:: FINSEQ_2 semantic presentation begin theorem :: FINSEQ_2:1 for i, j being Nat holds ( min (i,j) is Element of NAT & max (i,j) is Element of NAT ) by ORDINAL1:def_12; theorem Th2: :: FINSEQ_2:2 for l, i, j being Nat st l = min (i,j) holds (Seg i) /\ (Seg j) = Seg l proof let l, i, j be Nat; ::_thesis: ( l = min (i,j) implies (Seg i) /\ (Seg j) = Seg l ) ( i <= j or j <= i ) ; then ( ( i <= j & (Seg i) /\ (Seg j) = Seg i ) or ( j <= i & (Seg i) /\ (Seg j) = Seg j ) ) by FINSEQ_1:7; hence ( l = min (i,j) implies (Seg i) /\ (Seg j) = Seg l ) by XXREAL_0:def_9; ::_thesis: verum end; theorem Th3: :: FINSEQ_2:3 for i, j being Nat st i <= j holds max (0,(i - j)) = 0 proof let i, j be Nat; ::_thesis: ( i <= j implies max (0,(i - j)) = 0 ) assume i <= j ; ::_thesis: max (0,(i - j)) = 0 then i - i <= j - i by XREAL_1:9; then - (j - i) <= - 0 ; hence max (0,(i - j)) = 0 by XXREAL_0:def_10; ::_thesis: verum end; theorem Th4: :: FINSEQ_2:4 for j, i being Nat st j <= i holds max (0,(i - j)) = i - j proof let j, i be Nat; ::_thesis: ( j <= i implies max (0,(i - j)) = i - j ) assume j <= i ; ::_thesis: max (0,(i - j)) = i - j then j - j <= i - j by XREAL_1:9; hence max (0,(i - j)) = i - j by XXREAL_0:def_10; ::_thesis: verum end; theorem :: FINSEQ_2:5 for i, j being Nat holds max (0,(i - j)) is Element of NAT proof let i, j be Nat; ::_thesis: max (0,(i - j)) is Element of NAT percases ( i <= j or j <= i ) ; suppose i <= j ; ::_thesis: max (0,(i - j)) is Element of NAT hence max (0,(i - j)) is Element of NAT by Th3; ::_thesis: verum end; supposeA1: j <= i ; ::_thesis: max (0,(i - j)) is Element of NAT then i - j is Element of NAT by NAT_1:21; hence max (0,(i - j)) is Element of NAT by A1, Th4; ::_thesis: verum end; end; end; theorem :: FINSEQ_2:6 for i being Nat holds ( min (0,i) = 0 & min (i,0) = 0 & max (0,i) = i & max (i,0) = i ) by XXREAL_0:def_9, XXREAL_0:def_10; theorem :: FINSEQ_2:7 for i, l being Nat holds ( not i in Seg (l + 1) or i in Seg l or i = l + 1 ) proof let i, l be Nat; ::_thesis: ( not i in Seg (l + 1) or i in Seg l or i = l + 1 ) assume A1: i in Seg (l + 1) ; ::_thesis: ( i in Seg l or i = l + 1 ) then i <= l + 1 by FINSEQ_1:1; then A2: ( ( 1 <= i & i <= l ) or i = l + 1 ) by A1, FINSEQ_1:1, NAT_1:8; i is Element of NAT by ORDINAL1:def_12; hence ( i in Seg l or i = l + 1 ) by A2; ::_thesis: verum end; theorem :: FINSEQ_2:8 for i, l, j being Nat st i in Seg l holds i in Seg (l + j) proof let i, l, j be Nat; ::_thesis: ( i in Seg l implies i in Seg (l + j) ) l <= l + j by NAT_1:11; then Seg l c= Seg (l + j) by FINSEQ_1:5; hence ( i in Seg l implies i in Seg (l + j) ) ; ::_thesis: verum end; theorem :: FINSEQ_2:9 for p, q being FinSequence st len p = len q & ( for j being Nat st j in dom p holds p . j = q . j ) holds p = q proof let p, q be FinSequence; ::_thesis: ( len p = len q & ( for j being Nat st j in dom p holds p . j = q . j ) implies p = q ) assume that A1: len p = len q and A2: for j being Nat st j in dom p holds p . j = q . j ; ::_thesis: p = q ( dom p = Seg (len p) & dom q = Seg (len p) ) by A1, FINSEQ_1:def_3; hence p = q by A2, FINSEQ_1:13; ::_thesis: verum end; theorem Th10: :: FINSEQ_2:10 for b being set for p being FinSequence st b in rng p holds ex i being Nat st ( i in dom p & p . i = b ) proof let b be set ; ::_thesis: for p being FinSequence st b in rng p holds ex i being Nat st ( i in dom p & p . i = b ) let p be FinSequence; ::_thesis: ( b in rng p implies ex i being Nat st ( i in dom p & p . i = b ) ) assume b in rng p ; ::_thesis: ex i being Nat st ( i in dom p & p . i = b ) then ex a being set st ( a in dom p & b = p . a ) by FUNCT_1:def_3; hence ex i being Nat st ( i in dom p & p . i = b ) ; ::_thesis: verum end; theorem Th11: :: FINSEQ_2:11 for i being Nat for D being set for p being FinSequence of D st i in dom p holds p . i in D proof let i be Nat; ::_thesis: for D being set for p being FinSequence of D st i in dom p holds p . i in D let D be set ; ::_thesis: for p being FinSequence of D st i in dom p holds p . i in D let p be FinSequence of D; ::_thesis: ( i in dom p implies p . i in D ) assume i in dom p ; ::_thesis: p . i in D then A1: p . i in rng p by FUNCT_1:def_3; rng p c= D by FINSEQ_1:def_4; hence p . i in D by A1; ::_thesis: verum end; theorem Th12: :: FINSEQ_2:12 for p being FinSequence for D being set st ( for i being Nat st i in dom p holds p . i in D ) holds p is FinSequence of D proof let p be FinSequence; ::_thesis: for D being set st ( for i being Nat st i in dom p holds p . i in D ) holds p is FinSequence of D let D be set ; ::_thesis: ( ( for i being Nat st i in dom p holds p . i in D ) implies p is FinSequence of D ) assume A1: for i being Nat st i in dom p holds p . i in D ; ::_thesis: p is FinSequence of D let b be set ; :: according to TARSKI:def_3,FINSEQ_1:def_4 ::_thesis: ( not b in rng p or b in D ) assume b in rng p ; ::_thesis: b in D then ex i being Nat st ( i in dom p & p . i = b ) by Th10; hence b in D by A1; ::_thesis: verum end; Lm1: for x1, x2 being set holds rng <*x1,x2*> = {x1,x2} proof let x1, x2 be set ; ::_thesis: rng <*x1,x2*> = {x1,x2} thus rng <*x1,x2*> = (rng <*x1*>) \/ (rng <*x2*>) by FINSEQ_1:31 .= (rng <*x1*>) \/ {x2} by FINSEQ_1:38 .= {x1} \/ {x2} by FINSEQ_1:39 .= {x1,x2} by ENUMSET1:1 ; ::_thesis: verum end; theorem Th13: :: FINSEQ_2:13 for D being non empty set for d1, d2 being Element of D holds <*d1,d2*> is FinSequence of D proof let D be non empty set ; ::_thesis: for d1, d2 being Element of D holds <*d1,d2*> is FinSequence of D let d1, d2 be Element of D; ::_thesis: <*d1,d2*> is FinSequence of D rng <*d1,d2*> = {d1,d2} by Lm1; hence <*d1,d2*> is FinSequence of D by FINSEQ_1:def_4; ::_thesis: verum end; Lm2: for x1, x2, x3 being set holds rng <*x1,x2,x3*> = {x1,x2,x3} proof let x1, x2, x3 be set ; ::_thesis: rng <*x1,x2,x3*> = {x1,x2,x3} thus rng <*x1,x2,x3*> = rng (<*x1*> ^ <*x2,x3*>) by FINSEQ_1:43 .= (rng <*x1*>) \/ (rng <*x2,x3*>) by FINSEQ_1:31 .= {x1} \/ (rng <*x2,x3*>) by FINSEQ_1:38 .= {x1} \/ {x2,x3} by Lm1 .= {x1,x2,x3} by ENUMSET1:2 ; ::_thesis: verum end; theorem Th14: :: FINSEQ_2:14 for D being non empty set for d1, d2, d3 being Element of D holds <*d1,d2,d3*> is FinSequence of D proof let D be non empty set ; ::_thesis: for d1, d2, d3 being Element of D holds <*d1,d2,d3*> is FinSequence of D let d1, d2, d3 be Element of D; ::_thesis: <*d1,d2,d3*> is FinSequence of D rng <*d1,d2,d3*> = {d1,d2,d3} by Lm2; hence <*d1,d2,d3*> is FinSequence of D by FINSEQ_1:def_4; ::_thesis: verum end; theorem Th15: :: FINSEQ_2:15 for i being Nat for p, q being FinSequence st i in dom p holds i in dom (p ^ q) proof let i be Nat; ::_thesis: for p, q being FinSequence st i in dom p holds i in dom (p ^ q) let p, q be FinSequence; ::_thesis: ( i in dom p implies i in dom (p ^ q) ) A1: dom p c= dom (p ^ q) by FINSEQ_1:26; assume i in dom p ; ::_thesis: i in dom (p ^ q) hence i in dom (p ^ q) by A1; ::_thesis: verum end; theorem Th16: :: FINSEQ_2:16 for a being set for p being FinSequence holds len (p ^ <*a*>) = (len p) + 1 proof let a be set ; ::_thesis: for p being FinSequence holds len (p ^ <*a*>) = (len p) + 1 let p be FinSequence; ::_thesis: len (p ^ <*a*>) = (len p) + 1 len (p ^ <*a*>) = (len p) + (len <*a*>) by FINSEQ_1:22; hence len (p ^ <*a*>) = (len p) + 1 by FINSEQ_1:39; ::_thesis: verum end; theorem :: FINSEQ_2:17 for a, b being set for p, q being FinSequence st p ^ <*a*> = q ^ <*b*> holds ( p = q & a = b ) proof let a, b be set ; ::_thesis: for p, q being FinSequence st p ^ <*a*> = q ^ <*b*> holds ( p = q & a = b ) let p, q be FinSequence; ::_thesis: ( p ^ <*a*> = q ^ <*b*> implies ( p = q & a = b ) ) assume A1: p ^ <*a*> = q ^ <*b*> ; ::_thesis: ( p = q & a = b ) A2: ( (p ^ <*a*>) . ((len p) + 1) = a & (q ^ <*b*>) . ((len q) + 1) = b ) by FINSEQ_1:42; ( len (p ^ <*a*>) = (len p) + 1 & len (q ^ <*b*>) = (len q) + 1 ) by Th16; hence ( p = q & a = b ) by A1, A2, FINSEQ_1:33; ::_thesis: verum end; theorem :: FINSEQ_2:18 for i being Nat for p being FinSequence st len p = i + 1 holds ex q being FinSequence ex a being set st p = q ^ <*a*> by CARD_1:27, FINSEQ_1:46; theorem Th19: :: FINSEQ_2:19 for A being set for p being FinSequence of A st len p <> 0 holds ex q being FinSequence of A ex d being Element of A st p = q ^ <*d*> proof let A be set ; ::_thesis: for p being FinSequence of A st len p <> 0 holds ex q being FinSequence of A ex d being Element of A st p = q ^ <*d*> let p be FinSequence of A; ::_thesis: ( len p <> 0 implies ex q being FinSequence of A ex d being Element of A st p = q ^ <*d*> ) assume A1: len p <> 0 ; ::_thesis: ex q being FinSequence of A ex d being Element of A st p = q ^ <*d*> then p <> {} ; then consider q being FinSequence, d being set such that A2: p = q ^ <*d*> by FINSEQ_1:46; for i being Nat st i in dom q holds q . i in A proof let i be Nat; ::_thesis: ( i in dom q implies q . i in A ) assume i in dom q ; ::_thesis: q . i in A then ( p . i = q . i & i in dom p ) by A2, Th15, FINSEQ_1:def_7; hence q . i in A by Th11; ::_thesis: verum end; then A3: q is FinSequence of A by Th12; len p in Seg (len p) by A1, FINSEQ_1:3; then A4: len p in dom p by FINSEQ_1:def_3; len p = (len q) + 1 by A2, Th16; then p . (len p) = d by A2, FINSEQ_1:42; then d is Element of A by A4, Th11; hence ex q being FinSequence of A ex d being Element of A st p = q ^ <*d*> by A2, A3; ::_thesis: verum end; theorem Th20: :: FINSEQ_2:20 for i being Nat for q, p being FinSequence st q = p | (Seg i) & len p <= i holds p = q proof let i be Nat; ::_thesis: for q, p being FinSequence st q = p | (Seg i) & len p <= i holds p = q let q, p be FinSequence; ::_thesis: ( q = p | (Seg i) & len p <= i implies p = q ) assume A1: q = p | (Seg i) ; ::_thesis: ( not len p <= i or p = q ) assume len p <= i ; ::_thesis: p = q then Seg (len p) c= Seg i by FINSEQ_1:5; then dom p c= Seg i by FINSEQ_1:def_3; hence p = q by A1, RELAT_1:68; ::_thesis: verum end; theorem :: FINSEQ_2:21 for i being Nat for q, p being FinSequence st q = p | (Seg i) holds len q = min (i,(len p)) proof let i be Nat; ::_thesis: for q, p being FinSequence st q = p | (Seg i) holds len q = min (i,(len p)) let q, p be FinSequence; ::_thesis: ( q = p | (Seg i) implies len q = min (i,(len p)) ) assume A1: q = p | (Seg i) ; ::_thesis: len q = min (i,(len p)) now__::_thesis:_(_(_i_<=_len_p_&_len_q_=_i_)_or_(_not_i_<=_len_p_&_len_q_=_len_p_)_) percases ( i <= len p or not i <= len p ) ; case i <= len p ; ::_thesis: len q = i hence len q = i by A1, FINSEQ_1:17; ::_thesis: verum end; case not i <= len p ; ::_thesis: len q = len p hence len q = len p by A1, Th20; ::_thesis: verum end; end; end; hence len q = min (i,(len p)) by XXREAL_0:def_9; ::_thesis: verum end; theorem Th22: :: FINSEQ_2:22 for i, j being Nat for r being FinSequence st len r = i + j holds ex p, q being FinSequence st ( len p = i & len q = j & r = p ^ q ) proof let i, j be Nat; ::_thesis: for r being FinSequence st len r = i + j holds ex p, q being FinSequence st ( len p = i & len q = j & r = p ^ q ) let r be FinSequence; ::_thesis: ( len r = i + j implies ex p, q being FinSequence st ( len p = i & len q = j & r = p ^ q ) ) assume A1: len r = i + j ; ::_thesis: ex p, q being FinSequence st ( len p = i & len q = j & r = p ^ q ) reconsider z = i as Element of NAT by ORDINAL1:def_12; reconsider p = r | (Seg z) as FinSequence by FINSEQ_1:15; consider q being FinSequence such that A2: r = p ^ q by FINSEQ_1:80; take p ; ::_thesis: ex q being FinSequence st ( len p = i & len q = j & r = p ^ q ) take q ; ::_thesis: ( len p = i & len q = j & r = p ^ q ) i <= len r by A1, NAT_1:11; hence len p = i by FINSEQ_1:17; ::_thesis: ( len q = j & r = p ^ q ) then len (p ^ q) = i + (len q) by FINSEQ_1:22; hence len q = j by A1, A2; ::_thesis: r = p ^ q thus r = p ^ q by A2; ::_thesis: verum end; theorem Th23: :: FINSEQ_2:23 for i, j being Nat for D being non empty set for r being FinSequence of D st len r = i + j holds ex p, q being FinSequence of D st ( len p = i & len q = j & r = p ^ q ) proof let i, j be Nat; ::_thesis: for D being non empty set for r being FinSequence of D st len r = i + j holds ex p, q being FinSequence of D st ( len p = i & len q = j & r = p ^ q ) let D be non empty set ; ::_thesis: for r being FinSequence of D st len r = i + j holds ex p, q being FinSequence of D st ( len p = i & len q = j & r = p ^ q ) let r be FinSequence of D; ::_thesis: ( len r = i + j implies ex p, q being FinSequence of D st ( len p = i & len q = j & r = p ^ q ) ) assume len r = i + j ; ::_thesis: ex p, q being FinSequence of D st ( len p = i & len q = j & r = p ^ q ) then consider p, q being FinSequence such that A1: ( len p = i & len q = j ) and A2: r = p ^ q by Th22; ( p is FinSequence of D & q is FinSequence of D ) by A2, FINSEQ_1:36; hence ex p, q being FinSequence of D st ( len p = i & len q = j & r = p ^ q ) by A1, A2; ::_thesis: verum end; scheme :: FINSEQ_2:sch 1 SeqLambdaD{ F1() -> Nat, F2() -> non empty set , F3( set ) -> Element of F2() } : ex z being FinSequence of F2() st ( len z = F1() & ( for j being Nat st j in dom z holds z . j = F3(j) ) ) proof consider z being FinSequence such that A1: len z = F1() and A2: for i being Nat st i in dom z holds z . i = F3(i) from FINSEQ_1:sch_2(); A3: Seg F1() = dom z by A1, FINSEQ_1:def_3; for j being Nat st j in Seg F1() holds z . j in F2() proof let j be Nat; ::_thesis: ( j in Seg F1() implies z . j in F2() ) assume j in Seg F1() ; ::_thesis: z . j in F2() then z . j = F3(j) by A2, A3; hence z . j in F2() ; ::_thesis: verum end; then reconsider z = z as FinSequence of F2() by A3, Th12; take z ; ::_thesis: ( len z = F1() & ( for j being Nat st j in dom z holds z . j = F3(j) ) ) thus len z = F1() by A1; ::_thesis: for j being Nat st j in dom z holds z . j = F3(j) let j be Nat; ::_thesis: ( j in dom z implies z . j = F3(j) ) thus ( j in dom z implies z . j = F3(j) ) by A2; ::_thesis: verum end; scheme :: FINSEQ_2:sch 2 IndSeqD{ F1() -> set , P1[ set ] } : for p being FinSequence of F1() holds P1[p] provided A1: P1[ <*> F1()] and A2: for p being FinSequence of F1() for x being Element of F1() st P1[p] holds P1[p ^ <*x*>] proof defpred S1[ set ] means for p being FinSequence of F1() st len p = \$1 holds P1[p]; A3: for i being Nat st S1[i] holds S1[i + 1] proof let i be Nat; ::_thesis: ( S1[i] implies S1[i + 1] ) assume A4: for p being FinSequence of F1() st len p = i holds P1[p] ; ::_thesis: S1[i + 1] let p be FinSequence of F1(); ::_thesis: ( len p = i + 1 implies P1[p] ) assume A5: len p = i + 1 ; ::_thesis: P1[p] then consider q being FinSequence of F1(), x being Element of F1() such that A6: p = q ^ <*x*> by Th19; len p = (len q) + 1 by A6, Th16; hence P1[p] by A2, A4, A5, A6; ::_thesis: verum end; let p be FinSequence of F1(); ::_thesis: P1[p] A7: len p = len p ; A8: S1[ 0 ] proof let p be FinSequence of F1(); ::_thesis: ( len p = 0 implies P1[p] ) assume len p = 0 ; ::_thesis: P1[p] then p = <*> F1() ; hence P1[p] by A1; ::_thesis: verum end; for i being Nat holds S1[i] from NAT_1:sch_2(A8, A3); hence P1[p] by A7; ::_thesis: verum end; theorem Th24: :: FINSEQ_2:24 for D being set for D1 being Subset of D for p being FinSequence of D1 holds p is FinSequence of D proof let D be set ; ::_thesis: for D1 being Subset of D for p being FinSequence of D1 holds p is FinSequence of D let D1 be Subset of D; ::_thesis: for p being FinSequence of D1 holds p is FinSequence of D let p be FinSequence of D1; ::_thesis: p is FinSequence of D rng p c= D1 by FINSEQ_1:def_4; hence rng p c= D by XBOOLE_1:1; :: according to FINSEQ_1:def_4 ::_thesis: verum end; theorem Th25: :: FINSEQ_2:25 for i being Nat for D being non empty set for f being Function of (Seg i),D holds f is FinSequence of D proof let i be Nat; ::_thesis: for D being non empty set for f being Function of (Seg i),D holds f is FinSequence of D let D be non empty set ; ::_thesis: for f being Function of (Seg i),D holds f is FinSequence of D let f be Function of (Seg i),D; ::_thesis: f is FinSequence of D reconsider i = i as Element of NAT by ORDINAL1:def_12; dom f = Seg i by FUNCT_2:def_1; then A1: f is FinSequence by FINSEQ_1:def_2; rng f c= D by RELAT_1:def_19; hence f is FinSequence of D by A1, FINSEQ_1:def_4; ::_thesis: verum end; theorem :: FINSEQ_2:26 for D being non empty set for p being FinSequence of D holds p is Function of (dom p),D proof let D be non empty set ; ::_thesis: for p being FinSequence of D holds p is Function of (dom p),D let p be FinSequence of D; ::_thesis: p is Function of (dom p),D rng p c= D by FINSEQ_1:def_4; hence p is Function of (dom p),D by FUNCT_2:2; ::_thesis: verum end; theorem :: FINSEQ_2:27 for i being Nat for D being non empty set for f being Function of NAT,D holds f | (Seg i) is FinSequence of D proof let i be Nat; ::_thesis: for D being non empty set for f being Function of NAT,D holds f | (Seg i) is FinSequence of D let D be non empty set ; ::_thesis: for f being Function of NAT,D holds f | (Seg i) is FinSequence of D let f be Function of NAT,D; ::_thesis: f | (Seg i) is FinSequence of D f | (Seg i) is Function of (Seg i),D by FUNCT_2:32; hence f | (Seg i) is FinSequence of D by Th25; ::_thesis: verum end; theorem :: FINSEQ_2:28 for i being Nat for D being non empty set for q being FinSequence for f being Function of NAT,D st q = f | (Seg i) holds len q = i proof let i be Nat; ::_thesis: for D being non empty set for q being FinSequence for f being Function of NAT,D st q = f | (Seg i) holds len q = i let D be non empty set ; ::_thesis: for q being FinSequence for f being Function of NAT,D st q = f | (Seg i) holds len q = i let q be FinSequence; ::_thesis: for f being Function of NAT,D st q = f | (Seg i) holds len q = i let f be Function of NAT,D; ::_thesis: ( q = f | (Seg i) implies len q = i ) reconsider i = i as Element of NAT by ORDINAL1:def_12; f | (Seg i) is Function of (Seg i),D by FUNCT_2:32; then dom (f | (Seg i)) = Seg i by FUNCT_2:def_1; hence ( q = f | (Seg i) implies len q = i ) by FINSEQ_1:def_3; ::_thesis: verum end; theorem Th29: :: FINSEQ_2:29 for p, q being FinSequence for f being Function st rng p c= dom f & q = f * p holds len q = len p proof let p, q be FinSequence; ::_thesis: for f being Function st rng p c= dom f & q = f * p holds len q = len p let f be Function; ::_thesis: ( rng p c= dom f & q = f * p implies len q = len p ) assume rng p c= dom f ; ::_thesis: ( not q = f * p or len q = len p ) then dom (f * p) = dom p by RELAT_1:27; then dom (f * p) = Seg (len p) by FINSEQ_1:def_3; hence ( not q = f * p or len q = len p ) by FINSEQ_1:def_3; ::_thesis: verum end; theorem Th30: :: FINSEQ_2:30 for i being Nat for D being non empty set st D = Seg i holds for p being FinSequence for q being FinSequence of D st i <= len p holds p * q is FinSequence proof let i be Nat; ::_thesis: for D being non empty set st D = Seg i holds for p being FinSequence for q being FinSequence of D st i <= len p holds p * q is FinSequence let D be non empty set ; ::_thesis: ( D = Seg i implies for p being FinSequence for q being FinSequence of D st i <= len p holds p * q is FinSequence ) assume A1: D = Seg i ; ::_thesis: for p being FinSequence for q being FinSequence of D st i <= len p holds p * q is FinSequence let p be FinSequence; ::_thesis: for q being FinSequence of D st i <= len p holds p * q is FinSequence let q be FinSequence of D; ::_thesis: ( i <= len p implies p * q is FinSequence ) assume i <= len p ; ::_thesis: p * q is FinSequence then Seg i c= Seg (len p) by FINSEQ_1:5; then A2: Seg i c= dom p by FINSEQ_1:def_3; rng q c= Seg i by A1, FINSEQ_1:def_4; then dom (p * q) = dom q by A2, RELAT_1:27, XBOOLE_1:1; then dom (p * q) = Seg (len q) by FINSEQ_1:def_3; hence p * q is FinSequence by FINSEQ_1:def_2; ::_thesis: verum end; theorem :: FINSEQ_2:31 for i being Nat for D, D9 being non empty set st D = Seg i holds for p being FinSequence of D9 for q being FinSequence of D st i <= len p holds p * q is FinSequence of D9 proof let i be Nat; ::_thesis: for D, D9 being non empty set st D = Seg i holds for p being FinSequence of D9 for q being FinSequence of D st i <= len p holds p * q is FinSequence of D9 let D, D9 be non empty set ; ::_thesis: ( D = Seg i implies for p being FinSequence of D9 for q being FinSequence of D st i <= len p holds p * q is FinSequence of D9 ) assume A1: D = Seg i ; ::_thesis: for p being FinSequence of D9 for q being FinSequence of D st i <= len p holds p * q is FinSequence of D9 let p be FinSequence of D9; ::_thesis: for q being FinSequence of D st i <= len p holds p * q is FinSequence of D9 let q be FinSequence of D; ::_thesis: ( i <= len p implies p * q is FinSequence of D9 ) assume i <= len p ; ::_thesis: p * q is FinSequence of D9 then reconsider pq = p * q as FinSequence by A1, Th30; ( rng pq c= rng p & rng p c= D9 ) by FINSEQ_1:def_4, RELAT_1:26; then rng pq c= D9 by XBOOLE_1:1; hence p * q is FinSequence of D9 by FINSEQ_1:def_4; ::_thesis: verum end; theorem Th32: :: FINSEQ_2:32 for A, D being set for p being FinSequence of A for f being Function of A,D holds f * p is FinSequence of D proof let A, D be set ; ::_thesis: for p being FinSequence of A for f being Function of A,D holds f * p is FinSequence of D let p be FinSequence of A; ::_thesis: for f being Function of A,D holds f * p is FinSequence of D let f be Function of A,D; ::_thesis: f * p is FinSequence of D percases ( D = {} or D <> {} ) ; suppose D = {} ; ::_thesis: f * p is FinSequence of D then f * p = <*> D ; hence f * p is FinSequence of D ; ::_thesis: verum end; supposeA1: D <> {} ; ::_thesis: f * p is FinSequence of D A2: rng p c= A by RELAT_1:def_19; A3: rng (f * p) c= D by RELAT_1:def_19; dom f = A by A1, FUNCT_2:def_1; then f * p is FinSequence by A2, FINSEQ_1:16; hence f * p is FinSequence of D by A3, FINSEQ_1:def_4; ::_thesis: verum end; end; end; theorem Th33: :: FINSEQ_2:33 for A being set for D9 being non empty set for q being FinSequence for p being FinSequence of A for f being Function of A,D9 st q = f * p holds len q = len p proof let A be set ; ::_thesis: for D9 being non empty set for q being FinSequence for p being FinSequence of A for f being Function of A,D9 st q = f * p holds len q = len p let D9 be non empty set ; ::_thesis: for q being FinSequence for p being FinSequence of A for f being Function of A,D9 st q = f * p holds len q = len p let q be FinSequence; ::_thesis: for p being FinSequence of A for f being Function of A,D9 st q = f * p holds len q = len p let p be FinSequence of A; ::_thesis: for f being Function of A,D9 st q = f * p holds len q = len p let f be Function of A,D9; ::_thesis: ( q = f * p implies len q = len p ) ( dom f = A & rng p c= A ) by FINSEQ_1:def_4, FUNCT_2:def_1; hence ( q = f * p implies len q = len p ) by Th29; ::_thesis: verum end; theorem :: FINSEQ_2:34 for x being set for f being Function st x in dom f holds f * <*x*> = <*(f . x)*> proof let x be set ; ::_thesis: for f being Function st x in dom f holds f * <*x*> = <*(f . x)*> let f be Function; ::_thesis: ( x in dom f implies f * <*x*> = <*(f . x)*> ) assume A1: x in dom f ; ::_thesis: f * <*x*> = <*(f . x)*> then reconsider D = dom f, E = rng f as non empty set by FUNCT_1:3; rng <*x*> = {x} by FINSEQ_1:38; then rng <*x*> c= D by A1, ZFMISC_1:31; then reconsider p = <*x*> as FinSequence of D by FINSEQ_1:def_4; reconsider f = f as Function of D,E by FUNCT_2:def_1, RELSET_1:4; reconsider q = f * p as FinSequence of E by Th32; A2: p . 1 = x by FINSEQ_1:40; A3: len q = len p by Th33 .= 1 by FINSEQ_1:39 ; then 1 in Seg (len q) ; then 1 in dom q by FINSEQ_1:def_3; then q . 1 = f . x by A2, FUNCT_1:12; hence f * <*x*> = <*(f . x)*> by A3, FINSEQ_1:40; ::_thesis: verum end; theorem :: FINSEQ_2:35 for x1 being set for D, D9 being non empty set for p being FinSequence of D for f being Function of D,D9 st p = <*x1*> holds f * p = <*(f . x1)*> proof let x1 be set ; ::_thesis: for D, D9 being non empty set for p being FinSequence of D for f being Function of D,D9 st p = <*x1*> holds f * p = <*(f . x1)*> let D, D9 be non empty set ; ::_thesis: for p being FinSequence of D for f being Function of D,D9 st p = <*x1*> holds f * p = <*(f . x1)*> let p be FinSequence of D; ::_thesis: for f being Function of D,D9 st p = <*x1*> holds f * p = <*(f . x1)*> let f be Function of D,D9; ::_thesis: ( p = <*x1*> implies f * p = <*(f . x1)*> ) assume A1: p = <*x1*> ; ::_thesis: f * p = <*(f . x1)*> A2: p . 1 = x1 by A1, FINSEQ_1:40; reconsider q = f * p as FinSequence of D9 by Th32; len p = 1 by A1, FINSEQ_1:39; then A3: len q = 1 by Th33; then 1 in Seg (len q) ; then 1 in dom q by FINSEQ_1:def_3; then q . 1 = f . x1 by A2, FUNCT_1:12; hence f * p = <*(f . x1)*> by A3, FINSEQ_1:40; ::_thesis: verum end; theorem Th36: :: FINSEQ_2:36 for x1, x2 being set for D, D9 being non empty set for p being FinSequence of D for f being Function of D,D9 st p = <*x1,x2*> holds f * p = <*(f . x1),(f . x2)*> proof let x1, x2 be set ; ::_thesis: for D, D9 being non empty set for p being FinSequence of D for f being Function of D,D9 st p = <*x1,x2*> holds f * p = <*(f . x1),(f . x2)*> let D, D9 be non empty set ; ::_thesis: for p being FinSequence of D for f being Function of D,D9 st p = <*x1,x2*> holds f * p = <*(f . x1),(f . x2)*> let p be FinSequence of D; ::_thesis: for f being Function of D,D9 st p = <*x1,x2*> holds f * p = <*(f . x1),(f . x2)*> let f be Function of D,D9; ::_thesis: ( p = <*x1,x2*> implies f * p = <*(f . x1),(f . x2)*> ) assume A1: p = <*x1,x2*> ; ::_thesis: f * p = <*(f . x1),(f . x2)*> A2: p . 2 = x2 by A1, FINSEQ_1:44; reconsider q = f * p as FinSequence of D9 by Th32; len p = 2 by A1, FINSEQ_1:44; then A3: len q = 2 by Th33; then 2 in Seg (len q) ; then 2 in dom q by FINSEQ_1:def_3; then A4: q . 2 = f . x2 by A2, FUNCT_1:12; 1 in Seg (len q) by A3; then A5: 1 in dom q by FINSEQ_1:def_3; p . 1 = x1 by A1, FINSEQ_1:44; then q . 1 = f . x1 by A5, FUNCT_1:12; hence f * p = <*(f . x1),(f . x2)*> by A3, A4, FINSEQ_1:44; ::_thesis: verum end; theorem Th37: :: FINSEQ_2:37 for x1, x2, x3 being set for D, D9 being non empty set for p being FinSequence of D for f being Function of D,D9 st p = <*x1,x2,x3*> holds f * p = <*(f . x1),(f . x2),(f . x3)*> proof let x1, x2, x3 be set ; ::_thesis: for D, D9 being non empty set for p being FinSequence of D for f being Function of D,D9 st p = <*x1,x2,x3*> holds f * p = <*(f . x1),(f . x2),(f . x3)*> let D, D9 be non empty set ; ::_thesis: for p being FinSequence of D for f being Function of D,D9 st p = <*x1,x2,x3*> holds f * p = <*(f . x1),(f . x2),(f . x3)*> let p be FinSequence of D; ::_thesis: for f being Function of D,D9 st p = <*x1,x2,x3*> holds f * p = <*(f . x1),(f . x2),(f . x3)*> let f be Function of D,D9; ::_thesis: ( p = <*x1,x2,x3*> implies f * p = <*(f . x1),(f . x2),(f . x3)*> ) assume A1: p = <*x1,x2,x3*> ; ::_thesis: f * p = <*(f . x1),(f . x2),(f . x3)*> A2: p . 2 = x2 by A1, FINSEQ_1:45; reconsider q = f * p as FinSequence of D9 by Th32; len p = 3 by A1, FINSEQ_1:45; then A3: len q = 3 by Th33; then 2 in Seg (len q) ; then 2 in dom q by FINSEQ_1:def_3; then A4: q . 2 = f . x2 by A2, FUNCT_1:12; A5: p . 3 = x3 by A1, FINSEQ_1:45; A6: p . 1 = x1 by A1, FINSEQ_1:45; 3 in Seg (len q) by A3; then 3 in dom q by FINSEQ_1:def_3; then A7: q . 3 = f . x3 by A5, FUNCT_1:12; 1 in Seg (len q) by A3; then 1 in dom q by FINSEQ_1:def_3; then q . 1 = f . x1 by A6, FUNCT_1:12; hence f * p = <*(f . x1),(f . x2),(f . x3)*> by A3, A4, A7, FINSEQ_1:45; ::_thesis: verum end; theorem Th38: :: FINSEQ_2:38 for i, j being Nat for p being FinSequence for f being Function of (Seg i),(Seg j) st ( j = 0 implies i = 0 ) & j <= len p holds p * f is FinSequence proof let i, j be Nat; ::_thesis: for p being FinSequence for f being Function of (Seg i),(Seg j) st ( j = 0 implies i = 0 ) & j <= len p holds p * f is FinSequence let p be FinSequence; ::_thesis: for f being Function of (Seg i),(Seg j) st ( j = 0 implies i = 0 ) & j <= len p holds p * f is FinSequence let f be Function of (Seg i),(Seg j); ::_thesis: ( ( j = 0 implies i = 0 ) & j <= len p implies p * f is FinSequence ) assume ( j = 0 implies i = 0 ) ; ::_thesis: ( not j <= len p or p * f is FinSequence ) then ( Seg j = {} implies Seg i = {} ) ; then A1: dom f = Seg i by FUNCT_2:def_1; assume j <= len p ; ::_thesis: p * f is FinSequence then ( rng f c= Seg j & Seg j c= Seg (len p) ) by FINSEQ_1:5, RELAT_1:def_19; then rng f c= Seg (len p) by XBOOLE_1:1; then rng f c= dom p by FINSEQ_1:def_3; then dom (p * f) = dom f by RELAT_1:27; hence p * f is FinSequence by A1, FINSEQ_1:def_2; ::_thesis: verum end; theorem :: FINSEQ_2:39 for i being Nat for p being FinSequence for f being Function of (Seg i),(Seg i) st i <= len p holds p * f is FinSequence by Th38; theorem :: FINSEQ_2:40 for p being FinSequence for f being Function of (dom p),(dom p) holds p * f is FinSequence proof let p be FinSequence; ::_thesis: for f being Function of (dom p),(dom p) holds p * f is FinSequence dom p = Seg (len p) by FINSEQ_1:def_3; hence for f being Function of (dom p),(dom p) holds p * f is FinSequence by Th38; ::_thesis: verum end; theorem Th41: :: FINSEQ_2:41 for i being Nat for p, q being FinSequence for f being Function of (Seg i),(Seg i) st rng f = Seg i & i <= len p & q = p * f holds len q = i proof let i be Nat; ::_thesis: for p, q being FinSequence for f being Function of (Seg i),(Seg i) st rng f = Seg i & i <= len p & q = p * f holds len q = i let p, q be FinSequence; ::_thesis: for f being Function of (Seg i),(Seg i) st rng f = Seg i & i <= len p & q = p * f holds len q = i let f be Function of (Seg i),(Seg i); ::_thesis: ( rng f = Seg i & i <= len p & q = p * f implies len q = i ) assume ( rng f = Seg i & i <= len p ) ; ::_thesis: ( not q = p * f or len q = i ) then rng f c= Seg (len p) by FINSEQ_1:5; then rng f c= dom p by FINSEQ_1:def_3; then A1: dom (p * f) = dom f by RELAT_1:27; ( i is Element of NAT & dom f = Seg i ) by FUNCT_2:52, ORDINAL1:def_12; hence ( not q = p * f or len q = i ) by A1, FINSEQ_1:def_3; ::_thesis: verum end; theorem Th42: :: FINSEQ_2:42 for p, q being FinSequence for f being Function of (dom p),(dom p) st rng f = dom p & q = p * f holds len q = len p proof let p, q be FinSequence; ::_thesis: for f being Function of (dom p),(dom p) st rng f = dom p & q = p * f holds len q = len p Seg (len p) = dom p by FINSEQ_1:def_3; hence for f being Function of (dom p),(dom p) st rng f = dom p & q = p * f holds len q = len p by Th41; ::_thesis: verum end; theorem Th43: :: FINSEQ_2:43 for i being Nat for p, q being FinSequence for f being Permutation of (Seg i) st i <= len p & q = p * f holds len q = i proof let i be Nat; ::_thesis: for p, q being FinSequence for f being Permutation of (Seg i) st i <= len p & q = p * f holds len q = i let p, q be FinSequence; ::_thesis: for f being Permutation of (Seg i) st i <= len p & q = p * f holds len q = i let f be Permutation of (Seg i); ::_thesis: ( i <= len p & q = p * f implies len q = i ) rng f = Seg i by FUNCT_2:def_3; hence ( i <= len p & q = p * f implies len q = i ) by Th41; ::_thesis: verum end; theorem :: FINSEQ_2:44 for p, q being FinSequence for f being Permutation of (dom p) st q = p * f holds len q = len p proof let p, q be FinSequence; ::_thesis: for f being Permutation of (dom p) st q = p * f holds len q = len p Seg (len p) = dom p by FINSEQ_1:def_3; hence for f being Permutation of (dom p) st q = p * f holds len q = len p by Th43; ::_thesis: verum end; theorem Th45: :: FINSEQ_2:45 for i, j being Nat for D being non empty set for p being FinSequence of D for f being Function of (Seg i),(Seg j) st ( j = 0 implies i = 0 ) & j <= len p holds p * f is FinSequence of D proof let i, j be Nat; ::_thesis: for D being non empty set for p being FinSequence of D for f being Function of (Seg i),(Seg j) st ( j = 0 implies i = 0 ) & j <= len p holds p * f is FinSequence of D let D be non empty set ; ::_thesis: for p being FinSequence of D for f being Function of (Seg i),(Seg j) st ( j = 0 implies i = 0 ) & j <= len p holds p * f is FinSequence of D let p be FinSequence of D; ::_thesis: for f being Function of (Seg i),(Seg j) st ( j = 0 implies i = 0 ) & j <= len p holds p * f is FinSequence of D let f be Function of (Seg i),(Seg j); ::_thesis: ( ( j = 0 implies i = 0 ) & j <= len p implies p * f is FinSequence of D ) assume A1: ( ( j = 0 implies i = 0 ) & j <= len p ) ; ::_thesis: p * f is FinSequence of D set q = p * f; ( rng p c= D & rng (p * f) c= rng p ) by FINSEQ_1:def_4, RELAT_1:26; then A2: rng (p * f) c= D by XBOOLE_1:1; p * f is FinSequence by A1, Th38; hence p * f is FinSequence of D by A2, FINSEQ_1:def_4; ::_thesis: verum end; theorem :: FINSEQ_2:46 for i being Nat for D being non empty set for p being FinSequence of D for f being Function of (Seg i),(Seg i) st i <= len p holds p * f is FinSequence of D by Th45; theorem Th47: :: FINSEQ_2:47 for D being non empty set for p being FinSequence of D for f being Function of (dom p),(dom p) holds p * f is FinSequence of D proof let D be non empty set ; ::_thesis: for p being FinSequence of D for f being Function of (dom p),(dom p) holds p * f is FinSequence of D let p be FinSequence of D; ::_thesis: for f being Function of (dom p),(dom p) holds p * f is FinSequence of D Seg (len p) = dom p by FINSEQ_1:def_3; hence for f being Function of (dom p),(dom p) holds p * f is FinSequence of D by Th45; ::_thesis: verum end; theorem Th48: :: FINSEQ_2:48 for k being Nat holds id (Seg k) is FinSequence of NAT proof let k be Nat; ::_thesis: id (Seg k) is FinSequence of NAT set I = id (Seg k); reconsider k = k as Element of NAT by ORDINAL1:def_12; dom (id (Seg k)) = Seg k ; then ( rng (id (Seg k)) = Seg k & id (Seg k) is FinSequence ) by FINSEQ_1:def_2; hence id (Seg k) is FinSequence of NAT by FINSEQ_1:def_4; ::_thesis: verum end; definition let i be Nat; func idseq i -> FinSequence equals :: FINSEQ_2:def 1 id (Seg i); coherence id (Seg i) is FinSequence by Th48; end; :: deftheorem defines idseq FINSEQ_2:def_1_:_ for i being Nat holds idseq i = id (Seg i); registration let k be Nat; cluster idseq k -> k -element ; coherence idseq k is k -element proof ( k in NAT & dom (idseq k) = Seg k ) by ORDINAL1:def_12, RELAT_1:45; hence len (idseq k) = k by FINSEQ_1:def_3; :: according to CARD_1:def_7 ::_thesis: verum end; end; registration cluster idseq 0 -> empty ; coherence idseq 0 is empty ; end; theorem :: FINSEQ_2:49 for i being Nat for k being Element of Seg i holds (idseq i) . k = k proof let i be Nat; ::_thesis: for k being Element of Seg i holds (idseq i) . k = k let k be Element of Seg i; ::_thesis: (idseq i) . k = k percases ( i <> 0 or i = 0 ) ; suppose i <> 0 ; ::_thesis: (idseq i) . k = k hence (idseq i) . k = k by FUNCT_1:18; ::_thesis: verum end; suppose i = 0 ; ::_thesis: (idseq i) . k = k then ( k = 0 & dom (idseq i) = 0 ) by SUBSET_1:def_1; hence (idseq i) . k = k by FUNCT_1:def_2; ::_thesis: verum end; end; end; theorem Th50: :: FINSEQ_2:50 idseq 1 = <*1*> proof 1 in Seg 1 ; then ( len (idseq 1) = 1 & (idseq 1) . 1 = 1 ) by CARD_1:def_7, FUNCT_1:18; hence idseq 1 = <*1*> by FINSEQ_1:40; ::_thesis: verum end; theorem Th51: :: FINSEQ_2:51 for i being Nat holds idseq (i + 1) = (idseq i) ^ <*(i + 1)*> proof let i be Nat; ::_thesis: idseq (i + 1) = (idseq i) ^ <*(i + 1)*> set p = idseq (i + 1); consider q being FinSequence, a being set such that A1: idseq (i + 1) = q ^ <*a*> by FINSEQ_1:46; A2: ( len (idseq (i + 1)) = i + 1 & len (idseq (i + 1)) = (len q) + 1 ) by A1, Th16, CARD_1:def_7; A3: dom q = Seg (len q) by FINSEQ_1:def_3; A4: for a being set st a in Seg i holds q . a = a proof i <= i + 1 by NAT_1:11; then A5: Seg i c= Seg (i + 1) by FINSEQ_1:5; let a be set ; ::_thesis: ( a in Seg i implies q . a = a ) assume A6: a in Seg i ; ::_thesis: q . a = a then reconsider j = a as Element of NAT ; (idseq (i + 1)) . j = q . j by A1, A2, A3, A6, FINSEQ_1:def_7; hence q . a = a by A6, A5, FUNCT_1:18; ::_thesis: verum end; (idseq (i + 1)) . (i + 1) = i + 1 by FINSEQ_1:4, FUNCT_1:18; then a = i + 1 by A1, A2, FINSEQ_1:42; hence idseq (i + 1) = (idseq i) ^ <*(i + 1)*> by A1, A2, A3, A4, FUNCT_1:17; ::_thesis: verum end; theorem Th52: :: FINSEQ_2:52 idseq 2 = <*1,2*> by Th50, Th51; theorem :: FINSEQ_2:53 idseq 3 = <*1,2,3*> by Th51, Th52; theorem Th54: :: FINSEQ_2:54 for k being Nat for p being FinSequence st len p <= k holds p * (idseq k) = p proof let k be Nat; ::_thesis: for p being FinSequence st len p <= k holds p * (idseq k) = p let p be FinSequence; ::_thesis: ( len p <= k implies p * (idseq k) = p ) assume A1: len p <= k ; ::_thesis: p * (idseq k) = p reconsider k = k as Element of NAT by ORDINAL1:def_12; dom p = Seg (len p) by FINSEQ_1:def_3; then dom p c= Seg k by A1, FINSEQ_1:5; hence p * (idseq k) = p by RELAT_1:51; ::_thesis: verum end; theorem :: FINSEQ_2:55 for k being Nat holds idseq k is Permutation of (Seg k) ; theorem Th56: :: FINSEQ_2:56 for k being Nat for a being set holds (Seg k) --> a is FinSequence proof let k be Nat; ::_thesis: for a being set holds (Seg k) --> a is FinSequence let a be set ; ::_thesis: (Seg k) --> a is FinSequence set p = (Seg k) --> a; reconsider k = k as Element of NAT by ORDINAL1:def_12; dom ((Seg k) --> a) = Seg k by FUNCOP_1:13; hence (Seg k) --> a is FinSequence by FINSEQ_1:def_2; ::_thesis: verum end; registration let k be Nat; let a be set ; cluster(Seg k) --> a -> FinSequence-like ; coherence (Seg k) --> a is FinSequence-like by Th56; end; definition let i be Nat; let a be set ; funci |-> a -> FinSequence equals :: FINSEQ_2:def 2 (Seg i) --> a; coherence (Seg i) --> a is FinSequence ; end; :: deftheorem defines |-> FINSEQ_2:def_2_:_ for i being Nat for a being set holds i |-> a = (Seg i) --> a; registration let k be Nat; let a be set ; clusterk |-> a -> k -element ; coherence k |-> a is k -element proof ( k in NAT & dom (k |-> a) = Seg k ) by FUNCOP_1:13, ORDINAL1:def_12; hence len (k |-> a) = k by FINSEQ_1:def_3; :: according to CARD_1:def_7 ::_thesis: verum end; end; theorem Th57: :: FINSEQ_2:57 for k being Nat for d, w being set st w in Seg k holds (k |-> d) . w = d by FUNCOP_1:7; theorem :: FINSEQ_2:58 for a being set holds 0 |-> a = {} ; theorem Th59: :: FINSEQ_2:59 for a being set holds 1 |-> a = <*a*> proof let a be set ; ::_thesis: 1 |-> a = <*a*> 1 in Seg 1 ; then ( len (1 |-> a) = 1 & (1 |-> a) . 1 = a ) by CARD_1:def_7, FUNCOP_1:7; hence 1 |-> a = <*a*> by FINSEQ_1:40; ::_thesis: verum end; theorem Th60: :: FINSEQ_2:60 for i being Nat for a being set holds (i + 1) |-> a = (i |-> a) ^ <*a*> proof let i be Nat; ::_thesis: for a being set holds (i + 1) |-> a = (i |-> a) ^ <*a*> let a be set ; ::_thesis: (i + 1) |-> a = (i |-> a) ^ <*a*> set p = (i + 1) |-> a; consider q being FinSequence, x being set such that A1: (i + 1) |-> a = q ^ <*x*> by FINSEQ_1:46; A2: ( len ((i + 1) |-> a) = i + 1 & len ((i + 1) |-> a) = (len q) + 1 ) by A1, Th16, CARD_1:def_7; A3: now__::_thesis:_q_=_i_|->_a percases ( i = 0 or i <> 0 ) ; supposeA4: i = 0 ; ::_thesis: q = i |-> a then q = {} by A2; hence q = i |-> a by A4; ::_thesis: verum end; supposeA5: i <> 0 ; ::_thesis: q = i |-> a A6: ( rng q c= rng ((i + 1) |-> a) & rng ((i + 1) |-> a) = {a} ) by A1, FINSEQ_1:29, FUNCOP_1:8; A7: dom q = Seg (len q) by FINSEQ_1:def_3; then i in dom q by A2, A5, FINSEQ_1:3; then q . i in rng q by FUNCT_1:def_3; then rng q = {a} by A6, ZFMISC_1:33; hence q = i |-> a by A2, A7, FUNCOP_1:9; ::_thesis: verum end; end; end; ((i + 1) |-> a) . (i + 1) = a by FINSEQ_1:4, FUNCOP_1:7; hence (i + 1) |-> a = (i |-> a) ^ <*a*> by A1, A2, A3, FINSEQ_1:42; ::_thesis: verum end; theorem Th61: :: FINSEQ_2:61 for a being set holds 2 |-> a = <*a,a*> proof let a be set ; ::_thesis: 2 |-> a = <*a,a*> thus 2 |-> a = (1 + 1) |-> a .= (1 |-> a) ^ <*a*> by Th60 .= <*a,a*> by Th59 ; ::_thesis: verum end; theorem :: FINSEQ_2:62 for a being set holds 3 |-> a = <*a,a,a*> proof let a be set ; ::_thesis: 3 |-> a = <*a,a,a*> thus 3 |-> a = (2 + 1) |-> a .= (2 |-> a) ^ <*a*> by Th60 .= <*a,a*> ^ <*a*> by Th61 .= <*a,a,a*> ; ::_thesis: verum end; theorem Th63: :: FINSEQ_2:63 for k being Nat for D being non empty set for d being Element of D holds k |-> d is FinSequence of D proof let k be Nat; ::_thesis: for D being non empty set for d being Element of D holds k |-> d is FinSequence of D let D be non empty set ; ::_thesis: for d being Element of D holds k |-> d is FinSequence of D let d be Element of D; ::_thesis: k |-> d is FinSequence of D set s = k |-> d; rng (k |-> d) c= {d} by FUNCOP_1:13; then rng (k |-> d) c= D by XBOOLE_1:1; hence k |-> d is FinSequence of D by FINSEQ_1:def_4; ::_thesis: verum end; Lm3: for i being Nat for p, q being FinSequence for F being Function st [:(rng p),(rng q):] c= dom F & i = min ((len p),(len q)) holds dom (F .: (p,q)) = Seg i proof let i be Nat; ::_thesis: for p, q being FinSequence for F being Function st [:(rng p),(rng q):] c= dom F & i = min ((len p),(len q)) holds dom (F .: (p,q)) = Seg i let p, q be FinSequence; ::_thesis: for F being Function st [:(rng p),(rng q):] c= dom F & i = min ((len p),(len q)) holds dom (F .: (p,q)) = Seg i let F be Function; ::_thesis: ( [:(rng p),(rng q):] c= dom F & i = min ((len p),(len q)) implies dom (F .: (p,q)) = Seg i ) assume that A1: [:(rng p),(rng q):] c= dom F and A2: i = min ((len p),(len q)) ; ::_thesis: dom (F .: (p,q)) = Seg i A3: ( rng <:p,q:> c= [:(rng p),(rng q):] & dom <:p,q:> = (dom p) /\ (dom q) ) by FUNCT_3:51, FUNCT_3:def_7; ( dom p = Seg (len p) & dom q = Seg (len q) ) by FINSEQ_1:def_3; then (dom p) /\ (dom q) = Seg i by A2, Th2; hence dom (F .: (p,q)) = Seg i by A1, A3, RELAT_1:27, XBOOLE_1:1; ::_thesis: verum end; theorem Th64: :: FINSEQ_2:64 for p, q being FinSequence for F being Function st [:(rng p),(rng q):] c= dom F holds F .: (p,q) is FinSequence proof let p, q be FinSequence; ::_thesis: for F being Function st [:(rng p),(rng q):] c= dom F holds F .: (p,q) is FinSequence let F be Function; ::_thesis: ( [:(rng p),(rng q):] c= dom F implies F .: (p,q) is FinSequence ) reconsider k = min ((len p),(len q)) as Element of NAT by XXREAL_0:15; assume [:(rng p),(rng q):] c= dom F ; ::_thesis: F .: (p,q) is FinSequence then dom (F .: (p,q)) = Seg k by Lm3; hence F .: (p,q) is FinSequence by FINSEQ_1:def_2; ::_thesis: verum end; theorem Th65: :: FINSEQ_2:65 for p, q, r being FinSequence for F being Function st [:(rng p),(rng q):] c= dom F & r = F .: (p,q) holds len r = min ((len p),(len q)) proof let p, q, r be FinSequence; ::_thesis: for F being Function st [:(rng p),(rng q):] c= dom F & r = F .: (p,q) holds len r = min ((len p),(len q)) let F be Function; ::_thesis: ( [:(rng p),(rng q):] c= dom F & r = F .: (p,q) implies len r = min ((len p),(len q)) ) reconsider k = min ((len p),(len q)) as Element of NAT by XXREAL_0:15; assume [:(rng p),(rng q):] c= dom F ; ::_thesis: ( not r = F .: (p,q) or len r = min ((len p),(len q)) ) then dom (F .: (p,q)) = Seg k by Lm3; hence ( not r = F .: (p,q) or len r = min ((len p),(len q)) ) by FINSEQ_1:def_3; ::_thesis: verum end; Lm4: for a being set for p being FinSequence for F being Function st [:{a},(rng p):] c= dom F holds dom (F [;] (a,p)) = dom p proof let a be set ; ::_thesis: for p being FinSequence for F being Function st [:{a},(rng p):] c= dom F holds dom (F [;] (a,p)) = dom p let p be FinSequence; ::_thesis: for F being Function st [:{a},(rng p):] c= dom F holds dom (F [;] (a,p)) = dom p let F be Function; ::_thesis: ( [:{a},(rng p):] c= dom F implies dom (F [;] (a,p)) = dom p ) assume A1: [:{a},(rng p):] c= dom F ; ::_thesis: dom (F [;] (a,p)) = dom p set q = (dom p) --> a; dom ((dom p) --> a) = dom p by FUNCOP_1:13; then A2: dom <:((dom p) --> a),p:> = dom p by FUNCT_3:50; rng ((dom p) --> a) c= {a} by FUNCOP_1:13; then ( rng <:((dom p) --> a),p:> c= [:(rng ((dom p) --> a)),(rng p):] & [:(rng ((dom p) --> a)),(rng p):] c= [:{a},(rng p):] ) by FUNCT_3:51, ZFMISC_1:95; then A3: rng <:((dom p) --> a),p:> c= [:{a},(rng p):] by XBOOLE_1:1; thus dom (F [;] (a,p)) = dom p by A1, A3, A2, RELAT_1:27, XBOOLE_1:1; ::_thesis: verum end; theorem Th66: :: FINSEQ_2:66 for a being set for p being FinSequence for F being Function st [:{a},(rng p):] c= dom F holds F [;] (a,p) is FinSequence proof let a be set ; ::_thesis: for p being FinSequence for F being Function st [:{a},(rng p):] c= dom F holds F [;] (a,p) is FinSequence let p be FinSequence; ::_thesis: for F being Function st [:{a},(rng p):] c= dom F holds F [;] (a,p) is FinSequence let F be Function; ::_thesis: ( [:{a},(rng p):] c= dom F implies F [;] (a,p) is FinSequence ) assume [:{a},(rng p):] c= dom F ; ::_thesis: F [;] (a,p) is FinSequence then dom (F [;] (a,p)) = dom p by Lm4; then dom (F [;] (a,p)) = Seg (len p) by FINSEQ_1:def_3; hence F [;] (a,p) is FinSequence by FINSEQ_1:def_2; ::_thesis: verum end; theorem Th67: :: FINSEQ_2:67 for a being set for p, r being FinSequence for F being Function st [:{a},(rng p):] c= dom F & r = F [;] (a,p) holds len r = len p proof let a be set ; ::_thesis: for p, r being FinSequence for F being Function st [:{a},(rng p):] c= dom F & r = F [;] (a,p) holds len r = len p let p, r be FinSequence; ::_thesis: for F being Function st [:{a},(rng p):] c= dom F & r = F [;] (a,p) holds len r = len p let F be Function; ::_thesis: ( [:{a},(rng p):] c= dom F & r = F [;] (a,p) implies len r = len p ) assume [:{a},(rng p):] c= dom F ; ::_thesis: ( not r = F [;] (a,p) or len r = len p ) then dom (F [;] (a,p)) = dom p by Lm4; then dom (F [;] (a,p)) = Seg (len p) by FINSEQ_1:def_3; hence ( not r = F [;] (a,p) or len r = len p ) by FINSEQ_1:def_3; ::_thesis: verum end; Lm5: for a being set for p being FinSequence for F being Function st [:(rng p),{a}:] c= dom F holds dom (F [:] (p,a)) = dom p proof let a be set ; ::_thesis: for p being FinSequence for F being Function st [:(rng p),{a}:] c= dom F holds dom (F [:] (p,a)) = dom p let p be FinSequence; ::_thesis: for F being Function st [:(rng p),{a}:] c= dom F holds dom (F [:] (p,a)) = dom p let F be Function; ::_thesis: ( [:(rng p),{a}:] c= dom F implies dom (F [:] (p,a)) = dom p ) assume A1: [:(rng p),{a}:] c= dom F ; ::_thesis: dom (F [:] (p,a)) = dom p set q = (dom p) --> a; dom ((dom p) --> a) = dom p by FUNCOP_1:13; then A2: dom <:p,((dom p) --> a):> = dom p by FUNCT_3:50; rng ((dom p) --> a) c= {a} by FUNCOP_1:13; then ( rng <:p,((dom p) --> a):> c= [:(rng p),(rng ((dom p) --> a)):] & [:(rng p),(rng ((dom p) --> a)):] c= [:(rng p),{a}:] ) by FUNCT_3:51, ZFMISC_1:95; then A3: rng <:p,((dom p) --> a):> c= [:(rng p),{a}:] by XBOOLE_1:1; thus dom (F [:] (p,a)) = dom p by A1, A3, A2, RELAT_1:27, XBOOLE_1:1; ::_thesis: verum end; theorem Th68: :: FINSEQ_2:68 for a being set for p being FinSequence for F being Function st [:(rng p),{a}:] c= dom F holds F [:] (p,a) is FinSequence proof let a be set ; ::_thesis: for p being FinSequence for F being Function st [:(rng p),{a}:] c= dom F holds F [:] (p,a) is FinSequence let p be FinSequence; ::_thesis: for F being Function st [:(rng p),{a}:] c= dom F holds F [:] (p,a) is FinSequence let F be Function; ::_thesis: ( [:(rng p),{a}:] c= dom F implies F [:] (p,a) is FinSequence ) assume [:(rng p),{a}:] c= dom F ; ::_thesis: F [:] (p,a) is FinSequence then dom (F [:] (p,a)) = dom p by Lm5; then dom (F [:] (p,a)) = Seg (len p) by FINSEQ_1:def_3; hence F [:] (p,a) is FinSequence by FINSEQ_1:def_2; ::_thesis: verum end; theorem Th69: :: FINSEQ_2:69 for a being set for p, r being FinSequence for F being Function st [:(rng p),{a}:] c= dom F & r = F [:] (p,a) holds len r = len p proof let a be set ; ::_thesis: for p, r being FinSequence for F being Function st [:(rng p),{a}:] c= dom F & r = F [:] (p,a) holds len r = len p let p, r be FinSequence; ::_thesis: for F being Function st [:(rng p),{a}:] c= dom F & r = F [:] (p,a) holds len r = len p let F be Function; ::_thesis: ( [:(rng p),{a}:] c= dom F & r = F [:] (p,a) implies len r = len p ) assume [:(rng p),{a}:] c= dom F ; ::_thesis: ( not r = F [:] (p,a) or len r = len p ) then dom (F [:] (p,a)) = dom p by Lm5; then dom (F [:] (p,a)) = Seg (len p) by FINSEQ_1:def_3; hence ( not r = F [:] (p,a) or len r = len p ) by FINSEQ_1:def_3; ::_thesis: verum end; theorem Th70: :: FINSEQ_2:70 for D, D9, E being non empty set for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 holds F .: (p,q) is FinSequence of E proof let D, D9, E be non empty set ; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 holds F .: (p,q) is FinSequence of E let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D for q being FinSequence of D9 holds F .: (p,q) is FinSequence of E let p be FinSequence of D; ::_thesis: for q being FinSequence of D9 holds F .: (p,q) is FinSequence of E let q be FinSequence of D9; ::_thesis: F .: (p,q) is FinSequence of E A1: rng (F .: (p,q)) c= rng F by RELAT_1:26; ( rng p c= D & rng q c= D9 ) by FINSEQ_1:def_4; then [:(rng p),(rng q):] c= [:D,D9:] by ZFMISC_1:96; then [:(rng p),(rng q):] c= dom F by FUNCT_2:def_1; then A2: F .: (p,q) is FinSequence by Th64; rng F c= E by RELAT_1:def_19; then rng (F .: (p,q)) c= E by A1, XBOOLE_1:1; hence F .: (p,q) is FinSequence of E by A2, FINSEQ_1:def_4; ::_thesis: verum end; theorem Th71: :: FINSEQ_2:71 for D, D9, E being non empty set for r being FinSequence for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st r = F .: (p,q) holds len r = min ((len p),(len q)) proof let D, D9, E be non empty set ; ::_thesis: for r being FinSequence for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st r = F .: (p,q) holds len r = min ((len p),(len q)) let r be FinSequence; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st r = F .: (p,q) holds len r = min ((len p),(len q)) let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D for q being FinSequence of D9 st r = F .: (p,q) holds len r = min ((len p),(len q)) let p be FinSequence of D; ::_thesis: for q being FinSequence of D9 st r = F .: (p,q) holds len r = min ((len p),(len q)) let q be FinSequence of D9; ::_thesis: ( r = F .: (p,q) implies len r = min ((len p),(len q)) ) ( rng p c= D & rng q c= D9 ) by FINSEQ_1:def_4; then [:(rng p),(rng q):] c= [:D,D9:] by ZFMISC_1:96; then [:(rng p),(rng q):] c= dom F by FUNCT_2:def_1; hence ( r = F .: (p,q) implies len r = min ((len p),(len q)) ) by Th65; ::_thesis: verum end; theorem Th72: :: FINSEQ_2:72 for D, D9, E being non empty set for r being FinSequence for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st len p = len q & r = F .: (p,q) holds ( len r = len p & len r = len q ) proof let D, D9, E be non empty set ; ::_thesis: for r being FinSequence for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st len p = len q & r = F .: (p,q) holds ( len r = len p & len r = len q ) let r be FinSequence; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st len p = len q & r = F .: (p,q) holds ( len r = len p & len r = len q ) let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D for q being FinSequence of D9 st len p = len q & r = F .: (p,q) holds ( len r = len p & len r = len q ) let p be FinSequence of D; ::_thesis: for q being FinSequence of D9 st len p = len q & r = F .: (p,q) holds ( len r = len p & len r = len q ) let q be FinSequence of D9; ::_thesis: ( len p = len q & r = F .: (p,q) implies ( len r = len p & len r = len q ) ) assume that A1: len p = len q and A2: r = F .: (p,q) ; ::_thesis: ( len r = len p & len r = len q ) len r = min ((len p),(len q)) by A2, Th71; hence ( len r = len p & len r = len q ) by A1; ::_thesis: verum end; theorem :: FINSEQ_2:73 for D, D9, E being non empty set for F being Function of [:D,D9:],E for p being FinSequence of D for p9 being FinSequence of D9 holds ( F .: ((<*> D),p9) = <*> E & F .: (p,(<*> D9)) = <*> E ) proof let D, D9, E be non empty set ; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D for p9 being FinSequence of D9 holds ( F .: ((<*> D),p9) = <*> E & F .: (p,(<*> D9)) = <*> E ) let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D for p9 being FinSequence of D9 holds ( F .: ((<*> D),p9) = <*> E & F .: (p,(<*> D9)) = <*> E ) let p be FinSequence of D; ::_thesis: for p9 being FinSequence of D9 holds ( F .: ((<*> D),p9) = <*> E & F .: (p,(<*> D9)) = <*> E ) let p9 be FinSequence of D9; ::_thesis: ( F .: ((<*> D),p9) = <*> E & F .: (p,(<*> D9)) = <*> E ) reconsider r = F .: ((<*> D),p9), r9 = F .: (p,(<*> D9)) as FinSequence of E by Th70; len (<*> D) = 0 ; then ( len r = min (0,(len p9)) & len r9 = min ((len p),0) ) by Th71; hence ( F .: ((<*> D),p9) = <*> E & F .: (p,(<*> D9)) = <*> E ) by XXREAL_0:def_9; ::_thesis: verum end; theorem :: FINSEQ_2:74 for D, D9, E being non empty set for d1 being Element of D for d19 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st p = <*d1*> & q = <*d19*> holds F .: (p,q) = <*(F . (d1,d19))*> proof let D, D9, E be non empty set ; ::_thesis: for d1 being Element of D for d19 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st p = <*d1*> & q = <*d19*> holds F .: (p,q) = <*(F . (d1,d19))*> let d1 be Element of D; ::_thesis: for d19 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st p = <*d1*> & q = <*d19*> holds F .: (p,q) = <*(F . (d1,d19))*> let d19 be Element of D9; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st p = <*d1*> & q = <*d19*> holds F .: (p,q) = <*(F . (d1,d19))*> let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D for q being FinSequence of D9 st p = <*d1*> & q = <*d19*> holds F .: (p,q) = <*(F . (d1,d19))*> let p be FinSequence of D; ::_thesis: for q being FinSequence of D9 st p = <*d1*> & q = <*d19*> holds F .: (p,q) = <*(F . (d1,d19))*> let q be FinSequence of D9; ::_thesis: ( p = <*d1*> & q = <*d19*> implies F .: (p,q) = <*(F . (d1,d19))*> ) assume A1: ( p = <*d1*> & q = <*d19*> ) ; ::_thesis: F .: (p,q) = <*(F . (d1,d19))*> A2: ( p . 1 = d1 & q . 1 = d19 ) by A1, FINSEQ_1:40; reconsider r = F .: (p,q) as FinSequence of E by Th70; ( len p = 1 & len q = 1 ) by A1, FINSEQ_1:39; then A3: len r = 1 by Th72; then 1 in Seg (len r) ; then 1 in dom r by FINSEQ_1:def_3; then r . 1 = F . (d1,d19) by A2, FUNCOP_1:22; hence F .: (p,q) = <*(F . (d1,d19))*> by A3, FINSEQ_1:40; ::_thesis: verum end; theorem :: FINSEQ_2:75 for D, D9, E being non empty set for d1, d2 being Element of D for d19, d29 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st p = <*d1,d2*> & q = <*d19,d29*> holds F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*> proof let D, D9, E be non empty set ; ::_thesis: for d1, d2 being Element of D for d19, d29 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st p = <*d1,d2*> & q = <*d19,d29*> holds F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*> let d1, d2 be Element of D; ::_thesis: for d19, d29 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st p = <*d1,d2*> & q = <*d19,d29*> holds F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*> let d19, d29 be Element of D9; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st p = <*d1,d2*> & q = <*d19,d29*> holds F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*> let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D for q being FinSequence of D9 st p = <*d1,d2*> & q = <*d19,d29*> holds F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*> let p be FinSequence of D; ::_thesis: for q being FinSequence of D9 st p = <*d1,d2*> & q = <*d19,d29*> holds F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*> let q be FinSequence of D9; ::_thesis: ( p = <*d1,d2*> & q = <*d19,d29*> implies F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*> ) assume A1: ( p = <*d1,d2*> & q = <*d19,d29*> ) ; ::_thesis: F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*> A2: ( p . 2 = d2 & q . 2 = d29 ) by A1, FINSEQ_1:44; reconsider r = F .: (p,q) as FinSequence of E by Th70; ( len p = 2 & len q = 2 ) by A1, FINSEQ_1:44; then A3: len r = 2 by Th72; then 2 in Seg (len r) ; then 2 in dom r by FINSEQ_1:def_3; then A4: r . 2 = F . (d2,d29) by A2, FUNCOP_1:22; 1 in Seg (len r) by A3; then A5: 1 in dom r by FINSEQ_1:def_3; ( p . 1 = d1 & q . 1 = d19 ) by A1, FINSEQ_1:44; then r . 1 = F . (d1,d19) by A5, FUNCOP_1:22; hence F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*> by A3, A4, FINSEQ_1:44; ::_thesis: verum end; theorem :: FINSEQ_2:76 for D, D9, E being non empty set for d1, d2, d3 being Element of D for d19, d29, d39 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st p = <*d1,d2,d3*> & q = <*d19,d29,d39*> holds F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*> proof let D, D9, E be non empty set ; ::_thesis: for d1, d2, d3 being Element of D for d19, d29, d39 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st p = <*d1,d2,d3*> & q = <*d19,d29,d39*> holds F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*> let d1, d2, d3 be Element of D; ::_thesis: for d19, d29, d39 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st p = <*d1,d2,d3*> & q = <*d19,d29,d39*> holds F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*> let d19, d29, d39 be Element of D9; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D for q being FinSequence of D9 st p = <*d1,d2,d3*> & q = <*d19,d29,d39*> holds F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*> let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D for q being FinSequence of D9 st p = <*d1,d2,d3*> & q = <*d19,d29,d39*> holds F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*> let p be FinSequence of D; ::_thesis: for q being FinSequence of D9 st p = <*d1,d2,d3*> & q = <*d19,d29,d39*> holds F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*> let q be FinSequence of D9; ::_thesis: ( p = <*d1,d2,d3*> & q = <*d19,d29,d39*> implies F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*> ) assume A1: ( p = <*d1,d2,d3*> & q = <*d19,d29,d39*> ) ; ::_thesis: F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*> A2: ( p . 2 = d2 & q . 2 = d29 ) by A1, FINSEQ_1:45; reconsider r = F .: (p,q) as FinSequence of E by Th70; ( len p = 3 & len q = 3 ) by A1, FINSEQ_1:45; then A3: len r = 3 by Th72; then 2 in Seg (len r) ; then 2 in dom r by FINSEQ_1:def_3; then A4: r . 2 = F . (d2,d29) by A2, FUNCOP_1:22; A5: ( p . 3 = d3 & q . 3 = d39 ) by A1, FINSEQ_1:45; A6: ( p . 1 = d1 & q . 1 = d19 ) by A1, FINSEQ_1:45; 3 in Seg (len r) by A3; then 3 in dom r by FINSEQ_1:def_3; then A7: r . 3 = F . (d3,d39) by A5, FUNCOP_1:22; 1 in Seg (len r) by A3; then 1 in dom r by FINSEQ_1:def_3; then r . 1 = F . (d1,d19) by A6, FUNCOP_1:22; hence F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*> by A3, A4, A7, FINSEQ_1:45; ::_thesis: verum end; theorem Th77: :: FINSEQ_2:77 for D, D9, E being non empty set for d being Element of D for F being Function of [:D,D9:],E for p being FinSequence of D9 holds F [;] (d,p) is FinSequence of E proof let D, D9, E be non empty set ; ::_thesis: for d being Element of D for F being Function of [:D,D9:],E for p being FinSequence of D9 holds F [;] (d,p) is FinSequence of E let d be Element of D; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D9 holds F [;] (d,p) is FinSequence of E let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D9 holds F [;] (d,p) is FinSequence of E let p be FinSequence of D9; ::_thesis: F [;] (d,p) is FinSequence of E A1: rng (F [;] (d,p)) c= rng F by RELAT_1:26; rng p c= D9 by FINSEQ_1:def_4; then [:{d},(rng p):] c= [:D,D9:] by ZFMISC_1:96; then [:{d},(rng p):] c= dom F by FUNCT_2:def_1; then A2: F [;] (d,p) is FinSequence by Th66; rng F c= E by RELAT_1:def_19; then rng (F [;] (d,p)) c= E by A1, XBOOLE_1:1; hence F [;] (d,p) is FinSequence of E by A2, FINSEQ_1:def_4; ::_thesis: verum end; theorem Th78: :: FINSEQ_2:78 for D, D9, E being non empty set for d being Element of D for r being FinSequence for F being Function of [:D,D9:],E for p being FinSequence of D9 st r = F [;] (d,p) holds len r = len p proof let D, D9, E be non empty set ; ::_thesis: for d being Element of D for r being FinSequence for F being Function of [:D,D9:],E for p being FinSequence of D9 st r = F [;] (d,p) holds len r = len p let d be Element of D; ::_thesis: for r being FinSequence for F being Function of [:D,D9:],E for p being FinSequence of D9 st r = F [;] (d,p) holds len r = len p let r be FinSequence; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D9 st r = F [;] (d,p) holds len r = len p let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D9 st r = F [;] (d,p) holds len r = len p let p be FinSequence of D9; ::_thesis: ( r = F [;] (d,p) implies len r = len p ) rng p c= D9 by FINSEQ_1:def_4; then [:{d},(rng p):] c= [:D,D9:] by ZFMISC_1:96; then [:{d},(rng p):] c= dom F by FUNCT_2:def_1; hence ( r = F [;] (d,p) implies len r = len p ) by Th67; ::_thesis: verum end; theorem :: FINSEQ_2:79 for D, D9, E being non empty set for d being Element of D for F being Function of [:D,D9:],E holds F [;] (d,(<*> D9)) = <*> E proof let D, D9, E be non empty set ; ::_thesis: for d being Element of D for F being Function of [:D,D9:],E holds F [;] (d,(<*> D9)) = <*> E let d be Element of D; ::_thesis: for F being Function of [:D,D9:],E holds F [;] (d,(<*> D9)) = <*> E let F be Function of [:D,D9:],E; ::_thesis: F [;] (d,(<*> D9)) = <*> E reconsider r = F [;] (d,(<*> D9)) as FinSequence of E by Th77; len (<*> D9) = 0 ; then len r = 0 by Th78; hence F [;] (d,(<*> D9)) = <*> E ; ::_thesis: verum end; theorem :: FINSEQ_2:80 for D, D9, E being non empty set for d being Element of D for d19 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D9 st p = <*d19*> holds F [;] (d,p) = <*(F . (d,d19))*> proof let D, D9, E be non empty set ; ::_thesis: for d being Element of D for d19 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D9 st p = <*d19*> holds F [;] (d,p) = <*(F . (d,d19))*> let d be Element of D; ::_thesis: for d19 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D9 st p = <*d19*> holds F [;] (d,p) = <*(F . (d,d19))*> let d19 be Element of D9; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D9 st p = <*d19*> holds F [;] (d,p) = <*(F . (d,d19))*> let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D9 st p = <*d19*> holds F [;] (d,p) = <*(F . (d,d19))*> let p be FinSequence of D9; ::_thesis: ( p = <*d19*> implies F [;] (d,p) = <*(F . (d,d19))*> ) assume A1: p = <*d19*> ; ::_thesis: F [;] (d,p) = <*(F . (d,d19))*> A2: p . 1 = d19 by A1, FINSEQ_1:40; reconsider r = F [;] (d,p) as FinSequence of E by Th77; len p = 1 by A1, FINSEQ_1:39; then A3: len r = 1 by Th78; then 1 in Seg (len r) ; then 1 in dom r by FINSEQ_1:def_3; then r . 1 = F . (d,d19) by A2, FUNCOP_1:32; hence F [;] (d,p) = <*(F . (d,d19))*> by A3, FINSEQ_1:40; ::_thesis: verum end; theorem :: FINSEQ_2:81 for D, D9, E being non empty set for d being Element of D for d19, d29 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D9 st p = <*d19,d29*> holds F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29))*> proof let D, D9, E be non empty set ; ::_thesis: for d being Element of D for d19, d29 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D9 st p = <*d19,d29*> holds F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29))*> let d be Element of D; ::_thesis: for d19, d29 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D9 st p = <*d19,d29*> holds F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29))*> let d19, d29 be Element of D9; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D9 st p = <*d19,d29*> holds F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29))*> let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D9 st p = <*d19,d29*> holds F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29))*> let p be FinSequence of D9; ::_thesis: ( p = <*d19,d29*> implies F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29))*> ) assume A1: p = <*d19,d29*> ; ::_thesis: F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29))*> A2: p . 2 = d29 by A1, FINSEQ_1:44; reconsider r = F [;] (d,p) as FinSequence of E by Th77; len p = 2 by A1, FINSEQ_1:44; then A3: len r = 2 by Th78; then 2 in Seg (len r) ; then 2 in dom r by FINSEQ_1:def_3; then A4: r . 2 = F . (d,d29) by A2, FUNCOP_1:32; 1 in Seg (len r) by A3; then A5: 1 in dom r by FINSEQ_1:def_3; p . 1 = d19 by A1, FINSEQ_1:44; then r . 1 = F . (d,d19) by A5, FUNCOP_1:32; hence F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29))*> by A3, A4, FINSEQ_1:44; ::_thesis: verum end; theorem :: FINSEQ_2:82 for D, D9, E being non empty set for d being Element of D for d19, d29, d39 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D9 st p = <*d19,d29,d39*> holds F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29)),(F . (d,d39))*> proof let D, D9, E be non empty set ; ::_thesis: for d being Element of D for d19, d29, d39 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D9 st p = <*d19,d29,d39*> holds F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29)),(F . (d,d39))*> let d be Element of D; ::_thesis: for d19, d29, d39 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D9 st p = <*d19,d29,d39*> holds F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29)),(F . (d,d39))*> let d19, d29, d39 be Element of D9; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D9 st p = <*d19,d29,d39*> holds F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29)),(F . (d,d39))*> let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D9 st p = <*d19,d29,d39*> holds F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29)),(F . (d,d39))*> let p be FinSequence of D9; ::_thesis: ( p = <*d19,d29,d39*> implies F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29)),(F . (d,d39))*> ) assume A1: p = <*d19,d29,d39*> ; ::_thesis: F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29)),(F . (d,d39))*> A2: p . 2 = d29 by A1, FINSEQ_1:45; reconsider r = F [;] (d,p) as FinSequence of E by Th77; len p = 3 by A1, FINSEQ_1:45; then A3: len r = 3 by Th78; then 2 in Seg (len r) ; then 2 in dom r by FINSEQ_1:def_3; then A4: r . 2 = F . (d,d29) by A2, FUNCOP_1:32; A5: p . 3 = d39 by A1, FINSEQ_1:45; A6: p . 1 = d19 by A1, FINSEQ_1:45; 3 in Seg (len r) by A3; then 3 in dom r by FINSEQ_1:def_3; then A7: r . 3 = F . (d,d39) by A5, FUNCOP_1:32; 1 in Seg (len r) by A3; then 1 in dom r by FINSEQ_1:def_3; then r . 1 = F . (d,d19) by A6, FUNCOP_1:32; hence F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29)),(F . (d,d39))*> by A3, A4, A7, FINSEQ_1:45; ::_thesis: verum end; theorem Th83: :: FINSEQ_2:83 for D, D9, E being non empty set for d9 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D holds F [:] (p,d9) is FinSequence of E proof let D, D9, E be non empty set ; ::_thesis: for d9 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D holds F [:] (p,d9) is FinSequence of E let d9 be Element of D9; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D holds F [:] (p,d9) is FinSequence of E let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D holds F [:] (p,d9) is FinSequence of E let p be FinSequence of D; ::_thesis: F [:] (p,d9) is FinSequence of E A1: rng (F [:] (p,d9)) c= rng F by RELAT_1:26; rng p c= D by FINSEQ_1:def_4; then [:(rng p),{d9}:] c= [:D,D9:] by ZFMISC_1:96; then [:(rng p),{d9}:] c= dom F by FUNCT_2:def_1; then A2: F [:] (p,d9) is FinSequence by Th68; rng F c= E by RELAT_1:def_19; then rng (F [:] (p,d9)) c= E by A1, XBOOLE_1:1; hence F [:] (p,d9) is FinSequence of E by A2, FINSEQ_1:def_4; ::_thesis: verum end; theorem Th84: :: FINSEQ_2:84 for D, D9, E being non empty set for d9 being Element of D9 for r being FinSequence for F being Function of [:D,D9:],E for p being FinSequence of D st r = F [:] (p,d9) holds len r = len p proof let D, D9, E be non empty set ; ::_thesis: for d9 being Element of D9 for r being FinSequence for F being Function of [:D,D9:],E for p being FinSequence of D st r = F [:] (p,d9) holds len r = len p let d9 be Element of D9; ::_thesis: for r being FinSequence for F being Function of [:D,D9:],E for p being FinSequence of D st r = F [:] (p,d9) holds len r = len p let r be FinSequence; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D st r = F [:] (p,d9) holds len r = len p let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D st r = F [:] (p,d9) holds len r = len p let p be FinSequence of D; ::_thesis: ( r = F [:] (p,d9) implies len r = len p ) rng p c= D by FINSEQ_1:def_4; then [:(rng p),{d9}:] c= [:D,D9:] by ZFMISC_1:96; then [:(rng p),{d9}:] c= dom F by FUNCT_2:def_1; hence ( r = F [:] (p,d9) implies len r = len p ) by Th69; ::_thesis: verum end; theorem :: FINSEQ_2:85 for D, D9, E being non empty set for d9 being Element of D9 for F being Function of [:D,D9:],E holds F [:] ((<*> D),d9) = <*> E proof let D, D9, E be non empty set ; ::_thesis: for d9 being Element of D9 for F being Function of [:D,D9:],E holds F [:] ((<*> D),d9) = <*> E let d9 be Element of D9; ::_thesis: for F being Function of [:D,D9:],E holds F [:] ((<*> D),d9) = <*> E let F be Function of [:D,D9:],E; ::_thesis: F [:] ((<*> D),d9) = <*> E reconsider r = F [:] ((<*> D),d9) as FinSequence of E by Th83; len (<*> D) = 0 ; then len r = 0 by Th84; hence F [:] ((<*> D),d9) = <*> E ; ::_thesis: verum end; theorem :: FINSEQ_2:86 for D, D9, E being non empty set for d1 being Element of D for d9 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D st p = <*d1*> holds F [:] (p,d9) = <*(F . (d1,d9))*> proof let D, D9, E be non empty set ; ::_thesis: for d1 being Element of D for d9 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D st p = <*d1*> holds F [:] (p,d9) = <*(F . (d1,d9))*> let d1 be Element of D; ::_thesis: for d9 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D st p = <*d1*> holds F [:] (p,d9) = <*(F . (d1,d9))*> let d9 be Element of D9; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D st p = <*d1*> holds F [:] (p,d9) = <*(F . (d1,d9))*> let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D st p = <*d1*> holds F [:] (p,d9) = <*(F . (d1,d9))*> let p be FinSequence of D; ::_thesis: ( p = <*d1*> implies F [:] (p,d9) = <*(F . (d1,d9))*> ) assume A1: p = <*d1*> ; ::_thesis: F [:] (p,d9) = <*(F . (d1,d9))*> A2: p . 1 = d1 by A1, FINSEQ_1:40; reconsider r = F [:] (p,d9) as FinSequence of E by Th83; len p = 1 by A1, FINSEQ_1:39; then A3: len r = 1 by Th84; then 1 in Seg (len r) ; then 1 in dom r by FINSEQ_1:def_3; then r . 1 = F . (d1,d9) by A2, FUNCOP_1:27; hence F [:] (p,d9) = <*(F . (d1,d9))*> by A3, FINSEQ_1:40; ::_thesis: verum end; theorem :: FINSEQ_2:87 for D, D9, E being non empty set for d1, d2 being Element of D for d9 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D st p = <*d1,d2*> holds F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9))*> proof let D, D9, E be non empty set ; ::_thesis: for d1, d2 being Element of D for d9 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D st p = <*d1,d2*> holds F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9))*> let d1, d2 be Element of D; ::_thesis: for d9 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D st p = <*d1,d2*> holds F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9))*> let d9 be Element of D9; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D st p = <*d1,d2*> holds F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9))*> let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D st p = <*d1,d2*> holds F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9))*> let p be FinSequence of D; ::_thesis: ( p = <*d1,d2*> implies F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9))*> ) assume A1: p = <*d1,d2*> ; ::_thesis: F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9))*> A2: p . 2 = d2 by A1, FINSEQ_1:44; reconsider r = F [:] (p,d9) as FinSequence of E by Th83; len p = 2 by A1, FINSEQ_1:44; then A3: len r = 2 by Th84; then 2 in Seg (len r) ; then 2 in dom r by FINSEQ_1:def_3; then A4: r . 2 = F . (d2,d9) by A2, FUNCOP_1:27; 1 in Seg (len r) by A3; then A5: 1 in dom r by FINSEQ_1:def_3; p . 1 = d1 by A1, FINSEQ_1:44; then r . 1 = F . (d1,d9) by A5, FUNCOP_1:27; hence F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9))*> by A3, A4, FINSEQ_1:44; ::_thesis: verum end; theorem :: FINSEQ_2:88 for D, D9, E being non empty set for d1, d2, d3 being Element of D for d9 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D st p = <*d1,d2,d3*> holds F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9)),(F . (d3,d9))*> proof let D, D9, E be non empty set ; ::_thesis: for d1, d2, d3 being Element of D for d9 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D st p = <*d1,d2,d3*> holds F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9)),(F . (d3,d9))*> let d1, d2, d3 be Element of D; ::_thesis: for d9 being Element of D9 for F being Function of [:D,D9:],E for p being FinSequence of D st p = <*d1,d2,d3*> holds F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9)),(F . (d3,d9))*> let d9 be Element of D9; ::_thesis: for F being Function of [:D,D9:],E for p being FinSequence of D st p = <*d1,d2,d3*> holds F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9)),(F . (d3,d9))*> let F be Function of [:D,D9:],E; ::_thesis: for p being FinSequence of D st p = <*d1,d2,d3*> holds F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9)),(F . (d3,d9))*> let p be FinSequence of D; ::_thesis: ( p = <*d1,d2,d3*> implies F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9)),(F . (d3,d9))*> ) assume A1: p = <*d1,d2,d3*> ; ::_thesis: F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9)),(F . (d3,d9))*> A2: p . 2 = d2 by A1, FINSEQ_1:45; reconsider r = F [:] (p,d9) as FinSequence of E by Th83; len p = 3 by A1, FINSEQ_1:45; then A3: len r = 3 by Th84; then 2 in Seg (len r) ; then 2 in dom r by FINSEQ_1:def_3; then A4: r . 2 = F . (d2,d9) by A2, FUNCOP_1:27; A5: p . 3 = d3 by A1, FINSEQ_1:45; A6: p . 1 = d1 by A1, FINSEQ_1:45; 3 in Seg (len r) by A3; then 3 in dom r by FINSEQ_1:def_3; then A7: r . 3 = F . (d3,d9) by A5, FUNCOP_1:27; 1 in Seg (len r) by A3; then 1 in dom r by FINSEQ_1:def_3; then r . 1 = F . (d1,d9) by A6, FUNCOP_1:27; hence F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9)),(F . (d3,d9))*> by A3, A4, A7, FINSEQ_1:45; ::_thesis: verum end; definition let D be set ; mode FinSequenceSet of D -> set means :Def3: :: FINSEQ_2:def 3 for a being set st a in it holds a is FinSequence of D; existence ex b1 being set st for a being set st a in b1 holds a is FinSequence of D proof take D * ; ::_thesis: for a being set st a in D * holds a is FinSequence of D thus for a being set st a in D * holds a is FinSequence of D by FINSEQ_1:def_11; ::_thesis: verum end; end; :: deftheorem Def3 defines FinSequenceSet FINSEQ_2:def_3_:_ for D, b2 being set holds ( b2 is FinSequenceSet of D iff for a being set st a in b2 holds a is FinSequence of D ); definition let D be set ; let S be FinSequenceSet of D; :: original: Element redefine mode Element of S -> FinSequence of D; coherence for b1 being Element of S holds b1 is FinSequence of D proof percases ( not S is empty or S is empty ) ; suppose not S is empty ; ::_thesis: for b1 being Element of S holds b1 is FinSequence of D hence for b1 being Element of S holds b1 is FinSequence of D by Def3; ::_thesis: verum end; suppose S is empty ; ::_thesis: for b1 being Element of S holds b1 is FinSequence of D then S = <*> D ; hence for b1 being Element of S holds b1 is FinSequence of D by SUBSET_1:def_1; ::_thesis: verum end; end; end; end; registration let D be set ; cluster non empty for FinSequenceSet of D; existence not for b1 being FinSequenceSet of D holds b1 is empty proof {(<*> D)} is FinSequenceSet of D proof let a be set ; :: according to FINSEQ_2:def_3 ::_thesis: ( a in {(<*> D)} implies a is FinSequence of D ) thus ( a in {(<*> D)} implies a is FinSequence of D ) by TARSKI:def_1; ::_thesis: verum end; hence not for b1 being FinSequenceSet of D holds b1 is empty ; ::_thesis: verum end; end; theorem Th89: :: FINSEQ_2:89 for D being set holds D * is FinSequenceSet of D proof let D be set ; ::_thesis: D * is FinSequenceSet of D D * is FinSequenceSet of D proof let a be set ; :: according to FINSEQ_2:def_3 ::_thesis: ( a in D * implies a is FinSequence of D ) thus ( a in D * implies a is FinSequence of D ) by FINSEQ_1:def_11; ::_thesis: verum end; hence D * is FinSequenceSet of D ; ::_thesis: verum end; definition let D be set ; :: original: * redefine funcD * -> FinSequenceSet of D; coherence D * is FinSequenceSet of D by Th89; end; theorem :: FINSEQ_2:90 for D being set for D9 being FinSequenceSet of D holds D9 c= D * proof let D be set ; ::_thesis: for D9 being FinSequenceSet of D holds D9 c= D * let D9 be FinSequenceSet of D; ::_thesis: D9 c= D * let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in D9 or a in D * ) assume a in D9 ; ::_thesis: a in D * then a is FinSequence of D by Def3; hence a in D * by FINSEQ_1:def_11; ::_thesis: verum end; theorem :: FINSEQ_2:91 for D being non empty set for D9 being Subset of D for S being FinSequenceSet of D9 holds S is FinSequenceSet of D proof let D be non empty set ; ::_thesis: for D9 being Subset of D for S being FinSequenceSet of D9 holds S is FinSequenceSet of D let D9 be Subset of D; ::_thesis: for S being FinSequenceSet of D9 holds S is FinSequenceSet of D let S be FinSequenceSet of D9; ::_thesis: S is FinSequenceSet of D S is FinSequenceSet of D proof let a be set ; :: according to FINSEQ_2:def_3 ::_thesis: ( a in S implies a is FinSequence of D ) assume a in S ; ::_thesis: a is FinSequence of D then reconsider p = a as FinSequence of D9 by Def3; rng p c= D9 by FINSEQ_1:def_4; then rng p c= D by XBOOLE_1:1; hence a is FinSequence of D by FINSEQ_1:def_4; ::_thesis: verum end; hence S is FinSequenceSet of D ; ::_thesis: verum end; registration let i be Nat; let D be non empty set ; cluster Relation-like NAT -defined D -valued Function-like V46() i -element FinSequence-like FinSubsequence-like countable for FinSequence of D; existence ex b1 being FinSequence of D st b1 is i -element proof take i |-> the Element of D ; ::_thesis: ( i |-> the Element of D is Element of bool [:NAT,D:] & i |-> the Element of D is FinSequence of D & i |-> the Element of D is i -element ) thus ( i |-> the Element of D is Element of bool [:NAT,D:] & i |-> the Element of D is FinSequence of D & i |-> the Element of D is i -element ) by Th63; ::_thesis: verum end; end; definition let i be Nat; let D be non empty set ; mode Tuple of i,D is i -element FinSequence of D; end; definition let i be Nat; let D be set ; funci -tuples_on D -> FinSequenceSet of D equals :: FINSEQ_2:def 4 { s where s is Element of D * : len s = i } ; coherence { s where s is Element of D * : len s = i } is FinSequenceSet of D proof now__::_thesis:_for_a_being_set_st_a_in__{__s_where_s_is_Element_of_D_*_:_len_s_=_i__}__holds_ a_is_FinSequence_of_D let a be set ; ::_thesis: ( a in { s where s is Element of D * : len s = i } implies a is FinSequence of D ) assume a in { s where s is Element of D * : len s = i } ; ::_thesis: a is FinSequence of D then ex s being Element of D * st ( s = a & len s = i ) ; hence a is FinSequence of D ; ::_thesis: verum end; hence { s where s is Element of D * : len s = i } is FinSequenceSet of D by Def3; ::_thesis: verum end; end; :: deftheorem defines -tuples_on FINSEQ_2:def_4_:_ for i being Nat for D being set holds i -tuples_on D = { s where s is Element of D * : len s = i } ; registration let i be Nat; let D be non empty set ; clusteri -tuples_on D -> non empty ; coherence not i -tuples_on D is empty proof set d = the Element of D; i |-> the Element of D is FinSequence of D by Th63; then reconsider S = i |-> the Element of D as Element of D * by FINSEQ_1:def_11; len S = i by CARD_1:def_7; then S in { s where s is Element of D * : len s = i } ; hence not i -tuples_on D is empty ; ::_thesis: verum end; end; registration let D be non empty set ; let i be Nat; cluster -> i -element for Element of i -tuples_on D; coherence for b1 being Element of i -tuples_on D holds b1 is i -element proof let z be Element of i -tuples_on D; ::_thesis: z is i -element z in { p9 where p9 is Element of D * : len p9 = i } ; then ex p9 being Element of D * st ( p9 = z & len p9 = i ) ; hence len z = i ; :: according to CARD_1:def_7 ::_thesis: verum end; end; theorem Th92: :: FINSEQ_2:92 for D being set for z being FinSequence of D holds z is Element of (len z) -tuples_on D proof let D be set ; ::_thesis: for z being FinSequence of D holds z is Element of (len z) -tuples_on D let z be FinSequence of D; ::_thesis: z is Element of (len z) -tuples_on D z is Element of D * by FINSEQ_1:def_11; then z in { z9 where z9 is Element of D * : len z9 = len z } ; hence z is Element of (len z) -tuples_on D ; ::_thesis: verum end; theorem :: FINSEQ_2:93 for i being Nat for D being set holds i -tuples_on D = Funcs ((Seg i),D) proof let i be Nat; ::_thesis: for D being set holds i -tuples_on D = Funcs ((Seg i),D) let D be set ; ::_thesis: i -tuples_on D = Funcs ((Seg i),D) now__::_thesis:_for_z_being_set_holds_ (_(_z_in_i_-tuples_on_D_implies_z_in_Funcs_((Seg_i),D)_)_&_(_z_in_Funcs_((Seg_i),D)_implies_z_in_i_-tuples_on_D_)_) reconsider j = i as Element of NAT by ORDINAL1:def_12; let z be set ; ::_thesis: ( ( z in i -tuples_on D implies z in Funcs ((Seg i),D) ) & ( z in Funcs ((Seg i),D) implies z in i -tuples_on D ) ) thus ( z in i -tuples_on D implies z in Funcs ((Seg i),D) ) ::_thesis: ( z in Funcs ((Seg i),D) implies z in i -tuples_on D ) proof assume z in i -tuples_on D ; ::_thesis: z in Funcs ((Seg i),D) then consider s being Element of D * such that A1: z = s and A2: len s = i ; A3: rng s c= D by FINSEQ_1:def_4; dom s = Seg i by A2, FINSEQ_1:def_3; hence z in Funcs ((Seg i),D) by A1, A3, FUNCT_2:def_2; ::_thesis: verum end; assume z in Funcs ((Seg i),D) ; ::_thesis: z in i -tuples_on D then consider p being Function such that A4: z = p and A5: dom p = Seg j and A6: rng p c= D by FUNCT_2:def_2; p is FinSequence by A5, FINSEQ_1:def_2; then p is FinSequence of D by A6, FINSEQ_1:def_4; then reconsider p = p as Element of D * by FINSEQ_1:def_11; len p = i by A5, FINSEQ_1:def_3; hence z in i -tuples_on D by A4; ::_thesis: verum end; hence i -tuples_on D = Funcs ((Seg i),D) by TARSKI:1; ::_thesis: verum end; theorem :: FINSEQ_2:94 for D being set holds 0 -tuples_on D = {(<*> D)} proof let D be set ; ::_thesis: 0 -tuples_on D = {(<*> D)} now__::_thesis:_for_z_being_set_holds_ (_(_z_in_0_-tuples_on_D_implies_z_=_<*>_D_)_&_(_z_=_<*>_D_implies_z_in_0_-tuples_on_D_)_) let z be set ; ::_thesis: ( ( z in 0 -tuples_on D implies z = <*> D ) & ( z = <*> D implies z in 0 -tuples_on D ) ) thus ( z in 0 -tuples_on D implies z = <*> D ) ::_thesis: ( z = <*> D implies z in 0 -tuples_on D ) proof assume z in 0 -tuples_on D ; ::_thesis: z = <*> D then ex s being Element of D * st ( z = s & len s = 0 ) ; hence z = <*> D ; ::_thesis: verum end; ( <*> D is Element of D * & len (<*> D) = 0 ) by FINSEQ_1:def_11; hence ( z = <*> D implies z in 0 -tuples_on D ) ; ::_thesis: verum end; hence 0 -tuples_on D = {(<*> D)} by TARSKI:def_1; ::_thesis: verum end; theorem :: FINSEQ_2:95 for i being Nat for D being non empty set for z being Tuple of 0 ,D for t being Tuple of i,D holds ( z ^ t = t & t ^ z = t ) by FINSEQ_1:34; theorem Th96: :: FINSEQ_2:96 for D being non empty set holds 1 -tuples_on D = { <*d*> where d is Element of D : verum } proof let D be non empty set ; ::_thesis: 1 -tuples_on D = { <*d*> where d is Element of D : verum } now__::_thesis:_for_x_being_set_holds_ (_(_x_in_1_-tuples_on_D_implies_x_in__{__<*d*>_where_d_is_Element_of_D_:_verum__}__)_&_(_x_in__{__<*d*>_where_d_is_Element_of_D_:_verum__}__implies_x_in_1_-tuples_on_D_)_) let x be set ; ::_thesis: ( ( x in 1 -tuples_on D implies x in { <*d*> where d is Element of D : verum } ) & ( x in { <*d*> where d is Element of D : verum } implies x in 1 -tuples_on D ) ) thus ( x in 1 -tuples_on D implies x in { <*d*> where d is Element of D : verum } ) ::_thesis: ( x in { <*d*> where d is Element of D : verum } implies x in 1 -tuples_on D ) proof assume x in 1 -tuples_on D ; ::_thesis: x in { <*d*> where d is Element of D : verum } then consider s being Element of D * such that A1: x = s and A2: len s = 1 ; 1 in Seg 1 ; then 1 in dom s by A2, FINSEQ_1:def_3; then reconsider d1 = s . 1 as Element of D by Th11; s = <*d1*> by A2, FINSEQ_1:40; hence x in { <*d*> where d is Element of D : verum } by A1; ::_thesis: verum end; assume x in { <*d*> where d is Element of D : verum } ; ::_thesis: x in 1 -tuples_on D then consider d being Element of D such that A3: x = <*d*> ; ( len <*d*> = 1 & <*d*> is Element of D * ) by FINSEQ_1:40, FINSEQ_1:def_11; hence x in 1 -tuples_on D by A3; ::_thesis: verum end; hence 1 -tuples_on D = { <*d*> where d is Element of D : verum } by TARSKI:1; ::_thesis: verum end; Lm6: for i being Nat for D being non empty set for z being Tuple of i,D holds z in i -tuples_on D proof let i be Nat; ::_thesis: for D being non empty set for z being Tuple of i,D holds z in i -tuples_on D let D be non empty set ; ::_thesis: for z being Tuple of i,D holds z in i -tuples_on D let z be Tuple of i,D; ::_thesis: z in i -tuples_on D A1: z is Element of D * by FINSEQ_1:def_11; len z = i by CARD_1:def_7; hence z in i -tuples_on D by A1; ::_thesis: verum end; theorem Th97: :: FINSEQ_2:97 for D being non empty set for z being Tuple of 1,D ex d being Element of D st z = <*d*> proof let D be non empty set ; ::_thesis: for z being Tuple of 1,D ex d being Element of D st z = <*d*> let z be Tuple of 1,D; ::_thesis: ex d being Element of D st z = <*d*> z in 1 -tuples_on D by Lm6; then z in { <*d*> where d is Element of D : verum } by Th96; hence ex d being Element of D st z = <*d*> ; ::_thesis: verum end; theorem Th98: :: FINSEQ_2:98 for D being non empty set for d being Element of D holds <*d*> in 1 -tuples_on D proof let D be non empty set ; ::_thesis: for d being Element of D holds <*d*> in 1 -tuples_on D let d be Element of D; ::_thesis: <*d*> in 1 -tuples_on D <*d*> in { <*d1*> where d1 is Element of D : verum } ; hence <*d*> in 1 -tuples_on D by Th96; ::_thesis: verum end; theorem Th99: :: FINSEQ_2:99 for D being non empty set holds 2 -tuples_on D = { <*d1,d2*> where d1, d2 is Element of D : verum } proof let D be non empty set ; ::_thesis: 2 -tuples_on D = { <*d1,d2*> where d1, d2 is Element of D : verum } now__::_thesis:_for_x_being_set_holds_ (_(_x_in_2_-tuples_on_D_implies_x_in__{__<*d1,d2*>_where_d1,_d2_is_Element_of_D_:_verum__}__)_&_(_x_in__{__<*d1,d2*>_where_d1,_d2_is_Element_of_D_:_verum__}__implies_x_in_2_-tuples_on_D_)_) let x be set ; ::_thesis: ( ( x in 2 -tuples_on D implies x in { <*d1,d2*> where d1, d2 is Element of D : verum } ) & ( x in { <*d1,d2*> where d1, d2 is Element of D : verum } implies x in 2 -tuples_on D ) ) thus ( x in 2 -tuples_on D implies x in { <*d1,d2*> where d1, d2 is Element of D : verum } ) ::_thesis: ( x in { <*d1,d2*> where d1, d2 is Element of D : verum } implies x in 2 -tuples_on D ) proof assume x in 2 -tuples_on D ; ::_thesis: x in { <*d1,d2*> where d1, d2 is Element of D : verum } then consider s being Element of D * such that A1: x = s and A2: len s = 2 ; 2 in Seg 2 ; then A3: 2 in dom s by A2, FINSEQ_1:def_3; 1 in Seg 2 ; then 1 in dom s by A2, FINSEQ_1:def_3; then reconsider d19 = s . 1, d29 = s . 2 as Element of D by A3, Th11; s = <*d19,d29*> by A2, FINSEQ_1:44; hence x in { <*d1,d2*> where d1, d2 is Element of D : verum } by A1; ::_thesis: verum end; assume x in { <*d1,d2*> where d1, d2 is Element of D : verum } ; ::_thesis: x in 2 -tuples_on D then consider d1, d2 being Element of D such that A4: x = <*d1,d2*> ; <*d1,d2*> is FinSequence of D by Th13; then ( len <*d1,d2*> = 2 & <*d1,d2*> is Element of D * ) by FINSEQ_1:44, FINSEQ_1:def_11; hence x in 2 -tuples_on D by A4; ::_thesis: verum end; hence 2 -tuples_on D = { <*d1,d2*> where d1, d2 is Element of D : verum } by TARSKI:1; ::_thesis: verum end; theorem :: FINSEQ_2:100 for D being non empty set for z being Tuple of 2,D ex d1, d2 being Element of D st z = <*d1,d2*> proof let D be non empty set ; ::_thesis: for z being Tuple of 2,D ex d1, d2 being Element of D st z = <*d1,d2*> let z be Tuple of 2,D; ::_thesis: ex d1, d2 being Element of D st z = <*d1,d2*> A1: z is Element of D * by FINSEQ_1:def_11; len z = 2 by CARD_1:def_7; then z in 2 -tuples_on D by A1; then z in { <*d1,d2*> where d1, d2 is Element of D : verum } by Th99; hence ex d1, d2 being Element of D st z = <*d1,d2*> ; ::_thesis: verum end; theorem Th101: :: FINSEQ_2:101 for D being non empty set for d1, d2 being Element of D holds <*d1,d2*> in 2 -tuples_on D proof let D be non empty set ; ::_thesis: for d1, d2 being Element of D holds <*d1,d2*> in 2 -tuples_on D let d1, d2 be Element of D; ::_thesis: <*d1,d2*> in 2 -tuples_on D <*d1,d2*> in { <*c1,c2*> where c1, c2 is Element of D : verum } ; hence <*d1,d2*> in 2 -tuples_on D by Th99; ::_thesis: verum end; theorem Th102: :: FINSEQ_2:102 for D being non empty set holds 3 -tuples_on D = { <*d1,d2,d3*> where d1, d2, d3 is Element of D : verum } proof let D be non empty set ; ::_thesis: 3 -tuples_on D = { <*d1,d2,d3*> where d1, d2, d3 is Element of D : verum } now__::_thesis:_for_x_being_set_holds_ (_(_x_in_3_-tuples_on_D_implies_x_in__{__<*d1,d2,d3*>_where_d1,_d2,_d3_is_Element_of_D_:_verum__}__)_&_(_x_in__{__<*d1,d2,d3*>_where_d1,_d2,_d3_is_Element_of_D_:_verum__}__implies_x_in_3_-tuples_on_D_)_) let x be set ; ::_thesis: ( ( x in 3 -tuples_on D implies x in { <*d1,d2,d3*> where d1, d2, d3 is Element of D : verum } ) & ( x in { <*d1,d2,d3*> where d1, d2, d3 is Element of D : verum } implies x in 3 -tuples_on D ) ) thus ( x in 3 -tuples_on D implies x in { <*d1,d2,d3*> where d1, d2, d3 is Element of D : verum } ) ::_thesis: ( x in { <*d1,d2,d3*> where d1, d2, d3 is Element of D : verum } implies x in 3 -tuples_on D ) proof assume x in 3 -tuples_on D ; ::_thesis: x in { <*d1,d2,d3*> where d1, d2, d3 is Element of D : verum } then consider s being Element of D * such that A1: x = s and A2: len s = 3 ; 2 in Seg 3 ; then A3: 2 in dom s by A2, FINSEQ_1:def_3; 3 in Seg 3 ; then A4: 3 in dom s by A2, FINSEQ_1:def_3; 1 in Seg 3 ; then 1 in dom s by A2, FINSEQ_1:def_3; then reconsider d19 = s . 1, d29 = s . 2, d39 = s . 3 as Element of D by A3, A4, Th11; s = <*d19,d29,d39*> by A2, FINSEQ_1:45; hence x in { <*d1,d2,d3*> where d1, d2, d3 is Element of D : verum } by A1; ::_thesis: verum end; assume x in { <*d1,d2,d3*> where d1, d2, d3 is Element of D : verum } ; ::_thesis: x in 3 -tuples_on D then consider d1, d2, d3 being Element of D such that A5: x = <*d1,d2,d3*> ; <*d1,d2,d3*> is FinSequence of D by Th14; then ( len <*d1,d2,d3*> = 3 & <*d1,d2,d3*> is Element of D * ) by FINSEQ_1:45, FINSEQ_1:def_11; hence x in 3 -tuples_on D by A5; ::_thesis: verum end; hence 3 -tuples_on D = { <*d1,d2,d3*> where d1, d2, d3 is Element of D : verum } by TARSKI:1; ::_thesis: verum end; theorem :: FINSEQ_2:103 for D being non empty set for z being Tuple of 3,D ex d1, d2, d3 being Element of D st z = <*d1,d2,d3*> proof let D be non empty set ; ::_thesis: for z being Tuple of 3,D ex d1, d2, d3 being Element of D st z = <*d1,d2,d3*> let z be Tuple of 3,D; ::_thesis: ex d1, d2, d3 being Element of D st z = <*d1,d2,d3*> A1: z is Element of D * by FINSEQ_1:def_11; len z = 3 by CARD_1:def_7; then z in 3 -tuples_on D by A1; then z in { <*d1,d2,d3*> where d1, d2, d3 is Element of D : verum } by Th102; hence ex d1, d2, d3 being Element of D st z = <*d1,d2,d3*> ; ::_thesis: verum end; theorem Th104: :: FINSEQ_2:104 for D being non empty set for d1, d2, d3 being Element of D holds <*d1,d2,d3*> in 3 -tuples_on D proof let D be non empty set ; ::_thesis: for d1, d2, d3 being Element of D holds <*d1,d2,d3*> in 3 -tuples_on D let d1, d2, d3 be Element of D; ::_thesis: <*d1,d2,d3*> in 3 -tuples_on D <*d1,d2,d3*> in { <*c1,c2,c3*> where c1, c2, c3 is Element of D : verum } ; hence <*d1,d2,d3*> in 3 -tuples_on D by Th102; ::_thesis: verum end; theorem Th105: :: FINSEQ_2:105 for i, j being Nat for D being non empty set holds (i + j) -tuples_on D = { (z ^ t) where z is Tuple of i,D, t is Tuple of j,D : verum } proof let i, j be Nat; ::_thesis: for D being non empty set holds (i + j) -tuples_on D = { (z ^ t) where z is Tuple of i,D, t is Tuple of j,D : verum } let D be non empty set ; ::_thesis: (i + j) -tuples_on D = { (z ^ t) where z is Tuple of i,D, t is Tuple of j,D : verum } set T = { (z ^ t) where z is Tuple of i,D, t is Tuple of j,D : verum } ; now__::_thesis:_for_x_being_set_holds_ (_(_x_in_(i_+_j)_-tuples_on_D_implies_x_in__{__(z_^_t)_where_z_is_Tuple_of_i,D,_t_is_Tuple_of_j,D_:_verum__}__)_&_(_x_in__{__(z_^_t)_where_z_is_Tuple_of_i,D,_t_is_Tuple_of_j,D_:_verum__}__implies_x_in_(i_+_j)_-tuples_on_D_)_) let x be set ; ::_thesis: ( ( x in (i + j) -tuples_on D implies x in { (z ^ t) where z is Tuple of i,D, t is Tuple of j,D : verum } ) & ( x in { (z ^ t) where z is Tuple of i,D, t is Tuple of j,D : verum } implies x in (i + j) -tuples_on D ) ) thus ( x in (i + j) -tuples_on D implies x in { (z ^ t) where z is Tuple of i,D, t is Tuple of j,D : verum } ) ::_thesis: ( x in { (z ^ t) where z is Tuple of i,D, t is Tuple of j,D : verum } implies x in (i + j) -tuples_on D ) proof assume x in (i + j) -tuples_on D ; ::_thesis: x in { (z ^ t) where z is Tuple of i,D, t is Tuple of j,D : verum } then consider s being Element of D * such that A1: x = s and A2: len s = i + j ; consider z, t being FinSequence of D such that A3: ( len z = i & len t = j ) and A4: s = z ^ t by A2, Th23; ( z is Tuple of i,D & t is Tuple of j,D ) by A3, CARD_1:def_7; hence x in { (z ^ t) where z is Tuple of i,D, t is Tuple of j,D : verum } by A1, A4; ::_thesis: verum end; assume x in { (z ^ t) where z is Tuple of i,D, t is Tuple of j,D : verum } ; ::_thesis: x in (i + j) -tuples_on D then consider z being Tuple of i,D, t being Tuple of j,D such that A5: x = z ^ t ; ( len z = i & len t = j ) by CARD_1:def_7; then A6: len (z ^ t) = i + j by FINSEQ_1:22; z ^ t is Element of D * by FINSEQ_1:def_11; hence x in (i + j) -tuples_on D by A5, A6; ::_thesis: verum end; hence (i + j) -tuples_on D = { (z ^ t) where z is Tuple of i,D, t is Tuple of j,D : verum } by TARSKI:1; ::_thesis: verum end; theorem Th106: :: FINSEQ_2:106 for i, j being Nat for D being non empty set for s being Tuple of i + j,D ex z being Element of i -tuples_on D ex t being Element of j -tuples_on D st s = z ^ t proof let i, j be Nat; ::_thesis: for D being non empty set for s being Tuple of i + j,D ex z being Element of i -tuples_on D ex t being Element of j -tuples_on D st s = z ^ t let D be non empty set ; ::_thesis: for s being Tuple of i + j,D ex z being Element of i -tuples_on D ex t being Element of j -tuples_on D st s = z ^ t let s be Tuple of i + j,D; ::_thesis: ex z being Element of i -tuples_on D ex t being Element of j -tuples_on D st s = z ^ t A1: s is Element of D * by FINSEQ_1:def_11; len s = i + j by CARD_1:def_7; then s in (i + j) -tuples_on D by A1; then s in { (z ^ t) where z is Tuple of i,D, t is Tuple of j,D : verum } by Th105; then consider z being Tuple of i,D, t being Tuple of j,D such that A2: s = z ^ t ; reconsider z = z as Element of i -tuples_on D by Lm6; reconsider t = t as Element of j -tuples_on D by Lm6; s = z ^ t by A2; hence ex z being Element of i -tuples_on D ex t being Element of j -tuples_on D st s = z ^ t ; ::_thesis: verum end; theorem :: FINSEQ_2:107 for i, j being Nat for D being non empty set for z being Tuple of i,D for t being Tuple of j,D holds z ^ t is Tuple of i + j,D proof let i, j be Nat; ::_thesis: for D being non empty set for z being Tuple of i,D for t being Tuple of j,D holds z ^ t is Tuple of i + j,D let D be non empty set ; ::_thesis: for z being Tuple of i,D for t being Tuple of j,D holds z ^ t is Tuple of i + j,D let z be Tuple of i,D; ::_thesis: for t being Tuple of j,D holds z ^ t is Tuple of i + j,D let t be Tuple of j,D; ::_thesis: z ^ t is Tuple of i + j,D z ^ t in { (z9 ^ t9) where z9 is Tuple of i,D, t9 is Tuple of j,D : verum } ; then z ^ t in (i + j) -tuples_on D by Th105; hence z ^ t is Tuple of i + j,D ; ::_thesis: verum end; theorem :: FINSEQ_2:108 for D being non empty set holds D * = union { (i -tuples_on D) where i is Element of NAT : verum } proof let D be non empty set ; ::_thesis: D * = union { (i -tuples_on D) where i is Element of NAT : verum } for a being set holds ( a in D * iff a in union { (i -tuples_on D) where i is Element of NAT : verum } ) proof let a be set ; ::_thesis: ( a in D * iff a in union { (i -tuples_on D) where i is Element of NAT : verum } ) thus ( a in D * implies a in union { (i -tuples_on D) where i is Element of NAT : verum } ) ::_thesis: ( a in union { (i -tuples_on D) where i is Element of NAT : verum } implies a in D * ) proof assume a in D * ; ::_thesis: a in union { (i -tuples_on D) where i is Element of NAT : verum } then reconsider a = a as FinSequence of D by FINSEQ_1:def_11; ( a is Element of (len a) -tuples_on D & (len a) -tuples_on D in { (i -tuples_on D) where i is Element of NAT : verum } ) by Th92; hence a in union { (i -tuples_on D) where i is Element of NAT : verum } by TARSKI:def_4; ::_thesis: verum end; assume a in union { (i -tuples_on D) where i is Element of NAT : verum } ; ::_thesis: a in D * then consider Z being set such that A1: a in Z and A2: Z in { (i -tuples_on D) where i is Element of NAT : verum } by TARSKI:def_4; consider i being Element of NAT such that A3: Z = i -tuples_on D by A2; ex s being Element of D * st ( s = a & len s = i ) by A1, A3; hence a in D * ; ::_thesis: verum end; hence D * = union { (i -tuples_on D) where i is Element of NAT : verum } by TARSKI:1; ::_thesis: verum end; theorem :: FINSEQ_2:109 for i being Nat for D being non empty set for D9 being non empty Subset of D for z being Tuple of i,D9 holds z is Element of i -tuples_on D proof let i be Nat; ::_thesis: for D being non empty set for D9 being non empty Subset of D for z being Tuple of i,D9 holds z is Element of i -tuples_on D let D be non empty set ; ::_thesis: for D9 being non empty Subset of D for z being Tuple of i,D9 holds z is Element of i -tuples_on D let D9 be non empty Subset of D; ::_thesis: for z being Tuple of i,D9 holds z is Element of i -tuples_on D let z be Tuple of i,D9; ::_thesis: z is Element of i -tuples_on D z is Tuple of i,D by Th24; hence z is Element of i -tuples_on D by Lm6; ::_thesis: verum end; Lm7: for i being Nat for x, A being set st x in i -tuples_on A holds x is b1 -element FinSequence proof let i be Nat; ::_thesis: for x, A being set st x in i -tuples_on A holds x is i -element FinSequence let x, A be set ; ::_thesis: ( x in i -tuples_on A implies x is i -element FinSequence ) assume x in i -tuples_on A ; ::_thesis: x is i -element FinSequence then x in { s where s is Element of A * : len s = i } ; then ex s being Element of A * st ( x = s & len s = i ) ; hence x is i -element FinSequence by CARD_1:def_7; ::_thesis: verum end; theorem :: FINSEQ_2:110 for i, j being Nat for A being set for D being non empty set st i -tuples_on D = j -tuples_on A holds i = j proof let i, j be Nat; ::_thesis: for A being set for D being non empty set st i -tuples_on D = j -tuples_on A holds i = j let A be set ; ::_thesis: for D being non empty set st i -tuples_on D = j -tuples_on A holds i = j let D be non empty set ; ::_thesis: ( i -tuples_on D = j -tuples_on A implies i = j ) set f = i |-> the Element of D; i |-> the Element of D is Tuple of i,D by Th63; then A1: i |-> the Element of D in i -tuples_on D by Lm6; assume i -tuples_on D = j -tuples_on A ; ::_thesis: i = j then i |-> the Element of D in j -tuples_on A by A1; then i |-> the Element of D is j -element by Lm7; then len (i |-> the Element of D) = j by CARD_1:def_7; hence i = j by CARD_1:def_7; ::_thesis: verum end; theorem :: FINSEQ_2:111 for i being Nat holds idseq i is Element of i -tuples_on NAT proof let i be Nat; ::_thesis: idseq i is Element of i -tuples_on NAT ( idseq i is FinSequence of NAT & len (idseq i) = i ) by Th48, CARD_1:def_7; hence idseq i is Element of i -tuples_on NAT by Th92; ::_thesis: verum end; theorem Th112: :: FINSEQ_2:112 for i being Nat for D being non empty set for d being Element of D holds i |-> d is Element of i -tuples_on D proof let i be Nat; ::_thesis: for D being non empty set for d being Element of D holds i |-> d is Element of i -tuples_on D let D be non empty set ; ::_thesis: for d being Element of D holds i |-> d is Element of i -tuples_on D let d be Element of D; ::_thesis: i |-> d is Element of i -tuples_on D ( i |-> d is FinSequence of D & len (i |-> d) = i ) by Th63, CARD_1:def_7; hence i |-> d is Element of i -tuples_on D by Lm6; ::_thesis: verum end; theorem :: FINSEQ_2:113 for i being Nat for D, D9 being non empty set for z being Tuple of i,D for f being Function of D,D9 holds f * z is Element of i -tuples_on D9 proof let i be Nat; ::_thesis: for D, D9 being non empty set for z being Tuple of i,D for f being Function of D,D9 holds f * z is Element of i -tuples_on D9 let D, D9 be non empty set ; ::_thesis: for z being Tuple of i,D for f being Function of D,D9 holds f * z is Element of i -tuples_on D9 let z be Tuple of i,D; ::_thesis: for f being Function of D,D9 holds f * z is Element of i -tuples_on D9 let f be Function of D,D9; ::_thesis: f * z is Element of i -tuples_on D9 reconsider s = f * z as FinSequence of D9 by Th32; len z = i by CARD_1:def_7; then len s = i by Th33; hence f * z is Element of i -tuples_on D9 by Th92; ::_thesis: verum end; theorem Th114: :: FINSEQ_2:114 for i being Nat for D being non empty set for z being Tuple of i,D for f being Function of (Seg i),(Seg i) st rng f = Seg i holds z * f is Element of i -tuples_on D proof let i be Nat; ::_thesis: for D being non empty set for z being Tuple of i,D for f being Function of (Seg i),(Seg i) st rng f = Seg i holds z * f is Element of i -tuples_on D let D be non empty set ; ::_thesis: for z being Tuple of i,D for f being Function of (Seg i),(Seg i) st rng f = Seg i holds z * f is Element of i -tuples_on D let z be Tuple of i,D; ::_thesis: for f being Function of (Seg i),(Seg i) st rng f = Seg i holds z * f is Element of i -tuples_on D let f be Function of (Seg i),(Seg i); ::_thesis: ( rng f = Seg i implies z * f is Element of i -tuples_on D ) A1: len z = i by CARD_1:def_7; then A2: Seg i = dom z by FINSEQ_1:def_3; then reconsider t = z * f as FinSequence of D by Th47; assume rng f = Seg i ; ::_thesis: z * f is Element of i -tuples_on D then len t = len z by A2, Th42; hence z * f is Element of i -tuples_on D by A1, Th92; ::_thesis: verum end; theorem :: FINSEQ_2:115 for i being Nat for D being non empty set for z being Tuple of i,D for f being Permutation of (Seg i) holds z * f is Tuple of i,D proof let i be Nat; ::_thesis: for D being non empty set for z being Tuple of i,D for f being Permutation of (Seg i) holds z * f is Tuple of i,D let D be non empty set ; ::_thesis: for z being Tuple of i,D for f being Permutation of (Seg i) holds z * f is Tuple of i,D let z be Tuple of i,D; ::_thesis: for f being Permutation of (Seg i) holds z * f is Tuple of i,D let f be Permutation of (Seg i); ::_thesis: z * f is Tuple of i,D rng f = Seg i by FUNCT_2:def_3; hence z * f is Tuple of i,D by Th114; ::_thesis: verum end; theorem :: FINSEQ_2:116 for i being Nat for D being non empty set for z being Tuple of i,D for d being Element of D holds (z ^ <*d*>) . (i + 1) = d proof let i be Nat; ::_thesis: for D being non empty set for z being Tuple of i,D for d being Element of D holds (z ^ <*d*>) . (i + 1) = d let D be non empty set ; ::_thesis: for z being Tuple of i,D for d being Element of D holds (z ^ <*d*>) . (i + 1) = d let z be Tuple of i,D; ::_thesis: for d being Element of D holds (z ^ <*d*>) . (i + 1) = d let d be Element of D; ::_thesis: (z ^ <*d*>) . (i + 1) = d len z = i by CARD_1:def_7; hence (z ^ <*d*>) . (i + 1) = d by FINSEQ_1:42; ::_thesis: verum end; theorem :: FINSEQ_2:117 for i being Nat for D being non empty set for z being Tuple of i + 1,D ex t being Element of i -tuples_on D ex d being Element of D st z = t ^ <*d*> proof let i be Nat; ::_thesis: for D being non empty set for z being Tuple of i + 1,D ex t being Element of i -tuples_on D ex d being Element of D st z = t ^ <*d*> let D be non empty set ; ::_thesis: for z being Tuple of i + 1,D ex t being Element of i -tuples_on D ex d being Element of D st z = t ^ <*d*> let z be Tuple of i + 1,D; ::_thesis: ex t being Element of i -tuples_on D ex d being Element of D st z = t ^ <*d*> consider t being Element of i -tuples_on D, s being Element of 1 -tuples_on D such that A1: z = t ^ s by Th106; ex d being Element of D st s = <*d*> by Th97; hence ex t being Element of i -tuples_on D ex d being Element of D st z = t ^ <*d*> by A1; ::_thesis: verum end; theorem :: FINSEQ_2:118 for i being Nat for D being non empty set for z being Tuple of i,D holds z * (idseq i) = z proof let i be Nat; ::_thesis: for D being non empty set for z being Tuple of i,D holds z * (idseq i) = z let D be non empty set ; ::_thesis: for z being Tuple of i,D holds z * (idseq i) = z let z be Tuple of i,D; ::_thesis: z * (idseq i) = z len z = i by CARD_1:def_7; hence z * (idseq i) = z by Th54; ::_thesis: verum end; theorem :: FINSEQ_2:119 for i being Nat for D being non empty set for z1, z2 being Tuple of i,D st ( for j being Nat st j in Seg i holds z1 . j = z2 . j ) holds z1 = z2 proof let i be Nat; ::_thesis: for D being non empty set for z1, z2 being Tuple of i,D st ( for j being Nat st j in Seg i holds z1 . j = z2 . j ) holds z1 = z2 let D be non empty set ; ::_thesis: for z1, z2 being Tuple of i,D st ( for j being Nat st j in Seg i holds z1 . j = z2 . j ) holds z1 = z2 let z1, z2 be Tuple of i,D; ::_thesis: ( ( for j being Nat st j in Seg i holds z1 . j = z2 . j ) implies z1 = z2 ) len z2 = i by CARD_1:def_7; then A1: dom z2 = Seg i by FINSEQ_1:def_3; len z1 = i by CARD_1:def_7; then dom z1 = Seg i by FINSEQ_1:def_3; hence ( ( for j being Nat st j in Seg i holds z1 . j = z2 . j ) implies z1 = z2 ) by A1, FINSEQ_1:13; ::_thesis: verum end; theorem :: FINSEQ_2:120 for i being Nat for D, D9, E being non empty set for F being Function of [:D,D9:],E for z1 being Tuple of i,D for z2 being Tuple of i,D9 holds F .: (z1,z2) is Element of i -tuples_on E proof let i be Nat; ::_thesis: for D, D9, E being non empty set for F being Function of [:D,D9:],E for z1 being Tuple of i,D for z2 being Tuple of i,D9 holds F .: (z1,z2) is Element of i -tuples_on E let D, D9, E be non empty set ; ::_thesis: for F being Function of [:D,D9:],E for z1 being Tuple of i,D for z2 being Tuple of i,D9 holds F .: (z1,z2) is Element of i -tuples_on E let F be Function of [:D,D9:],E; ::_thesis: for z1 being Tuple of i,D for z2 being Tuple of i,D9 holds F .: (z1,z2) is Element of i -tuples_on E let z1 be Tuple of i,D; ::_thesis: for z2 being Tuple of i,D9 holds F .: (z1,z2) is Element of i -tuples_on E let z2 be Tuple of i,D9; ::_thesis: F .: (z1,z2) is Element of i -tuples_on E reconsider r = F .: (z1,z2) as FinSequence of E by Th70; ( len z1 = i & len z2 = i ) by CARD_1:def_7; then len r = i by Th72; hence F .: (z1,z2) is Element of i -tuples_on E by Th92; ::_thesis: verum end; theorem :: FINSEQ_2:121 for i being Nat for D, D9, E being non empty set for d being Element of D for F being Function of [:D,D9:],E for z being Tuple of i,D9 holds F [;] (d,z) is Element of i -tuples_on E proof let i be Nat; ::_thesis: for D, D9, E being non empty set for d being Element of D for F being Function of [:D,D9:],E for z being Tuple of i,D9 holds F [;] (d,z) is Element of i -tuples_on E let D, D9, E be non empty set ; ::_thesis: for d being Element of D for F being Function of [:D,D9:],E for z being Tuple of i,D9 holds F [;] (d,z) is Element of i -tuples_on E let d be Element of D; ::_thesis: for F being Function of [:D,D9:],E for z being Tuple of i,D9 holds F [;] (d,z) is Element of i -tuples_on E let F be Function of [:D,D9:],E; ::_thesis: for z being Tuple of i,D9 holds F [;] (d,z) is Element of i -tuples_on E let z be Tuple of i,D9; ::_thesis: F [;] (d,z) is Element of i -tuples_on E reconsider r = F [;] (d,z) as FinSequence of E by Th77; len z = i by CARD_1:def_7; then len r = i by Th78; hence F [;] (d,z) is Element of i -tuples_on E by Th92; ::_thesis: verum end; theorem :: FINSEQ_2:122 for i being Nat for D, D9, E being non empty set for d9 being Element of D9 for F being Function of [:D,D9:],E for z being Tuple of i,D holds F [:] (z,d9) is Element of i -tuples_on E proof let i be Nat; ::_thesis: for D, D9, E being non empty set for d9 being Element of D9 for F being Function of [:D,D9:],E for z being Tuple of i,D holds F [:] (z,d9) is Element of i -tuples_on E let D, D9, E be non empty set ; ::_thesis: for d9 being Element of D9 for F being Function of [:D,D9:],E for z being Tuple of i,D holds F [:] (z,d9) is Element of i -tuples_on E let d9 be Element of D9; ::_thesis: for F being Function of [:D,D9:],E for z being Tuple of i,D holds F [:] (z,d9) is Element of i -tuples_on E let F be Function of [:D,D9:],E; ::_thesis: for z being Tuple of i,D holds F [:] (z,d9) is Element of i -tuples_on E let z be Tuple of i,D; ::_thesis: F [:] (z,d9) is Element of i -tuples_on E reconsider r = F [:] (z,d9) as FinSequence of E by Th83; len z = i by CARD_1:def_7; then len r = i by Th84; hence F [:] (z,d9) is Element of i -tuples_on E by Th92; ::_thesis: verum end; theorem :: FINSEQ_2:123 for i, j being Nat for x being set holds (i + j) |-> x = (i |-> x) ^ (j |-> x) proof let i, j be Nat; ::_thesis: for x being set holds (i + j) |-> x = (i |-> x) ^ (j |-> x) let x be set ; ::_thesis: (i + j) |-> x = (i |-> x) ^ (j |-> x) defpred S1[ Nat] means (i + \$1) |-> x = (i |-> x) ^ (\$1 |-> x); A1: for j being Nat st S1[j] holds S1[j + 1] proof let j be Nat; ::_thesis: ( S1[j] implies S1[j + 1] ) assume A2: (i + j) |-> x = (i |-> x) ^ (j |-> x) ; ::_thesis: S1[j + 1] thus (i + (j + 1)) |-> x = ((i + j) + 1) |-> x .= ((i + j) |-> x) ^ <*x*> by Th60 .= (i |-> x) ^ ((j |-> x) ^ <*x*>) by A2, FINSEQ_1:32 .= (i |-> x) ^ ((j + 1) |-> x) by Th60 ; ::_thesis: verum end; (i + 0) |-> x = (i |-> x) ^ {} by FINSEQ_1:34 .= (i |-> x) ^ (0 |-> x) ; then A3: S1[ 0 ] ; for j being Nat holds S1[j] from NAT_1:sch_2(A3, A1); hence (i + j) |-> x = (i |-> x) ^ (j |-> x) ; ::_thesis: verum end; theorem :: FINSEQ_2:124 for i being Nat for D being non empty set for x being Tuple of i,D holds dom x = Seg i proof let i be Nat; ::_thesis: for D being non empty set for x being Tuple of i,D holds dom x = Seg i let D be non empty set ; ::_thesis: for x being Tuple of i,D holds dom x = Seg i let x be Tuple of i,D; ::_thesis: dom x = Seg i len x = i by CARD_1:def_7; hence dom x = Seg i by FINSEQ_1:def_3; ::_thesis: verum end; theorem :: FINSEQ_2:125 for f being Function for x, y being set st x in dom f & y in dom f holds f * <*x,y*> = <*(f . x),(f . y)*> proof let f be Function; ::_thesis: for x, y being set st x in dom f & y in dom f holds f * <*x,y*> = <*(f . x),(f . y)*> let x, y be set ; ::_thesis: ( x in dom f & y in dom f implies f * <*x,y*> = <*(f . x),(f . y)*> ) assume that A1: x in dom f and A2: y in dom f ; ::_thesis: f * <*x,y*> = <*(f . x),(f . y)*> reconsider D = dom f, E = rng f as non empty set by A1, FUNCT_1:3; rng <*x,y*> = {x,y} by Lm1; then rng <*x,y*> c= D by A1, A2, ZFMISC_1:32; then reconsider p = <*x,y*> as FinSequence of D by FINSEQ_1:def_4; reconsider g = f as Function of D,E by FUNCT_2:def_1, RELSET_1:4; thus f * <*x,y*> = g * p .= <*(f . x),(f . y)*> by Th36 ; ::_thesis: verum end; theorem :: FINSEQ_2:126 for f being Function for x, y, z being set st x in dom f & y in dom f & z in dom f holds f * <*x,y,z*> = <*(f . x),(f . y),(f . z)*> proof let f be Function; ::_thesis: for x, y, z being set st x in dom f & y in dom f & z in dom f holds f * <*x,y,z*> = <*(f . x),(f . y),(f . z)*> let x, y, z be set ; ::_thesis: ( x in dom f & y in dom f & z in dom f implies f * <*x,y,z*> = <*(f . x),(f . y),(f . z)*> ) assume that A1: x in dom f and A2: ( y in dom f & z in dom f ) ; ::_thesis: f * <*x,y,z*> = <*(f . x),(f . y),(f . z)*> reconsider D = dom f, E = rng f as non empty set by A1, FUNCT_1:3; rng <*x,y,z*> = {x,y,z} by Lm2; then A3: rng <*x,y,z*> = {x,y} \/ {z} by ENUMSET1:3; ( {x,y} c= D & {z} c= D ) by A1, A2, ZFMISC_1:31, ZFMISC_1:32; then rng <*x,y,z*> c= D by A3, XBOOLE_1:8; then reconsider p = <*x,y,z*> as FinSequence of D by FINSEQ_1:def_4; reconsider g = f as Function of D,E by FUNCT_2:def_1, RELSET_1:4; thus f * <*x,y,z*> = g * p .= <*(f . x),(f . y),(f . z)*> by Th37 ; ::_thesis: verum end; theorem :: FINSEQ_2:127 for x1, x2 being set holds rng <*x1,x2*> = {x1,x2} by Lm1; theorem :: FINSEQ_2:128 for x1, x2, x3 being set holds rng <*x1,x2,x3*> = {x1,x2,x3} by Lm2; begin theorem :: FINSEQ_2:129 for p1, p2, q being FinSequence st p1 c= q & p2 c= q & len p1 = len p2 holds p1 = p2 proof let p1, p2, q be FinSequence; ::_thesis: ( p1 c= q & p2 c= q & len p1 = len p2 implies p1 = p2 ) assume that A1: p1 c= q and A2: p2 c= q and A3: len p1 = len p2 ; ::_thesis: p1 = p2 reconsider i = len p1 as Element of NAT ; A4: ( dom p1 = Seg i & dom p2 = Seg i ) by A3, FINSEQ_1:def_3; now__::_thesis:_for_j_being_Nat_st_j_in_dom_p1_holds_ p1_._j_=_p2_._j let j be Nat; ::_thesis: ( j in dom p1 implies p1 . j = p2 . j ) assume A5: j in dom p1 ; ::_thesis: p1 . j = p2 . j hence p1 . j = q . j by A1, GRFUNC_1:2 .= p2 . j by A2, A4, A5, GRFUNC_1:2 ; ::_thesis: verum end; hence p1 = p2 by A4, FINSEQ_1:13; ::_thesis: verum end; theorem :: FINSEQ_2:130 for D being non empty set for s being FinSequence of D st s <> {} holds ex w being FinSequence of D ex n being Element of D st s = <*n*> ^ w proof let D be non empty set ; ::_thesis: for s being FinSequence of D st s <> {} holds ex w being FinSequence of D ex n being Element of D st s = <*n*> ^ w let s be FinSequence of D; ::_thesis: ( s <> {} implies ex w being FinSequence of D ex n being Element of D st s = <*n*> ^ w ) defpred S1[ FinSequence of D] means ( \$1 <> {} implies ex w being FinSequence of D ex n being Element of D st \$1 = <*n*> ^ w ); A1: for s being FinSequence of D for m being Element of D st S1[s] holds S1[s ^ <*m*>] proof let s be FinSequence of D; ::_thesis: for m being Element of D st S1[s] holds S1[s ^ <*m*>] let m be Element of D; ::_thesis: ( S1[s] implies S1[s ^ <*m*>] ) assume A2: ( s <> {} implies ex w being FinSequence of D ex n being Element of D st s = <*n*> ^ w ) ; ::_thesis: S1[s ^ <*m*>] assume s ^ <*m*> <> {} ; ::_thesis: ex w being FinSequence of D ex n being Element of D st s ^ <*m*> = <*n*> ^ w percases ( s = {} or s <> {} ) ; supposeA3: s = {} ; ::_thesis: ex w being FinSequence of D ex n being Element of D st s ^ <*m*> = <*n*> ^ w reconsider w = <*> D as FinSequence of D ; take w ; ::_thesis: ex n being Element of D st s ^ <*m*> = <*n*> ^ w take n = m; ::_thesis: s ^ <*m*> = <*n*> ^ w thus s ^ <*m*> = <*m*> by A3, FINSEQ_1:34 .= <*n*> ^ w by FINSEQ_1:34 ; ::_thesis: verum end; suppose s <> {} ; ::_thesis: ex w being FinSequence of D ex n being Element of D st s ^ <*m*> = <*n*> ^ w then consider w being FinSequence of D, n being Element of D such that A4: s = <*n*> ^ w by A2; take w ^ <*m*> ; ::_thesis: ex n being Element of D st s ^ <*m*> = <*n*> ^ (w ^ <*m*>) take n ; ::_thesis: s ^ <*m*> = <*n*> ^ (w ^ <*m*>) thus s ^ <*m*> = <*n*> ^ (w ^ <*m*>) by A4, FINSEQ_1:32; ::_thesis: verum end; end; end; A5: S1[ <*> D] ; for p being FinSequence of D holds S1[p] from FINSEQ_2:sch_2(A5, A1); hence ( s <> {} implies ex w being FinSequence of D ex n being Element of D st s = <*n*> ^ w ) ; ::_thesis: verum end; registration let D be set ; cluster -> functional for FinSequenceSet of D; coherence for b1 being FinSequenceSet of D holds b1 is functional proof let f be FinSequenceSet of D; ::_thesis: f is functional let x be set ; :: according to FUNCT_1:def_13 ::_thesis: ( not x in f or x is set ) thus ( not x in f or x is set ) by Def3; ::_thesis: verum end; end; definition let D be non empty set ; let n be Nat; let d be Element of D; :: original: |-> redefine funcn |-> d -> Element of n -tuples_on D; coherence n |-> d is Element of n -tuples_on D by Th112; end; theorem :: FINSEQ_2:131 for i being Nat for D being non empty set for z being set holds ( z is Tuple of i,D iff z in i -tuples_on D ) proof let i be Nat; ::_thesis: for D being non empty set for z being set holds ( z is Tuple of i,D iff z in i -tuples_on D ) let D be non empty set ; ::_thesis: for z being set holds ( z is Tuple of i,D iff z in i -tuples_on D ) let z be set ; ::_thesis: ( z is Tuple of i,D iff z in i -tuples_on D ) thus ( z is Tuple of i,D implies z in i -tuples_on D ) by Lm6; ::_thesis: ( z in i -tuples_on D implies z is Tuple of i,D ) assume z in i -tuples_on D ; ::_thesis: z is Tuple of i,D then ex s being Element of D * st ( z = s & len s = i ) ; hence z is Tuple of i,D by CARD_1:def_7; ::_thesis: verum end; theorem Th132: :: FINSEQ_2:132 for A being set for i being Element of NAT for p being FinSequence holds ( p in i -tuples_on A iff ( len p = i & rng p c= A ) ) proof let A be set ; ::_thesis: for i being Element of NAT for p being FinSequence holds ( p in i -tuples_on A iff ( len p = i & rng p c= A ) ) let i be Element of NAT ; ::_thesis: for p being FinSequence holds ( p in i -tuples_on A iff ( len p = i & rng p c= A ) ) let p be FinSequence; ::_thesis: ( p in i -tuples_on A iff ( len p = i & rng p c= A ) ) hereby ::_thesis: ( len p = i & rng p c= A implies p in i -tuples_on A ) assume p in i -tuples_on A ; ::_thesis: ( len p = i & rng p c= A ) then ex q being Element of A * st ( p = q & len q = i ) ; hence ( len p = i & rng p c= A ) by FINSEQ_1:def_4; ::_thesis: verum end; assume A1: len p = i ; ::_thesis: ( not rng p c= A or p in i -tuples_on A ) assume rng p c= A ; ::_thesis: p in i -tuples_on A then p is FinSequence of A by FINSEQ_1:def_4; then p in A * by FINSEQ_1:def_11; hence p in i -tuples_on A by A1; ::_thesis: verum end; theorem :: FINSEQ_2:133 for A being set for i being Element of NAT for p being FinSequence of A holds ( p in i -tuples_on A iff len p = i ) proof let A be set ; ::_thesis: for i being Element of NAT for p being FinSequence of A holds ( p in i -tuples_on A iff len p = i ) let i be Element of NAT ; ::_thesis: for p being FinSequence of A holds ( p in i -tuples_on A iff len p = i ) let p be FinSequence of A; ::_thesis: ( p in i -tuples_on A iff len p = i ) rng p c= A by RELAT_1:def_19; hence ( p in i -tuples_on A iff len p = i ) by Th132; ::_thesis: verum end; theorem :: FINSEQ_2:134 for A being set for i being Element of NAT holds i -tuples_on A c= A * proof let A be set ; ::_thesis: for i being Element of NAT holds i -tuples_on A c= A * let i be Element of NAT ; ::_thesis: i -tuples_on A c= A * let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in i -tuples_on A or x in A * ) assume x in i -tuples_on A ; ::_thesis: x in A * then x is FinSequence of A by Def3; hence x in A * by FINSEQ_1:def_11; ::_thesis: verum end; theorem Th135: :: FINSEQ_2:135 for A, x being set holds ( x in 1 -tuples_on A iff ex a being set st ( a in A & x = <*a*> ) ) proof let A, x be set ; ::_thesis: ( x in 1 -tuples_on A iff ex a being set st ( a in A & x = <*a*> ) ) hereby ::_thesis: ( ex a being set st ( a in A & x = <*a*> ) implies x in 1 -tuples_on A ) assume x in 1 -tuples_on A ; ::_thesis: ex a being set st ( a in A & x = <*a*> ) then x in { s where s is Element of A * : len s = 1 } ; then consider s being Element of A * such that A1: x = s and A2: len s = 1 ; take a = s . 1; ::_thesis: ( a in A & x = <*a*> ) A3: ( rng <*a*> = {a} & a in {a} ) by FINSEQ_1:39, TARSKI:def_1; A4: rng s c= A by RELAT_1:def_19; x = <*a*> by A1, A2, FINSEQ_1:40; hence ( a in A & x = <*a*> ) by A1, A3, A4; ::_thesis: verum end; given a being set such that A5: a in A and A6: x = <*a*> ; ::_thesis: x in 1 -tuples_on A reconsider A = A as non empty set by A5; reconsider a = a as Element of A by A5; <*a*> is Element of 1 -tuples_on A by Th98; hence x in 1 -tuples_on A by A6; ::_thesis: verum end; theorem :: FINSEQ_2:136 for A, a being set st <*a*> in 1 -tuples_on A holds a in A proof let A, a be set ; ::_thesis: ( <*a*> in 1 -tuples_on A implies a in A ) assume <*a*> in 1 -tuples_on A ; ::_thesis: a in A then A1: ex a9 being set st ( a9 in A & <*a*> = <*a9*> ) by Th135; <*a*> . 1 = a by FINSEQ_1:40; hence a in A by A1, FINSEQ_1:40; ::_thesis: verum end; theorem Th137: :: FINSEQ_2:137 for A, x being set holds ( x in 2 -tuples_on A iff ex a, b being set st ( a in A & b in A & x = <*a,b*> ) ) proof let A, x be set ; ::_thesis: ( x in 2 -tuples_on A iff ex a, b being set st ( a in A & b in A & x = <*a,b*> ) ) hereby ::_thesis: ( ex a, b being set st ( a in A & b in A & x = <*a,b*> ) implies x in 2 -tuples_on A ) assume x in 2 -tuples_on A ; ::_thesis: ex a, b being set st ( a in A & b in A & x = <*a,b*> ) then x in { s where s is Element of A * : len s = 2 } ; then consider s being Element of A * such that A1: x = s and A2: len s = 2 ; take a = s . 1; ::_thesis: ex b being set st ( a in A & b in A & x = <*a,b*> ) take b = s . 2; ::_thesis: ( a in A & b in A & x = <*a,b*> ) A3: ( rng <*a,b*> = {a,b} & a in {a,b} ) by Lm1, TARSKI:def_2; A4: ( b in {a,b} & rng s c= A ) by RELAT_1:def_19, TARSKI:def_2; x = <*a,b*> by A1, A2, FINSEQ_1:44; hence ( a in A & b in A & x = <*a,b*> ) by A1, A3, A4; ::_thesis: verum end; given a, b being set such that A5: a in A and A6: b in A and A7: x = <*a,b*> ; ::_thesis: x in 2 -tuples_on A reconsider A = A as non empty set by A5; reconsider a = a, b = b as Element of A by A5, A6; <*a,b*> is Element of 2 -tuples_on A by Th101; hence x in 2 -tuples_on A by A7; ::_thesis: verum end; theorem :: FINSEQ_2:138 for A, a, b being set st <*a,b*> in 2 -tuples_on A holds ( a in A & b in A ) proof let A, a, b be set ; ::_thesis: ( <*a,b*> in 2 -tuples_on A implies ( a in A & b in A ) ) assume <*a,b*> in 2 -tuples_on A ; ::_thesis: ( a in A & b in A ) then A1: ex a9, b9 being set st ( a9 in A & b9 in A & <*a,b*> = <*a9,b9*> ) by Th137; ( <*a,b*> . 1 = a & <*a,b*> . 2 = b ) by FINSEQ_1:44; hence ( a in A & b in A ) by A1, FINSEQ_1:44; ::_thesis: verum end; theorem Th139: :: FINSEQ_2:139 for A, x being set holds ( x in 3 -tuples_on A iff ex a, b, c being set st ( a in A & b in A & c in A & x = <*a,b,c*> ) ) proof let A, x be set ; ::_thesis: ( x in 3 -tuples_on A iff ex a, b, c being set st ( a in A & b in A & c in A & x = <*a,b,c*> ) ) hereby ::_thesis: ( ex a, b, c being set st ( a in A & b in A & c in A & x = <*a,b,c*> ) implies x in 3 -tuples_on A ) assume x in 3 -tuples_on A ; ::_thesis: ex a, b, c being set st ( a in A & b in A & c in A & x = <*a,b,c*> ) then x in { s where s is Element of A * : len s = 3 } ; then consider s being Element of A * such that A1: x = s and A2: len s = 3 ; take a = s . 1; ::_thesis: ex b, c being set st ( a in A & b in A & c in A & x = <*a,b,c*> ) take b = s . 2; ::_thesis: ex c being set st ( a in A & b in A & c in A & x = <*a,b,c*> ) take c = s . 3; ::_thesis: ( a in A & b in A & c in A & x = <*a,b,c*> ) A3: ( rng <*a,b,c*> = {a,b,c} & a in {a,b,c} ) by Lm2, ENUMSET1:def_1; A4: rng s c= A by RELAT_1:def_19; A5: ( b in {a,b,c} & c in {a,b,c} ) by ENUMSET1:def_1; x = <*a,b,c*> by A1, A2, FINSEQ_1:45; hence ( a in A & b in A & c in A & x = <*a,b,c*> ) by A1, A3, A5, A4; ::_thesis: verum end; given a, b, c being set such that A6: a in A and A7: ( b in A & c in A ) and A8: x = <*a,b,c*> ; ::_thesis: x in 3 -tuples_on A reconsider A = A as non empty set by A6; reconsider a = a, b = b, c = c as Element of A by A6, A7; <*a,b,c*> is Element of 3 -tuples_on A by Th104; hence x in 3 -tuples_on A by A8; ::_thesis: verum end; theorem :: FINSEQ_2:140 for A, a, b, c being set st <*a,b,c*> in 3 -tuples_on A holds ( a in A & b in A & c in A ) proof let A, a, b, c be set ; ::_thesis: ( <*a,b,c*> in 3 -tuples_on A implies ( a in A & b in A & c in A ) ) A1: <*a,b,c*> . 3 = c by FINSEQ_1:45; assume <*a,b,c*> in 3 -tuples_on A ; ::_thesis: ( a in A & b in A & c in A ) then A2: ex a9, b9, c9 being set st ( a9 in A & b9 in A & c9 in A & <*a,b,c*> = <*a9,b9,c9*> ) by Th139; ( <*a,b,c*> . 1 = a & <*a,b,c*> . 2 = b ) by FINSEQ_1:45; hence ( a in A & b in A & c in A ) by A2, A1, FINSEQ_1:45; ::_thesis: verum end; theorem :: FINSEQ_2:141 for i being Nat for x, A being set st x in i -tuples_on A holds x is b1 -element FinSequence by Lm7; theorem :: FINSEQ_2:142 for A being non empty set for n being Nat holds n -tuples_on A c= A * proof let A be non empty set ; ::_thesis: for n being Nat holds n -tuples_on A c= A * let n be Nat; ::_thesis: n -tuples_on A c= A * defpred S1[ Element of A * ] means len \$1 = n; { s where s is Element of A * : S1[s] } c= A * from FRAENKEL:sch_10(); hence n -tuples_on A c= A * ; ::_thesis: verum end; theorem :: FINSEQ_2:143 for x being set for n, m being Nat st n |-> x = m |-> x holds n = m proof let x be set ; ::_thesis: for n, m being Nat st n |-> x = m |-> x holds n = m let n, m be Nat; ::_thesis: ( n |-> x = m |-> x implies n = m ) len (n |-> x) = n by CARD_1:def_7; hence ( n |-> x = m |-> x implies n = m ) by CARD_1:def_7; ::_thesis: verum end; definition let I be set ; let M be ManySortedSet of I; funcM # -> ManySortedSet of I * means :Def5: :: FINSEQ_2:def 5 for i being Element of I * holds it . i = product (M * i); existence ex b1 being ManySortedSet of I * st for i being Element of I * holds b1 . i = product (M * i) proof defpred S1[ set , set ] means ex j being Element of I * st ( j = \$1 & \$2 = product (M * j) ); A1: for i being set st i in I * holds ex j being set st S1[i,j] proof let i be set ; ::_thesis: ( i in I * implies ex j being set st S1[i,j] ) assume i in I * ; ::_thesis: ex j being set st S1[i,j] then reconsider j = i as Element of I * ; reconsider e = product (M * j) as set ; take e ; ::_thesis: S1[i,e] take j ; ::_thesis: ( j = i & e = product (M * j) ) thus ( j = i & e = product (M * j) ) ; ::_thesis: verum end; consider f being ManySortedSet of I * such that A2: for i being set st i in I * holds S1[i,f . i] from PBOOLE:sch_3(A1); take f ; ::_thesis: for i being Element of I * holds f . i = product (M * i) let i be Element of I * ; ::_thesis: f . i = product (M * i) ex j being Element of I * st ( j = i & f . i = product (M * j) ) by A2; hence f . i = product (M * i) ; ::_thesis: verum end; uniqueness for b1, b2 being ManySortedSet of I * st ( for i being Element of I * holds b1 . i = product (M * i) ) & ( for i being Element of I * holds b2 . i = product (M * i) ) holds b1 = b2 proof let N1, N2 be ManySortedSet of I * ; ::_thesis: ( ( for i being Element of I * holds N1 . i = product (M * i) ) & ( for i being Element of I * holds N2 . i = product (M * i) ) implies N1 = N2 ) assume that A3: for i being Element of I * holds N1 . i = product (M * i) and A4: for i being Element of I * holds N2 . i = product (M * i) ; ::_thesis: N1 = N2 now__::_thesis:_for_i_being_set_st_i_in_I_*_holds_ N1_._i_=_N2_._i let i be set ; ::_thesis: ( i in I * implies N1 . i = N2 . i ) assume i in I * ; ::_thesis: N1 . i = N2 . i then reconsider e = i as Element of I * ; thus N1 . i = product (M * e) by A3 .= N2 . i by A4 ; ::_thesis: verum end; hence N1 = N2 by PBOOLE:3; ::_thesis: verum end; end; :: deftheorem Def5 defines # FINSEQ_2:def_5_:_ for I being set for M being ManySortedSet of I for b3 being ManySortedSet of I * holds ( b3 = M # iff for i being Element of I * holds b3 . i = product (M * i) ); registration let I be set ; let M be V2() ManySortedSet of I; clusterM # -> V2() ; coherence M # is non-empty proof M # is V2() proof let i be set ; :: according to PBOOLE:def_13 ::_thesis: ( not i in I * or not (M #) . i is empty ) assume i in I * ; ::_thesis: not (M #) . i is empty then reconsider f = i as Element of I * ; product (M * f) <> {} ; hence not (M #) . i is empty by Def5; ::_thesis: verum end; hence M # is non-empty ; ::_thesis: verum end; end; definition let a be set ; func *--> a -> Function of NAT,({a} *) means :Def6: :: FINSEQ_2:def 6 for n being Element of NAT holds it . n = n |-> a; existence ex b1 being Function of NAT,({a} *) st for n being Element of NAT holds b1 . n = n |-> a proof defpred S1[ Element of NAT , set ] means \$2 = \$1 |-> a; A1: for x being Element of NAT ex y being Element of {a} * st S1[x,y] proof let n be Element of NAT ; ::_thesis: ex y being Element of {a} * st S1[n,y] a is Element of {a} by TARSKI:def_1; then n |-> a is FinSequence of {a} by Th63; then reconsider u = n |-> a as Element of {a} * by FINSEQ_1:def_11; take u ; ::_thesis: S1[n,u] thus S1[n,u] ; ::_thesis: verum end; ex f being Function of NAT,({a} *) st for x being Element of NAT holds S1[x,f . x] from FUNCT_2:sch_3(A1); hence ex b1 being Function of NAT,({a} *) st for n being Element of NAT holds b1 . n = n |-> a ; ::_thesis: verum end; uniqueness for b1, b2 being Function of NAT,({a} *) st ( for n being Element of NAT holds b1 . n = n |-> a ) & ( for n being Element of NAT holds b2 . n = n |-> a ) holds b1 = b2 proof let f1, f2 be Function of NAT,({a} *); ::_thesis: ( ( for n being Element of NAT holds f1 . n = n |-> a ) & ( for n being Element of NAT holds f2 . n = n |-> a ) implies f1 = f2 ) assume that A2: for n being Element of NAT holds f1 . n = n |-> a and A3: for n being Element of NAT holds f2 . n = n |-> a ; ::_thesis: f1 = f2 now__::_thesis:_for_k_being_Element_of_NAT_holds_f1_._k_=_f2_._k let k be Element of NAT ; ::_thesis: f1 . k = f2 . k thus f1 . k = k |-> a by A2 .= f2 . k by A3 ; ::_thesis: verum end; hence f1 = f2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def6 defines *--> FINSEQ_2:def_6_:_ for a being set for b2 being Function of NAT,({a} *) holds ( b2 = *--> a iff for n being Element of NAT holds b2 . n = n |-> a ); theorem Th144: :: FINSEQ_2:144 for n being Element of NAT for a, b being set holds (a .--> b) * (n |-> a) = n |-> b proof let n be Element of NAT ; ::_thesis: for a, b being set holds (a .--> b) * (n |-> a) = n |-> b let a, b be set ; ::_thesis: (a .--> b) * (n |-> a) = n |-> b A1: now__::_thesis:_for_x_being_set_holds_ (_(_x_in_dom_(n_|->_b)_implies_(_x_in_dom_(n_|->_a)_&_(n_|->_a)_._x_in_dom_(a_.-->_b)_)_)_&_(_x_in_dom_(n_|->_a)_&_(n_|->_a)_._x_in_dom_(a_.-->_b)_implies_x_in_dom_(n_|->_b)_)_) let x be set ; ::_thesis: ( ( x in dom (n |-> b) implies ( x in dom (n |-> a) & (n |-> a) . x in dom (a .--> b) ) ) & ( x in dom (n |-> a) & (n |-> a) . x in dom (a .--> b) implies x in dom (n |-> b) ) ) hereby ::_thesis: ( x in dom (n |-> a) & (n |-> a) . x in dom (a .--> b) implies x in dom (n |-> b) ) assume x in dom (n |-> b) ; ::_thesis: ( x in dom (n |-> a) & (n |-> a) . x in dom (a .--> b) ) then A2: x in Seg n by FUNCOP_1:13; hence x in dom (n |-> a) by FUNCOP_1:13; ::_thesis: (n |-> a) . x in dom (a .--> b) ( dom (a .--> b) = {a} & (n |-> a) . x = a ) by A2, FUNCOP_1:7, FUNCOP_1:13; hence (n |-> a) . x in dom (a .--> b) by TARSKI:def_1; ::_thesis: verum end; assume that A3: x in dom (n |-> a) and (n |-> a) . x in dom (a .--> b) ; ::_thesis: x in dom (n |-> b) x in Seg n by A3, FUNCOP_1:13; hence x in dom (n |-> b) by FUNCOP_1:13; ::_thesis: verum end; now__::_thesis:_for_x_being_set_st_x_in_dom_(n_|->_b)_holds_ (n_|->_b)_._x_=_(a_.-->_b)_._((n_|->_a)_._x) let x be set ; ::_thesis: ( x in dom (n |-> b) implies (n |-> b) . x = (a .--> b) . ((n |-> a) . x) ) A4: a in {a} by TARSKI:def_1; assume x in dom (n |-> b) ; ::_thesis: (n |-> b) . x = (a .--> b) . ((n |-> a) . x) then A5: x in Seg n by FUNCOP_1:13; hence (n |-> b) . x = b by FUNCOP_1:7 .= (a .--> b) . a by A4, FUNCOP_1:7 .= (a .--> b) . ((n |-> a) . x) by A5, FUNCOP_1:7 ; ::_thesis: verum end; hence (a .--> b) * (n |-> a) = n |-> b by A1, FUNCT_1:10; ::_thesis: verum end; theorem :: FINSEQ_2:145 for D being non empty set for n being Element of NAT for a being set for M being ManySortedSet of {a} st M = a .--> D holds ((M #) * (*--> a)) . n = Funcs ((Seg n),D) proof let D be non empty set ; ::_thesis: for n being Element of NAT for a being set for M being ManySortedSet of {a} st M = a .--> D holds ((M #) * (*--> a)) . n = Funcs ((Seg n),D) let n be Element of NAT ; ::_thesis: for a being set for M being ManySortedSet of {a} st M = a .--> D holds ((M #) * (*--> a)) . n = Funcs ((Seg n),D) let a be set ; ::_thesis: for M being ManySortedSet of {a} st M = a .--> D holds ((M #) * (*--> a)) . n = Funcs ((Seg n),D) let M be ManySortedSet of {a}; ::_thesis: ( M = a .--> D implies ((M #) * (*--> a)) . n = Funcs ((Seg n),D) ) assume A1: M = a .--> D ; ::_thesis: ((M #) * (*--> a)) . n = Funcs ((Seg n),D) a is Element of {a} by TARSKI:def_1; then n |-> a is FinSequence of {a} by Th63; then A2: n |-> a in {a} * by FINSEQ_1:def_11; dom (*--> a) = NAT by FUNCT_2:def_1; hence ((M #) * (*--> a)) . n = (M #) . ((*--> a) . n) by FUNCT_1:13 .= (M #) . (n |-> a) by Def6 .= product ((a .--> D) * (n |-> a)) by A1, A2, Def5 .= product (n |-> D) by Th144 .= Funcs ((Seg n),D) by CARD_3:11 ; ::_thesis: verum end; theorem :: FINSEQ_2:146 for F being NAT -defined total Function for p being NAT -defined Function for n being Element of NAT st Shift (p,n) c= F holds for i being Element of NAT st i in dom p holds F . (n + i) = p . i proof let F be NAT -defined total Function; ::_thesis: for p being NAT -defined Function for n being Element of NAT st Shift (p,n) c= F holds for i being Element of NAT st i in dom p holds F . (n + i) = p . i let p be NAT -defined Function; ::_thesis: for n being Element of NAT st Shift (p,n) c= F holds for i being Element of NAT st i in dom p holds F . (n + i) = p . i let n be Element of NAT ; ::_thesis: ( Shift (p,n) c= F implies for i being Element of NAT st i in dom p holds F . (n + i) = p . i ) assume A1: Shift (p,n) c= F ; ::_thesis: for i being Element of NAT st i in dom p holds F . (n + i) = p . i let i be Element of NAT ; ::_thesis: ( i in dom p implies F . (n + i) = p . i ) assume A2: i in dom p ; ::_thesis: F . (n + i) = p . i then n + i in dom (Shift (p,n)) by VALUED_1:24; hence F . (n + i) = (Shift (p,n)) . (n + i) by A1, GRFUNC_1:2 .= p . i by A2, VALUED_1:def_12 ; ::_thesis: verum end; registration let i be Nat; clusteri |-> 0 -> empty-yielding ; coherence i |-> 0 is empty-yielding proof let x be set ; :: according to FUNCT_1:def_8 ::_thesis: ( not x in dom (i |-> 0) or (i |-> 0) . x is empty ) dom (i |-> 0) = Seg i by FUNCOP_1:13; hence ( not x in dom (i |-> 0) or (i |-> 0) . x is empty ) by Th57; ::_thesis: verum end; end; registration let D be set ; cluster -> FinSequence-membered for FinSequenceSet of D; coherence for b1 being FinSequenceSet of D holds b1 is FinSequence-membered proof let A be FinSequenceSet of D; ::_thesis: A is FinSequence-membered let x be set ; :: according to FINSEQ_1:def_18 ::_thesis: ( not x in A or x is set ) thus ( not x in A or x is set ) by Def3; ::_thesis: verum end; end;