### The Mizar article:

### Real Normed Space

**by****Jan Popiolek**

- Received September 20, 1990
Copyright (c) 1990 Association of Mizar Users

- MML identifier: NORMSP_1
- [ MML identifier index ]

environ vocabulary RLVECT_1, BINOP_1, FUNCT_1, PRE_TOPC, RLSUB_1, ABSVALUE, ARYTM_1, ARYTM_3, RELAT_1, SEQM_3, SEQ_2, SEQ_1, ORDINAL2, NORMSP_1, ARYTM; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XREAL_0, REAL_1, NAT_1, FUNCT_1, FUNCT_2, BINOP_1, RELAT_1, SEQ_1, SEQ_2, SEQM_3, ABSVALUE, STRUCT_0, PRE_TOPC, RLVECT_1, RLSUB_1; constructors REAL_1, NAT_1, DOMAIN_1, SEQ_2, SEQM_3, ABSVALUE, RLSUB_1, MEMBERED, XBOOLE_0; clusters STRUCT_0, XREAL_0, RELSET_1, ARYTM_3, MEMBERED, ZFMISC_1, XBOOLE_0, NUMBERS, ORDINAL2; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; definitions RLVECT_1, SEQM_3, STRUCT_0; theorems TARSKI, AXIOMS, REAL_1, NAT_1, FUNCT_1, SQUARE_1, SEQ_2, ABSVALUE, RLVECT_1, RLSUB_1, FUNCT_2, SEQM_3, STRUCT_0, RELSET_1, XREAL_0, XBOOLE_0, XCMPLX_0, XCMPLX_1; schemes FUNCT_1, FUNCT_2, SEQ_1, NAT_1; begin definition struct(RLSStruct) NORMSTR (# carrier -> set, Zero -> Element of the carrier, add -> BinOp of the carrier, Mult -> Function of [:REAL, the carrier:], the carrier, norm -> Function of the carrier, REAL #); end; definition cluster non empty NORMSTR; existence proof consider A being non empty set, Z being Element of A, a being BinOp of A, M being Function of [:REAL,A:],A, n being Function of A,REAL; take NORMSTR(#A,Z,a,M,n#); thus the carrier of NORMSTR(#A,Z,a,M,n#) is non empty; end; end; reserve X for non empty NORMSTR; reserve a, b for Real; reserve x for Point of X; deffunc 0'(NORMSTR) = 0.$1; definition let X, x; func ||.x.|| -> Real equals :Def1: ( the norm of X ).x; coherence; end; consider V being RealLinearSpace; Lm1: the carrier of (0).V = {0.V} by RLSUB_1:def 3; deffunc F(set) = 0; consider nil_to_nil being Function of the carrier of (0).V , REAL such that Lm2: for u being VECTOR of (0).V holds nil_to_nil.u = F(u) from LambdaD; Lm3: nil_to_nil.(0.V) = 0 proof 0.V is VECTOR of (0).V by Lm1,TARSKI:def 1; hence thesis by Lm2; end; Lm4: for u being VECTOR of (0).V , a holds nil_to_nil.(a * u) = abs(a) * nil_to_nil.u proof let u be VECTOR of (0).V; let a; nil_to_nil.u = 0 by Lm1,Lm3,TARSKI:def 1; hence thesis by Lm1,Lm3,TARSKI:def 1; end; Lm5: for u , v being VECTOR of (0).V holds nil_to_nil.(u + v) <= nil_to_nil.u + nil_to_nil.v proof let u, v be VECTOR of (0).V; u = 0.V & v = 0.V by Lm1,TARSKI:def 1; hence thesis by Lm1,Lm3,TARSKI:def 1; end; reconsider X = NORMSTR (# the carrier of (0).V, the Zero of (0).V, the add of (0).V, the Mult of (0).V, nil_to_nil #) as non empty NORMSTR by STRUCT_0:def 1; Lm6: now let x, y be Point of X; let a; A1: ||.x.|| = nil_to_nil.x & ||.y.|| = nil_to_nil.y & ||.a * x.|| = nil_to_nil.(a * x) & ||.x + y.|| = nil_to_nil.(x + y) by Def1; reconsider u = x, w = y as VECTOR of (0).V; thus ||.x.|| = 0 iff x = 0'(X) proof 0'(X) = the Zero of X by RLVECT_1:def 2 .= 0.(0).V by RLVECT_1:def 2 .= 0.V by RLSUB_1:19; hence thesis by Def1,Lm1,Lm3,TARSKI:def 1; end; a * x = (the Mult of X).[a,u] by RLVECT_1:def 4 .= a * u by RLVECT_1:def 4; hence ||.a * x.|| = abs(a) * ||.x.|| by A1,Lm4; x + y = (the add of X).[x,y] by RLVECT_1:def 3 .= u + w by RLVECT_1:def 3; hence ||.x + y.|| <= ||.x.|| + ||.y.|| by A1,Lm5; end; definition let IT be non empty NORMSTR; attr IT is RealNormSpace-like means :Def2: for x , y being Point of IT , a holds ( ||.x.|| = 0 iff x = 0.IT ) & ||.a * x.|| = abs(a) * ||.x.|| & ||.x + y.|| <= ||.x.|| + ||.y.||; end; definition cluster RealNormSpace-like RealLinearSpace-like Abelian add-associative right_zeroed right_complementable strict (non empty NORMSTR); existence proof take X; thus X is RealNormSpace-like by Def2,Lm6; A1: now let x,y be VECTOR of X; let x',y' be VECTOR of (0).V; assume A2: x = x' & y = y'; hence x + y= (the add of X).[x',y'] by RLVECT_1:def 3 .= x' + y' by RLVECT_1:def 3; let a; thus a * x = (the Mult of X).[a,x'] by A2,RLVECT_1:def 4 .= a * x' by RLVECT_1:def 4; end; A3: 0.X = the Zero of X by RLVECT_1:def 2 .= 0.(0).V by RLVECT_1:def 2; thus X is RealLinearSpace-like proof thus for a for v,w being VECTOR of X holds a * (v + w) = a * v + a * w proof let a; let v,w be VECTOR of X; reconsider v'= v, w' = w as VECTOR of (0).V; A4: a * v' = a * v by A1; A5: a * w' = a * w by A1; v + w = v'+ w' by A1; hence a * (v + w) = a *( v' + w') by A1 .= a * v' + a * w' by RLVECT_1:def 9 .= a * v + a * w by A1,A4,A5; end; thus for a,b for v being VECTOR of X holds (a + b) * v = a * v + b * v proof let a,b; let v be VECTOR of X; reconsider v'= v as VECTOR of (0).V; A6: a * v' = a * v by A1; A7: b * v' = b * v by A1; thus (a + b) * v = (a + b) * v' by A1 .= a * v' + b * v' by RLVECT_1:def 9 .= a * v + b * v by A1,A6,A7; end; thus for a,b for v being VECTOR of X holds (a * b) * v = a * (b * v) proof let a,b; let v be VECTOR of X; reconsider v'= v as VECTOR of (0).V; A8: b * v' = b * v by A1; thus (a * b) * v = (a * b) * v' by A1 .= a * (b * v') by RLVECT_1:def 9 .= a * (b * v) by A1,A8; end; let v be VECTOR of X; reconsider v'= v as VECTOR of (0).V; thus 1 * v = 1 * v' by A1 .= v by RLVECT_1:def 9; end; thus for v,w being VECTOR of X holds v + w = w + v proof let v,w be VECTOR of X; reconsider v'= v , w'= w as VECTOR of (0).V; thus v + w = w'+ v' by A1 .= w + v by A1; end; thus for u,v,w being VECTOR of X holds (u + v) + w = u + (v + w) proof let u,v,w be VECTOR of X; reconsider u'= u, v'= v, w'= w as VECTOR of (0).V; A9: u + v = u'+ v' by A1; A10: v + w = v' + w' by A1; thus (u + v) + w = (u' + v') + w' by A1,A9 .= u' + (v' + w') by RLVECT_1:def 6 .= u + (v + w) by A1,A10; end; thus for v being VECTOR of X holds v + 0.X = v proof let v be VECTOR of X; reconsider v'= v as VECTOR of (0).V; thus v + 0.X = v'+ 0.(0).V by A1,A3 .=v by RLVECT_1:10; end; thus for v being VECTOR of X ex w being VECTOR of X st v + w = 0.X proof let v be VECTOR of X; reconsider v'= v as VECTOR of (0).V; consider w' be VECTOR of (0).V such that A11: v' + w' = 0.(0).V by RLVECT_1:def 8; reconsider w = w' as VECTOR of X; take w; thus v + w = 0.X by A1,A3,A11; end; thus thesis; end; end; definition mode RealNormSpace is RealNormSpace-like RealLinearSpace-like Abelian add-associative right_zeroed right_complementable (non empty NORMSTR); end; reserve RNS for RealNormSpace; reserve x, y, z, g, g1, g2 for Point of RNS; canceled 4; theorem ||.0.RNS.|| = 0 by Def2; theorem Th6: ||.-x.|| = ||.x.|| proof A1: ||.-x.|| = ||.(-1) * x.|| by RLVECT_1:29 .= abs(-1 ) * ||.x.|| by Def2; abs(-1 ) = -(-1) by ABSVALUE:def 1 .= 1; hence thesis by A1; end; theorem Th7: ||.x - y.|| <= ||.x.|| + ||.y.|| proof x - y = x + (-y) by RLVECT_1:def 11; then ||.x - y.|| <= ||.x.|| + ||.(-y).|| by Def2; hence thesis by Th6; end; theorem Th8: 0 <= ||.x.|| proof ||.x - x.|| = ||.0'(RNS).|| by RLVECT_1:28 .= 0 by Def2; then 0 <= ||.x.|| + ||.x.|| by Th7; then 0 <= (||.x.|| + ||.x.||)/2 by SQUARE_1:27; hence thesis by XCMPLX_1:65; end; theorem ||.a * x + b * y.|| <= abs(a) * ||.x.|| + abs(b) * ||.y.|| proof ||.a * x + b * y.|| <= ||.a * x.|| + ||.b * y.|| by Def2; then ||.a * x + b * y.|| <= abs(a) * ||.x.|| + ||.b * y.|| by Def2; hence thesis by Def2; end; theorem Th10: ||.x - y.|| = 0 iff x = y proof ||.x - y.|| = 0 iff x - y = 0'(RNS) by Def2; hence thesis by RLVECT_1:28,35; end; theorem Th11: ||.x - y.|| = ||.y - x.|| proof x - y = (-y) + x by RLVECT_1:def 11 .= - (y - x) by RLVECT_1:47; hence thesis by Th6; end; theorem Th12: ||.x.|| - ||.y.|| <= ||.x - y.|| proof (x - y) + y = x - (y - y) by RLVECT_1:43 .= x - 0'(RNS) by RLVECT_1:28 .= x by RLVECT_1:26; then ||.x.|| <= ||.x - y.|| + ||.y.|| by Def2; hence thesis by REAL_1:86; end; theorem Th13: abs(||.x.|| - ||.y.||) <= ||.x - y.|| proof A1: ||.x.|| - ||.y.|| <= ||.x - y.|| by Th12; (y - x) + x = y - (x - x) by RLVECT_1:43 .= y - 0'(RNS) by RLVECT_1:28 .= y by RLVECT_1:26; then ||.y.|| <= ||.y - x.|| + ||.x.|| by Def2; then ||.y.|| - ||.x.|| <= ||.y - x.|| by REAL_1:86; then ||.y.|| - ||.x.|| <= ||.x - y.|| by Th11; then -(||.x.|| - ||.y.||) <= ||.x - y.|| by XCMPLX_1:143; then -||.x - y.|| <= -(-(||.x.|| - ||.y.||)) by REAL_1:50; hence thesis by A1,ABSVALUE:12; end; theorem Th14: ||.x - z.|| <= ||.x - y.|| + ||.y - z.|| proof x - z = x + (-z) by RLVECT_1:def 11 .= x + (0'(RNS) + (-z)) by RLVECT_1:10 .= x + (((-y) + y) + (-z)) by RLVECT_1:16 .= x + ((-y) + (y + (-z))) by RLVECT_1:def 6 .= (x + (-y)) + (y + (-z)) by RLVECT_1:def 6 .= (x - y) + (y + (-z)) by RLVECT_1:def 11 .= (x - y) + (y - z) by RLVECT_1:def 11; hence thesis by Def2; end; theorem x <> y implies ||.x - y.|| <> 0 by Th10; definition let RNS be 1-sorted; mode sequence of RNS means :Def3: it is Function of NAT, the carrier of RNS; existence proof consider f being Function of NAT, the carrier of RNS; take f; thus thesis; end; end; definition let RNS be 1-sorted; cluster -> Function-like Relation-like sequence of RNS; coherence by Def3; end; definition let RNS be non empty 1-sorted; redefine mode sequence of RNS -> Function of NAT, the carrier of RNS; coherence by Def3; end; reserve S, S1, S2 for sequence of RNS; reserve k, n, m, m1, m2 for Nat; reserve r for Real; reserve f for Function; reserve d, s, t for set; canceled; theorem Th17: for RNS being non empty 1-sorted, x being Element of RNS holds f is sequence of RNS iff ( dom f = NAT & for d st d in NAT holds f.d is Element of RNS ) proof let RNS be non empty 1-sorted; let x be Element of RNS; thus f is sequence of RNS implies ( dom f = NAT & for d st d in NAT holds f.d is Element of RNS ) proof assume A1: f is sequence of RNS; then A2: dom f = NAT by FUNCT_2:def 1; A3: rng f c= the carrier of RNS by A1,RELSET_1:12; for d st d in NAT holds f.d is Element of RNS proof let d; assume d in NAT; then f.d in rng f by A2,FUNCT_1:def 5; hence thesis by A3; end; hence thesis by A1,FUNCT_2:def 1; end; thus ( dom f = NAT & for d st d in NAT holds f.d is Element of RNS ) implies f is sequence of RNS proof assume that A4: dom f = NAT and A5: for d st d in NAT holds f.d is Element of RNS; for s st s in rng f holds s in the carrier of RNS proof let s; assume s in rng f; then consider d such that A6: d in dom f and A7: s = f.d by FUNCT_1:def 5; f.d is Element of RNS by A4,A5,A6; hence thesis by A7; end; then rng f c= the carrier of RNS by TARSKI:def 3; then f is Function of NAT, the carrier of RNS by A4,FUNCT_2:def 1, RELSET_1:11; hence thesis by Def3; end; end; canceled; theorem Th19: for RNS being non empty 1-sorted, x being Element of RNS ex S being sequence of RNS st rng S = {x} proof let RNS be non empty 1-sorted; let x be Element of RNS; consider f such that A1: dom f = NAT and A2: rng f = {x} by FUNCT_1:15; f is Function of NAT, the carrier of RNS by A1,A2,FUNCT_2:def 1,RELSET_1: 11 ; then reconsider f as sequence of RNS by Def3; take f; thus thesis by A2; end; theorem Th20: for RNS being non empty 1-sorted, S being sequence of RNS st (ex x being Element of RNS st for n holds S.n = x) ex x being Element of RNS st rng S={x} proof let RNS be non empty 1-sorted; let S be sequence of RNS; given z being Element of RNS such that A1: for n holds S.n = z; take x = z; now let t; assume t in rng S; then consider d such that A2: d in dom S and A3: S.d = t by FUNCT_1:def 5; d in NAT by A2,FUNCT_2:def 1; then t = z by A1,A3; hence t in {x} by TARSKI:def 1; end; then A4: rng S c= {x} by TARSKI:def 3; now let s; assume s in {x}; then A5: s = x by TARSKI:def 1; now assume A6: not s in rng S; now let n; n in NAT; then n in dom S by FUNCT_2:def 1; then S.n <> x by A5,A6,FUNCT_1:def 5; hence contradiction by A1; end; hence contradiction; end; hence s in rng S; end; then {x} c= rng S by TARSKI:def 3; hence thesis by A4,XBOOLE_0:def 10; end; theorem Th21: for RNS being non empty 1-sorted, S being sequence of RNS st ex x being Element of RNS st rng S = {x} holds for n holds S.n = S.(n+1) proof let RNS be non empty 1-sorted; let S be sequence of RNS; given z being Element of RNS such that A1: rng S = {z}; let n; n in NAT & (n+1) in NAT; then n in dom S & (n+1) in dom S by FUNCT_2:def 1; then S.n in {z} & S.(n+1) in {z} by A1,FUNCT_1:def 5; then S.n = z & S.(n+1) = z by TARSKI:def 1; hence thesis; end; theorem Th22: for RNS being non empty 1-sorted, S being sequence of RNS st for n holds S.n = S.(n+1) holds for n , k holds S.n = S.(n+k) proof let RNS be non empty 1-sorted; let S be sequence of RNS; assume A1: for n holds S.n = S.(n+1); let n; defpred P[Nat] means S.n = S.(n+$1); A2: P[0]; A3: now let k such that A4: P[k]; S.(n+k) = S.(n+k+1) by A1; hence P[k+1] by A4,XCMPLX_1:1; end; thus for k holds P[k] from Ind(A2,A3); end; theorem Th23: for RNS being non empty 1-sorted, S being sequence of RNS st for n , k holds S.n = S.(n+k) holds for n , m holds S.n = S.m proof let RNS be non empty 1-sorted; let S be sequence of RNS; assume A1: for n , k holds S.n = S.(n+k); let n , m; A2: now assume n <= m; then ex k st m = n + k by NAT_1:28; hence S.n = S.m by A1; end; now assume m <= n; then ex k st n = m + k by NAT_1:28; hence S.n = S.m by A1; end; hence thesis by A2; end; theorem Th24: for RNS being non empty 1-sorted, S being sequence of RNS st for n , m holds S.n = S.m ex x being Element of RNS st for n holds S.n = x proof let RNS be non empty 1-sorted; let S be sequence of RNS; assume that A1: for n , m holds S.n = S.m and A2: for x being Element of RNS ex n st S.n <> x; now let n; consider z being Element of RNS such that A3: S.n = z; ex k st S.k <> z by A2; hence contradiction by A1,A3; end; hence thesis; end; theorem ex S st rng S = {0.RNS} by Th19; definition let RNS be non empty 1-sorted; let S be sequence of RNS; redefine attr S is constant means :Def4: ex x being Element of RNS st for n holds S.n = x; compatibility proof A1: dom S = NAT by FUNCT_2:def 1; hereby assume A2: S is constant; consider n0 being Nat; take r = S.n0; let n be Nat; thus S.n = r by A1,A2,SEQM_3:def 5; end; given r being Element of RNS such that A3: for n holds S.n=r; let n1,n2 be set; assume A4: n1 in dom S & n2 in dom S; hence S.n1 = r by A1,A3 .= S.n2 by A1,A3,A4; end; end; canceled; theorem for RNS being non empty 1-sorted, S being sequence of RNS holds S is constant iff ex x being Element of RNS st rng S = {x} proof let RNS be non empty 1-sorted; let S be sequence of RNS; thus S is constant implies ex x being Element of RNS st rng S = {x} proof assume S is constant; then ex x being Element of RNS st for n holds S.n = x by Def4; hence thesis by Th20; end; assume ex x being Element of RNS st rng S = {x}; then ( for n holds S.n = S.(n+1) ) by Th21; then ( for n , k holds S.n = S.(n+k) ) by Th22; then ( for n , m holds S.n = S.m ) by Th23; then ( ex x being Element of RNS st for n holds S.n = x ) by Th24; hence thesis by Def4; end; Lm7: for RNS being non empty 1-sorted for S being sequence of RNS for n holds S.n is Element of RNS; definition let RNS be non empty 1-sorted; let S be sequence of RNS; let n; redefine func S.n -> Element of RNS; coherence by Lm7; end; scheme ExRNSSeq{RNS()->RealNormSpace, F(Nat) -> ( Point of RNS() ) } : ex S be sequence of RNS() st for n holds S.n = F(n) proof defpred P[set,set] means ex n st n =$1 & $2 = F(n); A1: for d st d in NAT ex s st P[d,s] proof let d; assume d in NAT; then consider n such that A2: n = d; reconsider z = F(n) as set; take z; thus thesis by A2; end; A3: for d , s , t st d in NAT & P[d,s] & P[d,t] holds s = t; consider f such that A4: dom f = NAT and A5: for d st d in NAT holds P[d,f.d] from FuncEx(A3,A1); for d st d in NAT holds f.d is Point of RNS() proof let d; assume d in NAT; then ex n st n = d & f.d = F(n) by A5; hence thesis; end; then reconsider f as sequence of RNS() by A4,Th17; take S = f; now let n; P[n,S.n] by A5; hence S.n = F(n); end; hence for n holds S.n = F(n); end; scheme ExRLSSeq{RNS()->RealLinearSpace, F(Nat) -> Element of RNS() } : ex S be sequence of RNS() st for n holds S.n = F(n) proof defpred P[set,set] means ex n st ( n =$1 & $2 = F(n)); A1: for d st d in NAT ex s st P[d,s] proof let d; assume d in NAT; then consider n such that A2: n = d; reconsider z = F(n) as set; take z; thus thesis by A2; end; A3: for d , s , t st d in NAT & P[d,s] & P[d,t] holds s = t; consider f such that A4: dom f = NAT and A5: for d st d in NAT holds P[d,f.d] from FuncEx(A3,A1); for d st d in NAT holds f.d is Element of RNS() proof let d; assume d in NAT; then ex n st n = d & f.d = F(n) by A5; hence thesis; end; then reconsider f as sequence of RNS() by A4,Th17; take S = f; now let n; P[n,S.n] by A5; hence S.n = F(n); end; hence for n holds S.n = F(n); end; definition let RNS be RealLinearSpace; let S1, S2 be sequence of RNS; func S1 + S2 -> sequence of RNS means :Def5: for n holds it.n = S1.n + S2.n; existence proof deffunc F(Nat) = S1.$1 + S2.$1; thus ex S being sequence of RNS st for n holds S.n = F(n) from ExRLSSeq; end; uniqueness proof let S, T be sequence of RNS; assume that A1: ( for n holds S.n = S1.n + S2.n ) and A2: ( for n holds T.n = S1.n + S2.n ); for n holds S.n = T.n proof let n; S.n = S1.n + S2.n by A1; hence thesis by A2; end; hence thesis by FUNCT_2:113; end; end; definition let RNS be RealLinearSpace; let S1 , S2 be sequence of RNS; func S1 - S2 -> sequence of RNS means :Def6: for n holds it.n = S1.n - S2.n; existence proof deffunc F(Nat) = S1.$1 - S2.$1; thus ex S being sequence of RNS st for n holds S.n = F(n) from ExRLSSeq; end; uniqueness proof let S , T be sequence of RNS; assume that A1: ( for n holds S.n = S1.n - S2.n ) and A2: ( for n holds T.n = S1.n - S2.n ); for n holds S.n = T.n proof let n; S.n = S1.n - S2.n by A1; hence thesis by A2; end; hence thesis by FUNCT_2:113; end; end; definition let RNS be RealLinearSpace; let S be sequence of RNS; let x be Element of RNS; func S - x -> sequence of RNS means :Def7: for n holds it.n = S.n - x; existence proof deffunc F(Nat) = S.$1 - x; thus ex S being sequence of RNS st for n holds S.n = F(n) from ExRLSSeq; end; uniqueness proof let S1 , S2 be sequence of RNS; assume that A1: ( for n holds S1.n = S.n - x ) and A2: ( for n holds S2.n = S.n - x ); for n holds S1.n = S2.n proof let n; S1.n = S.n - x by A1; hence thesis by A2; end; hence thesis by FUNCT_2:113; end; end; definition let RNS be RealLinearSpace; let S be sequence of RNS; let a; func a * S -> sequence of RNS means :Def8: for n holds it.n = a * S.n; existence proof deffunc F(Nat) = a * S.$1; thus ex S being sequence of RNS st for n holds S.n = F(n) from ExRLSSeq; end; uniqueness proof let S1 , S2 be sequence of RNS; assume that A1: ( for n holds S1.n = a * S.n ) and A2: ( for n holds S2.n = a * S.n ); for n holds S1.n = S2.n proof let n; S1.n = a * S.n by A1; hence thesis by A2; end; hence thesis by FUNCT_2:113; end; end; definition let RNS; let S; attr S is convergent means :Def9: ex g st for r st 0 < r ex m st for n st m <= n holds ||.(S.n) - g.|| < r; end; canceled 6; theorem Th34: S1 is convergent & S2 is convergent implies S1 + S2 is convergent proof assume that A1: S1 is convergent and A2: S2 is convergent; consider g1 such that A3: for r st 0 < r ex m st for n st m <= n holds ||.(S1.n) - g1.|| < r by A1,Def9; consider g2 such that A4: for r st 0 < r ex m st for n st m <= n holds ||.(S2.n) - g2.|| < r by A2,Def9; take g = g1 + g2; let r; assume 0 < r; then A5: 0< r/2 by SEQ_2:3; then consider m1 such that A6: for n st m1 <= n holds ||.(S1.n) - g1.|| < r/2 by A3; consider m2 such that A7: for n st m2 <= n holds ||.(S2.n) - g2.|| < r/2 by A4,A5; take k = m1 + m2; let n such that A8: k <= n; m1 <= m1 + m2 by NAT_1:37; then m1 <= n by A8,AXIOMS:22; then A9: ||.(S1.n) - g1.|| < r/2 by A6; m2 <= k by NAT_1:37; then m2 <= n by A8,AXIOMS:22; then ||.(S2.n) - g2.|| < r/2 by A7; then ||.(S1.n) - g1.|| + ||.(S2.n) - g2.|| < r/2 + r/2 by A9,REAL_1:67; then A10: ||.(S1.n) - g1.|| + ||.(S2.n) - g2.|| < r by XCMPLX_1:66; A11: ||.(S1 + S2).n - g.|| = ||.(S1.n) + (S2.n) - (g1+g2).|| by Def5 .= ||.(S1.n) + ((S2.n) - (g1+g2)).|| by RLVECT_1:42 .= ||.(S1.n) + (-(g1+g2) + (S2.n)).|| by RLVECT_1:def 11 .= ||.-(g1+g2) + ((S1.n) + (S2.n)).|| by RLVECT_1:def 6 .= ||.((-g1)+(-g2)) + ((S1.n) + (S2.n)).|| by RLVECT_1:45 .= ||.(S1.n) + ((-g1)+(-g2)) + (S2.n).|| by RLVECT_1:def 6 .= ||.((S1.n) + (-g1)) + (-g2) + (S2.n).|| by RLVECT_1:def 6 .= ||.(S1.n) - g1 + (-g2) + (S2.n).|| by RLVECT_1:def 11 .= ||.(S1.n)- g1 + ((S2.n) + -g2).|| by RLVECT_1:def 6 .= ||.(S1.n) - g1 + ((S2.n) - g2).|| by RLVECT_1:def 11; ||.(S1.n) - g1 + ((S2.n) - g2).|| <= ||.(S1.n) - g1.|| + ||.(S2.n) - g2.|| by Def2; hence ||.(S1 + S2).n - g.|| < r by A10,A11,AXIOMS:22; end; theorem Th35: S1 is convergent & S2 is convergent implies S1 - S2 is convergent proof assume that A1: S1 is convergent and A2: S2 is convergent; consider g1 such that A3: for r st 0 < r ex m st for n st m <= n holds ||.(S1.n) - g1.|| < r by A1,Def9; consider g2 such that A4: for r st 0 < r ex m st for n st m <= n holds ||.(S2.n) - g2.|| < r by A2,Def9; take g = g1 - g2; let r; assume 0 < r; then A5: 0< r/2 by SEQ_2:3; then consider m1 such that A6: for n st m1 <= n holds ||.(S1.n) - g1.|| < r/2 by A3; consider m2 such that A7: for n st m2 <= n holds ||.(S2.n) - g2.|| < r/2 by A4,A5; take k = m1 + m2; let n such that A8: k <= n; m1 <= m1 + m2 by NAT_1:37; then m1 <= n by A8,AXIOMS:22; then A9: ||.(S1.n) - g1.|| < r/2 by A6; m2 <= k by NAT_1:37; then m2 <= n by A8,AXIOMS:22; then ||.(S2.n) - g2.|| < r/2 by A7; then ||.(S1.n) - g1.|| + ||.(S2.n) - g2.|| < r/2 + r/2 by A9,REAL_1:67; then A10: ||.(S1.n) - g1.|| + ||.(S2.n) - g2.|| < r by XCMPLX_1:66; A11: ||.(S1 - S2).n - g.|| = ||.((S1.n) - (S2.n)) - (g1 - g2).|| by Def6 .= ||.(((S1.n) - (S2.n)) - g1) + g2.|| by RLVECT_1:43 .= ||. ((S1.n) - (g1 + (S2.n))) + g2.|| by RLVECT_1:41 .= ||. (((S1.n) - g1 ) - (S2.n)) + g2.|| by RLVECT_1:41 .= ||. ((S1.n) - g1 ) - ((S2.n) - g2).|| by RLVECT_1:43; ||.((S1.n) - g1) - ((S2.n) - g2).|| <= ||.(S1.n) - g1.|| + ||.(S2.n) - g2.|| by Th7; hence ||.(S1 - S2).n - g.|| < r by A10,A11,AXIOMS:22; end; theorem Th36: S is convergent implies S - x is convergent proof assume S is convergent; then consider g such that A1: for r st 0 < r ex m st for n st m <= n holds ||.(S.n) - g.|| < r by Def9; take h = g - x; let r; assume 0 < r; then consider m1 such that A2: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1; take k = m1; let n; assume k <= n; then A3: ||.(S.n) - g.|| < r by A2; ||.(S.n) - g.|| = ||.((S.n) - 0'(RNS)) - g.|| by RLVECT_1:26 .= ||.((S.n) - (x - x)) - g.|| by RLVECT_1:28 .= ||.(((S.n) - x) + x) - g.|| by RLVECT_1:43 .= ||.((S.n) - x) + (x - g).|| by RLVECT_1:42 .= ||.((S.n) - x) + ((-g) + x).|| by RLVECT_1:def 11 .= ||.((S.n) - x) + -h.|| by RLVECT_1:47 .= ||.((S.n) - x) - h.|| by RLVECT_1:def 11; hence ||.(S - x).n - h.|| < r by A3,Def7; end; theorem Th37: S is convergent implies a * S is convergent proof assume S is convergent; then consider g such that A1: for r st 0 < r ex m st for n st m <= n holds ||.(S.n) - g.|| < r by Def9; take h = a * g; A2: now assume A3: a = 0; let r; assume 0 < r; then consider m1 such that A4: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1; take k = m1; let n; assume k <= n; then A5: ||.(S.n) - g.|| < r by A4; ||.a * (S.n) - a * g.|| = ||.0 * (S.n) - 0'(RNS).|| by A3,RLVECT_1:23 .= ||.0'(RNS) - 0'(RNS).|| by RLVECT_1:23 .= ||.0'(RNS).|| by RLVECT_1:26 .= 0 by Def2; then ||.a * (S.n) - h.|| < r by A5,Th8; hence ||.(a * S).n - h.|| < r by Def8; end; now assume A6: a <> 0; then A7: 0 < abs(a) by ABSVALUE:6; let r such that A8: 0 < r; A9: 0 <> abs(a) by A6,ABSVALUE:6; 0/abs(a) =0; then 0 < r/abs(a) by A7,A8,REAL_1:73; then consider m1 such that A10: for n st m1 <= n holds ||.(S.n) - g.|| < r/abs(a) by A1; take k = m1; let n; assume k <= n; then A11: ||.(S.n) - g.|| < r/abs(a) by A10; A12: ||.(a * (S.n)) - (a * g).|| = ||.a * ((S.n) - g).|| by RLVECT_1:48 .= abs(a) * ||.(S.n) - g.|| by Def2; abs(a) * (r/abs(a)) = abs(a) * (abs(a)" * r) by XCMPLX_0:def 9 .= abs(a) *abs(a)" * r by XCMPLX_1:4 .= 1 * r by A9,XCMPLX_0:def 7 .= r; then ||.(a *(S.n)) - h.|| < r by A7,A11,A12,REAL_1:70; hence ||.(a * S).n - h.|| < r by Def8; end; hence thesis by A2; end; definition let RNS; let S; func ||.S.|| -> Real_Sequence means :Def10: for n holds it.n =||.(S.n).||; existence proof deffunc F(Nat) = ||.(S.$1).||; consider s being Real_Sequence such that A1: for n holds s.n = F(n) from ExRealSeq; take s; thus thesis by A1; end; uniqueness proof let s,t be Real_Sequence; assume A2: ( for n holds s.n = ||.(S.n).|| ) & ( for n holds t.n = ||.(S.n).|| ); now let n; s.n = ||.(S.n).|| by A2; hence s.n = t.n by A2; end; hence thesis by FUNCT_2:113; end; end; canceled; theorem Th39: S is convergent implies ||.S.|| is convergent proof assume S is convergent; then consider g such that A1: for r st 0 < r ex m st for n st m <= n holds ||.(S.n) - g.|| < r by Def9; now let r be real number; A2: r is Real by XREAL_0:def 1; assume 0 < r; then consider m1 such that A3: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1,A2; take k = m1; now let n; assume k <= n; then A4: ||.(S.n) - g.|| < r by A3; abs(||.(S.n).|| - ||.g.||) <= ||.(S.n) - g.|| by Th13; then abs(||.(S.n).|| - ||.g.||) < r by A4,AXIOMS:22; hence abs(||.S.||.n - ||.g.||) < r by Def10; end; hence for n st k <= n holds abs(||.S.||.n - ||.g.||) < r; end; hence thesis by SEQ_2:def 6; end; definition let RNS; let S; assume A1: S is convergent; func lim S -> Point of RNS means :Def11: for r st 0 < r ex m st for n st m <= n holds ||.(S.n) - it.|| < r; existence by A1,Def9; uniqueness proof for x , y st ( for r st 0 < r ex m st for n st m <= n holds ||.(S.n) - x.|| < r ) & ( for r st 0 < r ex m st for n st m <= n holds ||.(S.n) - y.|| < r ) holds x = y proof given x , y such that A2: for r st 0 < r ex m st for n st m <= n holds ||.(S.n) - x.|| < r and A3: for r st 0 < r ex m st for n st m <= n holds ||.(S.n) - y.|| < r and A4: x <> y; A5: ||.x - y.|| <> 0 by A4,Th10; A6: 0 <= ||.x - y.|| by Th8; then A7: 0/4 < ||.x - y.||/4 by A5,REAL_1:73; then consider m1 such that A8: for n st m1 <= n holds ||.(S.n) - x.|| < ||.x - y.||/4 by A2; consider m2 such that A9: for n st m2 <= n holds ||.(S.n) - y.|| < ||.x - y.||/4 by A3,A7; A10: now assume m1 <= m2; then A11: ||.(S.m2) - x.|| < ||.x - y.||/4 by A8; ||.(S.m2) - y.|| < ||.x - y.||/4 by A9; then ||.(S.m2) - x.|| + ||.(S.m2) - y.|| < ||.x - y.||/4 + ||.x - y.||/4 by A11,REAL_1:67; then A12: ||.(S.m2) - x.|| + ||.(S.m2) - y.|| <= ||.x - y.||/2 by XCMPLX_1:72; ||.x - y.|| <= ||.x - (S.m2).|| + ||.(S.m2) - y.|| by Th14; then ||.x - y.|| <= ||.(S.m2) - x.|| + ||.(S.m2) - y.|| by Th11; then not ||.x - y.||/2 < ||.x - y.|| by A12,AXIOMS:22; hence contradiction by A5,A6,SEQ_2:4; end; now assume m2 <= m1; then A13: ||.(S.m1) - y.|| < ||.x - y.||/4 by A9; ||.(S.m1) - x.|| < ||.x - y.||/4 by A8; then ||.(S.m1) - x.|| + ||.(S.m1) - y.|| < ||.x - y.||/4 + ||.x - y.||/4 by A13,REAL_1:67; then A14: ||.(S.m1) - x.|| + ||.(S.m1) - y.|| <= ||.x - y.||/2 by XCMPLX_1:72; ||.x - y.|| <= ||.x - (S.m1).|| + ||.(S.m1) - y.|| by Th14; then ||.x - y.|| <= ||.(S.m1) - x.|| + ||.(S.m1) - y.|| by Th11; then not ||.x - y.||/2 < ||.x - y.|| by A14,AXIOMS:22; hence contradiction by A5,A6,SEQ_2:4; end; hence contradiction by A10; end; hence thesis; end; end; canceled; theorem S is convergent & lim S = g implies ( ||.S - g.|| is convergent & lim ||.S - g.|| = 0 ) proof assume that A1: S is convergent and A2: lim S = g; S - g is convergent by A1,Th36; then A3: ||.S - g.|| is convergent by Th39; now let r be real number; A4: r is Real by XREAL_0:def 1; assume 0 < r; then consider m1 such that A5: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1,A2,A4,Def11; take k = m1; let n; assume k <= n; then ||.(S.n) - g.|| < r by A5; then A6: ||.((S.n) - g ) - 0'(RNS).|| < r by RLVECT_1:26; abs(||.(S.n) - g.|| - ||.0'(RNS).||) <= ||.((S.n) - g ) - 0'(RNS) .|| by Th13; then abs(||.(S.n) - g.|| - ||.0'(RNS).||) < r by A6,AXIOMS:22; then abs((||.(S.n) - g.|| - 0)) < r by Def2; then abs((||.(S - g).n.|| - 0)) < r by Def7; hence abs((||.S - g.||.n - 0)) < r by Def10; end; hence thesis by A3,SEQ_2:def 7; end; theorem S1 is convergent & S2 is convergent implies lim (S1 + S2) = (lim S1) + (lim S2) proof assume that A1: S1 is convergent and A2: S2 is convergent; A3: S1 + S2 is convergent by A1,A2,Th34; set g1 = lim S1; set g2 = lim S2; set g = g1 + g2; now let r; assume 0 < r; then A4: 0< r/2 by SEQ_2:3; then consider m1 such that A5: for n st m1 <= n holds ||.(S1.n) - g1.|| < r/2 by A1,Def11; consider m2 such that A6: for n st m2 <= n holds ||.(S2.n) - g2.|| < r/2 by A2,A4,Def11; take k = m1 + m2; let n such that A7: k <= n; m1 <= m1 + m2 by NAT_1:37; then m1 <= n by A7,AXIOMS:22; then A8: ||.(S1.n) - g1.|| < r/2 by A5; m2 <= k by NAT_1:37; then m2 <= n by A7,AXIOMS:22; then ||.(S2.n) - g2.|| < r/2 by A6; then ||.(S1.n) - g1.|| + ||.(S2.n) - g2.|| < r/2 + r/2 by A8,REAL_1:67; then A9: ||.(S1.n) - g1.|| + ||.(S2.n) - g2.|| < r by XCMPLX_1:66; A10: ||.(S1 + S2).n - g.|| = ||.(S1.n) + (S2.n) - (g1+g2).|| by Def5 .= ||.(S1.n) + ((S2.n) - (g1+g2)).|| by RLVECT_1:42 .= ||.(S1.n) + (-(g1+g2) + (S2.n)).|| by RLVECT_1:def 11 .= ||.-(g1+g2) + ((S1.n) + (S2.n)).|| by RLVECT_1:def 6 .= ||.((-g1)+(-g2)) + ((S1.n) + (S2.n)).|| by RLVECT_1:45 .= ||.(S1.n) + ((-g1)+(-g2)) + (S2.n).|| by RLVECT_1:def 6 .= ||.((S1.n) + (-g1)) + (-g2) + (S2.n).|| by RLVECT_1:def 6 .= ||.(S1.n) - g1 + (-g2) + (S2.n).|| by RLVECT_1:def 11 .= ||.(S1.n)- g1 + ((S2.n) + -g2).|| by RLVECT_1:def 6 .= ||.(S1.n) - g1 + ((S2.n) - g2).|| by RLVECT_1:def 11; ||.(S1.n) - g1 + ((S2.n) - g2).|| <= ||.(S1.n) - g1.|| + ||.(S2.n) - g2.|| by Def2; hence ||.(S1 + S2).n - g.|| < r by A9,A10,AXIOMS:22; end; hence thesis by A3,Def11; end; theorem S1 is convergent & S2 is convergent implies lim (S1 - S2) = (lim S1) - (lim S2) proof assume that A1: S1 is convergent and A2: S2 is convergent; A3: S1 - S2 is convergent by A1,A2,Th35; set g1 = lim S1; set g2 = lim S2; set g = g1 - g2; now let r; assume 0 < r; then A4: 0< r/2 by SEQ_2:3; then consider m1 such that A5: for n st m1 <= n holds ||.(S1.n) - g1.|| < r/2 by A1,Def11; consider m2 such that A6: for n st m2 <= n holds ||.(S2.n) - g2.|| < r/2 by A2,A4,Def11; take k = m1 + m2; let n such that A7: k <= n; m1 <= m1 + m2 by NAT_1:37; then m1 <= n by A7,AXIOMS:22; then A8: ||.(S1.n) - g1.|| < r/2 by A5; m2 <= k by NAT_1:37; then m2 <= n by A7,AXIOMS:22; then ||.(S2.n) - g2.|| < r/2 by A6; then ||.(S1.n) - g1.|| + ||.(S2.n) - g2.|| < r/2 + r/2 by A8,REAL_1:67; then A9: ||.(S1.n) - g1.|| + ||.(S2.n) - g2.|| < r by XCMPLX_1:66; A10: ||.(S1 - S2).n - g.|| = ||.((S1.n) - (S2.n)) - (g1 - g2).|| by Def6 .= ||.(((S1.n) - (S2.n)) - g1) + g2.|| by RLVECT_1:43 .= ||. ((S1.n) - (g1 + (S2.n))) + g2.|| by RLVECT_1:41 .= ||. (((S1.n) - g1 ) - (S2.n)) + g2.|| by RLVECT_1:41 .= ||. ((S1.n) - g1 ) - ((S2.n) - g2).|| by RLVECT_1:43; ||.((S1.n) - g1) - ((S2.n) - g2).|| <= ||.(S1.n) - g1.|| + ||.(S2.n) - g2.|| by Th7; hence ||.(S1 - S2).n - g.|| < r by A9,A10,AXIOMS:22; end; hence thesis by A3,Def11; end; theorem S is convergent implies lim (S - x) = (lim S) - x proof assume A1: S is convergent; then A2: S - x is convergent by Th36; set g = lim S; set h = g - x; now let r; assume 0 < r; then consider m1 such that A3: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1,Def11; take k = m1; let n; assume k <= n; then A4: ||.(S.n) - g.|| < r by A3; ||.(S.n) - g.|| = ||.((S.n) - 0'(RNS)) - g.|| by RLVECT_1:26 .= ||.((S.n) - (x - x)) - g.|| by RLVECT_1:28 .= ||.(((S.n) - x) + x) - g.|| by RLVECT_1:43 .= ||.((S.n) - x) + (x - g).|| by RLVECT_1:42 .= ||.((S.n) - x) + ((-g) + x).|| by RLVECT_1:def 11 .= ||.((S.n) - x) + -h.|| by RLVECT_1:47 .= ||.((S.n) - x) - h.|| by RLVECT_1:def 11; hence ||.(S - x).n - h.|| < r by A4,Def7; end; hence thesis by A2,Def11; end; theorem S is convergent implies lim (a * S) = a * (lim S) proof assume A1: S is convergent; then A2: a * S is convergent by Th37; set g = lim S; set h = a * g; A3: now assume A4: a = 0; let r; assume 0 < r; then consider m1 such that A5: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1,Def11; take k = m1; let n; assume k <= n; then A6: ||.(S.n) - g.|| < r by A5; ||.a * (S.n) - a * g.|| = ||.0 * (S.n) - 0'(RNS).|| by A4,RLVECT_1:23 .= ||.0'(RNS) - 0'(RNS).|| by RLVECT_1:23 .= ||.0'(RNS).|| by RLVECT_1:26 .= 0 by Def2; then ||.a * (S.n) - h.|| < r by A6,Th8; hence ||.(a * S).n - h.|| < r by Def8; end; now assume A7: a <> 0; then A8: 0 < abs(a) by ABSVALUE:6; let r such that A9: 0 < r; A10: 0 <> abs(a) by A7,ABSVALUE:6; 0/abs(a) =0; then 0 < r/abs(a) by A8,A9,REAL_1:73; then consider m1 such that A11: for n st m1 <= n holds ||.(S.n) - g.|| < r/abs(a) by A1,Def11; take k = m1; let n; assume k <= n; then A12: ||.(S.n) - g.|| < r/abs(a) by A11; A13: ||.(a * (S.n)) - (a * g).|| = ||.a * ((S.n) - g).|| by RLVECT_1:48 .= abs(a) * ||.(S.n) - g.|| by Def2; abs(a) * (r/abs(a)) = abs(a) * (abs(a)" * r) by XCMPLX_0:def 9 .= abs(a) *abs(a)" * r by XCMPLX_1:4 .= 1 * r by A10,XCMPLX_0:def 7 .= r; then ||.(a *(S.n)) - h.|| < r by A8,A12,A13,REAL_1:70; hence ||.(a * S).n - h.|| < r by Def8; end; hence thesis by A2,A3,Def11; end;

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