Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

### The abstract of the Mizar article:

### Lines in $n$-Dimensional Euclidean Spaces

**by****Akihiro Kubo**- Received August 8, 2003
- MML identifier: EUCLID_4

- [ Mizar article, MML identifier index ]

environ vocabulary EUCLID, FINSEQ_1, FUNCT_1, PRE_TOPC, ARYTM_1, ARYTM_3, COMPLEX1, ABSVALUE, ARYTM, INCSP_1, FINSEQ_2, AFF_1, EUCLID_2, BHSP_1, SQUARE_1, SEQ_4, SEQ_2, EUCLID_4; notation TARSKI, XBOOLE_0, FINSEQ_2, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, NAT_1, FUNCT_2, ABSVALUE, STRUCT_0, PRE_TOPC, FINSEQ_1, RVSUM_1, EUCLID_2, SQUARE_1, EUCLID, SEQ_4; constructors REAL_1, ABSVALUE, FINSEQOP, EUCLID_2, SEQ_4, SQUARE_1; clusters RELSET_1, STRUCT_0, EUCLID, XREAL_0, NAT_1, MEMBERED; requirements NUMERALS, BOOLE, SUBSET, ARITHM; begin reserve a,b,r,s,t,u,v,lambda for Real, i,j,n for Nat; reserve x,x1,x2,x3,y,y1,y2 for Element of REAL n; theorem :: EUCLID_4:1 :: EUCLID:31 0 * x + x = x & x + 0*n = x; theorem :: EUCLID_4:2 :: EUCLID:32 a*(0*n) = 0*n; theorem :: EUCLID_4:3 :: EUCLID:33 1*x = x & 0 * x = 0*n; theorem :: EUCLID_4:4 :: EUCLID:34 (a*b)*x = a*(b*x); theorem :: EUCLID_4:5 :: EUCLID:35 a*x = 0*n implies a = 0 or x = 0*n; theorem :: EUCLID_4:6 :: EUCLID:36 a*(x1 + x2) = a*x1 + a*x2; theorem :: EUCLID_4:7 :: EUCLID:37 (a + b)*x = a*x + b*x; theorem :: EUCLID_4:8 :: EUCLID:38 a*x1 = a*x2 implies a = 0 or x1 = x2; definition let n; let x1,x2 be Element of REAL n; func Line(x1,x2) -> Subset of REAL n equals :: EUCLID_4:def 1 { (1-lambda)*x1 + lambda*x2 : not contradiction }; end; definition let n; let x1,x2 be Element of REAL n; cluster Line(x1,x2) -> non empty; end; theorem :: EUCLID_4:9 :: AFF_1:25 Line(x1,x2)=Line(x2,x1); definition let n;let x1,x2 be Element of REAL n; redefine func Line(x1,x2); commutativity; end; theorem :: EUCLID_4:10 :: AFF_1:26 x1 in Line(x1,x2) & x2 in Line(x1,x2); theorem :: EUCLID_4:11 :: AFF_1:27 y1 in Line(x1,x2) & y2 in Line(x1,x2) implies Line(y1,y2) c= Line(x1,x2); theorem :: EUCLID_4:12 :: AFF_1:28 y1 in Line(x1,x2) & y2 in Line(x1,x2) & y1<>y2 implies Line(x1,x2) c= Line(y1,y2); definition let n; let A be Subset of REAL n; attr A is being_line means :: EUCLID_4:def 2 :: AFF_1:def 3 ex x1,x2 st x1<>x2 & A=Line(x1,x2); synonym A is_line; end; theorem :: EUCLID_4:13 :: AFF_1:30 for A,C be Subset of REAL n,x1,x2 holds A is_line & C is_line & x1 in A & x2 in A & x1 in C & x2 in C implies x1=x2 or A=C; theorem :: EUCLID_4:14 ::AFF_1:31 for A be Subset of REAL n st A is_line holds ex x1,x2 st x1 in A & x2 in A & x1<>x2; theorem :: EUCLID_4:15 :: AFF_1:32 for A be Subset of REAL n st A is_line holds ex x2 st x1<>x2 & x2 in A; definition let n;let x be Element of REAL n; func Rn2Fin (x) -> FinSequence of REAL equals :: EUCLID_4:def 3 x; end; definition let n;let x be Element of REAL n; func |. x .| -> Real equals :: EUCLID_4:def 4 |. Rn2Fin(x) .|; end; definition let n;let x1,x2 be Element of REAL n; func |( x1,x2 )| -> Real equals :: EUCLID_4:def 5 |( Rn2Fin(x1),Rn2Fin(x2) )|; commutativity; end; theorem :: EUCLID_4:16 :: EUCLID_2:10 for x1,x2 being Element of REAL n holds |(x1,x2)|=1/4*((|.(x1+x2).|)^2-(|.(x1-x2).|)^2); theorem :: EUCLID_4:17 :: EUCLID_2:11 for x being Element of REAL n holds |(x,x)| >= 0; theorem :: EUCLID_4:18 :: EUCLID_2:12 for x being Element of REAL n holds (|.x.|)^2=|(x,x)|; theorem :: EUCLID_4:19 :: EUCLID_2:14 for x being Element of REAL n holds 0 <= |.x.|; theorem :: EUCLID_4:20 :: EUCLID_2:13 for x being Element of REAL n holds |.x.| = sqrt |(x,x)|; theorem :: EUCLID_4:21 :: EUCLID_2:16 for x being Element of REAL n holds |(x,x)| = 0 iff |.x.| = 0; theorem :: EUCLID_4:22 :: EUCLID_2:15 for x being Element of REAL n holds |(x,x)|=0 iff x=0*n; theorem :: EUCLID_4:23 :: EUCLID_2:17 for x being Element of REAL n holds |(x, 0*n)| = 0; theorem :: EUCLID_4:24 :: EUCLID_2:18 for x being Element of REAL n holds |(0*n,x)| = 0; theorem :: EUCLID_4:25 :: EUCLID_2:19 for x1,x2,x3 being Element of REAL n holds |((x1+x2),x3)| = |(x1,x3)| + |(x2,x3)|; theorem :: EUCLID_4:26 ::EUCLID_2:20 for x1,x2 being Element of REAL n,a being real number holds |(a*x1,x2)| = a*|(x1,x2)|; theorem :: EUCLID_4:27 :: EUCLID_2:21 for x1,x2 being Element of REAL n,a being real number holds |(x1,a*x2)| = a*|(x1,x2)|; theorem :: EUCLID_4:28 :: EUCLID_2:22 for x1,x2 being Element of REAL n holds |(-x1, x2)| = - |(x1, x2)|; theorem :: EUCLID_4:29 :: EUCLID_2:23 for x1,x2 being Element of REAL n holds |(x1, -x2)| = - |(x1, x2)|; theorem :: EUCLID_4:30 :: EUCLID_2:24 for x1,x2 being Element of REAL n holds |(-x1, -x2)| = |(x1, x2)|; theorem :: EUCLID_4:31 :: EUCLID_2:25 for x1,x2,x3 being Element of REAL n holds |(x1-x2, x3)| = |(x1, x3)| - |(x2, x3)|; theorem :: EUCLID_4:32 :: EUCLID_2:26 for a,b being real number,x1,x2,x3 being Element of REAL n holds |( (a*x1+b*x2), x3 )| = a*|(x1,x3)| + b*|(x2,x3)|; theorem :: EUCLID_4:33 :: EUCLID_2:27 for x1,y1,y2 being Element of REAL n holds |(x1, y1+y2)| = |(x1, y1)| + |(x1, y2)|; theorem :: EUCLID_4:34 :: EUCLID_2:28 for x1,y1,y2 being Element of REAL n holds |(x1, y1-y2)| = |(x1, y1)| - |(x1, y2)|; theorem :: EUCLID_4:35 :: EUCLID_2:29 for x1,x2,y1,y2 being Element of REAL n holds |(x1+x2, y1+y2)| = |(x1, y1)| + |(x1, y2)| + |(x2, y1)| + |(x2, y2)|; theorem :: EUCLID_4:36 :: EUCLID_2:30 for x1,x2,y1,y2 being Element of REAL n holds |(x1-x2, y1-y2)| = |(x1, y1)| - |(x1, y2)| - |(x2, y1)| + |(x2, y2)|; theorem :: EUCLID_4:37 :: EUCLID_2:31 for x,y being Element of REAL n holds |(x+y, x+y)| = |(x, x)| + 2*|(x, y)| + |(y, y)|; theorem :: EUCLID_4:38 :: EUCLID_2:32 for x,y being Element of REAL n holds |(x-y, x-y)| = |(x, x)| - 2*|(x, y)| + |(y, y)|; theorem :: EUCLID_4:39 :: EUCLID_2:33 for x,y being Element of REAL n holds |.x+y.|^2 = |.x.|^2 + 2*|(x, y)| + |.y.|^2; theorem :: EUCLID_4:40 :: EUCLID_2:34 for x,y being Element of REAL n holds |.x-y.|^2 = |.x.|^2 - 2*|(x, y)| + |.y.|^2; theorem :: EUCLID_4:41 :: EUCLID_2:35 for x,y being Element of REAL n holds |.x+y.|^2 + |.x-y.|^2 = 2*(|.x.|^2 + |.y.|^2); theorem :: EUCLID_4:42 :: EUCLID_2:36 for x,y being Element of REAL n holds |.x+y.|^2 - |.x-y.|^2 = 4* |(x,y)|; theorem :: EUCLID_4:43 :: EUCLID_2:37 :: Schwartz for x,y being Element of REAL n holds abs |(x,y)| <= |.x.|*|.y.|; theorem :: EUCLID_4:44 :: EUCLID_2:38 :: Triangle for x,y being Element of REAL n holds |.x+y.| <= |.x.| + |.y.|; definition let n;let x1, x2 be Element of REAL n; pred x1,x2 are_orthogonal means :: EUCLID_4:def 6 :: EUCLID_2:def 3 |(x1,x2)| = 0; symmetry; end; theorem :: EUCLID_4:45 for R being Subset of REAL,x1,x2,y1 being Element of REAL n st R={|.y1-x.| where x is Element of REAL n: x in Line(x1,x2)} holds ex y2 being Element of REAL n st y2 in Line(x1,x2) & |.y1-y2.|=lower_bound R & x1-x2,y1-y2 are_orthogonal; definition let n; let p1,p2 be Point of TOP-REAL n; func Line(p1,p2) -> Subset of TOP-REAL n equals :: EUCLID_4:def 7 { (1-lambda)*p1 + lambda*p2 : not contradiction }; end; definition let n; let p1,p2 be Point of TOP-REAL n; cluster Line(p1,p2) -> non empty; end; reserve p,p1,p2,q1,q2 for Point of TOP-REAL n; theorem :: EUCLID_4:46 :: AFF_1:25 Line(p1,p2)=Line(p2,p1); definition let n;let p1,p2 be Point of TOP-REAL n; redefine func Line(p1,p2); commutativity; end; theorem :: EUCLID_4:47 :: AFF_1:26 p1 in Line(p1,p2) & p2 in Line(p1,p2); theorem :: EUCLID_4:48 :: AFF_1:27 q1 in Line(p1,p2) & q2 in Line(p1,p2) implies Line(q1,q2) c= Line(p1,p2); theorem :: EUCLID_4:49 :: AFF_1:28 q1 in Line(p1,p2) & q2 in Line(p1,p2) & q1<>q2 implies Line(p1,p2) c= Line(q1,q2); definition let n; let A be Subset of TOP-REAL n; attr A is being_line means :: EUCLID_4:def 8 :: AFF_1:def 3 ex p1,p2 st p1<>p2 & A=Line(p1,p2); synonym A is_line; end; theorem :: EUCLID_4:50 :: AFF_1:30 for A,C being Subset of TOP-REAL n holds A is_line & C is_line & p1 in A & p2 in A & p1 in C & p2 in C implies p1=p2 or A=C; theorem :: EUCLID_4:51 :: AFF_1:31 for A being Subset of TOP-REAL n st A is_line holds ex p1,p2 st p1 in A & p2 in A & p1<>p2; theorem :: EUCLID_4:52 :: AFF_1:32 for A being Subset of TOP-REAL n st A is_line holds ex p2 st p1<>p2 & p2 in A; definition let n;let p be Point of TOP-REAL n; func TPn2Rn (p) -> Element of REAL n equals :: EUCLID_4:def 9 p; end; definition let n;let p be Point of TOP-REAL n; func |. p .| -> Real equals :: EUCLID_4:def 10 |. TPn2Rn(p) .|; end; definition let n;let p1,p2 be Point of TOP-REAL n; func |( p1,p2 )| -> Real equals :: EUCLID_4:def 11 |( TPn2Rn(p1),TPn2Rn(p2) )|; commutativity; end; definition let n;let p1, p2 be Point of TOP-REAL n; pred p1,p2 are_orthogonal means :: EUCLID_4:def 12 :: EUCLID_2:def 3 |(p1,p2)| = 0; symmetry; end; theorem :: EUCLID_4:53 for R being Subset of REAL,p1,p2,q1 being Point of TOP-REAL n st R={|. q1-p .| where p is Point of TOP-REAL n: p in Line(p1,p2)} holds ex q2 being Point of TOP-REAL n st q2 in Line(p1,p2) & |. q1-q2 .| =lower_bound R & p1-p2,q1-q2 are_orthogonal;

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