A1: f,g are_fiberwise_equipotent; then dom f = dom g by Th16; hence thesis by A1,Th6; end; given P be Permutation of dom g such that A2: f = g*P; (dom g = {} implies dom g = {}) & P is Function of dom g,dom g; then dom P = dom g & rng P c= dom g by FUNCT_2:def 1,RELSET_1:12; then dom f = dom g by A2,RELAT_1:46; hence thesis by A2,Th6; end; definition let F be Function, X be finite set; cluster F|X -> finite Function-like; coherence proof dom(F|X) c= X by RELAT_1:87; then dom(F|X) is finite by FINSET_1:13; hence thesis by AMI_1:21; end; end; defpred P[Nat] means for X be finite set,F be Function st card(dom(F|X)) = $1 ex a be FinSequence st F|X, a are_fiberwise_equipotent; Lm2: P[0] proof let X be finite set, F be Function; assume card(dom(F|X)) = 0; then A1: dom(F|X) = {} by CARD_2:59; take A = {}; A2: dom A = {} by FINSEQ_1:26; let x; (F|X)"{x} c= dom(F|X) & A"{x} c= dom A by RELAT_1:167; then (F|X)"{x} = {} & A"{x} = {} by A1,A2,XBOOLE_1:3; hence Card((F|X)"{x}) = Card(A"{x}); end; Lm3: for n st P[n] holds P[n+1] proof let n be Nat; assume A1: P[n]; let X be finite set, F be Function; reconsider dx = dom(F|X) as finite set; assume A2: card dom(F|X) = n+1; then A3: dx <> {} by CARD_1:78; consider x being Element of dx; set Y = X\{x}, dy = dom(F|Y); A4: {x} c= dx & dy = dom F /\ Y & dx = dom F /\ X by A2,CARD_1:78,FUNCT_1:68,ZFMISC_1:37; dy = dx \ {x} proof thus dy c= dx \ {x} proof let y; assume y in dy; then A5: y in dom F & y in X \ {x} by A4,XBOOLE_0:def 3; then A6: y in X & not y in {x} by XBOOLE_0:def 4; then y in dx by A4,A5,XBOOLE_0:def 3; hence thesis by A6,XBOOLE_0:def 4; end; let y; assume y in dx \{x}; then A7: y in dx & not y in {x} by XBOOLE_0:def 4; then A8: y in dom F & y in X by A4,XBOOLE_0:def 3; then y in Y by A7,XBOOLE_0:def 4; hence thesis by A4,A8,XBOOLE_0:def 3; end; then card(dom(F|Y)) = card dx - card {x} by A4,CARD_2:63 .= n+1-1 by A2,CARD_1:79 .= n by XCMPLX_1:26; then consider a be FinSequence such that A9: F|Y, a are_fiberwise_equipotent by A1; take A = a^<* F.x *>; x in dx by A3; then x in dom F /\ X by FUNCT_1:68; then x in X & x in dom F by XBOOLE_0:def 3; then A10: {x} c= X & {x} c= dom F by ZFMISC_1:37; now let y; A11: card((F|Y)"{y}) = card(a"{y}) by A9,Def1; A12: Y misses {x} by XBOOLE_1:79; A13: (F|Y)"{y} /\ (F|{x})"{y} = Y /\ (F"{y}) /\ (F|{x})"{y} by FUNCT_1:139 .= Y /\ F"{y} /\ ({x} /\ F"{y}) by FUNCT_1:139 .= F"{y} /\ Y /\ {x} /\ F"{y} by XBOOLE_1:16 .= F"{y} /\ (Y /\ {x}) /\ F"{y} by XBOOLE_1:16 .= {} /\ F"{y} by A12,XBOOLE_0:def 7 .= {}; A14: (F|Y)"{y} \/ (F|{x})"{y} = (Y /\ F"{y}) \/ (F|{x})"{y} by FUNCT_1:139 .= (Y /\ F"{y}) \/ ({x} /\ F"{y}) by FUNCT_1:139 .= (Y \/ {x}) /\ F"{y} by XBOOLE_1:23 .= (X \/ {x}) /\ F"{y} by XBOOLE_1:39 .= X /\ F"{y} by A10,XBOOLE_1:12 .= (F|X)"{y} by FUNCT_1:139; A15: dom(F|{x}) = {x} by A10,RELAT_1:91; A16: dom<*F.x*> = {1} by FINSEQ_1:4,55; reconsider FF = <*F.x*>"{y} as finite set; now per cases; case A17: y = F.x; A18: (F|{x})"{y} c= {x} by A15,RELAT_1:167; {x} c= (F|{x})"{y} proof let z be set; assume A19: z in {x}; then z = x by TARSKI:def 1; then y = (F|{x}).z & y in {y} by A15,A17,A19,FUNCT_1:68,TARSKI:def 1; hence z in (F|{x})"{y} by A15,A19,FUNCT_1:def 13; end; then (F|{x})"{y} = {x} by A18,XBOOLE_0:def 10; then A20: card((F|{x})"{y}) = 1 by CARD_1:79; A21: <*F.x*>"{y} c= {1} by A16,RELAT_1:167; {1} c= <*F.x*>"{y} proof let z be set; assume A22: z in {1}; then z = 1 by TARSKI:def 1; then y = <*F.x*>.z & y in {y} by A17,FINSEQ_1:57,TARSKI:def 1; hence z in <*F.x*>"{y} by A16,A22,FUNCT_1:def 13; end; then <*F.x*>"{y} = {1} by A21,XBOOLE_0:def 10; hence card((F|{x})"{y}) = card FF by A20,CARD_1:79; case A23: y <> F.x; A24: now assume A25: (F|{x})"{y} <> {}; consider z being Element of (F|{x})"{y}; A26: z in {x} & (F|{x}).z in {y} by A15,A25,FUNCT_1:def 13; then z = x & (F|{x}).z = y by TARSKI:def 1; hence contradiction by A15,A23,A26,FUNCT_1:68; end; now assume A27: <*F.x*>"{y} <> {}; consider z being Element of <*F.x*>"{y}; z in {1} & <*F.x*>.z in {y} by A16,A27,FUNCT_1:def 13; then z = 1 & <*F.x*>.z = y by TARSKI:def 1; hence contradiction by A23,FINSEQ_1:57; end; hence card((F|{x})"{y}) = card FF by A24; end; hence card((F|X)"{y}) = card((F|Y)"{y}) + card FF - card {} by A13,A14,CARD_2:64 .= card(A"{y}) by A11,CARD_1:78,FINSEQ_3:63; end; hence thesis by Def1; end; theorem for F be Function, X be finite set ex f be FinSequence st F|X, f are_fiberwise_equipotent proof let F be Function, X be finite set; A1: for n holds P[n] from Ind(Lm2,Lm3); card(dom(F|X)) = card(dom(F|X)); hence thesis by A1; end; definition let D be set, f be FinSequence of D, n be Nat; func f /^ n -> FinSequence of D means :Def2: len it = len f - n & for m be Nat st m in dom it holds it.m = f.(m+n) if n<=len f otherwise it = <*>D; existence proof thus n<=len f implies ex p1 be FinSequence of D st len p1 = len f - n & for m st m in dom p1 holds p1.m = f.(m+n) proof assume n<=len f; then max(0,len f - n) = len f - n by FINSEQ_2:4; then reconsider k = len f - n as Nat by FINSEQ_2:5; deffunc F(Nat)=f.($1+n); consider p be FinSequence such that A1: len p = k & for m st m in Seg k holds p.m = F(m) from SeqLambda; A2: dom p = Seg len p & Seg len f = dom f & rng f c= D by FINSEQ_1:def 3,def 4; rng p c= rng f proof let x; assume x in rng p; then consider m such that A3: m in dom p & p.m = x by FINSEQ_2:11; m in Seg len p by A3,FINSEQ_1:def 3; then A4: p.m = f.(m+n) by A1; 1<=m & m<=len p & m<=m+n by A3,FINSEQ_3:27,NAT_1:29; then 1<=m+n & m+n<=len f by A1,AXIOMS:22,REAL_1:84; then m+n in dom f by FINSEQ_3:27; hence x in rng f by A3,A4,FUNCT_1:def 5; end; then rng p c= D by A2,XBOOLE_1:1; then reconsider p as FinSequence of D by FINSEQ_1:def 4; take p; thus len p = len f - n by A1; let m be Nat; assume m in dom p; then m in Seg k by A1,FINSEQ_1:def 3; hence p.m = f.(m+n) by A1; end; thus not n<=len f implies ex p1 be FinSequence of D st p1 = <*>D; end; uniqueness proof let p1,p2 be FinSequence of D; thus n<=len f & (len p1 = len f - n & for m st m in dom p1 holds p1.m = f.(m+n)) & (len p2 = len f - n & for m st m in dom p2 holds p2.m = f.(m+n)) implies p1 = p2 proof assume that n<=len f and A5: len p1 = len f - n & for m st m in dom p1 holds p1.m = f.(m+n) and A6: len p2 = len f - n & for m st m in dom p2 holds p2.m = f.(m+n); A7: dom p1 = Seg len p1 & Seg len p2 = dom p2 by FINSEQ_1:def 3; now let m; assume m in Seg len p1; then p1.m = f.(m+n) & p2.m = f.(m+n) by A5,A6,A7; hence p1.m = p2.m; end; hence p1 = p2 by A5,A6,FINSEQ_2:10; end; thus not n<=len f & p1 = <*>D & p2 = <*>D implies p1 = p2; end; consistency; end; Lm4: for D be non empty set, f be FinSequence of D st len f<=n holds (f|n) = f proof let D be non empty set,f be FinSequence of D; assume A1: len f<=n; A2: dom f = Seg len f & f|n = f|Seg n by FINSEQ_1:def 3,TOPREAL1:def 1; then dom f c= Seg n by A1,FINSEQ_1:7; hence (f|n) = f by A2,RELAT_1:97; end; theorem Th19: for D be non empty set, f be FinSequence of D, n,m be Nat holds n in dom f & m in Seg n implies (f|n).m = f.m & m in dom f proof let D be non empty set,f be FinSequence of D, n,m; assume A1: n in dom f & m in Seg n; A2: dom f = Seg len f & f|n = f|Seg n by FINSEQ_1:def 3,TOPREAL1:def 1; n<=len f by A1,FINSEQ_3:27; then A3: Seg n c= dom f by A2,FINSEQ_1:7; then Seg n = dom f /\ Seg n by XBOOLE_1:28 .= dom(f|n) by A2,FUNCT_1:68; hence (f|n).m = f.m & m in dom f by A1,A2,A3,FUNCT_1:68; end; theorem Th20: for D be non empty set, f be FinSequence of D, n be Nat, x be set st len f = n+1 & x = f.(n+1) holds f = (f|n) ^ <*x*> proof let D be non empty set, f be FinSequence of D, n,x; assume A1: len f = n+1 & x = f.(n+1); set fn = f|n; A2: n<=n+1 by NAT_1:29; then A3: len fn = n by A1,TOPREAL1:3; then A4: len (fn^<*x*>) = n + len <*x*> by FINSEQ_1:35 .= len f by A1,FINSEQ_1:57; now let m; assume m in Seg len f; then A5: 1<=m & m<=len f by FINSEQ_1:3; now per cases; case m = len f; hence f.m = (fn^<*x*>).m by A1,A3,FINSEQ_1:59; case m <> len f; then m<n+1 by A1,A5,REAL_1:def 5; then A6: m<=n by NAT_1:38; then A7: m in dom fn by A3,A5,FINSEQ_3:27; 1<=n by A5,A6,AXIOMS:22; then A8:n in dom f by A1,A2,FINSEQ_3:27; A9: Seg len fn = dom fn by FINSEQ_1:def 3; thus (fn^<*x*>).m = fn.m by A7,FINSEQ_1:def 7 .= f.m by A3,A7,A8,A9,Th19; end; hence f.m = (fn^<*x*>).m; end; hence thesis by A4,FINSEQ_2:10; end; theorem Th21: for D be non empty set, f be FinSequence of D, n be Nat holds (f|n)^(f/^n) = f proof let D be non empty set, f be FinSequence of D,n; set fn = f/^n; now per cases; case A1: len f<n; then A2: f/^n = <*>D by Def2; f|n = f by A1,Lm4; hence (f|n) ^ (f/^n) = f by A2,FINSEQ_1:47; case A3: n<=len f; then A4: len fn = len f - n & for m st m in dom fn holds fn.m = f.(m+n) by Def2; A5: len (f|n) = n by A3,TOPREAL1:3; then A6: len ((f|n)^(f/^n)) = n+(len f - n) by A4,FINSEQ_1:35 .= len f by XCMPLX_1:27; A7: dom(f|n) = Seg n & dom f = dom f & dom fn = dom fn by A5,FINSEQ_1:def 3; now let m; assume m in Seg len f; then A8: 1<=m & m<=len f by FINSEQ_1:3; now per cases; case m<=n; then A9: m in Seg n & 1<=n by A8,AXIOMS:22,FINSEQ_1:3; then A10: n in dom f by A3,FINSEQ_3:27; thus ((f|n)^(f/^n)).m = (f|n).m by A7,A9,FINSEQ_1:def 7 .= f.m by A9,A10,Th19; case A11: n<m; then max(0,m-n) = m-n by FINSEQ_2:4; then reconsider k = m-n as Nat by FINSEQ_2:5; n+1<=m by A11,NAT_1:38; then A12: 1<=k by REAL_1:84; k<=len fn by A4,A8,REAL_1:49; then A13: k in dom fn by A12,FINSEQ_3:27; thus ((f|n) ^ (f/^n)).m = fn.k by A5,A6,A8,A11,FINSEQ_1:37 .= f.(k+n) by A3,A13,Def2 .= f.m by XCMPLX_1:27; end; hence ((f|n) ^ (f/^n)).m = f.m; end; hence (f|n) ^ (f/^n) = f by A6,FINSEQ_2:10; end; hence thesis; end; theorem for R1,R2 be FinSequence of REAL st R1,R2 are_fiberwise_equipotent holds Sum R1 = Sum R2 proof defpred P[Nat] means for f,g be FinSequence of REAL st f,g are_fiberwise_equipotent & len f = $1 holds Sum f = Sum g; A1: P[0] proof let f,g be FinSequence of REAL; assume f,g are_fiberwise_equipotent & len f = 0; then len g = 0 & f = <*>REAL by Th16,FINSEQ_1:32; hence thesis by FINSEQ_1:32; end; A2: for n st P[n] holds P[n+1] proof let n; assume A3: P[n]; let f,g be FinSequence of REAL; assume A4: f,g are_fiberwise_equipotent & len f = n+1; then A5: rng f = rng g & len f = len g by Th1,Th16; 0<n+1 by NAT_1:19; then 0+1<=n+1 by NAT_1:38; then A6: n+1 in dom f by A4,FINSEQ_3:27; set a = f.(n+1); a in rng g by A5,A6,FUNCT_1:def 5; then consider m such that A7: m in dom g & g.m = a by FINSEQ_2:11; A8: g = (g|m)^(g/^m) by Th21; A9: 1<=m & m<=len g by A7,FINSEQ_3:27; then max(0,m-1) = m-1 by FINSEQ_2:4; then reconsider m1 = m-1 as Nat by FINSEQ_2:5; set gg = g/^m, gm = g|m; A10: len gm = m by A9,TOPREAL1:3; A11: m = m1+1 by XCMPLX_1:27; m in Seg m by A9,FINSEQ_1:3; then gm.m = a & m in dom g by A7,Th19; then A12: gm = (gm|m1)^<*a*> by A10,A11,Th20; m1<=m by A11,NAT_1:29; then A13: Seg m1 c= Seg m by FINSEQ_1:7; A14: gm|m1 = gm|(Seg m1) by TOPREAL1:def 1 .= (g|(Seg m))|(Seg m1) by TOPREAL1:def 1 .= g|((Seg m)/\(Seg m1)) by RELAT_1:100 .= g|(Seg m1) by A13,XBOOLE_1:28 .= g|m1 by TOPREAL1:def 1; set fn = f|n; n<=n+1 by NAT_1:29; then A15: len fn = n by A4,TOPREAL1:3; A16: f = fn ^ <*a*> by A4,Th20; now let x; card(f"{x}) = card(g"{x}) by A4,Def1; then card(fn"{x}) + card(<*a*>"{x}) = card(((g|m1)^<*a*>^(g/^m))"{x}) by A8,A12,A14,A16,FINSEQ_3:63 .= card(((g|m1)^<*a*>)"{x}) + card((g/^m)"{x}) by FINSEQ_3:63 .= card((g|m1)"{x})+ card(<*a*>"{x}) + card((g/^m)"{x}) by FINSEQ_3:63 .= card((g|m1)"{x}) + card((g/^m)"{x})+ card(<*a*>"{x}) by XCMPLX_1:1 .= card(((g|m1)^(g/^m))"{x})+ card(<*a*>"{x}) by FINSEQ_3:63; hence card(fn"{x}) = card(((g|m1)^(g/^m))"{x}) by XCMPLX_1:2; end; then fn, (g|m1)^(g/^m) are_fiberwise_equipotent by Def1; then Sum fn = Sum((g|m1)^gg) by A3,A15; hence Sum f = Sum((g|m1)^gg) + Sum <*a*> by A16,RVSUM_1:105 .= Sum(g|m1) + Sum gg+ Sum <*a*> by RVSUM_1:105 .= Sum(g|m1)+ Sum <*a*> + Sum gg by XCMPLX_1:1 .= Sum gm + Sum gg by A12,A14,RVSUM_1:105 .= Sum g by A8,RVSUM_1:105; end; A17: for n holds P[n] from Ind(A1,A2); let R1,R2 be FinSequence of REAL; assume A18: R1,R2 are_fiberwise_equipotent; len R1 = len R1; hence thesis by A17,A18; end; definition let R be FinSequence of REAL; func MIM(R) -> FinSequence of REAL means :Def3: len it = len R & it.(len it) = R.(len R) & for n be Nat st 1 <= n & n <= len it - 1 holds it.n = (R.n) - (R.(n+1)); existence proof per cases; suppose A1: len R = 0; take a = R; thus len a = len R & a.(len a) = R.(len R); let n; assume 1<=n & n<=len a - 1; hence a.n = R.n - R.(n+1) by A1,AXIOMS:22; suppose len R <> 0; then 0<len R by NAT_1:19; then 0+1<=len R by NAT_1:38; then max(0,len R -1) = len R - 1 by FINSEQ_2:4; then reconsider l = len R -1 as Nat by FINSEQ_2:5; set t = R.(len R); defpred P[Nat,set] means $2 = R.$1 - R.($1+1); A2: for n st n in Seg l ex x be Real st P[n,x]; consider p be FinSequence of REAL such that A3: dom p = Seg l & for n st n in Seg l holds P[n,p.n] from SeqDEx(A2); take a = p^<*t*>; thus A4: len a = len p + len <*t*> by FINSEQ_1:35 .= l+len <*t*> by A3,FINSEQ_1:def 3 .= l+1 by FINSEQ_1:56 .= len R by XCMPLX_1:27; hence a.(len a) = a.(l+1) by XCMPLX_1:27 .= a.(len p +1) by A3,FINSEQ_1:def 3 .= R.(len R) by FINSEQ_1:59; let n; assume 1<=n & n<=len a - 1; then A5: n in Seg l by A4,FINSEQ_1:3; then p.n = R.n - R.(n+1) by A3; hence thesis by A3,A5,FINSEQ_1:def 7; end; uniqueness proof let p1,p2 be FinSequence of REAL; assume that A6: len p1 = len R & p1.(len p1) = R.(len R) and A7: for n st 1<=n & n<=len p1 - 1 holds p1.n = R.n - R.(n+1) and A8: len p2 = len R & p2.(len p2) = R.(len R) and A9: for n st 1<=n & n<=len p2 - 1 holds p2.n = R.n - R.(n+1); now let n; assume n in Seg len R; then A10: 1<=n & n<=len R by FINSEQ_1:3; set r = R.n; now per cases; case n = len R; hence p1.n = p2.n by A6,A8; case n <> len R; then n<len R & n<=n+1 by A10,NAT_1:29,REAL_1:def 5; then A11: 1<=n+1 & n+1<=len R by A10,NAT_1:38; set s = R.(n+1); n<=len R - 1 by A11,REAL_1:84; then p1.n = r-s & p2.n = r-s by A6,A7,A8,A9,A10; hence p1.n = p2.n; end; hence p1.n = p2.n; end; hence thesis by A6,A8,FINSEQ_2:10; end; end; theorem Th23: for R be FinSequence of REAL, r be Real, n be Nat st len R = n+2 & R.(n+1) = r holds MIM(R|(n+1)) = MIM(R)|n ^ <* r *> proof let R be FinSequence of REAL, s,n; assume A1: len R = n+2 & R.(n+1) = s; set f1 = R|(n+1), m1 = MIM(f1), mf = MIM(R), fn = mf|n; A2: n+1+1 = n+(1+1) by XCMPLX_1:1; then A3: n+1<=n+2 by NAT_1:29; then A4: len f1 = n+1 by A1,TOPREAL1:3; then A5: len MIM(f1) = n+1 & len mf = n+2 by A1,Def3; A6: n<=n+2 by NAT_1:29; then A7: len fn = n by A5,TOPREAL1:3; then A8: len(fn^<*s*>) = n + len <*s*> by FINSEQ_1:35 .= n+1 by FINSEQ_1:57; A9: dom m1 = Seg len m1 & Seg len f1 = dom f1 & Seg len mf = dom mf & Seg len fn = dom fn & Seg len R = dom R by FINSEQ_1:def 3; 0<n+1 by NAT_1:19; then 0+1<=n+1 by NAT_1:38; then A10: n+1 in Seg(n+2) & n+1 in Seg(n+1) by A3,FINSEQ_1:3; then f1.(n+1) = s by A1,A9,Th19; then A11: m1.(n+1) = s by A4,A5,Def3; now let m; assume A12: m in Seg (n+1); then A13: 1<=m & m<=n+1 by FINSEQ_1:3; now per cases; case m = n+1; hence m1.m = (fn^<*s*>).m by A7,A11,FINSEQ_1:59; case m <> n+1; then A14: m<n+1 by A13,REAL_1:def 5; then A15: m<=n by NAT_1:38; then A16: m in Seg n by A13,FINSEQ_1:3; 1<=n by A13,A15,AXIOMS:22; then n in Seg(n+2) by A6,FINSEQ_1:3; then A17: fn.m = mf.m & m in dom mf by A5,A9,A16,Th19; A18: len mf -1 = n+(1+1-1) by A5,XCMPLX_1:29 .= n+1; set r1 = R.m; A19: 1<=m+1 & m+1<=n+2 by A2,A13,AXIOMS:24,NAT_1:29; set r2 = R.(m+1); A20: r1 - r2 = fn.m by A13,A17,A18,Def3 .= (fn^<*s*>).m by A7,A9,A16,FINSEQ_1:def 7; A21: len m1 -1 = n by A5,XCMPLX_1:26; A22: f1.m = r1 by A1,A9,A10,A12,Th19; m+1<=n+1 by A14,NAT_1:38; then m+1 in Seg(n+1) by A19,FINSEQ_1:3; then f1.(m+1) = r2 by A1,A9,A10,Th19; hence m1.m = (fn^<*s*>).m by A13,A15,A20,A21,A22,Def3; end; hence m1.m = (fn^<*s*>).m; end; hence thesis by A5,A8,FINSEQ_2:10; end; theorem Th24: for R be FinSequence of REAL, r,s be Real, n be Nat st len R = n+2 & R.(n+1) = r & R.(n+2) = s holds MIM(R) = MIM(R)|n ^ <* r-s,s *> proof let R be FinSequence of REAL, r,s,n; set mf = MIM(R), nf = mf|n; assume A1: len R = n+2 & R.(n+1) = r & R.(n+2) = s; then A2: len mf = n+2 by Def3; A3: n+(1+1) = n+1+1 by XCMPLX_1:1; then A4: n<=n+1 & n+1<=n+2 & n<=n+2 by NAT_1:29; then A5: len nf = n by A2,TOPREAL1:3; then len (nf^<* r-s,s *>) = n + len <* r-s,s *> by FINSEQ_1:35; then A6: len(nf^<* r-s,s *>) = n+2 & <*r-s,s*>.1 = r-s & <*r-s,s*>.2 = s by FINSEQ_1:61; A7: n<n+2 by A4,NAT_1:38; now let m; assume m in Seg(n+2); then A8: 1<=m & m<=n+2 by FINSEQ_1:3; now per cases; case A9: m = n+2; hence mf.m = s by A1,A2,Def3 .= <* r-s,s *>.(n+2-n) by A6,XCMPLX_1:26 .= (nf ^ <* r-s,s *>).m by A5,A6,A7,A9,FINSEQ_1:37; case m <> n+2; then m<n+2 by A8,REAL_1:def 5; then A10: m<=n+1 by A3,NAT_1:38; A11: len mf - 1 = n+1 by A2,A3,XCMPLX_1:26; now per cases; case A12: m = n+1; then A13: n<m by NAT_1:38; thus mf.m = r - s by A1,A3,A8,A11,A12,Def3 .= <* r-s,s *>.(n+1-n) by A6,XCMPLX_1:26 .= (nf ^ <* r-s,s *>).m by A5,A6,A8,A12,A13,FINSEQ_1:37; case m <> n+1; then m<n+1 by A10,REAL_1:def 5; then m<=n by NAT_1:38 ; then A14: m in Seg n & m<=m+1 & m+1<=n+1 & 1<=n by A8,AXIOMS:22,24,FINSEQ_1:3,NAT_1:29; then A15: n in Seg(n+2) by A4,FINSEQ_1:3; A16: dom nf = Seg len nf & dom mf = Seg len mf by FINSEQ_1:def 3; hence mf.m = nf.m by A2,A14,A15,Th19 .= (nf ^ <* r-s,s *>).m by A5,A14,A16,FINSEQ_1:def 7; end; hence mf.m = (nf ^ <* r-s,s *>).m; end; hence mf.m = (nf ^ <* r-s,s *>).m; end; hence thesis by A2,A6,FINSEQ_2:10; end; theorem Th25: MIM( <*>REAL ) = <*>REAL proof set o = <*>REAL, mo = MIM(o); A1: len mo = len o & len o = 0 by Def3,FINSEQ_1:32; for n st n in Seg 0 holds mo.n = o.n by FINSEQ_1:4; hence thesis by A1,FINSEQ_2:10; end; theorem Th26: for r be Real holds MIM(<*r*>) = <*r*> proof let r be Real; set f = <*r*>; A1: len f = 1 & f.1 = r by FINSEQ_1:57; then A2: len MIM(f) = 1 by Def3; now let n; assume n in Seg 1; then n = 1 by FINSEQ_1:4,TARSKI:def 1; hence MIM(f).n = f.n by A1,A2,Def3; end; hence thesis by A1,A2,FINSEQ_2:10; end; theorem for r,s be Real holds MIM(<*r,s*>) = <*r-s,s*> proof let r,s be Real; set f = <*r,s*>; 0+2 = 2 & 0+1 = 1 & len f = 2 & f.1 = r & f.2 = s by FINSEQ_1:61; then A1: MIM(f) = MIM(f)|0 ^ <*r-s,s*> by Th24; 0<=len MIM(f) by NAT_1:18; then len(MIM(f)|0) = 0 by TOPREAL1:3; then MIM(f)|0 = {} by FINSEQ_1:25; hence thesis by A1,FINSEQ_1:47; end; theorem for R be FinSequence of REAL, n be Nat holds MIM(R) /^ n = MIM(R/^n) proof let R be FinSequence of REAL,n; set mf = MIM(R), fn = R/^n, mn = MIM(fn); A1: len mf = len R & len mn = len fn by Def3; now per cases; case len R<n; then mf /^ n = <*>REAL & fn = <*>REAL by A1,Def2; hence mf /^ n = mn by Th25; case A2: n<=len R; then A3: len(mf/^n) = len R - n & for m st m in dom(mf/^n) holds (mf/^n).m = mf.(m+n) by A1,Def2; A4: len fn = len R - n & for m st m in dom fn holds fn.m = R.(m+n) by A2,Def2; A5: len mn = len fn by Def3; A6: Seg len fn = dom fn & Seg len(mf/^n) = dom(mf/^n) by FINSEQ_1:def 3; now let m; assume A7: m in Seg len fn; then A8: 1<=m & m<=len fn by FINSEQ_1:3; m<=m+n by NAT_1:29; then A9: 1<=m+n & m+n<=len R by A4,A8,AXIOMS:22,REAL_1:84; set r1 = R.(m+n); A10: fn.m = r1 by A2,A6,A7,Def2; now per cases; case A11: m = len fn; then A12: m+n = len R by A4,XCMPLX_1:27; thus (mf/^n).m = mf.(m+n) by A3,A4,A6,A7 .= r1 by A1,A12,Def3 .= mn.m by A5,A10,A11,Def3; case m <> len fn; then m<len fn by A8,REAL_1:def 5; then A13: m+1<=len fn by NAT_1:38; then A14: m+1+n<=len R & m<=m+1 & m+1<=m+1+n by A4,NAT_1:29,REAL_1:84; set r2 = R.(m+1+n); A15: m+1+n = m+n+1 by XCMPLX_1:1; then A16: m+n<=len R - 1 & m<=len fn - 1 by A13,A14,REAL_1:84; 1<=m+1 by NAT_1:29; then m+1 in dom fn by A13,FINSEQ_3:27; then A17: fn.(m+1) = r2 by A2,Def2; thus (mf/^n).m = mf.(m+n) by A3,A4,A6,A7 .= r1-r2 by A1,A9,A15,A16,Def3 .= mn.m by A1,A8,A10,A16,A17,Def3; end; hence (mf/^n).m = mn.m; end; hence thesis by A3,A4,A5,FINSEQ_2:10; end; hence thesis; end; theorem Th29: for R be FinSequence of REAL st len R <> 0 holds Sum MIM(R) = R.1 proof let R be FinSequence of REAL; defpred P[Nat] means for R be FinSequence of REAL st len R <> 0 & len R = $1 holds Sum MIM(R) = R.1; A1: P[0]; A2: for n st P[n] holds P[n+1] proof let n; assume A3: P[n]; let R be FinSequence of REAL; assume A4: len R <> 0 & len R = n+1; now per cases; case n = 0; then A5: R = <*R.1*> by A4,FINSEQ_1:57; then MIM(R) = R by Th26; hence Sum MIM(R) = R.1 by A5,RVSUM_1:103; case n <> 0; then 0<n by NAT_1:19; then 0+1<=n by NAT_1:38; then max(0,n-1) = n-1 by FINSEQ_2:4; then reconsider n1 = n-1 as Nat by FINSEQ_2:5; A6: n1+2 = n1+(1+1) .= n1+1+1 by XCMPLX_1:1; then A7: n1+2 = len R by A4,XCMPLX_1:27; A8: 0+1<=n1+1 & n1+1<=n1+2 by A6,NAT_1:29; then A9: n1+1 in Seg(n1+2) by FINSEQ_1:3; set r = R.(n1+1); set s = R.(n1+2); set f1 = R|(n1+1); A10: n1+1 = n by XCMPLX_1:27; A11: len f1 = n1+1 by A7,A8,TOPREAL1:3; A12: Seg len R = dom R & 1 in Seg(n1+1) by A8,FINSEQ_1:3,def 3; thus Sum MIM(R) = Sum((MIM(R)|n1) ^ <* r-s,s *>) by A7,Th24 .= Sum(MIM(R)|n1) + Sum(<* r-s,s *>) by RVSUM_1:105 .= Sum(MIM(R)|n1) + (r-s + s) by RVSUM_1:107 .= Sum(MIM(R)|n1) + r by XCMPLX_1:27 .= Sum((MIM(R)|n1) ^<*r*>) by RVSUM_1:104 .= Sum MIM(f1) by A7,Th23 .= f1.1 by A3,A10,A11 .= R.1 by A7,A9,A12,Th19; end; hence Sum MIM(R) = R.1; end; A13: for n holds P[n] from Ind(A1,A2); assume len R <> 0; hence thesis by A13; end; theorem for R be FinSequence of REAL, n be Nat st 1<=n & n<len R holds Sum MIM(R/^n) = R.(n+1) proof let R be FinSequence of REAL,n; assume A1: 1<=n & n<len R; then n+1<=len R & n<=len R & n <> len R by NAT_1:38; then A2: 1<=len R - n by REAL_1:84; len(R/^n) = len R - n & for m st m in dom(R/^n) holds (R/^n).m = R.(m+n) by A1,Def2; then A3: 1 in dom(R/^n) & Seg len(R/^n) = dom(R/^n) & len(R/^n) <> 0 by A2,FINSEQ_1:def 3,FINSEQ_3:27; hence Sum MIM(R/^n) = (R/^n).1 by Th29 .= R.(n+1) by A1,A3,Def2; end; definition let IT be FinSequence of REAL; attr IT is non-increasing means :Def4: for n be Nat st n in dom IT & n+1 in dom IT holds IT.n >= IT.(n+1); end; definition cluster non-increasing FinSequence of REAL; existence proof take f = <*>REAL; let n; assume n in dom f & n+1 in dom f; hence f.n>=f.(n+1) by FINSEQ_1:26; end; end; theorem Th31: for R be FinSequence of REAL st len R = 0 or len R = 1 holds R is non-increasing proof let R be FinSequence of REAL; assume A1: len R = 0 or len R = 1; now per cases by A1; case len R = 0; then R = <*>REAL by FINSEQ_1:32; then for n st n in dom R & n+1 in dom R holds R.n>=R.(n+1) by FINSEQ_1:26; hence thesis by Def4; case len R = 1; then A2: dom R = {1} by FINSEQ_1:4,def 3; now let n; assume n in dom R & n+1 in dom R; then n = 1 & n+1 = 1 by A2,TARSKI:def 1; hence R.n>=R.(n+1); end; hence thesis by Def4; end; hence thesis; end; theorem Th32: for R be FinSequence of REAL holds R is non-increasing iff for n,m be Nat st n in dom R & m in dom R & n<m holds R.n>=R.m proof let R be FinSequence of REAL; thus R is non-increasing implies for n,m st n in dom R & m in dom R & n<m holds R.n>=R.m proof assume A1: R is non-increasing; defpred P[Nat] means $1 in dom R implies for n st n in dom R & n<$1 holds R.n>=R.$1; Seg len R = dom R by FINSEQ_1:def 3; then A2: P[0] by FINSEQ_1:3; A3: for m st P[m] holds P[m+1] proof let m; assume A4: P[m]; assume A5: m+1 in dom R; let n; assume A6: n in dom R & n<m+1; then A7: 1<=m+1 & m+1<=len R & 1<=n & n<=len R & n<=m & m<=m+1 by A5,FINSEQ_3:27,NAT_1:38; then A8: 1<=m & m<=len R by AXIOMS:22; then A9: m in dom R by FINSEQ_3:27; set t = R.m, r = R.n, s = R.(m+1); now per cases; case n = m; hence r>=s by A1,A5,A6,Def4; case n <> m; then n<m by A7,REAL_1:def 5; then A10: r>=t by A4,A6,A8,FINSEQ_3:27; t>=s by A1,A5,A9,Def4; hence r>=s by A10,AXIOMS:22; end; hence r>=s; end; for m holds P[m] from Ind(A2,A3); hence thesis; end; assume A11: for n,m st n in dom R & m in dom R & n<m holds R.n>=R.m; let n; assume A12: n in dom R & n+1 in dom R; n<n+1 by NAT_1:38; hence thesis by A11,A12; end; theorem Th33: for R be non-increasing FinSequence of REAL, n be Nat holds R|n is non-increasing FinSequence of REAL proof let f be non-increasing FinSequence of REAL, n; set fn = f|n; now per cases; case A1: n = 0; then n<=len f by NAT_1:18; then len fn = 0 by A1,TOPREAL1:3; hence thesis by Th31; case n <> 0; then 0<n by NAT_1:19; then A2: 0+1<=n by NAT_1:38; now per cases; case len f<=n; hence thesis by Lm4; case n<len f; then A3: n in dom f & len fn = n by A2,FINSEQ_3:27,TOPREAL1:3; now let m; assume A4: m in dom fn & m+1 in dom fn; dom f = Seg len f & dom fn = Seg len fn by FINSEQ_1:def 3; then f.m = fn.m & f.(m+1) = fn.(m+1) & m in dom f & m+1 in dom f by A3,A4,Th19; hence fn.m>=fn.(m+1) by Def4; end; hence thesis by Def4; end; hence thesis; end; hence thesis; end; theorem for R be non-increasing FinSequence of REAL, n be Nat holds R /^ n is non-increasing FinSequence of REAL proof let f be non-increasing FinSequence of REAL, n; set fn = f /^ n; now let m; assume A1: m in dom fn & m+1 in dom fn; then A2: 1<=m & m<=len fn & 1<=m+1 & m+1<=len fn by FINSEQ_3:27; A3: m+1+n = m+n+1 & m<=m+n & m+1<=m+1+n by NAT_1:29,XCMPLX_1:1; then A4: 1<=m+n & 1<=m+n+1 & 1<=len fn by A2,AXIOMS:22; set r = fn.m, s = fn.(m+1); now per cases; case n<=len f; then A5: len fn = len f - n & r = f.(m+n) & s = f.(m+n+1) by A1,A3,Def2; then m+n<=len f & m+n+1<=len f by A2,A3,REAL_1:84; then m+n in dom f & m+n+1 in dom f by A4,FINSEQ_3:27; hence r>=s by A5,Def4; case len f<n; then fn = <*>REAL by Def2; hence r>=s by A1,FINSEQ_1:26; end; hence r>=s; end; hence thesis by Def4; end; Lm5: for f,g be non-increasing FinSequence of REAL, n st len f = n+1 & len f = len g & f,g are_fiberwise_equipotent holds f.(len f) = g.(len g) & f|n, g|n are_fiberwise_equipotent proof let f,g be non-increasing FinSequence of REAL, n; assume A1: len f = n+1 & len f = len g & f,g are_fiberwise_equipotent; 0<n+1 by NAT_1:19; then 0+1<=n+1 by NAT_1:38; then A2: n+1 in dom f & n+1 in dom g by A1,FINSEQ_3:27; set r = f.(n+1), s = g.(n+1); A3: now assume A4: r <> s; A5: rng f = rng g by A1,Th1; now per cases by A4,REAL_1:def 5; case A6: r>s; s in rng f by A2,A5,FUNCT_1:def 5; then consider m such that A7: m in dom f & f.m = s by FINSEQ_2:11; A8: 1<=m & m<=len f by A7,FINSEQ_3:27; now per cases; case m = len f; hence contradiction by A1,A6,A7; case m <> len f; then m<n+1 by A1,A8,REAL_1:def 5; hence contradiction by A2,A6,A7,Th32; end; hence contradiction; case A9: r<s; r in rng g by A2,A5,FUNCT_1:def 5; then consider m such that A10: m in dom g & g.m = r by FINSEQ_2:11; A11: 1<=m & m<=len g by A10,FINSEQ_3:27; now per cases; case m = len g; hence contradiction by A1,A9,A10; case m <> len g; then m<n+1 by A1,A11,REAL_1:def 5; hence contradiction by A2,A9,A10,Th32; end; hence contradiction; end; hence contradiction; end; hence f.(len f) = g.(len g) by A1; A12: f = (f|n)^<*r*> & g = (g|n)^<*r*> by A1,A3,Th20; now let x; card((f|n)"{x}) + card(<*r*>"{x}) = card (f"{x}) by A12,FINSEQ_3:63 .= card(g"{x}) by A1,Def1 .= card((g|n)"{x}) + card(<*r*>"{x}) by A12,FINSEQ_3:63; hence card((f|n)"{x}) = card((g|n)"{x}) by XCMPLX_1:2; end; hence thesis by Def1; end; defpred P[Nat] means for R be FinSequence of REAL st $1 = len R ex b be non-increasing FinSequence of REAL st R,b are_fiberwise_equipotent; Lm6: P[0] proof let R be FinSequence of REAL; assume len R = 0; then reconsider a = R as non-increasing FinSequence of REAL by Th31; take a; thus thesis; end; Lm7: for n st P[n] holds P[n+1] proof let n; assume A1: P[n]; let R be FinSequence of REAL; assume A2: n+1 = len R; then A3: dom R = Seg(n+1) by FINSEQ_1:def 3; set fn = R|(Seg n); A4: fn = R|n by TOPREAL1:def 1; A5: dom fn = dom R /\ Seg n by FUNCT_1:68; reconsider fn as FinSequence by FINSEQ_1:19; rng fn c= rng R & rng R c= REAL by FINSEQ_1:def 4,FUNCT_1:76; then rng fn c= REAL by XBOOLE_1:1; then reconsider fn as FinSequence of REAL by FINSEQ_1:def 4; n<=n+1 by NAT_1:29; then Seg n c= Seg(n+1) by FINSEQ_1:7; then dom fn = Seg n by A3,A5,XBOOLE_1:28; then A6: len fn = n by FINSEQ_1:def 3; then consider a be non-increasing FinSequence of REAL such that A7: fn,a are_fiberwise_equipotent by A1; A8: rng fn = rng a & len fn = len a by A7,Th1,Th16; 0<=n by NAT_1:18; then 0+1<=n+1 by REAL_1:55; then A9: n+1 in Seg(n+1) & Seg len a = dom a by FINSEQ_1:3,def 3; set q = R.(n+1); now per cases; case A10: for t be Real st t in rng a holds q<=t; set b = a^<*q*>; A11: len b = n + len <*q*> by A6,A8,FINSEQ_1:35 .= n+1 by FINSEQ_1:56; now let m; assume m in dom b & m+1 in dom b; then A12: 1<=m & m<=len b & 1<=m+1 & m+1<=len b by FINSEQ_3:27; then m<=n+1-1 by A11,REAL_1:84; then m<=n by XCMPLX_1:26; then m in Seg n by A12,FINSEQ_1:3; then A13: m in dom a by A6,A8,FINSEQ_1:def 3; set r = b.m, s = b.(m+1); A14: r = a.m & a.m in rng a by A13,FINSEQ_1:def 7,FUNCT_1:def 5; now per cases; case m+1 = len b; then s = q by A6,A8,A11,FINSEQ_1:59; hence r>=s by A10,A14; case m+1 <> len b; then m+1<len b by A12,REAL_1:def 5; then m+1+1<=n+1 by A11,NAT_1:38; then m+1<=n+1-1 by REAL_1:84; then m+1<=n by XCMPLX_1:26; then m+1 in Seg n by A12,FINSEQ_1:3; then A15: m+1 in dom a by A6,A8,FINSEQ_1:def 3; then a.(m+1) = s by FINSEQ_1:def 7; hence r>=s by A13,A14,A15,Def4; end; hence r>=s; end; then reconsider b as non-increasing FinSequence of REAL by Def4; take b; fn^<*q*> = R by A2,A4,Th20; hence R,b are_fiberwise_equipotent by A7,Th14; case A16: ex t be Real st t in rng a & not q<=t; defpred Q[Nat] means $1 in dom a & for r st r = a.$1 holds r<q; A17: ex k be Nat st Q[k] proof consider t be Real such that A18: t in rng a & t<q by A16; consider k be Nat such that A19: k in dom a & a.k = t by A18,FINSEQ_2:11; take k; thus k in dom a by A19; let r; assume r = a.k; hence r<q by A18,A19; end; consider mi be Nat such that A20: Q[mi] & for m st Q[m] holds mi <= m from Min(A17); 1<=mi by A20,FINSEQ_3:27; then max(0,mi-1) = mi-1 by FINSEQ_2:4; then reconsider k = mi-1 as Nat by FINSEQ_2:5; set ak = a/^k, b = (a|k)^<*q*>^ak; mi<=mi+1 & mi<mi+1 by NAT_1:38; then mi<=len a & k<=mi & k<mi by A20,FINSEQ_3:27,REAL_1:84; then A21: k<=len a & k<len a by AXIOMS:22; then A22: len(a|k) = k & len(a/^k) = len a - k & k+1<=len a by Def2,NAT_1:38,TOPREAL1:3; then A23: len((a|k)^<*q*>) = k+len <*q*> by FINSEQ_1:35 .= k+1 by FINSEQ_1:56; A24: dom (a|k) = Seg len (a|k) & dom((a|k)^<*q*>) = Seg len((a|k)^<*q*>) & dom(a/^k) = dom(a/^k) by FINSEQ_1:def 3; A25: dom (a|k) c= dom((a|k)^<*q*>) by FINSEQ_1:39; A26: 1<=len(a/^k) by A22,REAL_1:84; A27: len b = k+1 + len(a/^k) by A23,FINSEQ_1:35; now let m; assume m in dom b & m+1 in dom b; then A28: 1<=m & m<=len b & 1<=m+1 & m+1<=len b by FINSEQ_3:27; set r = b.m, s = b.(m+1); now per cases; case A29: m+1<=k; then A30: 1<=k & m<=k by A28,AXIOMS:22,NAT_1:38; dom a = Seg len a by FINSEQ_1:def 3; then A31: k in dom a & m in Seg k & m+1 in Seg k by A21,A28,A29,A30,FINSEQ_1:3; then A32: a.m = (a|k).m & a.(m+1) = (a|k).(m+1) & m in dom a & m+1 in dom a by Th19; A33: b.m = ((a|k)^<*q*>).m by A22,A24,A25,A31,FINSEQ_1:def 7 .= a.m by A22,A24,A31,A32,FINSEQ_1:def 7; A34: b.(m+1) = ((a|k)^<*q*>).(m+1) by A22,A24,A25,A31,FINSEQ_1:def 7 .= a.(m+1) by A22,A24,A31,A32,FINSEQ_1:def 7; dom(a|k) c= dom((a|k)^(a/^k)) by FINSEQ_1:39; then dom(a|k) c= dom a by Th21; hence r>=s by A22,A24,A31,A33,A34,Def4 ; case k<m+1; then A35: k<=m by NAT_1:38; now per cases; case A36: k = m; then A37: m in Seg k by A28,FINSEQ_1:3; A38: k in dom a by A9,A21,A28,A36,FINSEQ_1:3; then A39: a.m = (a|k).m & m in dom a by A37,Th19; A40: b.m = ((a|k)^<*q*>).m by A22,A24,A25,A37,FINSEQ_1:def 7 .= a.m by A22,A24,A37,A39,FINSEQ_1:def 7; m+1 in dom((a|k)^<*q*>) by A23,A24,A36,FINSEQ_1:6; then A41: b.(m+1) = ((a|k)^<*q*>).(k+1) by A36,FINSEQ_1:def 7 .= q by A22,FINSEQ_1:59; now assume s>r; then for t be Real st t = a.k holds t<q by A36,A40,A41; then mi<=k by A20,A38; hence contradiction by SQUARE_1:29; end; hence r>=s; case k <> m; then k<m by A35,REAL_1:def 5; then A42: k+1<=m by NAT_1:38; then A43: k+1<m+1 by NAT_1:38; max(0,m-(k+1)) = m-(k+1) by A42,FINSEQ_2:4; then reconsider l = m-(k+1) as Nat by FINSEQ_2:5; A44: l+1 = m+1-(k+1) by XCMPLX_1:29; A45: 1<=l+1 & l+1<=l+1+k by NAT_1:29; then l+1<=len b -(k+1) & 1<=l+1+k by A28,A44,AXIOMS:22,REAL_1:92; then A46: l<=l+1 & l+1<=len(a/^k) & 1<=k+l+1 by A27,NAT_1:29,XCMPLX_1:26; then A47: l+1 in dom(a/^k) & l<=len(a/^k) & k+(l+1)<=len a by A22,A45,AXIOMS:22,FINSEQ_3:27,REAL_1:84; then k+l+1<=len a & k+l<=k+l+1 by NAT_1:29,XCMPLX_1:1; then A48: k+l+1 in dom a & k+l<=len a by A46,AXIOMS:22,FINSEQ_3:27; now per cases; case A49: k+1 = m; then m in Seg (k+1) by A28,FINSEQ_1:3; then A50: b.m = ((a|k)^<*q*>).(k+1) by A23,A24,A49,FINSEQ_1:def 7 .= q by A22,FINSEQ_1:59; A51: 1 in dom(a/^k) by A26,FINSEQ_3:27; b.(m+1) = (a/^k).(k+1+1 - (k+1)) by A23,A28,A43,A49,FINSEQ_1:37 .= (a/^k).1 by XCMPLX_1:26 .= a.(1+k) by A21,A51,Def2 .= a.mi by XCMPLX_1:27; hence r>=s by A20,A50; case k+1 <> m; then A52: k+1<m by A42,REAL_1:def 5; then k+1+1<=m & l<=l+k by NAT_1:29,38; then A53: 1<=l & l<=k+l by REAL_1:84; then A54: l in dom(a/^k) by A47,FINSEQ_3:27; A55: b.m = (a/^k).l by A23,A28,A52,FINSEQ_1:37 .= a.(k+l) by A21,A54,Def2; A56: b.(m+1) = (a/^k).(l+1) by A23,A28,A43,A44,FINSEQ_1:37 .= a.(l+1+k) by A21,A47,Def2 .= a.(k+l+1) by XCMPLX_1:1; 1<=k+l by A53,AXIOMS:22; then k+l in dom a by A48,FINSEQ_3:27; hence r>=s by A48,A55,A56,Def4; end; hence r>=s; end; hence r>=s; end; hence r>=s; end; then reconsider b as non-increasing FinSequence of REAL by Def4; take b; now let x; A57: card(fn"{x}) = card(a"{x}) by A7,Def1; thus card (b"{x}) = card (((a|k)^<*q*>)"{x}) + card(ak"{x}) by FINSEQ_3:63 .= card((a|k)"{x}) + card(<*q*>"{x}) + card(ak"{x}) by FINSEQ_3:63 .= card((a|k)"{x}) + card(ak"{x}) + card(<*q*>"{x}) by XCMPLX_1:1 .= card(((a|k)^ak)"{x}) + card(<*q*>"{x}) by FINSEQ_3:63 .= card(fn"{x}) + card(<*q*>"{x}) by A57,Th21 .= card((fn^<*q*>)"{x}) by FINSEQ_3:63 .= card(R"{x}) by A2,A4,Th20; end; hence R,b are_fiberwise_equipotent by Def1; end; hence thesis; end; theorem for R be FinSequence of REAL ex R1 be non-increasing FinSequence of REAL st R,R1 are_fiberwise_equipotent proof let R be FinSequence of REAL; A1: for n holds P[n] from Ind(Lm6,Lm7); len R = len R; hence thesis by A1; end; Lm8: for n holds for g1,g2 be non-increasing FinSequence of REAL st n = len g1 & g1,g2 are_fiberwise_equipotent holds g1 = g2 proof defpred P[Nat] means for g1,g2 be non-increasing FinSequence of REAL st $1 = len g1 & g1,g2 are_fiberwise_equipotent holds g1 = g2; A1: P[0] proof let g1,g2 be non-increasing FinSequence of REAL; assume that A2: len g1 = 0 and A3: g1,g2 are_fiberwise_equipotent; len g2 = len g1 by A3,Th16; then g1 = <*>REAL & g2 = <*>REAL by A2,FINSEQ_1:32; hence thesis; end; A4: for n st P[n] holds P[n+1] proof let n; assume A5: P[n]; let g1,g2 be non-increasing FinSequence of REAL; assume that A6: len g1 = n+1 and A7: g1,g2 are_fiberwise_equipotent; A8: len g2 = len g1 by A7,Th16; then A9: g1.(len g1) = g2.(len g2) & g1|n, g2|n are_fiberwise_equipotent by A6,A7,Lm5; n<=len g1 by A6,NAT_1:29; then A10: len(g1|n) = n & len(g2|n) = n by A8,TOPREAL1:3; reconsider g1n = g1|n, g2n = g2|n as non-increasing FinSequence of REAL by Th33; set r = g1.(n+1); A11: g1n = g2n by A5,A9,A10; (g1|n)^<*r*> = g1 & (g2|n)^<*r*> = g2 by A6,A8,A9,Th20; hence thesis by A11; end; thus for n holds P[n] from Ind(A1,A4); end; theorem for R1,R2 be non-increasing FinSequence of REAL st R1,R2 are_fiberwise_equipotent holds R1 = R2 proof let g1,g2 be non-increasing FinSequence of REAL; assume A1: g1,g2 are_fiberwise_equipotent; len g1 = len g1; hence thesis by A1,Lm8; end; theorem for R be FinSequence of REAL, r,s be Real st r <> 0 holds R"{s/r} = (r*R)"{s} proof let R be FinSequence of REAL, r,s; assume A1: r <> 0; A2: r*R = (r multreal) * R by RVSUM_1:def 7; A3: Seg len R = dom R & dom(r*R) = Seg len(r*R) by FINSEQ_1:def 3; thus R"{s/r} c= (r*R)"{s} proof let x; assume A4: x in R"{s/r}; then reconsider i = x as Nat; A5: i in dom R & R.i in {s/r} by A4,FUNCT_1:def 13; then A6: i in Seg len(r*R) & R.i = s/r by A2,A3,FINSEQ_2:37,TARSKI:def 1; A7:i in dom (r*R) by A2,A3,A5,FINSEQ_2:37; r*(R.i) = s by A1,A6,XCMPLX_1:88; then (r*R).i = s by RVSUM_1:66; then (r*R).i in {s} by TARSKI:def 1; hence thesis by A7,FUNCT_1:def 13; end; let x; assume A8: x in (r*R)"{s}; then reconsider i = x as Nat; i in dom(r*R) & (r*R).i in {s} by A8,FUNCT_1:def 13; then A9: i in dom R & (r*R).i = s by A2,A3,FINSEQ_2:37,TARSKI:def 1; then r*R.i = s by RVSUM_1:66; then R.i = s/r by A1,XCMPLX_1:90; then R.i in {s/r} by TARSKI:def 1; hence thesis by A9,FUNCT_1:def 13; end; theorem for R be FinSequence of REAL holds (0*R)"{0} = dom R proof let R be FinSequence of REAL; 0*R = ((0 qua Real) multreal) * R by RVSUM_1:def 7; then A1: len(0*R) = len R by FINSEQ_2:37; A2: dom R = Seg len R & Seg len(0*R) = dom(0*R) by FINSEQ_1:def 3; hence (0*R)"{0} c= dom R by A1,RELAT_1:167; let x; assume A3: x in dom R; then reconsider i = x as Nat; (0*R).i = 0*R.i by RVSUM_1:66 .= 0; then (0*R).i in {0} by TARSKI:def 1; hence thesis by A1,A2,A3,FUNCT_1:def 13; end;

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