:: On the Geometry of a Go-board
:: by Andrzej Trybulec
::
:: Received July 9, 1995
:: Copyright (c) 1995-2026 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, SUBSET_1, REAL_1, PRE_TOPC, EUCLID, GOBOARD1, ARYTM_3,
MCART_1, ARYTM_1, RELAT_1, XBOOLE_0, METRIC_1, XXREAL_0, CARD_1,
PCOMPS_1, RCOMP_1, STRUCT_0, CONNSP_2, TOPS_1, TARSKI, SQUARE_1, FUNCT_1,
FINSEQ_1, TREES_1, GOBOARD5, COMPLEX1, RLTOPSP1, SUPINF_2, NAT_1;
notations TARSKI, XBOOLE_0, ORDINAL1, SUBSET_1, NUMBERS, XXREAL_0, XREAL_0,
XCMPLX_0, COMPLEX1, REAL_1, NAT_1, SQUARE_1, NAT_D, BINOP_1, FINSEQ_1,
MATRIX_0, STRUCT_0, METRIC_1, PRE_TOPC, TOPS_1, CONNSP_2, PCOMPS_1,
RLVECT_1, RLTOPSP1, EUCLID, GOBOARD1, GOBOARD5;
constructors REAL_1, SQUARE_1, NAT_1, COMPLEX1, NAT_D, TOPS_1, CONNSP_2,
PCOMPS_1, GOBOARD1, GOBOARD5, FUNCSDOM, BINOP_2;
registrations RELSET_1, XXREAL_0, XREAL_0, STRUCT_0, METRIC_1, PCOMPS_1,
EUCLID, TOPS_1, PRE_TOPC, SQUARE_1, ORDINAL1;
requirements REAL, NUMERALS, SUBSET, ARITHM;
definitions TARSKI, XBOOLE_0;
equalities EUCLID, SQUARE_1, XCMPLX_0, RLTOPSP1, RLVECT_1;
expansions TARSKI;
theorems GOBOARD5, TOPS_1, SPPOL_2, TOPMETR, METRIC_1, PCOMPS_1, PRE_TOPC,
EUCLID, TOPREAL3, SQUARE_1, ABSVALUE, TARSKI, TOPREAL1, NAT_1, XBOOLE_0,
XBOOLE_1, XCMPLX_1, COMPLEX1, XREAL_1, CONNSP_2, XXREAL_0, RLTOPSP1,
MATRIX_0, RLVECT_1;
begin
reserve n for Nat,
i,j for Nat,
r,s,r1,s1,r2,s2,r9,s9 for Real,
p,q for Point of TOP-REAL 2,
G for Go-board,
x,y for set,
v for Point of Euclid 2;
Lm1: (p+q)`1 = p`1+q`1 & (p+q)`2 = p`2+q`2
proof
p + q = |[ p`1 + q`1, p`2 + q`2]| by EUCLID:55;
hence thesis by EUCLID:52;
end;
Lm2: (p-q)`1 = p`1-q`1 & (p-q)`2 = p`2-q`2
proof
p - q = |[ p`1 - q`1, p`2 - q`2]| by EUCLID:61;
hence thesis by EUCLID:52;
end;
Lm3: (r*p)`1 = r*(p`1) & (r*p)`2 = r*(p`2)
proof
r*p = |[ r*p`1 ,r*p`2 ]| by EUCLID:57;
hence thesis by EUCLID:52;
end;
theorem Th1:
for M being non empty Reflexive MetrStruct, u being Point of M, r
being Real holds r > 0 implies u in Ball(u,r)
proof
let M be non empty Reflexive MetrStruct, u be Point of M, r be Real;
A1: Ball(u,r) = {q where q is Point of M:dist(u,q)<r} & dist(u,u) = 0 by
METRIC_1:1,17;
assume r > 0;
hence thesis by A1;
end;
Lm4: for M being MetrSpace, B being Subset of TopSpaceMetr(M),r being Real
, u being Point of M st B = Ball(u,r) holds B is open
proof
let M be MetrSpace, B be Subset of TopSpaceMetr(M), r be Real, u be
Point of M;
A1: TopSpaceMetr M = TopStruct (#the carrier of M,Family_open_set M#) & Ball
(u,r ) in Family_open_set M by PCOMPS_1:29,def 5;
assume B = Ball(u,r);
hence thesis by A1,PRE_TOPC:def 2;
end;
theorem Th2:
for p being Point of Euclid n, q being Point of TOP-REAL n, r
being Real st p = q & r > 0 holds Ball (p, r) is a_neighborhood of q
proof
let p be Point of Euclid n, q be Point of TOP-REAL n, r be Real;
reconsider A = Ball (p, r) as Subset of TOP-REAL n by TOPREAL3:8;
A1: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
then reconsider AA = A as Subset of TopSpaceMetr Euclid n;
AA is open by TOPMETR:14;
then
A2: A is open by A1,PRE_TOPC:30;
assume p = q & r > 0;
hence thesis by A2,Th1,CONNSP_2:3;
end;
theorem Th3:
for B being Subset of TOP-REAL n, u being Point of Euclid n st B
= Ball(u,r) holds B is open
proof
let B be Subset of TOP-REAL n, u be Point of Euclid n;
A1: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
then reconsider BB = B as Subset of TopSpaceMetr Euclid n;
assume B = Ball(u,r);
then BB is open by Lm4;
hence thesis by A1,PRE_TOPC:30;
end;
theorem Th4:
for M being non empty MetrSpace, u being Point of M, P being
Subset of TopSpaceMetr(M) holds u in Int P iff ex r being Real st r > 0
& Ball(u,r) c= P
proof
let M be non empty MetrSpace, u be Point of M, P be Subset of TopSpaceMetr(M
);
hereby
assume u in Int P;
then consider r be Real such that
A1: r > 0 and
A2: Ball(u,r) c= Int P by TOPMETR:15;
take r;
thus r > 0 by A1;
Int P c= P by TOPS_1:16;
hence Ball(u,r) c= P by A2;
end;
given r being Real such that
A3: r > 0 and
A4: Ball(u,r) c= P;
TopSpaceMetr M = TopStruct (#the carrier of M,Family_open_set M#) by
PCOMPS_1:def 5;
then reconsider B = Ball(u,r) as Subset of TopSpaceMetr(M);
A5: B is open by Lm4;
u in Ball(u,r) by A3,Th1;
hence thesis by A4,A5,TOPS_1:22;
end;
Lm5: for T being TopSpace, A being Subset of T, B being Subset of the
TopStruct of T st A = B holds Int A = Int B
proof
let T be TopSpace, A be Subset of T, B be Subset of the TopStruct of T such
that
A1: A = B;
reconsider AA = Int A as Subset of the TopStruct of T;
AA is open by PRE_TOPC:30;
hence Int A c= Int B by A1,TOPS_1:16,24;
reconsider BB = Int B as Subset of T;
BB is open by PRE_TOPC:30;
hence Int B c= Int A by A1,TOPS_1:16,24;
end;
theorem Th5:
for u being Point of Euclid n, P being Subset of TOP-REAL n holds
u in Int P iff ex r being Real st r > 0 & Ball(u,r) c= P
proof
let u be Point of Euclid n, P be Subset of TOP-REAL n;
A1: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
then reconsider PP = P as Subset of TopSpaceMetr Euclid n;
u in Int PP iff ex r being Real st r > 0 & Ball(u,r) c= PP by Th4;
hence thesis by A1,Lm5;
end;
theorem Th6: :: TOPREAL3:12
for u,v being Point of Euclid 2 st u = |[r1,s1]| & v = |[r2,s2]|
holds dist(u,v) =sqrt ((r1 - r2)^2 + (s1 - s2)^2)
proof
let u,v be Point of Euclid 2 such that
A1: u = |[r1,s1]| & v = |[r2,s2]|;
A2: |[r1,s1]|`1 = r1 & |[r1,s1]|`2 = s1 by EUCLID:52;
A3: |[r2,s2]|`1 = r2 & |[r2,s2]|`2 = s2 by EUCLID:52;
thus dist(u,v) = (Pitag_dist 2).(u,v) by METRIC_1:def 1
.= sqrt ((r1 - r2)^2 + (s1 - s2)^2) by A1,A2,A3,TOPREAL3:7;
end;
theorem Th7:
for u being Point of Euclid 2 st u = |[r,s]| holds 0 <= r2 & r2
< r1 implies |[r+r2,s]| in Ball(u,r1)
proof
let u be Point of Euclid 2 such that
A1: u = |[r,s]| and
A2: 0 <= r2 and
A3: r2 < r1;
reconsider v = |[r+r2,s]| as Point of Euclid 2 by TOPREAL3:8;
dist(u,v) = sqrt ((r - (r+r2))^2 + (s - s)^2) by A1,Th6
.= sqrt ((-(r - (r+r2)))^2)
.= r2 by A2,SQUARE_1:22;
hence thesis by A3,METRIC_1:11;
end;
theorem Th8:
for u being Point of Euclid 2 st u = |[r,s]| holds 0 <= s2 & s2
< s1 implies |[r,s+s2]| in Ball(u,s1)
proof
let u be Point of Euclid 2 such that
A1: u = |[r,s]| and
A2: 0 <= s2 and
A3: s2 < s1;
reconsider v = |[r,s+s2]| as Point of Euclid 2 by TOPREAL3:8;
dist(u,v) = sqrt ((r - r)^2 + (s - (s+s2))^2) by A1,Th6
.= sqrt ((-(s - (s+s2)))^2)
.= s2 by A2,SQUARE_1:22;
hence thesis by A3,METRIC_1:11;
end;
theorem Th9:
for u being Point of Euclid 2 st u = |[r,s]| holds 0 <= r2 & r2
< r1 implies |[r-r2,s]| in Ball(u,r1)
proof
let u be Point of Euclid 2 such that
A1: u = |[r,s]| and
A2: 0 <= r2 and
A3: r2 < r1;
reconsider v = |[r-r2,s]| as Point of Euclid 2 by TOPREAL3:8;
dist(u,v) = sqrt ((r - (r-r2))^2 + (s - s)^2) by A1,Th6
.= r2 by A2,SQUARE_1:22;
hence thesis by A3,METRIC_1:11;
end;
theorem Th10:
for u being Point of Euclid 2 st u = |[r,s]| holds 0 <= s2 & s2
< s1 implies |[r,s-s2]| in Ball(u,s1)
proof
let u be Point of Euclid 2 such that
A1: u = |[r,s]| and
A2: 0 <= s2 and
A3: s2 < s1;
reconsider v = |[r,s-s2]| as Point of Euclid 2 by TOPREAL3:8;
dist(u,v) = sqrt ((s - (s-s2))^2 + (r - r)^2) by A1,Th6
.= s2 by A2,SQUARE_1:22;
hence thesis by A3,METRIC_1:11;
end;
theorem Th11:
1 <= i & i < len G & 1 <= j & j < width G implies G*(i,j)+G*(i+1
,j+1) = G*(i,j+1)+G*(i+1,j)
proof
assume that
A1: 1 <= i & i < len G and
A2: 1 <= j & j < width G;
A3: 1 <= j+1 & j+1 <= width G by A2,NAT_1:13;
A4: 1 <= i+1 & i+1 <= len G by A1,NAT_1:13;
then
A5: G*(i+1,j+1)`1 = G*(i+1,1)`1 by A3,GOBOARD5:2
.= G*(i+1,j)`1 by A2,A4,GOBOARD5:2;
A6: G*(i+1,j+1)`2 = G*(1,j+1)`2 by A4,A3,GOBOARD5:1
.= G*(i,j+1)`2 by A1,A3,GOBOARD5:1;
A7: G*(i,j)`2 = G*(1,j)`2 by A1,A2,GOBOARD5:1
.= G*(i+1,j)`2 by A2,A4,GOBOARD5:1;
A8: (G*(i,j)+G*(i+1,j+1))`2 = G*(i,j)`2+G*(i+1,j+1)`2 by Lm1
.= (G*(i,j+1)+G*(i+1,j))`2 by A7,A6,Lm1;
A9: G*(i,j)`1 = G*(i,1)`1 by A1,A2,GOBOARD5:2
.= G*(i,j+1)`1 by A1,A3,GOBOARD5:2;
(G*(i,j)+G*(i+1,j+1))`1 = G*(i,j)`1+G*(i+1,j+1)`1 by Lm1
.= (G*(i,j+1)+G*(i+1,j))`1 by A9,A5,Lm1;
hence
G*(i,j)+G*(i+1,j+1) = |[(G*(i,j+1)+G*(i+1,j))`1,(G*(i,j+1)+G*(i+1,j))`2
]| by A8,EUCLID:53
.= G*(i,j+1)+G*(i+1,j) by EUCLID:53;
end;
Lm6: the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8
.= TopStruct (#the carrier of Euclid 2,Family_open_set Euclid 2#) by
PCOMPS_1:def 5;
theorem Th12:
Int v_strip(G,0) = { |[r,s]| : r < G*(1,1)`1 }
proof
0 <> width G by MATRIX_0:def 10;
then 1 <= width G by NAT_1:14;
then
A1: v_strip(G,0) = { |[r,s]| : r <= G*(1,1)`1 } by GOBOARD5:10;
thus Int v_strip(G,0) c= { |[r,s]| : r < G*(1,1)`1 }
proof
let x be object;
assume
A2: x in Int v_strip(G,0);
then reconsider u = x as Point of Euclid 2 by Lm6;
consider r1 being Real such that
A3: r1 > 0 and
A4: Ball(u,r1) c= v_strip(G,0) by A2,Th5;
reconsider p = u as Point of TOP-REAL 2 by Lm6;
A5: p = |[p`1,p`2]| by EUCLID:53;
set q = |[p`1+r1/2,p`2+0]|;
r1/2 < r1 by A3,XREAL_1:216;
then q in Ball(u,r1) by A3,A5,Th7;
then q in v_strip(G,0) by A4;
then ex r2,s2 st q = |[r2,s2]| & r2 <= G*(1,1)`1 by A1;
then
A6: p`1+r1/2 <= G*(1,1)`1 by SPPOL_2:1;
p`1 < p`1 + r1/2 by A3,XREAL_1:29,215;
then p`1 < G*(1,1)`1 by A6,XXREAL_0:2;
hence thesis by A5;
end;
let x be object;
assume x in { |[r,s]| : r < G*(1,1)`1 };
then consider r,s such that
A7: x = |[r,s]| and
A8: r < G*(1,1)`1;
reconsider u = |[r,s]| as Point of Euclid 2 by TOPREAL3:8;
A9: Ball(u,G*(1,1)`1-r) c= v_strip(G,0)
proof
let y be object;
A10: Ball(u,G*(1,1)`1-r) = { v : dist(u,v)<G*(1,1)`1-r} by METRIC_1:17;
assume y in Ball(u,G*(1,1)`1-r);
then consider v such that
A11: v = y and
A12: dist(u,v)<G*(1,1)`1-r by A10;
reconsider q = v as Point of TOP-REAL 2 by TOPREAL3:8;
(r - q`1)^2 >= 0 & (r - q`1)^2 + 0 <= (r - q`1)^2 + (s - q`2)^2 by
XREAL_1:6,63;
then
A13: sqrt (r - q`1)^2 <= sqrt ((r - q`1)^2 + (s - q`2)^2) by SQUARE_1:26;
A14: q = |[q`1,q`2]| by EUCLID:53;
then sqrt ((r - q`1)^2 + (s - q`2)^2) < G*(1,1)`1-r by A12,Th6;
then sqrt (r - q`1)^2 <= G*(1,1)`1-r by A13,XXREAL_0:2;
then
A15: |.r-q`1.| <= G*(1,1)`1-r by COMPLEX1:72;
per cases;
suppose
r <= q`1;
then
A16: q`1-r >= 0 by XREAL_1:48;
|.r-q`1.| = |.-(r-q`1).| by COMPLEX1:52
.= q`1 - r by A16,ABSVALUE:def 1;
then q`1 <= G*(1,1)`1 by A15,XREAL_1:9;
hence thesis by A1,A11,A14;
end;
suppose
r >= q`1;
then q`1 <= G*(1,1)`1 by A8,XXREAL_0:2;
hence thesis by A1,A11,A14;
end;
end;
reconsider B = Ball(u,G*(1,1)`1-r) as Subset of TOP-REAL2 by TOPREAL3:8;
A17: B is open by Th3;
u in Ball(u,G*(1,1)`1-r) by A8,Th1,XREAL_1:50;
hence thesis by A7,A9,A17,TOPS_1:22;
end;
theorem Th13:
Int v_strip(G,len G) = { |[r,s]| : G*(len G,1)`1 < r }
proof
0 <> width G by MATRIX_0:def 10;
then 1 <= width G by NAT_1:14;
then
A1: v_strip(G,len G) = { |[r,s]| : G*(len G,1)`1 <= r } by GOBOARD5:9;
thus Int v_strip(G,len G) c= { |[r,s]| : G*(len G,1)`1 < r }
proof
let x be object;
assume
A2: x in Int v_strip(G,len G);
then reconsider u = x as Point of Euclid 2 by Lm6;
consider r1 being Real such that
A3: r1 > 0 and
A4: Ball(u,r1) c= v_strip(G,len G) by A2,Th5;
reconsider p = u as Point of TOP-REAL 2 by Lm6;
A5: p = |[p`1,p`2]| by EUCLID:53;
set q = |[p`1-r1/2,p`2+0]|;
r1/2 < r1 by A3,XREAL_1:216;
then q in Ball(u,r1) by A3,A5,Th9;
then q in v_strip(G,len G) by A4;
then ex r2,s2 st q = |[r2,s2]| & G*(len G,1)`1 <= r2 by A1;
then G*(len G,1)`1 <= p`1-r1/2 by SPPOL_2:1;
then
A6: G*(len G,1)`1+r1/2 <= p`1 by XREAL_1:19;
G*(len G,1)`1 < G*(len G,1)`1 + r1/2 by A3,XREAL_1:29,215;
then G*(len G,1)`1 < p`1 by A6,XXREAL_0:2;
hence thesis by A5;
end;
let x be object;
assume x in { |[r,s]| : G*(len G,1)`1 < r };
then consider r,s such that
A7: x = |[r,s]| and
A8: G*(len G,1)`1 < r;
reconsider u = |[r,s]| as Point of Euclid 2 by TOPREAL3:8;
A9: Ball(u,r-G*(len G,1)`1) c= v_strip(G,len G)
proof
let y be object;
A10: Ball(u,r-G*(len G,1)`1) = { v : dist(u,v)<r-G*(len G,1)`1} by METRIC_1:17;
assume y in Ball(u,r-G*(len G,1)`1);
then consider v such that
A11: v = y and
A12: dist(u,v)<r-G*(len G,1)`1 by A10;
reconsider q = v as Point of TOP-REAL 2 by TOPREAL3:8;
(r - q`1)^2 >= 0 & (r - q`1)^2 + 0 <= (r - q`1)^2 + (s - q`2)^2 by
XREAL_1:6,63;
then
A13: sqrt (r - q`1)^2 <= sqrt ((r - q`1)^2 + (s - q`2)^2) by SQUARE_1:26;
A14: q = |[q`1,q`2]| by EUCLID:53;
then sqrt ((r - q`1)^2 + (s - q`2)^2) < r-G*(len G,1)`1 by A12,Th6;
then sqrt (r - q`1)^2 <= r-G*(len G,1)`1 by A13,XXREAL_0:2;
then
A15: |.r-q`1.| <= r-G*(len G,1)`1 by COMPLEX1:72;
per cases;
suppose
r >= q`1;
then r-q`1 >= 0 by XREAL_1:48;
then |.r-q`1.| = r - q`1 by ABSVALUE:def 1;
then G*(len G,1)`1 <= q`1 by A15,XREAL_1:10;
hence thesis by A1,A11,A14;
end;
suppose
r <= q`1;
then G*(len G,1)`1 <= q`1 by A8,XXREAL_0:2;
hence thesis by A1,A11,A14;
end;
end;
reconsider B = Ball(u,r-G*(len G,1)`1) as Subset of TOP-REAL2 by TOPREAL3:8;
A16: B is open by Th3;
u in Ball(u,r-G*(len G,1)`1) by A8,Th1,XREAL_1:50;
hence thesis by A7,A9,A16,TOPS_1:22;
end;
theorem Th14:
1 <= i & i < len G implies Int v_strip(G,i) = { |[r,s]| : G*(i,1
)`1 < r & r < G*(i+1,1)`1 }
proof
0 <> width G by MATRIX_0:def 10;
then
A1: 1 <= width G by NAT_1:14;
assume 1 <= i & i < len G;
then
A2: v_strip(G,i) = { |[r,s]| : G*(i,1)`1 <= r & r <= G*(i+1,1)`1 } by A1,
GOBOARD5:8;
thus Int v_strip(G,i) c= { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 }
proof
let x be object;
assume
A3: x in Int v_strip(G,i);
then reconsider u = x as Point of Euclid 2 by Lm6;
consider r1 being Real such that
A4: r1 > 0 and
A5: Ball(u,r1) c= v_strip(G,i) by A3,Th5;
reconsider p = u as Point of TOP-REAL 2 by Lm6;
A6: p = |[p`1,p`2]| by EUCLID:53;
set q2 = |[p`1-r1/2,p`2+0]|;
A7: r1/2 < r1 by A4,XREAL_1:216;
then q2 in Ball(u,r1) by A4,A6,Th9;
then q2 in v_strip(G,i) by A5;
then ex r2,s2 st q2 = |[r2,s2]| & G*(i,1)`1 <= r2 & r2 <= G*(i+1,1)`1 by A2
;
then G*(i,1)`1 <= p`1-r1/2 by SPPOL_2:1;
then
A8: G*(i,1)`1+r1/2 <= p`1 by XREAL_1:19;
set q1 = |[p`1+r1/2,p`2+0]|;
q1 in Ball(u,r1) by A4,A6,A7,Th7;
then q1 in v_strip(G,i) by A5;
then ex r2,s2 st q1 = |[r2,s2]| & G*(i,1)`1 <= r2 & r2 <= G*(i+1,1)`1 by A2
;
then
A9: p`1+r1/2 <= G*(i+1,1)`1 by SPPOL_2:1;
G*(i,1)`1 < G*(i,1)`1 + r1/2 by A4,XREAL_1:29,215;
then
A10: G*(i,1)`1 < p`1 by A8,XXREAL_0:2;
p`1 < p`1 + r1/2 by A4,XREAL_1:29,215;
then p`1 < G*(i+1,1)`1 by A9,XXREAL_0:2;
hence thesis by A6,A10;
end;
let x be object;
assume x in { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 };
then consider r,s such that
A11: x = |[r,s]| and
A12: G*(i,1)`1 < r and
A13: r < G*(i+1,1)`1;
reconsider u = |[r,s]| as Point of Euclid 2 by TOPREAL3:8;
G*(i+1,1)`1-r > 0 & r - G*(i,1)`1 > 0 by A12,A13,XREAL_1:50;
then min(r-G*(i,1)`1,G*(i+1,1)`1-r) > 0 by XXREAL_0:15;
then
A14: u in Ball(u,min(r-G*(i,1)`1,G*(i+1,1)`1-r)) by Th1;
A15: Ball(u,min(r-G*(i,1)`1,G*(i+1,1)`1-r)) c= v_strip(G,i)
proof
let y be object;
A16: Ball(u,min(r-G*(i,1)`1,G*(i+1,1)`1-r)) = { v : dist(u,v)<min(r-G*(i,1
)`1,G*(i+1,1)`1-r)} by METRIC_1:17;
assume y in Ball(u,min(r-G*(i,1)`1,G*(i+1,1)`1-r));
then consider v such that
A17: v = y and
A18: dist(u,v)<min(r-G*(i,1)`1,G*(i+1,1)`1-r) by A16;
reconsider q = v as Point of TOP-REAL 2 by TOPREAL3:8;
(r - q`1)^2 >= 0 & (r - q`1)^2 + 0 <= (r - q`1)^2 + (s - q`2)^2 by
XREAL_1:6,63;
then
A19: sqrt (r - q`1)^2 <= sqrt ((r - q`1)^2 + (s - q`2)^2) by SQUARE_1:26;
A20: q = |[q`1,q`2]| by EUCLID:53;
then sqrt ((r - q`1)^2 + (s - q`2)^2) < min(r-G*(i,1)`1,G*(i+1,1 )`1-r )
by A18,Th6;
then sqrt (r - q`1)^2 <= min(r-G*(i,1)`1,G* (i+1,1)`1-r) by A19,XXREAL_0:2;
then
A21: |.r-q`1.| <= min(r-G*(i,1)`1,G*(i+1,1)`1-r) by COMPLEX1:72;
then
A22: |.r-q`1.| <= r-G*(i,1)`1 by XXREAL_0:22;
A23: |.r-q`1.| <= G* (i+1,1)`1-r by A21,XXREAL_0:22;
per cases;
suppose
A24: r <= q`1;
then
A25: q`1-r >= 0 by XREAL_1:48;
|.r-q`1.| = |.-(r-q`1).| by COMPLEX1:52
.= q`1 - r by A25,ABSVALUE:def 1;
then
A26: q`1 <= G*(i+1,1)`1 by A23,XREAL_1:9;
G*(i,1)`1 <= q`1 by A12,A24,XXREAL_0:2;
hence thesis by A2,A17,A20,A26;
end;
suppose
A27: r >= q`1;
then r-q`1 >= 0 by XREAL_1:48;
then |.r-q`1.| = r - q`1 by ABSVALUE:def 1;
then
A28: G*(i,1)`1 <= q`1 by A22,XREAL_1:10;
q`1 <= G*(i+1,1)`1 by A13,A27,XXREAL_0:2;
hence thesis by A2,A17,A20,A28;
end;
end;
reconsider B = Ball(u,min(r-G*(i,1)`1,G*(i+1,1)`1-r)) as Subset of TOP-REAL2
by TOPREAL3:8;
B is open by Th3;
hence thesis by A11,A14,A15,TOPS_1:22;
end;
theorem Th15:
Int h_strip(G,0) = { |[r,s]| : s < G*(1,1)`2 }
proof
0 <> len G by MATRIX_0:def 10;
then 1 <= len G by NAT_1:14;
then
A1: h_strip(G,0) = { |[r,s]| : s <= G*(1,1)`2 } by GOBOARD5:7;
thus Int h_strip(G,0) c= { |[r,s]| : s < G*(1,1)`2 }
proof
let x be object;
assume
A2: x in Int h_strip(G,0);
then reconsider u = x as Point of Euclid 2 by Lm6;
consider s1 being Real such that
A3: s1 > 0 and
A4: Ball(u,s1) c= h_strip(G,0) by A2,Th5;
reconsider p = u as Point of TOP-REAL 2 by Lm6;
A5: p = |[p`1,p`2]| by EUCLID:53;
set q = |[p`1+0,p`2+s1/2]|;
s1/2 < s1 by A3,XREAL_1:216;
then q in Ball(u,s1) by A3,A5,Th8;
then q in h_strip(G,0) by A4;
then ex r2,s2 st q = |[r2,s2]| & s2 <= G*(1,1)`2 by A1;
then
A6: p`2+s1/2 <= G*(1,1)`2 by SPPOL_2:1;
p`2 < p`2 + s1/2 by A3,XREAL_1:29,215;
then p`2 < G*(1,1)`2 by A6,XXREAL_0:2;
hence thesis by A5;
end;
let x be object;
assume x in { |[r,s]| : s < G*(1,1)`2 };
then consider r,s such that
A7: x = |[r,s]| and
A8: s < G*(1,1)`2;
reconsider u = |[r,s]| as Point of Euclid 2 by TOPREAL3:8;
A9: Ball(u,G*(1,1)`2-s) c= h_strip(G,0)
proof
let y be object;
A10: Ball(u,G*(1,1)`2-s) = { v : dist(u,v)<G*(1,1)`2-s} by METRIC_1:17;
assume y in Ball(u,G*(1,1)`2-s);
then consider v such that
A11: v = y and
A12: dist(u,v)<G*(1,1)`2-s by A10;
reconsider q = v as Point of TOP-REAL 2 by TOPREAL3:8;
(s - q`2)^2 >= 0 & (s - q`2)^2 + 0 <= (r - q`1)^2 + (s - q`2)^2 by
XREAL_1:6,63;
then
A13: sqrt (s - q`2)^2 <= sqrt ((r - q`1)^2 + (s - q`2)^2) by SQUARE_1:26;
A14: q = |[q`1,q`2]| by EUCLID:53;
then sqrt ((r - q`1)^2 + (s - q`2)^2) < G*(1,1)`2-s by A12,Th6;
then sqrt (s - q`2)^2 <= G*(1,1)`2-s by A13,XXREAL_0:2;
then
A15: |.s-q`2.| <= G*(1,1)`2-s by COMPLEX1:72;
per cases;
suppose
s <= q`2;
then
A16: q`2-s >= 0 by XREAL_1:48;
|.s-q`2.| = |.-(s-q`2).| by COMPLEX1:52
.= q`2 - s by A16,ABSVALUE:def 1;
then q`2 <= G*(1,1)`2 by A15,XREAL_1:9;
hence thesis by A1,A11,A14;
end;
suppose
s >= q`2;
then q`2 <= G*(1,1)`2 by A8,XXREAL_0:2;
hence thesis by A1,A11,A14;
end;
end;
reconsider B = Ball(u,G*(1,1)`2-s) as Subset of TOP-REAL2 by TOPREAL3:8;
A17: B is open by Th3;
u in Ball(u,G*(1,1)`2-s) by A8,Th1,XREAL_1:50;
hence thesis by A7,A9,A17,TOPS_1:22;
end;
theorem Th16:
Int h_strip(G,width G) = { |[r,s]| : G*(1,width G)`2 < s }
proof
0 <> len G by MATRIX_0:def 10;
then 1 <= len G by NAT_1:14;
then
A1: h_strip(G,width G) = { |[r,s]| : G*(1,width G)`2 <= s } by GOBOARD5:6;
thus Int h_strip(G,width G) c= { |[r,s]| : G*(1,width G)`2 < s }
proof
let x be object;
assume
A2: x in Int h_strip(G,width G);
then reconsider u = x as Point of Euclid 2 by Lm6;
consider s1 being Real such that
A3: s1 > 0 and
A4: Ball(u,s1) c= h_strip(G,width G) by A2,Th5;
reconsider p = u as Point of TOP-REAL 2 by Lm6;
A5: p = |[p`1,p`2]| by EUCLID:53;
set q = |[p`1+0,p`2-s1/2]|;
s1/2 < s1 by A3,XREAL_1:216;
then q in Ball(u,s1) by A3,A5,Th10;
then q in h_strip(G,width G) by A4;
then ex r2,s2 st q = |[r2,s2]| & G*(1,width G)`2 <= s2 by A1;
then G*(1,width G)`2 <= p`2-s1/2 by SPPOL_2:1;
then
A6: G*(1,width G)`2+s1/2 <= p`2 by XREAL_1:19;
G*(1,width G)`2 < G*(1,width G)`2 + s1/2 by A3,XREAL_1:29,215;
then G*(1,width G)`2 < p`2 by A6,XXREAL_0:2;
hence thesis by A5;
end;
let x be object;
assume x in { |[r,s]| : G*(1,width G)`2 < s };
then consider r,s such that
A7: x = |[r,s]| and
A8: G*(1,width G)`2 < s;
reconsider u = |[r,s]| as Point of Euclid 2 by TOPREAL3:8;
A9: Ball(u,s-G*(1,width G)`2) c= h_strip(G,width G)
proof
let y be object;
A10: Ball(u,s-G*(1,width G)`2) = { v : dist(u,v)<s-G*(1,width G)`2} by
METRIC_1:17;
assume y in Ball(u,s-G*(1,width G)`2);
then consider v such that
A11: v = y and
A12: dist(u,v)<s-G*(1,width G)`2 by A10;
reconsider q = v as Point of TOP-REAL 2 by TOPREAL3:8;
(s - q`2)^2 >= 0 & (s - q`2)^2 + 0 <= (r - q`1)^2 + (s - q`2)^2 by
XREAL_1:6,63;
then
A13: sqrt (s - q`2)^2 <= sqrt ((r - q`1)^2 + (s - q`2)^2) by SQUARE_1:26;
A14: q = |[q`1,q`2]| by EUCLID:53;
then sqrt ((r - q`1)^2 + (s - q`2)^2) < s-G*(1,width G)`2 by A12,Th6;
then sqrt (s - q`2)^2 <= s-G*(1,width G)`2 by A13,XXREAL_0:2;
then
A15: |.s-q`2.| <= s-G*(1,width G)`2 by COMPLEX1:72;
per cases;
suppose
s >= q`2;
then s-q`2 >= 0 by XREAL_1:48;
then |.s-q`2.| = s - q`2 by ABSVALUE:def 1;
then G*(1,width G)`2 <= q`2 by A15,XREAL_1:10;
hence thesis by A1,A11,A14;
end;
suppose
s <= q`2;
then G*(1,width G)`2 <= q`2 by A8,XXREAL_0:2;
hence thesis by A1,A11,A14;
end;
end;
reconsider B = Ball(u,s-G*(1,width G)`2) as Subset of TOP-REAL2 by TOPREAL3:8
;
A16: B is open by Th3;
u in Ball(u,s-G*(1,width G)`2) by A8,Th1,XREAL_1:50;
hence thesis by A7,A9,A16,TOPS_1:22;
end;
theorem Th17:
1 <= j & j < width G implies Int h_strip(G,j) = { |[r,s]| : G*(1
,j)`2 < s & s < G*(1,j+1)`2 }
proof
0 <> len G by MATRIX_0:def 10;
then
A1: 1 <= len G by NAT_1:14;
assume 1 <= j & j < width G;
then
A2: h_strip(G,j) = { |[r,s]| : G*(1,j)`2 <= s & s <= G*(1,j+1)`2 } by A1,
GOBOARD5:5;
thus Int h_strip(G,j) c= { |[r,s]| : G*(1,j)`2 < s & s < G*(1,j+1)`2 }
proof
let x be object;
assume
A3: x in Int h_strip(G,j);
then reconsider u = x as Point of Euclid 2 by Lm6;
consider s1 being Real such that
A4: s1 > 0 and
A5: Ball(u,s1) c= h_strip(G,j) by A3,Th5;
reconsider p = u as Point of TOP-REAL 2 by Lm6;
A6: p = |[p`1,p`2]| by EUCLID:53;
set q2 = |[p`1+0,p`2-s1/2]|;
A7: s1/2 < s1 by A4,XREAL_1:216;
then q2 in Ball(u,s1) by A4,A6,Th10;
then q2 in h_strip(G,j) by A5;
then ex r2,s2 st q2 = |[r2,s2]| & G*(1,j)`2 <= s2 & s2 <= G*(1,j+1)`2 by A2
;
then G*(1,j)`2 <= p`2-s1/2 by SPPOL_2:1;
then
A8: G*(1,j)`2+s1/2 <= p`2 by XREAL_1:19;
set q1 = |[p`1+0,p`2+s1/2]|;
q1 in Ball(u,s1) by A4,A6,A7,Th8;
then q1 in h_strip(G,j) by A5;
then ex r2,s2 st q1 = |[r2,s2]| & G*(1,j)`2 <= s2 & s2 <= G*(1,j+1)`2 by A2
;
then
A9: p`2+s1/2 <= G*(1,j+1)`2 by SPPOL_2:1;
G*(1,j)`2 < G*(1,j)`2 + s1/2 by A4,XREAL_1:29,215;
then
A10: G*(1,j)`2 < p`2 by A8,XXREAL_0:2;
p`2 < p`2 + s1/2 by A4,XREAL_1:29,215;
then p`2 < G*(1,j+1)`2 by A9,XXREAL_0:2;
hence thesis by A6,A10;
end;
let x be object;
assume x in { |[r,s]| : G*(1,j)`2 < s & s < G*(1,j+1)`2 };
then consider r,s such that
A11: x = |[r,s]| and
A12: G*(1,j)`2 < s and
A13: s < G*(1,j+1)`2;
reconsider u = |[r,s]| as Point of Euclid 2 by TOPREAL3:8;
G*(1,j+1)`2-s > 0 & s - G*(1,j)`2 > 0 by A12,A13,XREAL_1:50;
then min(s-G*(1,j)`2,G*(1,j+1)`2-s) > 0 by XXREAL_0:15;
then
A14: u in Ball(u,min(s-G*(1,j)`2,G*(1,j+1)`2-s)) by Th1;
A15: Ball(u,min(s-G*(1,j)`2,G*(1,j+1)`2-s)) c= h_strip(G,j)
proof
let y be object;
A16: Ball(u,min(s-G*(1,j)`2,G*(1,j+1)`2-s)) = { v : dist(u,v)<min(s-G*(1,j
)`2,G*(1,j+1)`2-s)} by METRIC_1:17;
assume y in Ball(u,min(s-G*(1,j)`2,G*(1,j+1)`2-s));
then consider v such that
A17: v = y and
A18: dist(u,v)<min(s-G*(1,j)`2,G*(1,j+1)`2-s) by A16;
reconsider q = v as Point of TOP-REAL 2 by TOPREAL3:8;
(s - q`2)^2 >= 0 & (s - q`2)^2 + 0 <= (r - q`1)^2 + (s - q`2)^2 by
XREAL_1:6,63;
then
A19: sqrt (s - q`2)^2 <= sqrt ((r - q`1)^2 + (s - q`2)^2) by SQUARE_1:26;
A20: q = |[q`1,q`2]| by EUCLID:53;
then sqrt ((r - q`1)^2 + (s - q`2)^2) < min(s-G*(1,j)`2,G*(1,j+1 )`2-s )
by A18,Th6;
then sqrt (s - q`2)^2 <= min(s-G*(1,j)`2,G* (1,j+1)`2-s) by A19,XXREAL_0:2;
then
A21: |.s-q`2.| <= min(s-G*(1,j)`2,G*(1,j+1)`2-s) by COMPLEX1:72;
then
A22: |.s-q`2.| <= s-G*(1,j)`2 by XXREAL_0:22;
A23: |.s-q`2.| <= G* (1,j+1)`2-s by A21,XXREAL_0:22;
per cases;
suppose
A24: s <= q`2;
then
A25: q`2-s >= 0 by XREAL_1:48;
|.s-q`2.| = |.-(s-q`2).| by COMPLEX1:52
.= q`2 - s by A25,ABSVALUE:def 1;
then
A26: q`2 <= G*(1,j+1)`2 by A23,XREAL_1:9;
G*(1,j)`2 <= q`2 by A12,A24,XXREAL_0:2;
hence thesis by A2,A17,A20,A26;
end;
suppose
A27: s >= q`2;
then s-q`2 >= 0 by XREAL_1:48;
then |.s-q`2.| = s - q`2 by ABSVALUE:def 1;
then
A28: G*(1,j)`2 <= q`2 by A22,XREAL_1:10;
q`2 <= G*(1,j+1)`2 by A13,A27,XXREAL_0:2;
hence thesis by A2,A17,A20,A28;
end;
end;
reconsider B = Ball(u,min(s-G*(1,j)`2,G*(1,j+1)`2-s)) as Subset of TOP-REAL2
by TOPREAL3:8;
B is open by Th3;
hence thesis by A11,A14,A15,TOPS_1:22;
end;
theorem Th18:
Int cell(G,0,0) = { |[r,s]| : r < G*(1,1)`1 & s < G*(1,1)`2 }
proof
cell(G,0,0) = v_strip(G,0) /\ h_strip(G,0) by GOBOARD5:def 3;
then
A1: Int cell(G,0,0) = Int v_strip(G,0) /\ Int h_strip(G,0) by TOPS_1:17;
A2: Int h_strip(G,0) = { |[r,s]| : s < G*(1,1)`2 } by Th15;
A3: Int v_strip(G,0) = { |[r,s]| : r < G*(1,1)`1 } by Th12;
thus Int cell(G,0,0) c= { |[r,s]| : r < G*(1,1)`1 & s < G*(1,1)`2 }
proof
let x be object;
assume
A4: x in Int cell(G,0,0);
then x in Int v_strip(G,0) by A1,XBOOLE_0:def 4;
then consider r1,s1 such that
A5: x = |[r1,s1]| and
A6: r1 < G*(1,1)`1 by A3;
x in Int h_strip(G,0) by A1,A4,XBOOLE_0:def 4;
then consider r2,s2 such that
A7: x = |[r2,s2]| and
A8: s2 < G*(1,1)`2 by A2;
s1 = s2 by A5,A7,SPPOL_2:1;
hence thesis by A5,A6,A8;
end;
let x be object;
assume x in { |[r,s]| : r < G*(1,1)`1 & s < G*(1,1)`2 };
then
A9: ex r,s st x = |[r,s]| & r < G*(1,1)`1 & s < G*(1,1)`2;
then
A10: x in Int h_strip(G,0) by A2;
x in Int v_strip(G,0) by A3,A9;
hence thesis by A1,A10,XBOOLE_0:def 4;
end;
theorem Th19:
Int cell(G,0,width G) = { |[r,s]| : r < G*(1,1)`1 & G*(1,width G )`2 < s }
proof
cell(G,0,width G) = v_strip(G,0) /\ h_strip(G,width G) by GOBOARD5:def 3;
then
A1: Int cell(G,0,width G) = Int v_strip(G,0) /\ Int h_strip(G,width G) by
TOPS_1:17;
A2: Int h_strip(G,width G) = { |[r,s]| : G*(1,width G)`2 < s } by Th16;
A3: Int v_strip(G,0) = { |[r,s]| : r < G*(1,1)`1 } by Th12;
thus Int cell(G,0,width G) c= { |[r,s]| : r < G*(1,1)`1 & G*(1,width G)`2 <
s }
proof
let x be object;
assume
A4: x in Int cell(G,0,width G);
then x in Int v_strip(G,0) by A1,XBOOLE_0:def 4;
then consider r1,s1 such that
A5: x = |[r1,s1]| and
A6: r1 < G*(1,1)`1 by A3;
x in Int h_strip(G,width G) by A1,A4,XBOOLE_0:def 4;
then consider r2,s2 such that
A7: x = |[r2,s2]| and
A8: G*(1,width G)`2 < s2 by A2;
s1 = s2 by A5,A7,SPPOL_2:1;
hence thesis by A5,A6,A8;
end;
let x be object;
assume x in { |[r,s]| : r < G*(1,1)`1 & G*(1,width G)`2 < s };
then
A9: ex r,s st x = |[r,s]| & r < G*(1,1)`1 & G*(1,width G)`2 < s;
then
A10: x in Int h_strip(G,width G) by A2;
x in Int v_strip(G,0) by A3,A9;
hence thesis by A1,A10,XBOOLE_0:def 4;
end;
theorem Th20:
1 <= j & j < width G implies Int cell(G,0,j) = { |[r,s]| : r < G
*(1,1)`1 & G*(1,j)`2 < s & s < G*(1,j+1)`2 }
proof
cell(G,0,j) = v_strip(G,0) /\ h_strip(G,j) by GOBOARD5:def 3;
then
A1: Int cell(G,0,j) = Int v_strip(G,0) /\ Int h_strip(G,j) by TOPS_1:17;
assume 1 <= j & j < width G;
then
A2: Int h_strip(G,j) = { |[r,s]| : G*(1,j)`2 < s & s < G* (1,j+1)`2 } by Th17;
A3: Int v_strip(G,0) = { |[r,s]| : r < G*(1,1)`1 } by Th12;
thus Int cell(G,0,j) c= { |[r,s]| : r < G*(1,1)`1 & G*(1,j)`2 < s & s < G*(1
,j+1)`2 }
proof
let x be object;
assume
A4: x in Int cell(G,0,j);
then x in Int v_strip(G,0) by A1,XBOOLE_0:def 4;
then consider r1,s1 such that
A5: x = |[r1,s1]| and
A6: r1 < G*(1,1)`1 by A3;
x in Int h_strip(G,j) by A1,A4,XBOOLE_0:def 4;
then consider r2,s2 such that
A7: x = |[r2,s2]| and
A8: G*(1,j)`2 < s2 & s2 < G*(1,j+1)`2 by A2;
s1 = s2 by A5,A7,SPPOL_2:1;
hence thesis by A5,A6,A8;
end;
let x be object;
assume x in { |[r,s]| : r < G*(1,1)`1 & G*(1,j)`2 < s & s < G*(1,j+1)`2 };
then
A9: ex r,s st x = |[r,s]| & r < G*(1,1)`1 & G*(1,j)`2 < s & s < G*(1,j+1)`2;
then
A10: x in Int h_strip(G,j) by A2;
x in Int v_strip(G,0) by A3,A9;
hence thesis by A1,A10,XBOOLE_0:def 4;
end;
theorem Th21:
Int cell(G,len G,0) = { |[r,s]| : G*(len G,1)`1 < r & s < G*(1,1 )`2 }
proof
cell(G,len G,0) = v_strip(G,len G) /\ h_strip(G,0) by GOBOARD5:def 3;
then
A1: Int cell(G,len G,0) = Int v_strip(G,len G) /\ Int h_strip(G,0) by TOPS_1:17
;
A2: Int h_strip(G,0) = { |[r,s]| : s < G*(1,1)`2 } by Th15;
A3: Int v_strip(G,len G) = { |[r,s]| : G*(len G,1)`1 < r } by Th13;
thus Int cell(G,len G,0) c= { |[r,s]| : G*(len G,1)`1 < r & s < G*(1,1)`2 }
proof
let x be object;
assume
A4: x in Int cell(G,len G,0);
then x in Int v_strip(G,len G) by A1,XBOOLE_0:def 4;
then consider r1,s1 such that
A5: x = |[r1,s1]| and
A6: G*(len G,1)`1 < r1 by A3;
x in Int h_strip(G,0) by A1,A4,XBOOLE_0:def 4;
then consider r2,s2 such that
A7: x = |[r2,s2]| and
A8: s2 < G*(1,1)`2 by A2;
s1 = s2 by A5,A7,SPPOL_2:1;
hence thesis by A5,A6,A8;
end;
let x be object;
assume x in { |[r,s]| : G*(len G,1)`1 < r & s < G*(1,1)`2 };
then
A9: ex r,s st x = |[r,s]| & G*(len G,1)`1 < r & s < G*(1,1)`2;
then
A10: x in Int h_strip(G,0) by A2;
x in Int v_strip(G,len G) by A3,A9;
hence thesis by A1,A10,XBOOLE_0:def 4;
end;
theorem Th22:
Int cell(G,len G,width G) = { |[r,s]| : G*(len G,1)`1 < r & G*(1
,width G)`2 < s }
proof
cell(G,len G,width G) = v_strip(G,len G) /\ h_strip(G,width G) by
GOBOARD5:def 3;
then
A1: Int cell(G,len G,width G) = Int v_strip(G,len G) /\ Int h_strip(G, width
G) by TOPS_1:17;
A2: Int h_strip(G,width G) = { |[r,s]| : G*(1,width G)`2 < s } by Th16;
A3: Int v_strip(G,len G) = { |[r,s]| : G*(len G,1)`1 < r } by Th13;
thus Int cell(G,len G,width G) c= { |[r,s]| : G*(len G,1)`1 < r & G*(1,width
G)`2 < s }
proof
let x be object;
assume
A4: x in Int cell(G,len G,width G);
then x in Int v_strip(G,len G) by A1,XBOOLE_0:def 4;
then consider r1,s1 such that
A5: x = |[r1,s1]| and
A6: G*(len G,1)`1 < r1 by A3;
x in Int h_strip(G,width G) by A1,A4,XBOOLE_0:def 4;
then consider r2,s2 such that
A7: x = |[r2,s2]| and
A8: G*(1,width G)`2 < s2 by A2;
s1 = s2 by A5,A7,SPPOL_2:1;
hence thesis by A5,A6,A8;
end;
let x be object;
assume x in { |[r,s]| : G*(len G,1)`1 < r & G*(1,width G)`2 < s };
then
A9: ex r,s st x = |[r,s]| & G*(len G,1)`1 < r & G*(1,width G)`2 < s;
then
A10: x in Int h_strip(G,width G) by A2;
x in Int v_strip(G,len G) by A3,A9;
hence thesis by A1,A10,XBOOLE_0:def 4;
end;
theorem Th23:
1 <= j & j < width G implies Int cell(G,len G,j) = { |[r,s]| : G
*(len G,1)`1 < r & G*(1,j)`2 < s & s < G*(1,j+1)`2 }
proof
cell(G,len G,j) = v_strip(G,len G) /\ h_strip(G,j) by GOBOARD5:def 3;
then
A1: Int cell(G,len G,j) = Int v_strip(G,len G) /\ Int h_strip(G,j) by TOPS_1:17
;
assume 1 <= j & j < width G;
then
A2: Int h_strip(G,j) = { |[r,s]| : G*(1,j)`2 < s & s < G* (1,j+1)`2 } by Th17;
A3: Int v_strip(G,len G) = { |[r,s]| : G*(len G,1)`1 < r } by Th13;
thus Int cell(G,len G,j) c= { |[r,s]| : G*(len G,1)`1 < r & G*(1,j)`2 < s &
s < G*(1,j+1)`2 }
proof
let x be object;
assume
A4: x in Int cell(G,len G,j);
then x in Int v_strip(G,len G) by A1,XBOOLE_0:def 4;
then consider r1,s1 such that
A5: x = |[r1,s1]| and
A6: G*(len G,1)`1 < r1 by A3;
x in Int h_strip(G,j) by A1,A4,XBOOLE_0:def 4;
then consider r2,s2 such that
A7: x = |[r2,s2]| and
A8: G*(1,j)`2 < s2 & s2 < G*(1,j+1)`2 by A2;
s1 = s2 by A5,A7,SPPOL_2:1;
hence thesis by A5,A6,A8;
end;
let x be object;
assume
x in { |[r,s]| : G*(len G,1)`1 < r & G*(1,j)`2 < s & s < G* (1,j+1) `2 };
then
A9: ex r,s st x = |[r,s]| & G*(len G,1)`1 < r & G*(1,j)`2 < s & s < G*(1,j+
1)`2;
then
A10: x in Int h_strip(G,j) by A2;
x in Int v_strip(G,len G) by A3,A9;
hence thesis by A1,A10,XBOOLE_0:def 4;
end;
theorem Th24:
1 <= i & i < len G implies Int cell(G,i,0) = { |[r,s]| : G*(i,1)
`1 < r & r < G*(i+1,1)`1 & s < G*(1,1)`2 }
proof
cell(G,i,0) = v_strip(G,i) /\ h_strip(G,0) by GOBOARD5:def 3;
then
A1: Int cell(G,i,0) = Int v_strip(G,i) /\ Int h_strip(G,0) by TOPS_1:17;
assume 1 <= i & i < len G;
then
A2: Int v_strip(G,i) = { |[r,s]| : G*(i,1)`1 < r & r < G* (i+1,1)`1 } by Th14;
A3: Int h_strip(G,0) = { |[r,s]| : s < G*(1,1)`2 } by Th15;
thus Int cell(G,i,0) c= { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & s < G*
(1,1)`2 }
proof
let x be object;
assume
A4: x in Int cell(G,i,0);
then x in Int v_strip(G,i) by A1,XBOOLE_0:def 4;
then consider r1,s1 such that
A5: x = |[r1,s1]| and
A6: G*(i,1)`1 < r1 & r1 < G*(i+1,1)`1 by A2;
x in Int h_strip(G,0) by A1,A4,XBOOLE_0:def 4;
then consider r2,s2 such that
A7: x = |[r2,s2]| and
A8: s2 < G*(1,1)`2 by A3;
s1 = s2 by A5,A7,SPPOL_2:1;
hence thesis by A5,A6,A8;
end;
let x be object;
assume x in { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & s < G*(1,1)`2 };
then
A9: ex r,s st x = |[r,s]| & G*(i,1)`1 < r & r < G*(i+1,1)`1 & s < G*(1,1)`2;
then
A10: x in Int h_strip(G,0) by A3;
x in Int v_strip(G,i) by A2,A9;
hence thesis by A1,A10,XBOOLE_0:def 4;
end;
theorem Th25:
1 <= i & i < len G implies Int cell(G,i,width G) = { |[r,s]| : G
*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,width G)`2 < s }
proof
cell(G,i,width G) = v_strip(G,i) /\ h_strip(G,width G) by GOBOARD5:def 3;
then
A1: Int cell(G,i,width G) = Int v_strip(G,i) /\ Int h_strip(G,width G) by
TOPS_1:17;
assume 1 <= i & i < len G;
then
A2: Int v_strip(G,i) = { |[r,s]| : G*(i,1)`1 < r & r < G* (i+1,1)`1 } by Th14;
A3: Int h_strip(G,width G) = { |[r,s]| : G*(1,width G)`2 < s } by Th16;
thus Int cell(G,i,width G) c= { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 &
G*(1,width G)`2 < s }
proof
let x be object;
assume
A4: x in Int cell(G,i,width G);
then x in Int v_strip(G,i) by A1,XBOOLE_0:def 4;
then consider r1,s1 such that
A5: x = |[r1,s1]| and
A6: G*(i,1)`1 < r1 & r1 < G*(i+1,1)`1 by A2;
x in Int h_strip(G,width G) by A1,A4,XBOOLE_0:def 4;
then consider r2,s2 such that
A7: x = |[r2,s2]| and
A8: G*(1,width G)`2 < s2 by A3;
s1 = s2 by A5,A7,SPPOL_2:1;
hence thesis by A5,A6,A8;
end;
let x be object;
assume
x in { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,width G)`2 < s };
then
A9: ex r,s st x = |[r,s]| & G*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,width G)
`2 < s;
then
A10: x in Int h_strip(G,width G) by A3;
x in Int v_strip(G,i) by A2,A9;
hence thesis by A1,A10,XBOOLE_0:def 4;
end;
theorem Th26:
1 <= i & i < len G & 1 <= j & j < width G implies Int cell(G,i,j
) = { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,j)`2 < s & s < G*(1,j+1)
`2 }
proof
assume that
A1: 1 <= i & i < len G and
A2: 1 <= j & j < width G;
A3: Int h_strip(G,j) = { |[r,s]| : G*(1,j)`2 < s & s < G* (1,j+1)`2 } by A2
,Th17;
cell(G,i,j) = v_strip(G,i) /\ h_strip(G,j) by GOBOARD5:def 3;
then
A4: Int cell(G,i,j) = Int v_strip(G,i) /\ Int h_strip(G,j) by TOPS_1:17;
A5: Int v_strip(G,i) = { |[r,s]| : G*(i,1)`1 < r & r < G* (i+1,1)`1 } by A1
,Th14;
thus Int cell(G,i,j) c= { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,j
)`2 < s & s < G*(1,j+1)`2 }
proof
let x be object;
assume
A6: x in Int cell(G,i,j);
then x in Int v_strip(G,i) by A4,XBOOLE_0:def 4;
then consider r1,s1 such that
A7: x = |[r1,s1]| and
A8: G*(i,1)`1 < r1 & r1 < G*(i+1,1)`1 by A5;
x in Int h_strip(G,j) by A4,A6,XBOOLE_0:def 4;
then consider r2,s2 such that
A9: x = |[r2,s2]| and
A10: G*(1,j)`2 < s2 & s2 < G*(1,j+1)`2 by A3;
s1 = s2 by A7,A9,SPPOL_2:1;
hence thesis by A7,A8,A10;
end;
let x be object;
assume x in { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,j)`2 < s &
s < G*(1,j+1)`2 };
then
A11: ex r,s st x = |[r,s]| & G*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,j)`2 < s
& s < G*(1,j+1)`2;
then
A12: x in Int h_strip(G,j) by A3;
x in Int v_strip(G,i) by A5,A11;
hence thesis by A4,A12,XBOOLE_0:def 4;
end;
theorem
1 <= j & j <= width G & p in Int h_strip(G,j) implies p`2 > G*(1,j)`2
proof
assume that
A1: 1 <= j and
A2: j <= width G and
A3: p in Int h_strip(G,j);
per cases by A2,XXREAL_0:1;
suppose
j = width G;
then Int h_strip(G,j) = { |[r,s]| : G*(1,j)`2 < s } by Th16;
then ex r,s st p = |[r,s]| & G*(1,j)`2 < s by A3;
hence thesis by EUCLID:52;
end;
suppose
j < width G;
then
Int h_strip(G,j) = { |[r,s]| : G*(1,j)`2 < s & s < G*(1,j+1)`2 } by A1,Th17
;
then ex r,s st p = |[r,s]| & G*(1,j)`2 < s & s < G*(1,j+1)`2 by A3;
hence thesis by EUCLID:52;
end;
end;
theorem
j < width G & p in Int h_strip(G,j) implies p`2 < G*(1,j+1)`2
proof
assume that
A1: j < width G and
A2: p in Int h_strip(G,j);
per cases by NAT_1:14;
suppose
j = 0;
then Int h_strip(G,j) = { |[r,s]| : s < G*(1,j+1)`2 } by Th15;
then ex r,s st p = |[r,s]| & G*(1,j+1)`2 > s by A2;
hence thesis by EUCLID:52;
end;
suppose
j >= 1;
then
Int h_strip(G,j) = { |[r,s]| : G*(1,j)`2 < s & s < G*(1,j+1)`2 } by A1,Th17
;
then ex r,s st p = |[r,s]| & G*(1,j)`2 < s & s < G*(1,j+1)`2 by A2;
hence thesis by EUCLID:52;
end;
end;
theorem
1 <= i & i <= len G & p in Int v_strip(G,i) implies p`1 > G*(i,1)`1
proof
assume that
A1: 1 <= i and
A2: i <= len G and
A3: p in Int v_strip(G,i);
per cases by A2,XXREAL_0:1;
suppose
i = len G;
then Int v_strip(G,i) = { |[r,s]| : G*(i,1)`1 < r } by Th13;
then ex r,s st p = |[r,s]| & G*(i,1)`1 < r by A3;
hence thesis by EUCLID:52;
end;
suppose
i < len G;
then
Int v_strip(G,i) = { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 } by A1,Th14
;
then ex r,s st p = |[r,s]| & G*(i,1)`1 < r & r < G*(i+1,1)`1 by A3;
hence thesis by EUCLID:52;
end;
end;
theorem
i < len G & p in Int v_strip(G,i) implies p`1 < G*(i+1,1)`1
proof
assume that
A1: i < len G and
A2: p in Int v_strip(G,i);
per cases by NAT_1:14;
suppose
i = 0;
then Int v_strip(G,i) = { |[r,s]| : r < G*(i+1,1)`1 } by Th12;
then ex r,s st p = |[r,s]| & G*(i+1,1)`1 > r by A2;
hence thesis by EUCLID:52;
end;
suppose
i >= 1;
then
Int v_strip(G,i) = { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 } by A1,Th14
;
then ex r,s st p = |[r,s]| & G*(i,1)`1 < r & r < G*(i+1,1)`1 by A2;
hence thesis by EUCLID:52;
end;
end;
theorem Th31:
1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G implies 1/2*(G*(
i,j)+G*(i+1,j+1)) in Int cell(G,i,j)
proof
assume that
A1: 1 <= i and
A2: i+1 <= len G and
A3: 1 <= j and
A4: j+1 <= width G;
A5: j < j+1 by XREAL_1:29;
set r1 = G*(i,j)`1, s1 = G*(i,j)`2, r2 = G*(i+1,j+1)`1, s2 = G* (i+1,j+1)`2;
A6: 1 <= i+1 & 1 <= j+1 by NAT_1:11;
then
A7: G*(1,j+1)`2 = s2 by A2,A4,GOBOARD5:1;
i < len G & j < width G by A2,A4,NAT_1:13;
then
A8: Int cell(G,i,j) = { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,j)
`2 < s & s < G*(1,j+1)`2 } by A1,A3,Th26;
G*(i,j) = |[r1,s1]| & G*(i+1,j+1) = |[r2,s2]| by EUCLID:53;
then G*(i,j)+G*(i+1,j+1) = |[r1+r2,s1+s2]| by EUCLID:56;
then
A9: 1/2*(G*(i,j)+G*(i+1,j+1)) = |[1/2*(r1+r2),1/2*(s1+s2)]| by EUCLID:58;
i <= i+1 by NAT_1:11;
then
A10: i <= len G by A2,XXREAL_0:2;
then
A11: 1 <= len G by A1,XXREAL_0:2;
j <= j+1 by NAT_1:11;
then
A12: j <= width G by A4,XXREAL_0:2;
then
A13: 1 <= width G by A3,XXREAL_0:2;
A14: G*(i,1)`1 = r1 by A1,A3,A10,A12,GOBOARD5:2;
G*(1,j)`2 = s1 by A1,A3,A10,A12,GOBOARD5:1;
then
A15: s1 < s2 by A3,A4,A7,A11,A5,GOBOARD5:4;
then s1+s1 < s1+s2 by XREAL_1:6;
then 1/2*(s1+s1) < 1/2*(s1+s2) by XREAL_1:68;
then
A16: G*(1,j)`2 < 1/2*(s1+s2) by A1,A3,A10,A12,GOBOARD5:1;
A17: i < i+1 by XREAL_1:29;
G*(i+1,1)`1 = r2 by A2,A4,A6,GOBOARD5:2;
then
A18: r1 < r2 by A1,A2,A14,A13,A17,GOBOARD5:3;
then r1+r2 < r2+r2 by XREAL_1:6;
then 1/2*(r1+r2) < 1/2*(r2+r2) by XREAL_1:68;
then
A19: 1/2*(r1+r2) < G*(i+1,1)`1 by A2,A4,A6,GOBOARD5:2;
s1+s2 < s2+s2 by A15,XREAL_1:6;
then 1/2*(s1+s2) < 1/2*(s2+s2) by XREAL_1:68;
then
A20: 1/2*(s1+s2) < G*(1,j+1)`2 by A2,A4,A6,GOBOARD5:1;
r1+r1 < r1+r2 by A18,XREAL_1:6;
then 1/2*(r1+r1) < 1/2*(r1+r2) by XREAL_1:68;
hence thesis by A9,A14,A19,A16,A20,A8;
end;
theorem Th32:
1 <= i & i+1 <= len G implies 1/2*(G*(i,width G)+G*(i+1,width G)
)+|[0,1]| in Int cell(G,i,width G)
proof
assume that
A1: 1 <= i and
A2: i+1 <= len G;
set r1 = G*(i,width G)`1, s1 = G*(i,width G)`2, r2 = G*(i+1,width G)`1;
width G <> 0 by MATRIX_0:def 10;
then
A3: 1 <= width G by NAT_1:14;
i < len G by A2,NAT_1:13;
then
A4: Int cell(G,i,width G) = { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & G
* (1,width G)`2 < s } by A1,Th25;
width G <> 0 by MATRIX_0:def 10;
then
A5: 1 <= width G by NAT_1:14;
i < i+1 by XREAL_1:29;
then
A6: r1 < r2 by A1,A2,A5,GOBOARD5:3;
then r1+r1 < r1+r2 by XREAL_1:6;
then
A7: 1/2*(r1+r1) < 1/2*(r1+r2) by XREAL_1:68;
A8: i < len G by A2,NAT_1:13;
then
A9: G*(1,width G)`2 = s1 by A1,A3,GOBOARD5:1;
then
A10: G*(1,width G)`2 < s1+1 by XREAL_1:29;
A11: 1 <= i+1 by NAT_1:11;
then G*(1,width G)`2 = G*(i+1,width G)`2 by A2,A3,GOBOARD5:1;
then G*(i,width G) = |[r1,s1]| & G*(i+1,width G) = |[r2,s1]| by A9,EUCLID:53;
then 1/2*(s1+s1) = s1 & G*(i,width G)+G*(i+1,width G) = |[r1+r2,s1+s1]| by
EUCLID:56;
then 1/2*(G*(i,width G)+G*(i+1,width G))= |[1/2*(r1+r2),s1]| by EUCLID:58;
then
A12: 1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]| = |[1/2*(r1+r2)+0,s1+1 ]|
by EUCLID:56;
r1+r2 < r2+r2 by A6,XREAL_1:6;
then 1/2*(r1+r2) < 1/2*(r2+r2) by XREAL_1:68;
then
A13: 1/2*(r1+r2) < G*(i+1,1)`1 by A2,A11,A3,GOBOARD5:2;
G*(i,1)`1 = r1 by A1,A8,A3,GOBOARD5:2;
hence thesis by A12,A7,A13,A10,A4;
end;
theorem Th33:
1 <= i & i+1 <= len G implies 1/2*(G*(i,1)+G*(i+1,1))-|[0,1]| in
Int cell(G,i,0)
proof
assume that
A1: 1 <= i and
A2: i+1 <= len G;
set r1 = G*(i,1)`1, s1 = G*(i,1)`2, r2 = G*(i+1,1)`1;
width G <> 0 by MATRIX_0:def 10;
then
A3: 1 <= width G by NAT_1:14;
width G <> 0 by MATRIX_0:def 10;
then
A4: 1 <= width G by NAT_1:14;
i < i+1 by XREAL_1:29;
then
A5: r1 < r2 by A1,A2,A4,GOBOARD5:3;
then r1+r1 < r1+r2 by XREAL_1:6;
then
A6: 1/2*(r1+r1) < 1/2*(r1+r2) by XREAL_1:68;
i < len G by A2,NAT_1:13;
then
A7: G*(1,1)`2 = s1 by A1,A3,GOBOARD5:1;
then s1 < G*(1,1)`2+1 by XREAL_1:29;
then
A8: s1-1 < G*(1,1)`2 by XREAL_1:19;
1 <= i+1 by NAT_1:11;
then G*(1,1)`2 = G*(i+1,1)`2 by A2,A3,GOBOARD5:1;
then G*(i,1) = |[r1,s1]| & G*(i+1,1) = |[r2,s1]| by A7,EUCLID:53;
then 1/2*(s1+s1) = s1 & G*(i,1)+G*(i+1,1) = |[r1+r2,s1+s1]| by EUCLID:56;
then 1/2*(G*(i,1)+G*(i+1,1))= |[1/2*(r1+r2),s1]| by EUCLID:58;
then
A9: 1/2*(G*(i,1)+G*(i+1,1))-|[0,1]| = |[1/2*(r1+r2)-0,s1-1]| by EUCLID:62
.= |[1/2*(r1+r2),s1-1]|;
r1+r2 < r2+r2 by A5,XREAL_1:6;
then
A10: 1/2*(r1+r2) < 1/2*(r2+r2) by XREAL_1:68;
i < len G by A2,NAT_1:13;
then
Int cell(G,i,0) = { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & s < G*(
1,1)`2 } by A1,Th24;
hence thesis by A9,A6,A10,A8;
end;
theorem Th34:
1 <= j & j+1 <= width G implies 1/2*(G*(len G,j)+G*(len G,j+1))+
|[1,0]| in Int cell(G,len G,j)
proof
assume that
A1: 1 <= j and
A2: j+1 <= width G;
set s1 = G*(len G,j)`2, r1 = G*(len G,j)`1, s2 = G*(len G,j+1)`2;
len G <> 0 by MATRIX_0:def 10;
then
A3: 1 <= len G by NAT_1:14;
j < width G by A2,NAT_1:13;
then
A4: Int cell(G,len G,j) = { |[r,s]| : G*(len G,1)`1 < r & G*(1,j)`2 < s & s
< G* (1,j+1)`2 } by A1,Th23;
len G <> 0 by MATRIX_0:def 10;
then
A5: 1 <= len G by NAT_1:14;
j < j+1 by XREAL_1:29;
then
A6: s1 < s2 by A1,A2,A5,GOBOARD5:4;
then s1+s1 < s1+s2 by XREAL_1:6;
then
A7: 1/2*(s1+s1) < 1/2*(s1+s2) by XREAL_1:68;
A8: j < width G by A2,NAT_1:13;
then
A9: G*(len G,1)`1 = r1 by A1,A3,GOBOARD5:2;
then
A10: G*(len G,1)`1 < r1+1 by XREAL_1:29;
A11: 1 <= j+1 by NAT_1:11;
then G*(len G,1)`1 = G*(len G,j+1)`1 by A2,A3,GOBOARD5:2;
then G*(len G,j) = |[r1,s1]| & G*(len G,j+1) = |[r1,s2]| by A9,EUCLID:53;
then 1/2*(r1+r1) = r1 & G*(len G,j)+G*(len G,j+1) = |[r1+r1,s1+s2]| by
EUCLID:56;
then 1/2*(G*(len G,j)+G*(len G,j+1))= |[r1,1/2*(s1+s2)]| by EUCLID:58;
then
A12: 1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]| = |[r1+1,1/2*(s1+s2)+0]| by
EUCLID:56;
s1+s2 < s2+s2 by A6,XREAL_1:6;
then 1/2*(s1+s2) < 1/2*(s2+s2) by XREAL_1:68;
then
A13: 1/2*(s1+s2) < G*(1,j+1)`2 by A2,A11,A3,GOBOARD5:1;
G*(1,j)`2 = s1 by A1,A8,A3,GOBOARD5:1;
hence thesis by A12,A7,A13,A10,A4;
end;
theorem Th35:
1 <= j & j+1 <= width G implies 1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|
in Int cell(G,0,j)
proof
assume that
A1: 1 <= j and
A2: j+1 <= width G;
set s1 = G*(1,j)`2, r1 = G*(1,j)`1, s2 = G*(1,j+1)`2;
len G <> 0 by MATRIX_0:def 10;
then
A3: 1 <= len G by NAT_1:14;
len G <> 0 by MATRIX_0:def 10;
then
A4: 1 <= len G by NAT_1:14;
j < j+1 by XREAL_1:29;
then
A5: s1 < s2 by A1,A2,A4,GOBOARD5:4;
then s1+s1 < s1+s2 by XREAL_1:6;
then
A6: 1/2*(s1+s1) < 1/2*(s1+s2) by XREAL_1:68;
j < width G by A2,NAT_1:13;
then
A7: G*(1,1)`1 = r1 by A1,A3,GOBOARD5:2;
then r1 < G*(1,1)`1+1 by XREAL_1:29;
then
A8: r1-1 < G*(1,1)`1 by XREAL_1:19;
1 <= j+1 by NAT_1:11;
then G*(1,1)`1 = G*(1,j+1)`1 by A2,A3,GOBOARD5:2;
then G*(1,j) = |[r1,s1]| & G*(1,j+1) = |[r1,s2]| by A7,EUCLID:53;
then 1/2*(r1+r1) = r1 & G*(1,j)+G*(1,j+1) = |[r1+r1,s1+s2]| by EUCLID:56;
then 1/2*(G*(1,j)+G*(1,j+1))= |[r1,1/2*(s1+s2)]| by EUCLID:58;
then
A9: 1/2*(G*(1,j)+G*(1,j+1))-|[1,0]| = |[r1-1,1/2*(s1+s2)-0]| by EUCLID:62
.= |[r1-1,1/2*(s1+s2)]|;
s1+s2 < s2+s2 by A5,XREAL_1:6;
then
A10: 1/2*(s1+s2) < 1/2*(s2+s2) by XREAL_1:68;
j < width G by A2,NAT_1:13;
then
Int cell(G,0,j) = { |[r,s]| : r < G*(1,1)`1 & G*(1,j)`2 < s & s < G*(1,
j+1)`2 } by A1,Th20;
hence thesis by A9,A6,A10,A8;
end;
theorem Th36:
G*(1,1)-|[1,1]| in Int cell(G,0,0)
proof
set s1 = G*(1,1)`2, r1 = G*(1,1)`1;
G*(1,1) = |[r1,s1]| by EUCLID:53;
then
A1: G*(1,1)-|[1,1]| = |[r1-1,s1-1]| by EUCLID:62;
s1 < G*(1,1)`2+1 by XREAL_1:29;
then
A2: s1-1 < G*(1,1)`2 by XREAL_1:19;
r1 < G*(1,1)`1+1 by XREAL_1:29;
then
A3: r1-1 < G*(1,1)`1 by XREAL_1:19;
Int cell(G,0,0) = { |[r,s]| : r < G*(1,1)`1 & s < G*(1,1)`2 } by Th18;
hence thesis by A1,A2,A3;
end;
theorem Th37:
G*(len G,width G)+|[1,1]| in Int cell(G,len G,width G)
proof
set s1 = G*(len G,width G)`2, r1 = G*(len G,width G)`1;
len G <> 0 by MATRIX_0:def 10;
then
A1: 1 <= len G by NAT_1:14;
width G <> 0 by MATRIX_0:def 10;
then
A2: 1 <= width G by NAT_1:14;
then G*(len G,1)`1 = r1 by A1,GOBOARD5:2;
then
A3: r1+1 > G*(len G,1)`1 by XREAL_1:29;
G*(len G,width G) = |[r1,s1]| by EUCLID:53;
then
A4: G*(len G,width G)+|[1,1]| = |[r1+1,s1+1]| by EUCLID:56;
G*(1,width G)`2 = s1 by A2,A1,GOBOARD5:1;
then
A5: s1+1 > G*(1,width G)`2 by XREAL_1:29;
Int cell(G,len G,width G) = { |[r,s]| : G*(len G,1)`1 < r & G*(1,width
G)`2 < s } by Th22;
hence thesis by A4,A5,A3;
end;
theorem Th38:
G*(1,width G)+|[-1,1]| in Int cell(G,0,width G)
proof
set s1 = G*(1,width G)`2, r1 = G*(1,width G)`1;
len G <> 0 by MATRIX_0:def 10;
then
A1: 1 <= len G by NAT_1:14;
width G <> 0 by MATRIX_0:def 10;
then 1 <= width G by NAT_1:14;
then G*(1,1)`1 = r1 by A1,GOBOARD5:2;
then r1 < G*(1,1)`1+1 by XREAL_1:29;
then
A2: s1+1 > G*(1,width G)`2 & r1-1 < G*(1,1)`1 by XREAL_1:19,29;
G*(1,width G) = |[r1,s1]| by EUCLID:53;
then
A3: G*(1,width G)+|[-1,1]| = |[r1+-1,s1+1]| by EUCLID:56
.= |[r1-1,s1+1]|;
Int cell(G,0,width G) = { |[r,s]| : r < G*(1,1)`1 & G*(1,width G)`2 < s
} by Th19;
hence thesis by A3,A2;
end;
theorem Th39:
G*(len G,1)+|[1,-1]| in Int cell(G,len G,0)
proof
set s1 = G*(len G,1)`2, r1 = G*(len G,1)`1;
A1: r1+1 > G*(len G,1)`1 by XREAL_1:29;
len G <> 0 by MATRIX_0:def 10;
then
A2: 1 <= len G by NAT_1:14;
width G <> 0 by MATRIX_0:def 10;
then 1 <= width G by NAT_1:14;
then G*(1,1)`2 = s1 by A2,GOBOARD5:1;
then s1 < G*(1,1)`2+1 by XREAL_1:29;
then
A3: s1-1 < G*(1,1)`2 by XREAL_1:19;
G*(len G,1) = |[r1,s1]| by EUCLID:53;
then
A4: G*(len G,1)+|[1,-1]| = |[r1+1,s1+-1]| by EUCLID:56
.= |[r1+1,s1-1]|;
Int cell(G,len G,0) = { |[r,s]| : G*(len G,1)`1 < r & s < G* (1,1)`2 }
by Th21;
hence thesis by A4,A3,A1;
end;
theorem Th40:
1 <= i & i < len G & 1 <= j & j < width G implies LSeg(1/2*(G*(i
,j)+G*(i+1,j+1)),1/2*(G*(i,j)+G*(i,j+1))) c= Int cell(G,i,j) \/ { 1/2*(G*(i,j)+
G*(i,j+1)) }
proof
assume that
A1: 1 <= i and
A2: i < len G and
A3: 1 <= j and
A4: j < width G;
let x be object;
assume
A5: x in LSeg(1/2*(G*(i,j)+G*(i+1,j+1)),1/2*(G*(i,j)+G*(i,j+1)));
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A6: p = (1-r)*(1/2*(G*(i,j)+G*(i+1,j+1)))+r*(1/2*(G*(i,j)+G*(i,j+1))) and
A7: 0<=r and
A8: r<=1 by A5;
now
per cases by A8,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(1/2*(G*(i,j)+G*(i,j+1))) by A6,RLVECT_1:10
.= 1*(1/2*(G*(i,j)+G*(i,j+1))) by RLVECT_1:4
.= 1/2*(G*(i,j)+G*(i,j+1)) by RLVECT_1:def 8;
hence p in { 1/2*(G*(i,j)+G*(i,j+1)) } by TARSKI:def 1;
end;
case
A9: r < 1;
set r3 = (1-r)*(1/2), s3 = r*(1/2);
set r1 = G*(i,1)`1, r2 = G*(i+1,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2;
A10: r3*(s1+s1)+s3*(s1+s1) = s1;
0 <> len G by MATRIX_0:def 10;
then
A11: 1 <= len G by NAT_1:14;
A12: j+1 <= width G by A4,NAT_1:13;
j < j+1 by XREAL_1:29;
then
A13: s1 < s2 by A3,A12,A11,GOBOARD5:4;
then
A14: s1+s1 < s1+s2 by XREAL_1:6;
then
A15: s3*(s1+s1) <= s3*(s1+s2) by A7,XREAL_1:64;
1 - r > 0 by A9,XREAL_1:50;
then
A16: r3 > (1/2)*0 by XREAL_1:68;
then r3*(s1+s1) < r3*(s1+s2) by A14,XREAL_1:68;
then
A17: s1 < r3*(s1+s2)+s3*(s1+s2) by A15,A10,XREAL_1:8;
A18: s1+s2 < s2+s2 by A13,XREAL_1:6;
then
A19: s3*(s1+s2) <= s3*(s2+s2) by A7,XREAL_1:64;
0 <> width G by MATRIX_0:def 10;
then
A20: 1 <= width G by NAT_1:14;
A21: 1 <= i+1 by A1,NAT_1:13;
A22: Int cell(G,i,j) = { |[r9,s9]| : r1 < r9 & r9 < r2 & s1 < s9 & s9 <
s2 } by A1,A2,A3,A4,Th26;
A23: 1 <= j+1 by A3,NAT_1:13;
A24: G*(i,j+1) = |[G*(i,j+1)`1,G*(i,j+1)`2]| by EUCLID:53
.= |[r1,G*(i,j+1)`2]| by A1,A2,A23,A12,GOBOARD5:2
.= |[r1,s2]| by A1,A2,A23,A12,GOBOARD5:1;
A25: r3*(s2+s2)+s3*(s2+s2) = s2;
r3*(s1+s2) < r3*(s2+s2) by A16,A18,XREAL_1:68;
then
A26: r3*(s1+s2)+s3*(s1+s2) < s2 by A19,A25,XREAL_1:8;
A27: i+1 <= len G by A2,NAT_1:13;
i < i+1 by XREAL_1:29;
then
A28: r1 < r2 by A1,A27,A20,GOBOARD5:3;
then r1+r1 < r2+r2 by XREAL_1:8;
then
A29: s3*(r1+r1) <= s3*(r2+r2) by A7,XREAL_1:64;
r1+r2 < r2+r2 by A28,XREAL_1:6;
then
A30: r3*(r1+r2) < r3*(r2+r2) by A16,XREAL_1:68;
r3*(r2+r2)+s3*(r2+r2) = r2;
then
A31: r3*(r1+r2)+s3*(r1+r1) < r2 by A30,A29,XREAL_1:8;
A32: G*(i,j) = |[G*(i,j)`1,G*(i,j)`2]| by EUCLID:53
.= |[r1,G*(i,j)`2]| by A1,A2,A3,A4,GOBOARD5:2
.= |[r1,s1]| by A1,A2,A3,A4,GOBOARD5:1;
A33: G*(i+1,j+1) = |[G*(i+1,j+1)`1,G*(i+1,j+1)`2]| by EUCLID:53
.= |[r2,G*(i+1,j+1)`2]| by A23,A12,A21,A27,GOBOARD5:2
.= |[r2,s2]| by A23,A12,A21,A27,GOBOARD5:1;
A34: r3*(r1+r1)+s3*(r1+r1) = r1;
r1+r1 < r1+r2 by A28,XREAL_1:6;
then r3*(r1+r1) < r3*(r1+r2) by A16,XREAL_1:68;
then
A35: r1 < r3*(r1+r2)+s3*(r1+r1) by A34,XREAL_1:6;
p = r3*(G*(i,j)+G*(i+1,j+1))+r*(1/2*(G*(i,j)+G* (i,j+1))) by A6,
RLVECT_1:def 7
.= r3*(G*(i,j)+G*(i+1,j+1))+s3*(G*(i,j)+G*(i,j+1)) by RLVECT_1:def 7
.= r3*|[r1+r2,s1+s2]|+s3*(G*(i,j)+G*(i,j+1)) by A32,A33,EUCLID:56
.= r3*|[r1+r2,s1+s2]|+s3*|[r1+r1,s1+s2]| by A32,A24,EUCLID:56
.= |[r3*(r1+r2),r3*(s1+s2)]|+s3*|[r1+r1,s1+s2]| by EUCLID:58
.= |[r3*(r1+r2),r3*(s1+s2)]|+|[s3*(r1+r1),s3*(s1+s2)]| by EUCLID:58
.= |[r3*(r1+r2)+s3*(r1+r1),r3*(s1+s2)+s3*(s1+s2)]| by EUCLID:56;
hence p in Int cell(G,i,j) by A35,A31,A17,A26,A22;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th41:
1 <= i & i < len G & 1 <= j & j < width G implies LSeg(1/2*(G*(i
,j)+G*(i+1,j+1)),1/2*(G*(i,j+1)+G*(i+1,j+1))) c= Int cell(G,i,j) \/ { 1/2*(G*(i
,j+1)+G*(i+1,j+1)) }
proof
assume that
A1: 1 <= i and
A2: i < len G and
A3: 1 <= j and
A4: j < width G;
let x be object;
assume
A5: x in LSeg(1/2*(G*(i,j)+G*(i+1,j+1)),1/2*(G*(i,j+1)+G*(i+1,j+1)));
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A6: p = (1-r)*(1/2*(G*(i,j)+G*(i+1,j+1)))+r*(1/2*(G*(i,j+1)+G*(i+1,j+1)) ) and
A7: 0<=r and
A8: r<=1 by A5;
now
per cases by A8,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(1/2*(G*(i,j+1)+G*(i+1,j+1))) by A6,RLVECT_1:10
.= 1*(1/2*(G*(i,j+1)+G*(i+1,j+1))) by RLVECT_1:4
.= 1/2*(G*(i,j+1)+G*(i+1,j+1)) by RLVECT_1:def 8;
hence p in { 1/2*(G*(i,j+1)+G*(i+1,j+1)) } by TARSKI:def 1;
end;
case
A9: r < 1;
set r3 = (1-r)*(1/2), s3 = r*(1/2);
set r1 = G*(i,1)`1, r2 = G*(i+1,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2;
A10: r3*(r1+r1)+s3*(r1+r1) = r1;
0 <> width G by MATRIX_0:def 10;
then
A11: 1 <= width G by NAT_1:14;
A12: i+1 <= len G by A2,NAT_1:13;
i < i+1 by XREAL_1:29;
then
A13: r1 < r2 by A1,A12,A11,GOBOARD5:3;
then
A14: r1+r1 < r1+r2 by XREAL_1:6;
then
A15: s3*(r1+r1) <= s3*(r1+r2) by A7,XREAL_1:64;
1 - r > 0 by A9,XREAL_1:50;
then
A16: r3 > (1/2)*0 by XREAL_1:68;
then r3*(r1+r1) < r3*(r1+r2) by A14,XREAL_1:68;
then
A17: r1 < r3*(r1+r2)+s3*(r1+r2) by A15,A10,XREAL_1:8;
0 <> len G by MATRIX_0:def 10;
then
A18: 1 <= len G by NAT_1:14;
A19: 1 <= i+1 by A1,NAT_1:13;
r1+r2 < r2+r2 by A13,XREAL_1:8;
then
A20: s3*(r1+r2) <= s3*(r2+r2) by A7,XREAL_1:64;
A21: j+1 <= width G by A4,NAT_1:13;
r1+r2 < r2+r2 by A13,XREAL_1:6;
then
A22: r3*(r1+r2) < r3*(r2+r2) by A16,XREAL_1:68;
r3*(r2+r2)+s3*(r2+r2) = r2;
then
A23: r3*(r1+r2)+s3*(r1+r2) < r2 by A22,A20,XREAL_1:8;
A24: Int cell(G,i,j) = { |[r9,s9]| : r1 < r9 & r9 < r2 & s1 < s9 & s9 <
s2 } by A1,A2,A3,A4,Th26;
A25: 1 <= j+1 by A3,NAT_1:13;
j < j+1 by XREAL_1:29;
then
A26: s1 < s2 by A3,A21,A18,GOBOARD5:4;
then
A27: s1+s1 < s1+s2 by XREAL_1:6;
A28: G*(i+1,j+1) = |[G*(i+1,j+1)`1,G*(i+1,j+1)`2]| by EUCLID:53
.= |[r2,G*(i+1,j+1)`2]| by A25,A21,A19,A12,GOBOARD5:2
.= |[r2,s2]| by A25,A21,A19,A12,GOBOARD5:1;
s1+s2 < s2+s2 by A26,XREAL_1:6;
then s1+s1 < s2+s2 by A27,XXREAL_0:2;
then
A29: s3*(s1+s1) <= s3*(s2+s2) by A7,XREAL_1:64;
A30: G*(i,j) = |[G*(i,j)`1,G*(i,j)`2]| by EUCLID:53
.= |[r1,G*(i,j)`2]| by A1,A2,A3,A4,GOBOARD5:2
.= |[r1,s1]| by A1,A2,A3,A4,GOBOARD5:1;
A31: r3*(s2+s2)+s3*(s2+s2) = s2;
A32: G*(i,j+1) = |[G*(i,j+1)`1,G*(i,j+1)`2]| by EUCLID:53
.= |[r1,G*(i,j+1)`2]| by A1,A2,A25,A21,GOBOARD5:2
.= |[r1,s2]| by A1,A2,A25,A21,GOBOARD5:1;
A33: r3*(s1+s1)+s3*(s1+s1) = s1;
s1+s2 < s2+s2 by A26,XREAL_1:6;
then r3*(s1+s2) < r3*(s2+s2) by A16,XREAL_1:68;
then
A34: r3*(s1+s2)+s3*(s2+s2) < s2 by A31,XREAL_1:8;
r3*(s1+s1) < r3*(s1+s2) by A16,A27,XREAL_1:68;
then
A35: s1 < r3*(s1+s2)+s3*(s2+s2) by A29,A33,XREAL_1:8;
p = r3*(G*(i,j)+G*(i+1,j+1))+r*(1/2*(G*(i,j+1)+G*(i+1,j+1))) by A6,
RLVECT_1:def 7
.= r3*(G*(i,j)+G*(i+1,j+1))+s3*(G*(i,j+1)+G*(i+1,j+1)) by
RLVECT_1:def 7
.= r3*|[r1+r2,s1+s2]|+s3*(G*(i,j+1)+G*(i+1,j+1)) by A30,A28,EUCLID:56
.= r3*|[r1+r2,s1+s2]|+s3*|[r1+r2,s2+s2]| by A28,A32,EUCLID:56
.= |[r3*(r1+r2),r3*(s1+s2)]|+s3*|[r1+r2,s2+s2]| by EUCLID:58
.= |[r3*(r1+r2),r3*(s1+s2)]|+|[s3*(r1+r2),s3*(s2+s2)]| by EUCLID:58
.= |[r3*(r1+r2)+s3*(r1+r2),r3*(s1+s2)+s3*(s2+s2)]| by EUCLID:56;
hence p in Int cell(G,i,j) by A17,A23,A35,A34,A24;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th42:
1 <= i & i < len G & 1 <= j & j < width G implies LSeg(1/2*(G*(i
,j)+G*(i+1,j+1)),1/2*(G*(i+1,j)+G*(i+1,j+1))) c= Int cell(G,i,j) \/ { 1/2*(G*(i
+1,j)+G*(i+1,j+1)) }
proof
assume that
A1: 1 <= i and
A2: i < len G and
A3: 1 <= j and
A4: j < width G;
let x be object;
assume
A5: x in LSeg(1/2*(G*(i,j)+G*(i+1,j+1)),1/2*(G*(i+1,j)+G*(i+1,j+1)));
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A6: p = (1-r)*(1/2*(G*(i,j)+G*(i+1,j+1)))+r*(1/2*(G*(i+1,j)+G*(i+1,j+1)) ) and
A7: 0<=r and
A8: r<=1 by A5;
now
per cases by A8,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(1/2*(G*(i+1,j)+G*(i+1,j+1))) by A6,RLVECT_1:10
.= 1*(1/2*(G*(i+1,j)+G*(i+1,j+1))) by RLVECT_1:4
.= 1/2*(G*(i+1,j)+G*(i+1,j+1)) by RLVECT_1:def 8;
hence p in { 1/2*(G*(i+1,j)+G*(i+1,j+1)) } by TARSKI:def 1;
end;
case
A9: r < 1;
set r3 = (1-r)*(1/2), s3 = r*(1/2);
set r1 = G*(i,1)`1, r2 = G*(i+1,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2;
A10: r3*(r1+r1)+s3*(r1+r1) = r1;
0 <> width G by MATRIX_0:def 10;
then
A11: 1 <= width G by NAT_1:14;
A12: i+1 <= len G by A2,NAT_1:13;
i < i+1 by XREAL_1:29;
then
A13: r1 < r2 by A1,A12,A11,GOBOARD5:3;
then
A14: r1+r1 < r1+r2 by XREAL_1:6;
r1+r2 < r2+r2 by A13,XREAL_1:6;
then r1+r1 < r2+r2 by A14,XXREAL_0:2;
then
A15: s3*(r1+r1) <= s3*(r2+r2) by A7,XREAL_1:64;
1 - r > 0 by A9,XREAL_1:50;
then
A16: r3 > (1/2)*0 by XREAL_1:68;
then r3*(r1+r1) < r3*(r1+r2) by A14,XREAL_1:68;
then
A17: r1 < r3*(r1+r2)+s3*(r2+r2) by A15,A10,XREAL_1:8;
0 <> len G by MATRIX_0:def 10;
then
A18: 1 <= len G by NAT_1:14;
A19: 1 <= j+1 by A3,NAT_1:13;
A20: Int cell(G,i,j) = { |[r9,s9]| : r1 < r9 & r9 < r2 & s1 < s9 & s9 <
s2 } by A1,A2,A3,A4,Th26;
A21: r3*(s2+s2)+s3*(s2+s2) = s2;
A22: G*(i,j) = |[G*(i,j)`1,G*(i,j)`2]| by EUCLID:53
.= |[r1,G*(i,j)`2]| by A1,A2,A3,A4,GOBOARD5:2
.= |[r1,s1]| by A1,A2,A3,A4,GOBOARD5:1;
A23: r3*(s1+s1)+s3*(s1+s1) = s1;
A24: 1 <= i+1 by A1,NAT_1:13;
A25: G*(i+1,j) = |[G*(i+1,j)`1,G*(i+1,j)`2]| by EUCLID:53
.= |[r2,G*(i+1,j)`2]| by A3,A4,A24,A12,GOBOARD5:2
.= |[r2,s1]| by A3,A4,A24,A12,GOBOARD5:1;
A26: r3*(r2+r2)+s3*(r2+r2) = r2;
r1+r2 < r2+r2 by A13,XREAL_1:6;
then r3*(r1+r2) < r3*(r2+r2) by A16,XREAL_1:68;
then
A27: r3*(r1+r2)+s3*(r2+r2) < r2 by A26,XREAL_1:8;
A28: j+1 <= width G by A4,NAT_1:13;
A29: G*(i+1,j+1) = |[G*(i+1,j+1)`1,G*(i+1,j+1)`2]| by EUCLID:53
.= |[r2,G*(i+1,j+1)`2]| by A19,A28,A24,A12,GOBOARD5:2
.= |[r2,s2]| by A19,A28,A24,A12,GOBOARD5:1;
j < j+1 by XREAL_1:29;
then
A30: s1 < s2 by A3,A28,A18,GOBOARD5:4;
then
A31: s1+s1 < s1+s2 by XREAL_1:6;
then
A32: s3*(s1+s1) <= s3*(s1+s2) by A7,XREAL_1:64;
r3*(s1+s1) < r3*(s1+s2) by A16,A31,XREAL_1:68;
then
A33: s1 < r3*(s1+s2)+s3*(s1+s2) by A32,A23,XREAL_1:8;
A34: s1+s2 < s2+s2 by A30,XREAL_1:6;
then
A35: s3*(s1+s2) <= s3*(s2+s2) by A7,XREAL_1:64;
r3*(s1+s2) < r3*(s2+s2) by A16,A34,XREAL_1:68;
then
A36: r3*(s1+s2)+s3*(s1+s2) < s2 by A35,A21,XREAL_1:8;
p = r3*(G*(i,j)+G*(i+1,j+1))+r*(1/2*(G*(i+1,j)+G*(i+1,j+1))) by A6,
RLVECT_1:def 7
.= r3*(G*(i,j)+G*(i+1,j+1))+s3*(G*(i+1,j)+G*(i+1,j+1)) by
RLVECT_1:def 7
.= r3*|[r1+r2,s1+s2]|+s3*(G*(i+1,j)+G*(i+1,j+1)) by A22,A29,EUCLID:56
.= r3*|[r1+r2,s1+s2]|+s3*|[r2+r2,s1+s2]| by A29,A25,EUCLID:56
.= |[r3*(r1+r2),r3*(s1+s2)]|+s3*|[r2+r2,s1+s2]| by EUCLID:58
.= |[r3*(r1+r2),r3*(s1+s2)]|+|[s3*(r2+r2),s3*(s1+s2)]| by EUCLID:58
.= |[r3*(r1+r2)+s3*(r2+r2),r3*(s1+s2)+s3*(s1+s2)]| by EUCLID:56;
hence p in Int cell(G,i,j) by A17,A27,A33,A36,A20;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th43:
1 <= i & i < len G & 1 <= j & j < width G implies LSeg(1/2*(G*(i
,j)+G*(i+1,j+1)),1/2*(G*(i,j)+G*(i+1,j))) c= Int cell(G,i,j) \/ { 1/2*(G*(i,j)+
G*(i+1,j)) }
proof
assume that
A1: 1 <= i and
A2: i < len G and
A3: 1 <= j and
A4: j < width G;
let x be object;
assume
A5: x in LSeg(1/2*(G*(i,j)+G*(i+1,j+1)),1/2*(G*(i,j)+G*(i+1,j)));
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A6: p = (1-r)*(1/2*(G*(i,j)+G*(i+1,j+1)))+r*(1/2*(G*(i,j)+G*(i+1,j))) and
A7: 0<=r and
A8: r<=1 by A5;
now
per cases by A8,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(1/2*(G*(i,j)+G*(i+1,j))) by A6,RLVECT_1:10
.= 1*(1/2*(G*(i,j)+G*(i+1,j))) by RLVECT_1:4
.= 1/2*(G*(i,j)+G*(i+1,j)) by RLVECT_1:def 8;
hence p in { 1/2*(G*(i,j)+G*(i+1,j)) } by TARSKI:def 1;
end;
case
A9: r < 1;
set r3 = (1-r)*(1/2), s3 = r*(1/2);
set r1 = G*(i,1)`1, r2 = G*(i+1,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2;
A10: r3*(r1+r1)+s3*(r1+r1) = r1;
0 <> width G by MATRIX_0:def 10;
then
A11: 1 <= width G by NAT_1:14;
A12: i+1 <= len G by A2,NAT_1:13;
i < i+1 by XREAL_1:29;
then
A13: r1 < r2 by A1,A12,A11,GOBOARD5:3;
then
A14: r1+r1 < r1+r2 by XREAL_1:6;
then
A15: s3*(r1+r1) <= s3*(r1+r2) by A7,XREAL_1:64;
r1+r2 < r2+r2 by A13,XREAL_1:8;
then
A16: s3*(r1+r2) <= s3*(r2+r2) by A7,XREAL_1:64;
A17: 1 <= i+1 by A1,NAT_1:13;
1 - r > 0 by A9,XREAL_1:50;
then
A18: r3 > (1/2)*0 by XREAL_1:68;
then r3*(r1+r1) < r3*(r1+r2) by A14,XREAL_1:68;
then
A19: r1 < r3*(r1+r2)+s3*(r1+r2) by A15,A10,XREAL_1:8;
r1+r2 < r2+r2 by A13,XREAL_1:6;
then
A20: r3*(r1+r2) < r3*(r2+r2) by A18,XREAL_1:68;
r3*(r2+r2)+s3*(r2+r2) = r2;
then
A21: r3*(r1+r2)+s3*(r1+r2) < r2 by A20,A16,XREAL_1:8;
A22: Int cell(G,i,j) = { |[r9,s9]| : r1 < r9 & r9 < r2 & s1 < s9 & s9 <
s2 } by A1,A2,A3,A4,Th26;
A23: j+1 <= width G by A4,NAT_1:13;
A24: G*(i+1,j) = |[G*(i+1,j)`1,G*(i+1,j)`2]| by EUCLID:53
.= |[r2,G*(i+1,j)`2]| by A3,A4,A17,A12,GOBOARD5:2
.= |[r2,s1]| by A3,A4,A17,A12,GOBOARD5:1;
A25: 1 <= j+1 by A3,NAT_1:13;
0 <> len G by MATRIX_0:def 10;
then
A26: 1 <= len G by NAT_1:14;
A27: G*(i,j) = |[G*(i,j)`1,G*(i,j)`2]| by EUCLID:53
.= |[r1,G*(i,j)`2]| by A1,A2,A3,A4,GOBOARD5:2
.= |[r1,s1]| by A1,A2,A3,A4,GOBOARD5:1;
j < j+1 by XREAL_1:29;
then
A28: s1 < s2 by A3,A23,A26,GOBOARD5:4;
then s1+s2 < s2+s2 by XREAL_1:6;
then
A29: r3*(s1+s2) < r3*(s2+s2) by A18,XREAL_1:68;
A30: G*(i+1,j+1) = |[G*(i+1,j+1)`1,G*(i+1,j+1)`2]| by EUCLID:53
.= |[r2,G*(i+1,j+1)`2]| by A25,A23,A17,A12,GOBOARD5:2
.= |[r2,s2]| by A25,A23,A17,A12,GOBOARD5:1;
A31: r3*(s1+s1)+s3*(s1+s1) = s1;
A32: s1+s1 < s1+s2 by A28,XREAL_1:6;
then r3*(s1+s1) < r3*(s1+s2) by A18,XREAL_1:68;
then
A33: s1 < r3*(s1+s2)+s3*(s1+s1) by A31,XREAL_1:8;
s1+s2 < s2+s2 by A28,XREAL_1:6;
then s1+s1 < s2+s2 by A32,XXREAL_0:2;
then
A34: s3*(s1+s1) <= s3*(s2+s2) by A7,XREAL_1:64;
r3*(s2+s2)+s3*(s2+s2) = s2;
then
A35: r3*(s1+s2)+s3*(s1+s1) < s2 by A29,A34,XREAL_1:8;
p = r3*(G*(i,j)+G*(i+1,j+1))+r*(1/2*(G*(i,j)+G* (i+1,j))) by A6,
RLVECT_1:def 7
.= r3*(G*(i,j)+G*(i+1,j+1))+s3*(G*(i,j)+G*(i+1,j)) by RLVECT_1:def 7
.= r3*|[r1+r2,s1+s2]|+s3*(G*(i,j)+G*(i+1,j)) by A27,A30,EUCLID:56
.= r3*|[r1+r2,s1+s2]|+s3*|[r1+r2,s1+s1]| by A27,A24,EUCLID:56
.= |[r3*(r1+r2),r3*(s1+s2)]|+s3*|[r1+r2,s1+s1]| by EUCLID:58
.= |[r3*(r1+r2),r3*(s1+s2)]|+|[s3*(r1+r2),s3*(s1+s1)]| by EUCLID:58
.= |[r3*(r1+r2)+s3*(r1+r2),r3*(s1+s2)+s3*(s1+s1)]| by EUCLID:56;
hence p in Int cell(G,i,j) by A19,A21,A33,A35,A22;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th44:
1 <= j & j < width G implies LSeg(1/2*(G*(1,j)+G*(1,j+1)) - |[1,
0]|,1/2*(G*(1,j)+G*(1,j+1))) c= Int cell(G,0,j) \/ { 1/2*(G*(1,j)+G*(1,j+1)) }
proof
assume that
A1: 1 <= j and
A2: j < width G;
let x be object;
assume
A3: x in LSeg(1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|,1/2*(G*(1,j)+G*(1,j+1)));
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A4: p = (1-r)*(1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|)+r*(1/2*(G*(1,j)+G* (1,j+
1))) and
A5: 0<=r and
A6: r<=1 by A3;
now
per cases by A6,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(1/2*(G*(1,j)+G*(1,j+1))) by A4,RLVECT_1:10
.= 1*(1/2*(G*(1,j)+G*(1,j+1))) by RLVECT_1:4
.= 1/2*(G*(1,j)+G*(1,j+1)) by RLVECT_1:def 8;
hence p in { 1/2*(G*(1,j)+G*(1,j+1)) } by TARSKI:def 1;
end;
case
A7: r < 1;
set r3 = (1-r)*(1/2), s3 = r*(1/2);
set r2 = G*(1,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2;
A8: r3*(s1+s1)+s3*(s1+s1) = s1;
A9: j+1 <= width G by A2,NAT_1:13;
0 <> len G by MATRIX_0:def 10;
then
A10: 1 <= len G by NAT_1:14;
j < j+1 by XREAL_1:29;
then
A11: s1 < s2 by A1,A9,A10,GOBOARD5:4;
then
A12: s1+s1 < s1+s2 by XREAL_1:6;
then
A13: s3*(s1+s1) <= s3*(s1+s2) by A5,XREAL_1:64;
A14: 1 - r > 0 by A7,XREAL_1:50;
then
A15: r3 > (1/2)*0 by XREAL_1:68;
then r3*(s1+s1) < r3*(s1+s2) by A12,XREAL_1:68;
then
A16: s1 < r3*(s1+s2)+s3*(s1+s2) by A13,A8,XREAL_1:8;
r2 < r2+(1-r) by A14,XREAL_1:29;
then
A17: r2-(1-r) < r2 by XREAL_1:19;
A18: 1 <= j+1 by A1,NAT_1:13;
A19: G*(1,j+1) = |[G*(1,j+1)`1,G*(1,j+1)`2]| by EUCLID:53
.= |[r2,s2]| by A18,A9,A10,GOBOARD5:2;
A20: s1+s2 < s2+s2 by A11,XREAL_1:6;
then
A21: s3*(s1+s2) <= s3*(s2+s2) by A5,XREAL_1:64;
A22: Int cell(G,0,j) = { |[r9,s9]| : r9 < G*(1,1)`1 & G*(1,j)`2 < s9 &
s9 < G*(1,j+1)`2 } by A1,A2,Th20;
A23: r3*(s2+s2)+s3*(s2+s2) = s2;
r3*(s1+s2) < r3*(s2+s2) by A15,A20,XREAL_1:68;
then
A24: r3*(s1+s2)+s3*(s1+s2) < s2 by A21,A23,XREAL_1:8;
A25: G*(1,j) = |[G*(1,j)`1,G*(1,j)`2]| by EUCLID:53
.= |[r2,s1]| by A1,A2,A10,GOBOARD5:2;
p = (1-r)*(1/2*(G*(1,j)+G*(1,j+1)))-(1-r)*|[1,0]| +r*(1/2*(G*(1,j)+
G*(1,j+1))) by A4,RLVECT_1:34
.= r3*(G*(1,j)+G*(1,j+1))-(1-r)*|[1,0]|+r*(1/2*(G*(1,j)+G*(1,j+1)))
by RLVECT_1:def 7
.= r3*(G*(1,j)+G*(1,j+1))-|[(1-r)*1,(1-r)*0]|+r*(1/2*(G*(1,j)+G* (1,
j+1))) by EUCLID:58
.= r3*(G*(1,j)+G*(1,j+1))-|[1-r,0]|+s3*(G*(1,j)+G* (1,j+1)) by
RLVECT_1:def 7
.= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+s3*(G*(1,j)+G*(1,j+1)) by A19,A25,
EUCLID:56
.= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+s3*|[r2+r2,s1+s2]| by A19,A25,EUCLID:56
.= |[r3*(r2+r2),r3*(s1+s2)]|-|[1-r,0]|+s3*|[r2+r2,s1+s2]| by EUCLID:58
.= |[r3*(r2+r2),r3*(s1+s2)]|-|[1-r,0]|+|[s3*(r2+r2),s3*(s1+s2)]| by
EUCLID:58
.= |[r3*(r2+r2)-(1-r),r3*(s1+s2)-0]|+|[s3*(r2+r2),s3*(s1+s2)]| by
EUCLID:62
.= |[r3*(r2+r2)-(1-r)+s3*(r2+r2),r3*(s1+s2)+s3*(s1+s2)]| by EUCLID:56;
hence p in Int cell(G,0,j) by A17,A16,A24,A22;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th45:
1 <= j & j < width G implies LSeg(1/2*(G*(len G,j)+G*(len G,j+1)
) + |[1,0]|,1/2*(G*(len G,j)+G*(len G,j+1))) c= Int cell(G,len G,j) \/ { 1/2*(G
*(len G,j)+G*(len G,j+1)) }
proof
assume that
A1: 1 <= j and
A2: j < width G;
let x be object;
assume
A3: x in LSeg(1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|, 1/2*(G*(len G,j)+
G*(len G,j+1)));
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A4: p = (1-r)*(1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|) +r*(1/2*(G*(len
G,j)+G*(len G,j+1))) and
A5: 0<=r and
A6: r<=1 by A3;
now
per cases by A6,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(1/2*(G*(len G,j)+G*(len G,j+1))) by A4,
RLVECT_1:10
.= 1*(1/2*(G*(len G,j)+G*(len G,j+1))) by RLVECT_1:4
.= 1/2*(G*(len G,j)+G*(len G,j+1)) by RLVECT_1:def 8;
hence p in { 1/2*(G*(len G,j)+G*(len G,j+1)) } by TARSKI:def 1;
end;
case
A7: r < 1;
set r3 = (1-r)*(1/2), s3 = r*(1/2);
set r2 = G*(len G,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2;
A8: r3*(s1+s1)+s3*(s1+s1) = s1;
A9: j+1 <= width G by A2,NAT_1:13;
0 <> len G by MATRIX_0:def 10;
then
A10: 1 <= len G by NAT_1:14;
j < j+1 by XREAL_1:29;
then
A11: s1 < s2 by A1,A9,A10,GOBOARD5:4;
then
A12: s1+s1 < s1+s2 by XREAL_1:6;
then
A13: s3*(s1+s1) <= s3*(s1+s2) by A5,XREAL_1:64;
A14: 1 - r > 0 by A7,XREAL_1:50;
then
A15: r3 > (1/2)*0 by XREAL_1:68;
then r3*(s1+s1) < r3*(s1+s2) by A12,XREAL_1:68;
then
A16: s1 < r3*(s1+s2)+s3*(s1+s2) by A13,A8,XREAL_1:8;
A17: r2+(1-r) > r2 by A14,XREAL_1:29;
A18: 1 <= j+1 by A1,NAT_1:13;
A19: s1+s2 < s2+s2 by A11,XREAL_1:6;
then
A20: s3*(s1+s2) <= s3*(s2+s2) by A5,XREAL_1:64;
A21: Int cell(G,len G,j) = { |[r9,s9]| : G*(len G,1)`1 < r9 & G*(1,j)`2
< s9 & s9 < G*(1,j+1)`2 } by A1,A2,Th23;
A22: r3*(s2+s2)+s3*(s2+s2) = s2;
r3*(s1+s2) < r3*(s2+s2) by A15,A19,XREAL_1:68;
then
A23: r3*(s1+s2)+s3*(s1+s2) < s2 by A20,A22,XREAL_1:8;
A24: G*(len G,j) = |[G*(len G,j)`1,G*(len G,j)`2]| by EUCLID:53
.= |[r2,G*(len G,j)`2]| by A1,A2,A10,GOBOARD5:2
.= |[r2,s1]| by A1,A2,A10,GOBOARD5:1;
A25: G*(len G,j+1) = |[G*(len G,j+1)`1,G*(len G,j+1)`2]| by EUCLID:53
.= |[r2,G*(len G,j+1)`2]| by A18,A9,A10,GOBOARD5:2
.= |[r2,s2]| by A18,A9,A10,GOBOARD5:1;
p = (1-r)*(1/2*(G*(len G,j)+G*(len G,j+1)))+(1-r)*|[1,0]| +r*(1/2*(
G*(len G,j)+G*(len G,j+1))) by A4,RLVECT_1:def 5
.= r3*(G*(len G,j)+G*(len G,j+1))+(1-r)*|[1,0]|+ r*(1/2*(G*(len G,j)
+G*(len G,j+1))) by RLVECT_1:def 7
.= r3*(G*(len G,j)+G*(len G,j+1))+|[(1-r)*1,(1-r)*0]|+ r*(1/2*(G*(
len G,j)+G*(len G,j+1))) by EUCLID:58
.= r3*(G*(len G,j)+G*(len G,j+1))+|[1-r,0]|+ s3*(G*(len G,j)+G*(len
G,j+1)) by RLVECT_1:def 7
.= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+s3*(G*(len G,j)+G*(len G,j+1)) by A25
,A24,EUCLID:56
.= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+s3*|[r2+r2,s1+s2]| by A25,A24,EUCLID:56
.= |[r3*(r2+r2),r3*(s1+s2)]|+|[1-r,0]|+s3*|[r2+r2,s1+s2]| by EUCLID:58
.= |[r3*(r2+r2),r3*(s1+s2)]|+|[1-r,0]|+|[s3*(r2+r2),s3*(s1+s2)]| by
EUCLID:58
.= |[r3*(r2+r2)+(1-r),r3*(s1+s2)+0]|+|[s3*(r2+r2),s3*(s1+s2)]| by
EUCLID:56
.= |[r3*(r2+r2)+(1-r)+s3*(r2+r2),r3*(s1+s2)+s3*(s1+s2)]| by EUCLID:56;
hence p in Int cell(G,len G,j) by A17,A16,A23,A21;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th46:
1 <= i & i < len G implies LSeg(1/2*(G*(i,1)+G*(i+1,1)) - |[0,1
]|,1/2*(G*(i,1)+G*(i+1,1))) c= Int cell(G,i,0) \/ { 1/2*(G*(i,1)+G*(i+1,1)) }
proof
assume that
A1: 1 <= i and
A2: i < len G;
let x be object;
assume
A3: x in LSeg(1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|,1/2*(G*(i,1)+G*(i+1,1)));
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A4: p = (1-r)*(1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|)+r*(1/2*(G*(i,1)+G* (i+1,
1))) and
A5: 0<=r and
A6: r<=1 by A3;
now
per cases by A6,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(1/2*(G*(i,1)+G*(i+1,1))) by A4,RLVECT_1:10
.= 1*(1/2*(G*(i,1)+G*(i+1,1))) by RLVECT_1:4
.= 1/2*(G*(i,1)+G*(i+1,1)) by RLVECT_1:def 8;
hence p in { 1/2*(G*(i,1)+G*(i+1,1)) } by TARSKI:def 1;
end;
case
A7: r < 1;
set r3 = (1-r)*(1/2), s3 = r*(1/2);
set s2 = G*(1,1)`2, r1 = G*(i,1)`1, r2 = G*(i+1,1)`1;
A8: r3*(r1+r1)+s3*(r1+r1) = r1;
A9: i+1 <= len G by A2,NAT_1:13;
0 <> width G by MATRIX_0:def 10;
then
A10: 1 <= width G by NAT_1:14;
i < i+1 by XREAL_1:29;
then
A11: r1 < r2 by A1,A9,A10,GOBOARD5:3;
then
A12: r1+r1 < r1+r2 by XREAL_1:6;
then
A13: s3*(r1+r1) <= s3*(r1+r2) by A5,XREAL_1:64;
A14: 1 - r > 0 by A7,XREAL_1:50;
then
A15: r3 > (1/2)*0 by XREAL_1:68;
then r3*(r1+r1) < r3*(r1+r2) by A12,XREAL_1:68;
then
A16: r1 < r3*(r1+r2)+s3*(r1+r2) by A13,A8,XREAL_1:8;
s2 < s2+(1-r) by A14,XREAL_1:29;
then
A17: s2-(1-r) < s2 by XREAL_1:19;
A18: 1 <= i+1 by A1,NAT_1:13;
A19: G*(i+1,1) = |[G*(i+1,1)`1,G*(i+1,1)`2]| by EUCLID:53
.= |[r2,s2]| by A18,A9,A10,GOBOARD5:1;
A20: r1+r2 < r2+r2 by A11,XREAL_1:6;
then
A21: s3*(r1+r2) <= s3*(r2+r2) by A5,XREAL_1:64;
A22: Int cell(G,i,0) = { |[r9,s9]| : G*(i,1)`1 < r9 & r9 < G*(i+1,1)`1 &
s9 < G*(1,1)`2 } by A1,A2,Th24;
A23: r3*(r2+r2)+s3*(r2+r2) = r2;
r3*(r1+r2) < r3*(r2+r2) by A15,A20,XREAL_1:68;
then
A24: r3*(r1+r2)+s3*(r1+r2) < r2 by A21,A23,XREAL_1:8;
A25: G*(i,1) = |[G*(i,1)`1,G*(i,1)`2]| by EUCLID:53
.= |[r1,s2]| by A1,A2,A10,GOBOARD5:1;
p = (1-r)*(1/2*(G*(i,1)+G*(i+1,1)))-(1-r)*|[0,1]| +r*(1/2*(G*(i,1)+
G*(i+1,1))) by A4,RLVECT_1:34
.= r3*(G*(i,1)+G*(i+1,1))-(1-r)*|[0,1]|+r*(1/2*(G*(i,1)+G*(i+1,1)))
by RLVECT_1:def 7
.= r3*(G*(i,1)+G*(i+1,1))-|[(1-r)*0,(1-r)*1]|+r*(1/2*(G*(i,1)+G* (i+
1,1))) by EUCLID:58
.= r3*(G*(i,1)+G*(i+1,1))-|[0,1-r]|+s3*(G*(i,1)+G* (i+1,1)) by
RLVECT_1:def 7
.= r3*|[r1+r2,s2+s2]|-|[0,1-r]|+s3*(G*(i,1)+G*(i+1,1)) by A19,A25,
EUCLID:56
.= r3*|[r1+r2,s2+s2]|-|[0,1-r]|+s3*|[r1+r2,s2+s2]| by A19,A25,EUCLID:56
.= |[r3*(r1+r2),r3*(s2+s2)]|-|[0,1-r]|+s3*|[r1+r2,s2+s2]| by EUCLID:58
.= |[r3*(r1+r2),r3*(s2+s2)]|-|[0,1-r]|+|[s3*(r1+r2),s3*(s2+s2)]| by
EUCLID:58
.= |[r3*(r1+r2)-0,r3*(s2+s2)-(1-r)]|+|[s3*(r1+r2),s3*(s2+s2)]| by
EUCLID:62
.= |[r3*(r1+r2)+s3*(r1+r2),r3*(s2+s2)-(1-r)+s3*(s2+s2)]| by EUCLID:56;
hence p in Int cell(G,i,0) by A17,A16,A24,A22;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th47:
1 <= i & i < len G implies LSeg(1/2*(G*(i,width G)+G*(i+1,width
G))+ |[0,1]|, 1/2*(G*(i,width G)+G*(i+1,width G))) c= Int cell(G,i,width G) \/
{ 1/2*(G*(i,width G)+G*(i+1,width G)) }
proof
assume that
A1: 1 <= i and
A2: i < len G;
let x be object;
assume
A3: x in LSeg(1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]|, 1/2*(G*(i,
width G)+G*(i+1,width G)));
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A4: p = (1-r)*(1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]|) +r*(1/2*(G*(
i,width G)+G*(i+1,width G))) and
A5: 0<=r and
A6: r<=1 by A3;
now
per cases by A6,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(1/2*(G*(i,width G)+G*(i+1,width G))) by A4,
RLVECT_1:10
.= 1*(1/2*(G*(i,width G)+G*(i+1,width G))) by RLVECT_1:4
.= 1/2*(G*(i,width G)+G*(i+1,width G)) by RLVECT_1:def 8;
hence p in { 1/2*(G*(i,width G)+G*(i+1,width G)) } by TARSKI:def 1;
end;
case
A7: r < 1;
set r3 = (1-r)*(1/2), s3 = r*(1/2);
set s2 = G*(1,width G)`2, r1 = G*(i,1)`1, r2 = G*(i+1,1)`1;
A8: r3*(r1+r1)+s3*(r1+r1) = r1;
A9: i+1 <= len G by A2,NAT_1:13;
0 <> width G by MATRIX_0:def 10;
then
A10: 1 <= width G by NAT_1:14;
i < i+1 by XREAL_1:29;
then
A11: r1 < r2 by A1,A9,A10,GOBOARD5:3;
then
A12: r1+r1 < r1+r2 by XREAL_1:6;
then
A13: s3*(r1+r1) <= s3*(r1+r2) by A5,XREAL_1:64;
A14: 1 - r > 0 by A7,XREAL_1:50;
then
A15: r3 > (1/2)*0 by XREAL_1:68;
then r3*(r1+r1) < r3*(r1+r2) by A12,XREAL_1:68;
then
A16: r1 < r3*(r1+r2)+s3*(r1+r2) by A13,A8,XREAL_1:8;
A17: s2+(1-r) > s2 by A14,XREAL_1:29;
A18: 1 <= i+1 by A1,NAT_1:13;
A19: r1+r2 < r2+r2 by A11,XREAL_1:6;
then
A20: s3*(r1+r2) <= s3*(r2+r2) by A5,XREAL_1:64;
A21: Int cell(G,i,width G) = { |[r9,s9]| : G*(i,1)`1 < r9 & r9 < G*(i+1,
1)`1 & G* (1,width G)`2 < s9 } by A1,A2,Th25;
A22: r3*(r2+r2)+s3*(r2+r2) = r2;
r3*(r1+r2) < r3*(r2+r2) by A15,A19,XREAL_1:68;
then
A23: r3*(r1+r2)+s3*(r1+r2) < r2 by A20,A22,XREAL_1:8;
A24: G*(i,width G) = |[G*(i,width G)`1,G*(i,width G)`2]| by EUCLID:53
.= |[G*(i,width G)`1,s2]| by A1,A2,A10,GOBOARD5:1
.= |[r1,s2]| by A1,A2,A10,GOBOARD5:2;
A25: G*(i+1,width G) = |[G*(i+1,width G)`1,G*(i+1,width G)`2]| by EUCLID:53
.= |[G*(i+1,width G)`1,s2]| by A18,A9,A10,GOBOARD5:1
.= |[r2,s2]| by A18,A9,A10,GOBOARD5:2;
p = (1-r)*(1/2*(G*(i,width G)+G*(i+1,width G)))+(1-r)*|[0,1]| +r*(1
/2*(G*(i,width G)+G* (i+1,width G))) by A4,RLVECT_1:def 5
.= r3*(G*(i,width G)+G*(i+1,width G))+(1-r)*|[0,1]|+ r*(1/2*(G*(i,
width G)+G*(i+1,width G))) by RLVECT_1:def 7
.= r3*(G*(i,width G)+G*(i+1,width G))+|[(1-r)*0,(1-r)*1]|+ r*(1/2*(G
*(i,width G)+G*(i+1,width G))) by EUCLID:58
.= r3*(G*(i,width G)+G*(i+1,width G))+|[0,1-r]|+ s3*(G*(i,width G)+G
*(i+1,width G)) by RLVECT_1:def 7
.= r3*|[r1+r2,s2+s2]|+|[0,1-r]|+s3*(G*(i,width G)+G*(i+1,width G))
by A25,A24,EUCLID:56
.= r3*|[r1+r2,s2+s2]|+|[0,1-r]|+s3*|[r1+r2,s2+s2]| by A25,A24,EUCLID:56
.= |[r3*(r1+r2),r3*(s2+s2)]|+|[0,1-r]|+s3*|[r1+r2,s2+s2]| by EUCLID:58
.= |[r3*(r1+r2),r3*(s2+s2)]|+|[0,1-r]|+|[s3*(r1+r2),s3*(s2+s2)]| by
EUCLID:58
.= |[r3*(r1+r2)+0,r3*(s2+s2)+(1-r)]|+|[s3*(r1+r2),s3*(s2+s2)]| by
EUCLID:56
.= |[r3*(r1+r2)+s3*(r1+r2),r3*(s2+s2)+(1-r)+s3*(s2+s2)]| by EUCLID:56;
hence p in Int cell(G,i,width G) by A17,A16,A23,A21;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th48:
1 <= j & j < width G implies LSeg(1/2*(G*(1,j)+G*(1,j+1)) - |[1,
0]|,G*(1,j) - |[1,0]|) c= Int cell(G,0,j) \/ { G*(1,j) - |[1,0]| }
proof
assume that
A1: 1 <= j and
A2: j < width G;
let x be object;
assume
A3: x in LSeg(1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|,G*(1,j) - |[1,0]|);
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A4: p = (1-r)*(1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|)+r*(G*(1,j) - |[1,0]|) and
A5: 0<=r and
A6: r<=1 by A3;
now
per cases by A6,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(G*(1,j) - |[1,0]|) by A4,RLVECT_1:10
.= 1*(G*(1,j) - |[1,0]|) by RLVECT_1:4
.= (G*(1,j) - |[1,0]|) by RLVECT_1:def 8;
hence p in { G*(1,j) - |[1,0]| } by TARSKI:def 1;
end;
case
A7: r < 1;
set r3 = (1-r)*(1/2);
1 - r > 0 by A7,XREAL_1:50;
then
A8: r3 > (1/2)*0 by XREAL_1:68;
set r2 = G*(1,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2;
A9: r3*(s1+s1)+r*s1 = s1;
A10: j+1 <= width G by A2,NAT_1:13;
r2 < r2+1 by XREAL_1:29;
then
A11: r2-1 < r2 by XREAL_1:19;
A12: Int cell(G,0,j) = { |[r9,s9]| : r9 < G*(1,1)`1 & G*(1,j)`2 < s9 &
s9 < G*(1,j+1)`2 } by A1,A2,Th20;
0 <> len G by MATRIX_0:def 10;
then
A13: 1 <= len G by NAT_1:14;
j < j+1 by XREAL_1:29;
then
A14: s1 < s2 by A1,A10,A13,GOBOARD5:4;
then s1+s2 < s2+s2 by XREAL_1:6;
then
A15: r3*(s1+s2) < r3*(s2+s2) by A8,XREAL_1:68;
s1+s1 < s1+s2 by A14,XREAL_1:6;
then r3*(s1+s1) < r3*(s1+s2) by A8,XREAL_1:68;
then
A16: s1 < r3*(s1+s2)+r*s1 by A9,XREAL_1:6;
A17: r3*(s2+s2)+r*s2 = s2;
r*s1 <= r*s2 by A5,A14,XREAL_1:64;
then
A18: r3*(s1+s2)+r*s1 < s2 by A15,A17,XREAL_1:8;
A19: G*(1,j) = |[G*(1,j)`1,G*(1,j)`2]| by EUCLID:53
.= |[r2,s1]| by A1,A2,A13,GOBOARD5:2;
A20: 1 <= j+1 by A1,NAT_1:13;
A21: G*(1,j+1) = |[G*(1,j+1)`1,G*(1,j+1)`2]| by EUCLID:53
.= |[r2,s2]| by A20,A10,A13,GOBOARD5:2;
p = (1-r)*(1/2*(G*(1,j)+G*(1,j+1)))-(1-r)*|[1,0]| +r*(G*(1,j) - |[1
,0]|) by A4,RLVECT_1:34
.= r3*(G*(1,j)+G*(1,j+1))-(1-r)*|[1,0]|+r*(G* (1,j) - |[1,0]|) by
RLVECT_1:def 7
.= r3*(G*(1,j)+G*(1,j+1))-|[(1-r)*1,(1-r)*0]|+r*(G*(1,j) - |[1,0]|)
by EUCLID:58
.= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+r*(|[r2,s1]| - |[1,0]|) by A21,A19,
EUCLID:56
.= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+(r*|[r2,s1]| - r*|[1,0]|) by
RLVECT_1:34
.= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+(|[r*r2,r*s1]| - r*|[1,0]|) by
EUCLID:58
.= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+(|[r*r2,r*s1]| - |[r*1,r*0]|) by
EUCLID:58
.= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+|[r*r2-r,r*s1-0]| by EUCLID:62
.= |[r3*(r2+r2),r3*(s1+s2)]|-|[1-r,0]|+|[r*r2-r,r*s1-0]| by EUCLID:58
.= |[r3*(r2+r2)-(1-r),r3*(s1+s2)-0]|+|[r*r2-r,r*s1-0]| by EUCLID:62
.= |[r3*(r2+r2)-(1-r)+(r*r2-r),r3*(s1+s2)+r*s1]| by EUCLID:56;
hence p in Int cell(G,0,j) by A11,A16,A18,A12;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th49:
1 <= j & j < width G implies LSeg(1/2*(G*(1,j)+G*(1,j+1)) - |[1,
0]|,G*(1,j+1) - |[1,0]|) c= Int cell(G,0,j) \/ { G*(1,j+1) - |[1,0]| }
proof
assume that
A1: 1 <= j and
A2: j < width G;
let x be object;
assume
A3: x in LSeg(1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|,G*(1,j+1) - |[1,0]|);
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A4: p = (1-r)*(1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|)+r*(G*(1,j+1) - |[1,0]|) and
A5: 0<=r and
A6: r<=1 by A3;
now
per cases by A6,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(G*(1,j+1) - |[1,0]|) by A4,RLVECT_1:10
.= 1*(G*(1,j+1) - |[1,0]|) by RLVECT_1:4
.= (G*(1,j+1) - |[1,0]|) by RLVECT_1:def 8;
hence p in { G*(1,j+1) - |[1,0]| } by TARSKI:def 1;
end;
case
A7: r < 1;
set r3 = (1-r)*(1/2);
1 - r > 0 by A7,XREAL_1:50;
then
A8: r3 > (1/2)*0 by XREAL_1:68;
set r2 = G*(1,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2;
A9: r3*(s1+s1)+r*s1 = s1;
A10: j+1 <= width G by A2,NAT_1:13;
0 <> len G by MATRIX_0:def 10;
then
A11: 1 <= len G by NAT_1:14;
j < j+1 by XREAL_1:29;
then
A12: s1 < s2 by A1,A10,A11,GOBOARD5:4;
then s1+s1 < s1+s2 by XREAL_1:6;
then
A13: r3*(s1+s1) < r3*(s1+s2) by A8,XREAL_1:68;
r*s1 <= r*s2 by A5,A12,XREAL_1:64;
then
A14: s1 < r3*(s1+s2)+r*s2 by A13,A9,XREAL_1:8;
A15: 1 <= j+1 by A1,NAT_1:13;
A16: G*(1,j) = |[G*(1,j)`1,G*(1,j)`2]| by EUCLID:53
.= |[r2,s1]| by A1,A2,A11,GOBOARD5:2;
r2 < r2+1 by XREAL_1:29;
then
A17: r2-1 < r2 by XREAL_1:19;
A18: r3*(s2+s2)+r*s2 = s2;
s1+s2 < s2+s2 by A12,XREAL_1:6;
then r3*(s1+s2) < r3*(s2+s2) by A8,XREAL_1:68;
then
A19: r3*(s1+s2)+r*s2 < s2 by A18,XREAL_1:8;
A20: Int cell(G,0,j) = { |[r9,s9]| : r9 < G*(1,1)`1 & G*(1,j)`2 < s9 &
s9 < G*(1,j+1)`2 } by A1,A2,Th20;
A21: G*(1,j+1) = |[G*(1,j+1)`1,G*(1,j+1)`2]| by EUCLID:53
.= |[r2,s2]| by A15,A10,A11,GOBOARD5:2;
p = (1-r)*(1/2*(G*(1,j)+G*(1,j+1)))-(1-r)*|[1,0]| +r*(G*(1,j+1) -
|[1,0]|) by A4,RLVECT_1:34
.= r3*(G*(1,j)+G*(1,j+1))-(1-r)*|[1,0]|+r*(G*(1,j+1) - |[1,0]|) by
RLVECT_1:def 7
.= r3*(G*(1,j)+G*(1,j+1))-|[(1-r)*1,(1-r)*0]|+r*(G*(1,j+1) - |[1,0]|
) by EUCLID:58
.= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+r*(|[r2,s2]| - |[1,0]|) by A21,A16,
EUCLID:56
.= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+(r*|[r2,s2]| - r*|[1,0]|) by
RLVECT_1:34
.= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+(|[r*r2,r*s2]| - r*|[1,0]|) by
EUCLID:58
.= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+(|[r*r2,r*s2]| - |[r*1,r*0]|) by
EUCLID:58
.= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+|[r*r2-r,r*s2-0]| by EUCLID:62
.= |[r3*(r2+r2),r3*(s1+s2)]|-|[1-r,0]|+|[r*r2-r,r*s2-0]| by EUCLID:58
.= |[r3*(r2+r2)-(1-r),r3*(s1+s2)-0]|+|[r*r2-r,r*s2-0]| by EUCLID:62
.= |[r3*(r2+r2)-(1-r)+(r*r2-r),r3*(s1+s2)+r*s2]| by EUCLID:56;
hence p in Int cell(G,0,j) by A17,A14,A19,A20;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th50:
1 <= j & j < width G implies LSeg(1/2*(G*(len G,j)+G*(len G,j+1)
) + |[1,0]|,G*(len G,j) + |[1,0]|) c= Int cell(G,len G,j) \/ { G*(len G,j) + |[
1,0]| }
proof
assume that
A1: 1 <= j and
A2: j < width G;
let x be object;
assume
A3: x in LSeg(1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|,G* (len G,j) + |[1 ,0]|);
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A4: p = (1-r)*(1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|)+ r*(G*(len G,j)
+ |[1,0]|) and
A5: 0<=r and
A6: r<=1 by A3;
now
per cases by A6,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(G*(len G,j) + |[1,0]|) by A4,RLVECT_1:10
.= 1*(G*(len G,j) + |[1,0]|) by RLVECT_1:4
.= (G*(len G,j) + |[1,0]|) by RLVECT_1:def 8;
hence p in { G*(len G,j) + |[1,0]| } by TARSKI:def 1;
end;
case
A7: r < 1;
set r3 = (1-r)*(1/2);
1 - r > 0 by A7,XREAL_1:50;
then
A8: r3 > (1/2)*0 by XREAL_1:68;
set r2 = G*(len G,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2;
A9: r3*(s1+s1)+r*s1 = s1;
A10: j+1 <= width G by A2,NAT_1:13;
0 <> len G by MATRIX_0:def 10;
then
A11: 1 <= len G by NAT_1:14;
A12: G*(len G,j) = |[G*(len G,j)`1,G*(len G,j)`2]| by EUCLID:53
.= |[r2,G*(len G,j)`2]| by A1,A2,A11,GOBOARD5:2
.= |[r2,s1]| by A1,A2,A11,GOBOARD5:1;
A13: 1 <= j+1 by A1,NAT_1:13;
j < j+1 by XREAL_1:29;
then
A14: s1 < s2 by A1,A10,A11,GOBOARD5:4;
then s1+s2 < s2+s2 by XREAL_1:6;
then
A15: r3*(s1+s2) < r3*(s2+s2) by A8,XREAL_1:68;
s1+s1 < s1+s2 by A14,XREAL_1:6;
then r3*(s1+s1) < r3*(s1+s2) by A8,XREAL_1:68;
then
A16: r2 < r2+1 & s1 < r3*(s1+s2)+r*s1 by A9,XREAL_1:6,29;
A17: Int cell(G,len G,j) = { |[r9,s9]| : G*(len G,1)`1 < r9 & G*(1,j)`2
< s9 & s9 < G*(1,j+1)`2 } by A1,A2,Th23;
A18: G*(len G,j+1) = |[G*(len G,j+1)`1,G*(len G,j+1)`2]| by EUCLID:53
.= |[r2,G*(len G,j+1)`2]| by A13,A10,A11,GOBOARD5:2
.= |[r2,s2]| by A13,A10,A11,GOBOARD5:1;
A19: r3*(s2+s2)+r*s2 = s2;
r*s1 <= r*s2 by A5,A14,XREAL_1:64;
then
A20: r3*(s1+s2)+r*s1 < s2 by A15,A19,XREAL_1:8;
p = (1-r)*(1/2*(G*(len G,j)+G*(len G,j+1)))+(1-r)*|[1,0]| +r*(G*(
len G,j) + |[1,0]|) by A4,RLVECT_1:def 5
.= r3*(G*(len G,j)+G*(len G,j+1))+(1-r)*|[1,0]|+r*(G* (len G,j) + |[
1,0]|) by RLVECT_1:def 7
.= r3*(G*(len G,j)+G*(len G,j+1))+|[(1-r)*1,(1-r)*0]|+ r*(G*(len G,j
) + |[1,0]|) by EUCLID:58
.= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+r*(|[r2,s1]| + |[1,0]|) by A18,A12,
EUCLID:56
.= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+(r*|[r2,s1]| + r*|[1,0]|) by
RLVECT_1:def 5
.= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+(|[r*r2,r*s1]| + r*|[1,0]|) by
EUCLID:58
.= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+(|[r*r2,r*s1]| + |[r*1,r*0]|) by
EUCLID:58
.= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+|[r*r2+r,r*s1+0]| by EUCLID:56
.= |[r3*(r2+r2),r3*(s1+s2)]|+|[1-r,0]|+|[r*r2+r,r*s1+0]| by EUCLID:58
.= |[r3*(r2+r2)+(1-r),r3*(s1+s2)+0]|+|[r*r2+r,r*s1+0]| by EUCLID:56
.= |[r3*(r2+r2)+(1-r)+(r*r2+r),r3*(s1+s2)+r*s1]| by EUCLID:56;
hence p in Int cell(G,len G,j) by A16,A20,A17;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th51:
1 <= j & j < width G implies LSeg(1/2*(G*(len G,j)+G*(len G,j+1)
) + |[1,0]|,G*(len G,j+1) + |[1,0]|) c= Int cell(G,len G,j) \/ { G*(len G,j+1)
+ |[1,0]| }
proof
assume that
A1: 1 <= j and
A2: j < width G;
let x be object;
assume
A3: x in LSeg(1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|,G* (len G,j+1) +
|[1,0]|);
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A4: p = (1-r)*(1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|)+ r*(G*(len G,j+1
) + |[1,0]|) and
A5: 0<=r and
A6: r<=1 by A3;
now
per cases by A6,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(G*(len G,j+1) + |[1,0]|) by A4,RLVECT_1:10
.= 1*(G*(len G,j+1) + |[1,0]|) by RLVECT_1:4
.= (G*(len G,j+1) + |[1,0]|) by RLVECT_1:def 8;
hence p in { G*(len G,j+1) + |[1,0]| } by TARSKI:def 1;
end;
case
A7: r < 1;
set r3 = (1-r)*(1/2);
1 - r > 0 by A7,XREAL_1:50;
then
A8: r3 > (1/2)*0 by XREAL_1:68;
set r2 = G*(len G,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2;
A9: r3*(s1+s1)+r*s1 = s1;
A10: j+1 <= width G by A2,NAT_1:13;
0 <> len G by MATRIX_0:def 10;
then
A11: 1 <= len G by NAT_1:14;
j < j+1 by XREAL_1:29;
then
A12: s1 < s2 by A1,A10,A11,GOBOARD5:4;
then s1+s1 < s1+s2 by XREAL_1:6;
then
A13: r3*(s1+s1) < r3*(s1+s2) by A8,XREAL_1:68;
A14: r3*(s2+s2)+r*s2 = s2;
s1+s2 < s2+s2 by A12,XREAL_1:6;
then r3*(s1+s2) < r3*(s2+s2) by A8,XREAL_1:68;
then
A15: r3*(s1+s2)+r*s2 < s2 by A14,XREAL_1:8;
A16: G*(len G,j) = |[G*(len G,j)`1,G*(len G,j)`2]| by EUCLID:53
.= |[r2,G*(len G,j)`2]| by A1,A2,A11,GOBOARD5:2
.= |[r2,s1]| by A1,A2,A11,GOBOARD5:1;
A17: 1 <= j+1 by A1,NAT_1:13;
r*s1 <= r*s2 by A5,A12,XREAL_1:64;
then
A18: r2+1 > r2 & s1 < r3*(s1+s2)+r*s2 by A13,A9,XREAL_1:8,29;
A19: Int cell(G,len G,j) = { |[r9,s9]| : G*(len G,1)`1 < r9 & G*(1,j)`2
< s9 & s9 < G*(1,j+1)`2 } by A1,A2,Th23;
A20: G*(len G,j+1) = |[G*(len G,j+1)`1,G*(len G,j+1)`2]| by EUCLID:53
.= |[r2,G*(len G,j+1)`2]| by A17,A10,A11,GOBOARD5:2
.= |[r2,s2]| by A17,A10,A11,GOBOARD5:1;
p = (1-r)*(1/2*(G*(len G,j)+G*(len G,j+1)))+(1-r)*|[1,0]| +r*(G*(
len G,j+1) + |[1,0]|) by A4,RLVECT_1:def 5
.= r3*(G*(len G,j)+G*(len G,j+1))+(1-r)*|[1,0]|+ r*(G*(len G,j+1) +
|[1,0]|) by RLVECT_1:def 7
.= r3*(G*(len G,j)+G*(len G,j+1))+|[(1-r)*1,(1-r)*0]|+ r*(G*(len G,j
+1) + |[1,0]|) by EUCLID:58
.= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+r*(|[r2,s2]| + |[1,0]|) by A20,A16,
EUCLID:56
.= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+(r*|[r2,s2]| + r*|[1,0]|) by
RLVECT_1:def 5
.= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+(|[r*r2,r*s2]| + r*|[1,0]|) by
EUCLID:58
.= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+(|[r*r2,r*s2]| + |[r*1,r*0]|) by
EUCLID:58
.= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+|[r*r2+r,r*s2+0]| by EUCLID:56
.= |[r3*(r2+r2),r3*(s1+s2)]|+|[1-r,0]|+|[r*r2+r,r*s2+0]| by EUCLID:58
.= |[r3*(r2+r2)+(1-r),r3*(s1+s2)+0]|+|[r*r2+r,r*s2+0]| by EUCLID:56
.= |[r3*(r2+r2)+(1-r)+(r*r2+r),r3*(s1+s2)+r*s2]| by EUCLID:56;
hence p in Int cell(G,len G,j) by A18,A15,A19;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th52:
1 <= i & i < len G implies LSeg(1/2*(G*(i,1)+G*(i+1,1)) - |[0,1
]|,G*(i,1) - |[0,1]|) c= Int cell(G,i,0) \/ { G*(i,1) - |[0,1]| }
proof
assume that
A1: 1 <= i and
A2: i < len G;
let x be object;
assume
A3: x in LSeg(1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|,G*(i,1) - |[0,1]|);
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A4: p = (1-r)*(1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|)+r*(G*(i,1) - |[0,1]|) and
A5: 0<=r and
A6: r<=1 by A3;
now
per cases by A6,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(G*(i,1) - |[0,1]|) by A4,RLVECT_1:10
.= 1*(G*(i,1) - |[0,1]|) by RLVECT_1:4
.= (G*(i,1) - |[0,1]|) by RLVECT_1:def 8;
hence p in { G*(i,1) - |[0,1]| } by TARSKI:def 1;
end;
case
A7: r < 1;
set r3 = (1-r)*(1/2);
1 - r > 0 by A7,XREAL_1:50;
then
A8: r3 > (1/2)*0 by XREAL_1:68;
set s1 = G*(1,1)`2, r1 = G*(i,1)`1, r2 = G*(i+1,1)`1;
A9: r3*(r1+r1)+r*r1 = r1;
A10: i+1 <= len G by A2,NAT_1:13;
s1 < s1+1 by XREAL_1:29;
then
A11: s1-1 < s1 by XREAL_1:19;
A12: Int cell(G,i,0) = { |[r9,s9]| : G*(i,1)`1 < r9 & r9 < G*(i+1,1)`1 &
s9 < G*(1,1)`2 } by A1,A2,Th24;
0 <> width G by MATRIX_0:def 10;
then
A13: 1 <= width G by NAT_1:14;
i < i+1 by XREAL_1:29;
then
A14: r1 < r2 by A1,A10,A13,GOBOARD5:3;
then r1+r2 < r2+r2 by XREAL_1:6;
then
A15: r3*(r1+r2) < r3*(r2+r2) by A8,XREAL_1:68;
r1+r1 < r1+r2 by A14,XREAL_1:6;
then r3*(r1+r1) < r3*(r1+r2) by A8,XREAL_1:68;
then
A16: r1 < r3*(r1+r2)+r*r1 by A9,XREAL_1:6;
A17: r3*(r2+r2)+r*r2 = r2;
r*r1 <= r*r2 by A5,A14,XREAL_1:64;
then
A18: r3*(r1+r2)+r*r1 < r2 by A15,A17,XREAL_1:8;
A19: G*(i,1) = |[G*(i,1)`1,G*(i,1)`2]| by EUCLID:53
.= |[r1,s1]| by A1,A2,A13,GOBOARD5:1;
A20: 1 <= i+1 by A1,NAT_1:13;
A21: G*(i+1,1) = |[G*(i+1,1)`1,G*(i+1,1)`2]| by EUCLID:53
.= |[r2,s1]| by A20,A10,A13,GOBOARD5:1;
p = (1-r)*(1/2*(G*(i,1)+G*(i+1,1)))-(1-r)*|[0,1]| +r*(G*(i,1) - |[0
,1]|) by A4,RLVECT_1:34
.= r3*(G*(i,1)+G*(i+1,1))-(1-r)*|[0,1]|+r*(G* (i,1) - |[0,1]|) by
RLVECT_1:def 7
.= r3*(G*(i,1)+G*(i+1,1))-|[(1-r)*0,(1-r)*1]|+r*(G*(i,1) - |[0,1]|)
by EUCLID:58
.= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+r*(|[r1,s1]| - |[0,1]|) by A21,A19,
EUCLID:56
.= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+(r*|[r1,s1]| - r*|[0,1]|) by
RLVECT_1:34
.= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+(|[r*r1,r*s1]| - r*|[0,1]|) by
EUCLID:58
.= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+(|[r*r1,r*s1]| - |[r*0,r*1]|) by
EUCLID:58
.= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+|[r*r1-0,r*s1-r]| by EUCLID:62
.= |[r3*(r1+r2),r3*(s1+s1)]|-|[0,1-r]|+|[r*r1-0,r*s1-r]| by EUCLID:58
.= |[r3*(r1+r2)-0,r3*(s1+s1)-(1-r)]|+|[r*r1-0,r*s1-r]| by EUCLID:62
.= |[r3*(r1+r2)+r*r1,r3*(s1+s1)-(1-r)+(r*s1-r)]| by EUCLID:56;
hence p in Int cell(G,i,0) by A11,A16,A18,A12;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th53:
1 <= i & i < len G implies LSeg(1/2*(G*(i,1)+G*(i+1,1)) - |[0,1
]|,G*(i+1,1) - |[0,1]|) c= Int cell(G,i,0) \/ { G*(i+1,1) - |[0,1]| }
proof
assume that
A1: 1 <= i and
A2: i < len G;
let x be object;
assume
A3: x in LSeg(1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|,G*(i+1,1) - |[0,1]|);
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A4: p = (1-r)*(1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|)+r*(G*(i+1,1) - |[0,1]|) and
A5: 0<=r and
A6: r<=1 by A3;
now
per cases by A6,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(G*(i+1,1) - |[0,1]|) by A4,RLVECT_1:10
.= 1*(G*(i+1,1) - |[0,1]|) by RLVECT_1:4
.= (G*(i+1,1) - |[0,1]|) by RLVECT_1:def 8;
hence p in { G*(i+1,1) - |[0,1]| } by TARSKI:def 1;
end;
case
A7: r < 1;
set r3 = (1-r)*(1/2);
1 - r > 0 by A7,XREAL_1:50;
then
A8: r3 > (1/2)*0 by XREAL_1:68;
set s1 = G*(1,1)`2, r1 = G*(i,1)`1, r2 = G*(i+1,1)`1;
A9: r3*(r1+r1)+r*r1 = r1;
A10: i+1 <= len G by A2,NAT_1:13;
0 <> width G by MATRIX_0:def 10;
then
A11: 1 <= width G by NAT_1:14;
i < i+1 by XREAL_1:29;
then
A12: r1 < r2 by A1,A10,A11,GOBOARD5:3;
then r1+r1 < r1+r2 by XREAL_1:6;
then
A13: r3*(r1+r1) < r3*(r1+r2) by A8,XREAL_1:68;
r*r1 <= r*r2 by A5,A12,XREAL_1:64;
then
A14: r1 < r3*(r1+r2)+r*r2 by A13,A9,XREAL_1:8;
A15: 1 <= i+1 by A1,NAT_1:13;
A16: G*(i,1) = |[G*(i,1)`1,G*(i,1)`2]| by EUCLID:53
.= |[r1,s1]| by A1,A2,A11,GOBOARD5:1;
s1 < s1+1 by XREAL_1:29;
then
A17: s1-1 < s1 by XREAL_1:19;
A18: r3*(r2+r2)+r*r2 = r2;
r1+r2 < r2+r2 by A12,XREAL_1:6;
then r3*(r1+r2) < r3*(r2+r2) by A8,XREAL_1:68;
then
A19: r3*(r1+r2)+r*r2 < r2 by A18,XREAL_1:8;
A20: Int cell(G,i,0) = { |[r9,s9]| : G*(i,1)`1 < r9 & r9 < G*(i+1,1)`1 &
s9 < G*(1,1)`2 } by A1,A2,Th24;
A21: G*(i+1,1) = |[G*(i+1,1)`1,G*(i+1,1)`2]| by EUCLID:53
.= |[r2,s1]| by A15,A10,A11,GOBOARD5:1;
p = (1-r)*(1/2*(G*(i,1)+G*(i+1,1)))-(1-r)*|[0,1]| +r*(G*(i+1,1) -
|[0,1]|) by A4,RLVECT_1:34
.= r3*(G*(i,1)+G*(i+1,1))-(1-r)*|[0,1]|+r*(G*(i+1,1) - |[0,1]|) by
RLVECT_1:def 7
.= r3*(G*(i,1)+G*(i+1,1))-|[(1-r)*0,(1-r)*1]|+r*(G*(i+1,1) - |[0,1]|
) by EUCLID:58
.= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+r*(|[r2,s1]| - |[0,1]|) by A21,A16,
EUCLID:56
.= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+(r*|[r2,s1]| - r*|[0,1]|) by
RLVECT_1:34
.= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+(|[r*r2,r*s1]| - r*|[0,1]|) by
EUCLID:58
.= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+(|[r*r2,r*s1]| - |[r*0,r*1]|) by
EUCLID:58
.= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+|[r*r2-0,r*s1-r]| by EUCLID:62
.= |[r3*(r1+r2),r3*(s1+s1)]|-|[0,1-r]|+|[r*r2-0,r*s1-r]| by EUCLID:58
.= |[r3*(r1+r2)-0,r3*(s1+s1)-(1-r)]|+|[r*r2-0,r*s1-r]| by EUCLID:62
.= |[r3*(r1+r2)+r*r2,r3*(s1+s1)-(1-r)+(r*s1-r)]| by EUCLID:56;
hence p in Int cell(G,i,0) by A17,A14,A19,A20;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th54:
1 <= i & i < len G implies LSeg(1/2*(G*(i,width G)+G*(i+1,width
G)) + |[0,1]|,G* (i,width G) + |[0,1]|) c= Int cell(G,i,width G) \/ { G*(i,
width G) + |[0,1]| }
proof
assume that
A1: 1 <= i and
A2: i < len G;
let x be object;
assume
A3: x in LSeg(1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]|, G*(i,width G)
+ |[0,1]|);
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A4: p = (1-r)*(1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]|)+ r*(G*(i,
width G) + |[0,1]|) and
A5: 0<=r and
A6: r<=1 by A3;
now
per cases by A6,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(G*(i,width G) + |[0,1]|) by A4,RLVECT_1:10
.= 1*(G*(i,width G) + |[0,1]|) by RLVECT_1:4
.= (G*(i,width G) + |[0,1]|) by RLVECT_1:def 8;
hence p in { G*(i,width G) + |[0,1]| } by TARSKI:def 1;
end;
case
A7: r < 1;
set r3 = (1-r)*(1/2);
1 - r > 0 by A7,XREAL_1:50;
then
A8: r3 > (1/2)*0 by XREAL_1:68;
set s1 = G*(1,width G)`2, r1 = G*(i,1)`1, r2 = G*(i+1,1)`1;
A9: r3*(r1+r1)+r*r1 = r1;
A10: i+1 <= len G by A2,NAT_1:13;
0 <> width G by MATRIX_0:def 10;
then
A11: 1 <= width G by NAT_1:14;
A12: G*(i,width G) = |[G*(i,width G)`1,G*(i,width G)`2]| by EUCLID:53
.= |[G*(i,width G)`1,s1]| by A1,A2,A11,GOBOARD5:1
.= |[r1,s1]| by A1,A2,A11,GOBOARD5:2;
A13: 1 <= i+1 by A1,NAT_1:13;
i < i+1 by XREAL_1:29;
then
A14: r1 < r2 by A1,A10,A11,GOBOARD5:3;
then r1+r2 < r2+r2 by XREAL_1:6;
then
A15: r3*(r1+r2) < r3*(r2+r2) by A8,XREAL_1:68;
r1+r1 < r1+r2 by A14,XREAL_1:6;
then r3*(r1+r1) < r3*(r1+r2) by A8,XREAL_1:68;
then
A16: s1 < s1+1 & r1 < r3*(r1+r2)+r*r1 by A9,XREAL_1:6,29;
A17: Int cell(G,i,width G) = { |[r9,s9]| : G*(i,1)`1 < r9 & r9 < G*(i+1,
1)`1 & G*(1,width G)`2 < s9 } by A1,A2,Th25;
A18: G*(i+1,width G) = |[G*(i+1,width G)`1,G*(i+1,width G)`2]| by EUCLID:53
.= |[G*(i+1,width G)`1,s1]| by A13,A10,A11,GOBOARD5:1
.= |[r2,s1]| by A13,A10,A11,GOBOARD5:2;
A19: r3*(r2+r2)+r*r2 = r2;
r*r1 <= r*r2 by A5,A14,XREAL_1:64;
then
A20: r3*(r1+r2)+r*r1 < r2 by A15,A19,XREAL_1:8;
p = (1-r)*(1/2*(G*(i,width G)+G*(i+1,width G)))+(1-r)*|[0,1]| +r*(G
*(i,width G) + |[0,1]|) by A4,RLVECT_1:def 5
.= r3*(G*(i,width G)+G*(i+1,width G))+(1-r)*|[0,1]|+ r*(G*(i,width G
) + |[0,1]|) by RLVECT_1:def 7
.= r3*(G*(i,width G)+G*(i+1,width G))+|[(1-r)*0,(1-r)*1]|+ r*(G*(i,
width G) + |[0,1]|) by EUCLID:58
.= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+r*(|[r1,s1]| + |[0,1]|) by A18,A12,
EUCLID:56
.= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+(r*|[r1,s1]| + r*|[0,1]|) by
RLVECT_1:def 5
.= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+(|[r*r1,r*s1]| + r*|[0,1]|) by
EUCLID:58
.= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+(|[r*r1,r*s1]| + |[r*0,r*1]|) by
EUCLID:58
.= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+|[r*r1+0,r*s1+r]| by EUCLID:56
.= |[r3*(r1+r2),r3*(s1+s1)]|+|[0,1-r]|+|[r*r1+0,r*s1+r]| by EUCLID:58
.= |[r3*(r1+r2)+0,r3*(s1+s1)+(1-r)]|+|[r*r1+0,r*s1+r]| by EUCLID:56
.= |[r3*(r1+r2)+r*r1,r3*(s1+s1)+(1-r)+(r*s1+r)]| by EUCLID:56;
hence p in Int cell(G,i,width G) by A16,A20,A17;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th55:
1 <= i & i < len G implies LSeg(1/2*(G*(i,width G)+G*(i+1,width
G)) + |[0,1]|,G* (i+1,width G) + |[0,1]|) c= Int cell(G,i,width G) \/ { G*(i+1,
width G) + |[0,1]| }
proof
assume that
A1: 1 <= i and
A2: i < len G;
let x be object;
assume
A3: x in LSeg(1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]|, G*(i+1,width
G) + |[0,1]|);
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A4: p = (1-r)*(1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]|)+ r*(G*(i+1,
width G) + |[0,1]|) and
A5: 0<=r and
A6: r<=1 by A3;
now
per cases by A6,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(G*(i+1,width G) + |[0,1]|) by A4,RLVECT_1:10
.= 1*(G*(i+1,width G) + |[0,1]|) by RLVECT_1:4
.= (G*(i+1,width G) + |[0,1]|) by RLVECT_1:def 8;
hence p in { G*(i+1,width G) + |[0,1]| } by TARSKI:def 1;
end;
case
A7: r < 1;
set r3 = (1-r)*(1/2);
1 - r > 0 by A7,XREAL_1:50;
then
A8: r3 > (1/2)*0 by XREAL_1:68;
set s1 = G*(1,width G)`2, r1 = G*(i,1)`1, r2 = G*(i+1,1)`1;
A9: r3*(r1+r1)+r*r1 = r1;
A10: i+1 <= len G by A2,NAT_1:13;
0 <> width G by MATRIX_0:def 10;
then
A11: 1 <= width G by NAT_1:14;
i < i+1 by XREAL_1:29;
then
A12: r1 < r2 by A1,A10,A11,GOBOARD5:3;
then r1+r1 < r1+r2 by XREAL_1:6;
then
A13: r3*(r1+r1) < r3*(r1+r2) by A8,XREAL_1:68;
A14: r3*(r2+r2)+r*r2 = r2;
r1+r2 < r2+r2 by A12,XREAL_1:6;
then r3*(r1+r2) < r3*(r2+r2) by A8,XREAL_1:68;
then
A15: r3*(r1+r2)+r*r2 < r2 by A14,XREAL_1:8;
A16: G*(i,width G) = |[G*(i,width G)`1,G*(i,width G)`2]| by EUCLID:53
.= |[G*(i,width G)`1,s1]| by A1,A2,A11,GOBOARD5:1
.= |[r1,s1]| by A1,A2,A11,GOBOARD5:2;
A17: 1 <= i+1 by A1,NAT_1:13;
r*r1 <= r*r2 by A5,A12,XREAL_1:64;
then
A18: s1+1 > s1 & r1 < r3*(r1+r2)+r*r2 by A13,A9,XREAL_1:8,29;
A19: Int cell(G,i,width G) = { |[r9,s9]| : G*(i,1)`1 < r9 & r9 < G*(i+1,
1)`1 & G*(1,width G)`2 < s9 } by A1,A2,Th25;
A20: G*(i+1,width G) = |[G*(i+1,width G)`1,G*(i+1,width G)`2]| by EUCLID:53
.= |[G*(i+1,width G)`1,s1]| by A17,A10,A11,GOBOARD5:1
.= |[r2,s1]| by A17,A10,A11,GOBOARD5:2;
p = (1-r)*(1/2*(G*(i,width G)+G*(i+1,width G)))+(1-r)*|[0,1]| +r*(G
*(i+1,width G) + |[0,1]|) by A4,RLVECT_1:def 5
.= r3*(G*(i,width G)+G*(i+1,width G))+(1-r)*|[0,1]|+ r*(G*(i+1,width
G) + |[0,1]|) by RLVECT_1:def 7
.= r3*(G*(i,width G)+G*(i+1,width G))+|[(1-r)*0,(1-r)*1]|+ r*(G*(i+1
,width G) + |[0,1]|) by EUCLID:58
.= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+r*(|[r2,s1]| + |[0,1]|) by A20,A16,
EUCLID:56
.= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+(r*|[r2,s1]| + r*|[0,1]|) by
RLVECT_1:def 5
.= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+(|[r*r2,r*s1]| + r*|[0,1]|) by
EUCLID:58
.= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+(|[r*r2,r*s1]| + |[r*0,r*1]|) by
EUCLID:58
.= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+|[r*r2+0,r*s1+r]| by EUCLID:56
.= |[r3*(r1+r2),r3*(s1+s1)]|+|[0,1-r]|+|[r*r2+0,r*s1+r]| by EUCLID:58
.= |[r3*(r1+r2)+0,r3*(s1+s1)+(1-r)]|+|[r*r2+0,r*s1+r]| by EUCLID:56
.= |[r3*(r1+r2)+r*r2,r3*(s1+s1)+(1-r)+(r*s1+r)]| by EUCLID:56;
hence p in Int cell(G,i,width G) by A18,A15,A19;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th56:
LSeg(G*(1,1) - |[1,1]|,G*(1,1) - |[1,0]|) c= Int cell(G,0,0) \/
{ G*(1,1) - |[1,0]| }
proof
let x be object;
set r1 = G*(1,1)`1, s1 = G*(1,1)`2;
assume
A1: x in LSeg(G*(1,1)-|[1,1]|,G*(1,1) - |[1,0]|);
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A2: p = (1-r)*(G*(1,1)-|[1,1]|)+r*(G*(1,1) - |[1,0]|) and
0<=r and
A3: r<=1 by A1;
now
per cases by A3,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(G*(1,1) - |[1,0]|) by A2,RLVECT_1:10
.= 1*(G*(1,1) - |[1,0]|) by RLVECT_1:4
.= G*(1,1) - |[1,0]| by RLVECT_1:def 8;
hence p in { G*(1,1) - |[1,0]| } by TARSKI:def 1;
end;
case
r < 1;
then 1 - r > 0 by XREAL_1:50;
then s1 < s1 +(1-r) by XREAL_1:29;
then
A4: s1-(1-r) < s1 by XREAL_1:19;
A5: G*(1,1) = |[r1,s1]| by EUCLID:53;
r1 < r1+1 by XREAL_1:29;
then
A6: r1-1 < r1 by XREAL_1:19;
A7: Int cell(G,0,0) = { |[r9,s9]| : r9 < G*(1,1)`1 & s9 < G* (1,1)`2 }
by Th18;
p = (1-r)*(G*(1,1))-(1-r)*|[1,1]|+r*(G*(1,1) - |[1,0]|) by A2,RLVECT_1:34
.= (1-r)*(G*(1,1))-(1-r)*|[1,1]|+(r*(G*(1,1)) - r*|[1,0]|) by
RLVECT_1:34
.= r*(G*(1,1)) + ((1-r)*(G*(1,1))-(1-r)*|[1,1]|) - r*|[1,0]| by
RLVECT_1:def 3
.= r*(G*(1,1)) + (1-r)*(G*(1,1))-(1-r)*|[1,1]| - r*|[1,0]| by
RLVECT_1:def 3
.= (r+(1-r))*(G*(1,1)) -(1-r)*|[1,1]| - r*|[1,0]| by RLVECT_1:def 6
.= G*(1,1) -(1-r)*|[1,1]| - r*|[1,0]| by RLVECT_1:def 8
.= G*(1,1)-|[(1-r)*1,(1-r)*1]| - r*|[1,0]| by EUCLID:58
.= G*(1,1)-|[1-r,1-r]| - |[r*1,r*0]| by EUCLID:58
.= |[r1-(1-r),s1-(1-r)]| - |[r,0]| by A5,EUCLID:62
.= |[r1-(1-r)-r,s1-(1-r)-0]| by EUCLID:62
.= |[r1-1,s1-(1-r)]|;
hence p in Int cell(G,0,0) by A4,A6,A7;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th57:
LSeg(G*(len G,1) + |[1,-1]|,G*(len G,1) + |[1,0]|) c= Int cell(G
,len G,0) \/ { G*(len G,1) + |[1,0]| }
proof
let x be object;
set r1 = G*(len G,1)`1, s1 = G*(1,1)`2;
assume
A1: x in LSeg(G*(len G,1)+|[1,-1]|,G*(len G,1) + |[1,0]|);
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A2: p = (1-r)*(G*(len G,1)+|[1,-1]|)+r*(G*(len G,1) + |[1,0]|) and
0<=r and
A3: r<=1 by A1;
now
per cases by A3,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(G*(len G,1) + |[1,0]|) by A2,RLVECT_1:10
.= 1*(G*(len G,1) + |[1,0]|) by RLVECT_1:4
.= G*(len G,1) + |[1,0]| by RLVECT_1:def 8;
hence p in { G*(len G,1) + |[1,0]| } by TARSKI:def 1;
end;
case
r < 1;
then 1 - r > 0 by XREAL_1:50;
then
A4: s1 < s1 +(1-r) by XREAL_1:29;
s1+(r-1) = s1-(1-r);
then
A5: s1+(r-1) < s1 by A4,XREAL_1:19;
A6: r1 < r1+1 by XREAL_1:29;
0 <> len G by MATRIX_0:def 10;
then
A7: 1 <= len G by NAT_1:14;
0 <> width G by MATRIX_0:def 10;
then
A8: 1 <= width G by NAT_1:14;
A9: G*(len G,1) = |[r1,G*(len G,1)`2]| by EUCLID:53
.= |[r1,s1]| by A8,A7,GOBOARD5:1;
A10: Int cell(G,len G,0) = { |[r9,s9]| : G*(len G,1)`1 < r9 & s9 < G*(1,
1)`2 } by Th21;
p = (1-r)*(G*(len G,1))+(1-r)*|[1,-1]|+r*(G*(len G,1) + |[1,0]|) by A2,
RLVECT_1:def 5
.= (1-r)*(G*(len G,1))+(1-r)*|[1,-1]|+(r*(G*(len G,1)) + r*|[1,0]|)
by RLVECT_1:def 5
.= r*(G*(len G,1)) + ((1-r)*(G*(len G,1))+(1-r)*|[1,-1]|) + r*|[1,0
]| by RLVECT_1:def 3
.= r*(G*(len G,1)) + (1-r)*(G*(len G,1))+(1-r)*|[1,-1]| + r*|[1,0]|
by RLVECT_1:def 3
.= (r+(1-r))*(G*(len G,1)) +(1-r)*|[1,-1]| + r*|[1,0]| by
RLVECT_1:def 6
.= G*(len G,1) +(1-r)*|[1,-1]| + r*|[1,0]| by RLVECT_1:def 8
.= G*(len G,1)+|[(1-r)*1,(1-r)*(-1)]| + r*|[1,0]| by EUCLID:58
.= G*(len G,1)+|[1-r,r-1]| + |[r*1,r*0]| by EUCLID:58
.= |[r1+(1-r),s1+(r-1)]| + |[r,0]| by A9,EUCLID:56
.= |[r1+(1-r)+r,s1+(r-1)+0]| by EUCLID:56
.= |[r1+1,s1+(r-1)]|;
hence p in Int cell(G,len G,0) by A5,A6,A10;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th58:
LSeg(G*(1,width G) + |[-1,1]|,G*(1,width G) - |[1,0]|) c= Int
cell(G,0,width G) \/ { G*(1,width G) - |[1,0]| }
proof
let x be object;
set r1 = G*(1,1)`1, s1 = G*(1,width G)`2;
assume
A1: x in LSeg(G*(1,width G)+|[-1,1]|,G*(1,width G) - |[1,0]|);
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A2: p = (1-r)*(G*(1,width G)+|[-1,1]|)+r*(G*(1,width G) - |[1,0]|) and
0<=r and
A3: r<=1 by A1;
now
per cases by A3,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(G*(1,width G) - |[1,0]|) by A2,RLVECT_1:10
.= 1*(G*(1,width G) - |[1,0]|) by RLVECT_1:4
.= G*(1,width G) - |[1,0]| by RLVECT_1:def 8;
hence p in { G*(1,width G) - |[1,0]| } by TARSKI:def 1;
end;
case
r < 1;
then 1 - r > 0 by XREAL_1:50;
then
A4: s1 < s1 +(1-r) by XREAL_1:29;
0 <> width G by MATRIX_0:def 10;
then
A5: 1 <= width G by NAT_1:14;
0 <> len G by MATRIX_0:def 10;
then
A6: 1 <= len G by NAT_1:14;
A7: G*(1,width G) = |[G*(1,width G)`1,s1]| by EUCLID:53
.= |[r1,s1]| by A5,A6,GOBOARD5:2;
A8: Int cell(G,0,width G) = { |[r9,s9]| : r9 < G*(1,1)`1 & G*(1,width G
)`2 < s9 } by Th19;
r1 < r1+1 by XREAL_1:29;
then
A9: r1-1 < r1 by XREAL_1:19;
p = (1-r)*(G*(1,width G))+(1-r)*|[-1,1]|+r*(G*(1,width G) - |[1,0]|
) by A2,RLVECT_1:def 5
.= (1-r)*(G*(1,width G))+(1-r)*|[-1,1]|+(r*(G*(1,width G)) - r*|[1,0
]|) by RLVECT_1:34
.= r*(G*(1,width G)) + ((1-r)*(G* (1,width G))+(1-r)*|[-1,1]|) - r*
|[1,0]| by RLVECT_1:def 3
.= r*(G*(1,width G)) + (1-r)*(G*(1,width G))+(1-r)*|[-1,1]| - r*|[1,
0]| by RLVECT_1:def 3
.= (r+(1-r))*(G*(1,width G)) +(1-r)*|[-1,1]| - r*|[1,0]| by
RLVECT_1:def 6
.= G*(1,width G) +(1-r)*|[-1,1]| - r*|[1,0]| by RLVECT_1:def 8
.= G*(1,width G)+|[(1-r)*(-1),(1-r)*1]| - r*|[1,0]| by EUCLID:58
.= G*(1,width G)+|[r-1,1-r]| - |[r*1,r*0]| by EUCLID:58
.= |[r1+(r-1),s1+(1-r)]| - |[r,0]| by A7,EUCLID:56
.= |[r1+(r-1)-r,s1+(1-r)-0]| by EUCLID:62
.= |[r1-1,s1+(1-r)]|;
hence p in Int cell(G,0,width G) by A4,A9,A8;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th59:
LSeg(G*(len G,width G) + |[1,1]|,G*(len G,width G) + |[1,0]|) c=
Int cell(G,len G,width G) \/ { G*(len G,width G) + |[1,0]| }
proof
let x be object;
set r1 = G*(len G,1)`1, s1 = G*(1,width G)`2;
assume
A1: x in LSeg(G*(len G,width G)+|[1,1]|,G*(len G,width G) + |[1,0]|);
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A2: p = (1-r)*(G*(len G,width G)+|[1,1]|)+r*(G*(len G,width G) + |[1,0]| ) and
0<=r and
A3: r<=1 by A1;
now
per cases by A3,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(G*(len G,width G) + |[1,0]|) by A2,RLVECT_1:10
.= 1*(G*(len G,width G) + |[1,0]|) by RLVECT_1:4
.= G*(len G,width G) + |[1,0]| by RLVECT_1:def 8;
hence p in { G*(len G,width G) + |[1,0]| } by TARSKI:def 1;
end;
case
r < 1;
then 1 - r > 0 by XREAL_1:50;
then
A4: s1 < s1 +(1-r) by XREAL_1:29;
A5: r1 < r1+1 by XREAL_1:29;
0 <> width G by MATRIX_0:def 10;
then
A6: 1 <= width G by NAT_1:14;
0 <> len G by MATRIX_0:def 10;
then
A7: 1 <= len G by NAT_1:14;
A8: G*(len G,width G) = |[G*(len G,width G)`1,G*(len G,width G)`2]| by
EUCLID:53
.= |[r1,G*(len G,width G)`2]| by A6,A7,GOBOARD5:2
.= |[r1,s1]| by A6,A7,GOBOARD5:1;
A9: Int cell(G,len G,width G) = { |[r9,s9]| : G*(len G,1)`1 < r9 & G*(1
,width G)`2 < s9 } by Th22;
p = (1-r)*(G*(len G,width G))+(1-r)*|[1,1]|+r*(G* (len G,width G) +
|[1,0]|) by A2,RLVECT_1:def 5
.= (1-r)*(G*(len G,width G))+(1-r)*|[1,1]|+(r*(G* (len G,width G)) +
r*|[1,0]|) by RLVECT_1:def 5
.= r*(G*(len G,width G))+((1-r)*(G* (len G,width G))+(1-r)*|[1,1]|)
+ r*|[1,0]| by RLVECT_1:def 3
.= r*(G*(len G,width G)) + (1-r)*(G* (len G,width G))+(1-r)*|[1,1]|
+ r*|[1,0]| by RLVECT_1:def 3
.= (r+(1-r))*(G*(len G,width G)) +(1-r)*|[1,1]| + r*|[1,0]| by
RLVECT_1:def 6
.= G*(len G,width G) +(1-r)*|[1,1]| + r*|[1,0]| by RLVECT_1:def 8
.= G*(len G,width G)+|[(1-r)*1,(1-r)*1]| + r*|[1,0]| by EUCLID:58
.= G*(len G,width G)+|[1-r,1-r]| + |[r*1,r*0]| by EUCLID:58
.= |[r1+(1-r),s1+(1-r)]| + |[r,0]| by A8,EUCLID:56
.= |[r1+(1-r)+r,s1+(1-r)+0]| by EUCLID:56
.= |[r1+1,s1+(1-r)]|;
hence p in Int cell(G,len G,width G) by A4,A5,A9;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th60:
LSeg(G*(1,1) - |[1,1]|,G*(1,1) - |[0,1]|) c= Int cell(G,0,0) \/
{ G*(1,1) - |[0,1]| }
proof
let x be object;
set r1 = G*(1,1)`1, s1 = G*(1,1)`2;
assume
A1: x in LSeg(G*(1,1)-|[1,1]|,G*(1,1) - |[0,1]|);
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A2: p = (1-r)*(G*(1,1)-|[1,1]|)+r*(G*(1,1) - |[0,1]|) and
0<=r and
A3: r<=1 by A1;
now
per cases by A3,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(G*(1,1) - |[0,1]|) by A2,RLVECT_1:10
.= 1*(G*(1,1) - |[0,1]|) by RLVECT_1:4
.= G*(1,1) - |[0,1]| by RLVECT_1:def 8;
hence p in { G*(1,1) - |[0,1]| } by TARSKI:def 1;
end;
case
r < 1;
then 1 - r > 0 by XREAL_1:50;
then r1 < r1+(1-r) by XREAL_1:29;
then
A4: r1-(1-r) < r1 by XREAL_1:19;
A5: G*(1,1) = |[r1,s1]| by EUCLID:53;
s1 < s1 +1 by XREAL_1:29;
then
A6: s1-1 < s1 by XREAL_1:19;
A7: Int cell(G,0,0) = { |[r9,s9]| : r9 < G*(1,1)`1 & s9 < G* (1,1)`2 }
by Th18;
p = (1-r)*(G*(1,1))-(1-r)*|[1,1]|+r*(G*(1,1) - |[0,1]|) by A2,RLVECT_1:34
.= (1-r)*(G*(1,1))-(1-r)*|[1,1]|+(r*(G*(1,1)) - r*|[0,1]|) by
RLVECT_1:34
.= r*(G*(1,1)) + ((1-r)*(G*(1,1))-(1-r)*|[1,1]|) - r*|[0,1]| by
RLVECT_1:def 3
.= r*(G*(1,1)) + (1-r)*(G*(1,1))-(1-r)*|[1,1]| - r*|[0,1]| by
RLVECT_1:def 3
.= (r+(1-r))*(G*(1,1)) -(1-r)*|[1,1]| - r*|[0,1]| by RLVECT_1:def 6
.= G*(1,1) -(1-r)*|[1,1]| - r*|[0,1]| by RLVECT_1:def 8
.= G*(1,1)-|[(1-r)*1,(1-r)*1]| - r*|[0,1]| by EUCLID:58
.= G*(1,1)-|[1-r,1-r]| - |[r*0,r*1]| by EUCLID:58
.= |[r1-(1-r),s1-(1-r)]| - |[0,r]| by A5,EUCLID:62
.= |[r1-(1-r)-0,s1-(1-r)-r]| by EUCLID:62
.= |[r1-(1-r),s1-1]|;
hence p in Int cell(G,0,0) by A6,A4,A7;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th61:
LSeg(G*(len G,1) + |[1,-1]|,G*(len G,1) - |[0,1]|) c= Int cell(G
,len G,0) \/ { G*(len G,1) - |[0,1]| }
proof
let x be object;
set r1 = G*(len G,1)`1, s1 = G*(1,1)`2;
assume
A1: x in LSeg(G*(len G,1)+|[1,-1]|,G*(len G,1) - |[0,1]|);
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A2: p = (1-r)*(G*(len G,1)+|[1,-1]|)+r*(G*(len G,1) - |[0,1]|) and
0<=r and
A3: r<=1 by A1;
now
per cases by A3,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(G*(len G,1) - |[0,1]|) by A2,RLVECT_1:10
.= 1*(G*(len G,1) - |[0,1]|) by RLVECT_1:4
.= G*(len G,1) - |[0,1]| by RLVECT_1:def 8;
hence p in { G*(len G,1) - |[0,1]| } by TARSKI:def 1;
end;
case
r < 1;
then 1 - r > 0 by XREAL_1:50;
then
A4: r1 < r1+(1-r) by XREAL_1:29;
s1 < s1+1 by XREAL_1:29;
then
A5: s1-1 < s1 by XREAL_1:19;
0 <> len G by MATRIX_0:def 10;
then
A6: 1 <= len G by NAT_1:14;
0 <> width G by MATRIX_0:def 10;
then
A7: 1 <= width G by NAT_1:14;
A8: G*(len G,1) = |[r1,G*(len G,1)`2]| by EUCLID:53
.= |[r1,s1]| by A7,A6,GOBOARD5:1;
A9: Int cell(G,len G,0) = { |[r9,s9]| : G*(len G,1)`1 < r9 & s9 < G*(1,
1)`2 } by Th21;
p = (1-r)*(G*(len G,1))+(1-r)*|[1,-1]|+r*(G*(len G,1) - |[0,1]|) by A2,
RLVECT_1:def 5
.= (1-r)*(G*(len G,1))+(1-r)*|[1,-1]|+(r*(G*(len G,1)) - r*|[0,1]|)
by RLVECT_1:34
.= r*(G*(len G,1)) + ((1-r)*(G*(len G,1))+(1-r)*|[1,-1]|) - r*|[0,1
]| by RLVECT_1:def 3
.= r*(G*(len G,1)) + (1-r)*(G*(len G,1))+(1-r)*|[1,-1]| - r*|[0,1]|
by RLVECT_1:def 3
.= (r+(1-r))*(G*(len G,1)) +(1-r)*|[1,-1]| - r*|[0,1]| by
RLVECT_1:def 6
.= G*(len G,1) +(1-r)*|[1,-1]| - r*|[0,1]| by RLVECT_1:def 8
.= G*(len G,1)+|[(1-r)*1,(1-r)*(-1)]| - r*|[0,1]| by EUCLID:58
.= G*(len G,1)+|[1-r,r-1]| - |[r*0,r*1]| by EUCLID:58
.= |[r1+(1-r),s1+(r-1)]| - |[0,r]| by A8,EUCLID:56
.= |[r1+(1-r)-0,s1+(r-1)-r]| by EUCLID:62
.= |[r1+(1-r),s1-1]|;
hence p in Int cell(G,len G,0) by A5,A4,A9;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th62:
LSeg(G*(1,width G) + |[-1,1]|,G*(1,width G) + |[0,1]|) c= Int
cell(G,0,width G) \/ { G*(1,width G) + |[0,1]| }
proof
let x be object;
set r1 = G*(1,1)`1, s1 = G*(1,width G)`2;
assume
A1: x in LSeg(G*(1,width G)+|[-1,1]|,G*(1,width G) + |[0,1]|);
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A2: p = (1-r)*(G*(1,width G)+|[-1,1]|)+r*(G*(1,width G) + |[0,1]|) and
0<=r and
A3: r<=1 by A1;
now
per cases by A3,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(G*(1,width G) + |[0,1]|) by A2,RLVECT_1:10
.= 1*(G*(1,width G) + |[0,1]|) by RLVECT_1:4
.= G*(1,width G) + |[0,1]| by RLVECT_1:def 8;
hence p in { G*(1,width G) + |[0,1]| } by TARSKI:def 1;
end;
case
r < 1;
then 1 - r > 0 by XREAL_1:50;
then r1 < r1+(1-r) by XREAL_1:29;
then
A4: s1 < s1 +1 & r1-(1-r) < r1 by XREAL_1:19,29;
0 <> width G by MATRIX_0:def 10;
then
A5: 1 <= width G by NAT_1:14;
0 <> len G by MATRIX_0:def 10;
then
A6: 1 <= len G by NAT_1:14;
A7: G*(1,width G) = |[G*(1,width G)`1,s1]| by EUCLID:53
.= |[r1,s1]| by A5,A6,GOBOARD5:2;
A8: Int cell(G,0,width G) = { |[r9,s9]| : r9 < G*(1,1)`1 & G*(1,width G
)`2 < s9 } by Th19;
p = (1-r)*(G*(1,width G))+(1-r)*|[-1,1]|+r*(G*(1,width G) + |[0,1]|
) by A2,RLVECT_1:def 5
.= (1-r)*(G*(1,width G))+(1-r)*|[-1,1]|+(r*(G*(1,width G)) + r*|[0,1
]|) by RLVECT_1:def 5
.= r*(G*(1,width G)) + ((1-r)*(G* (1,width G))+(1-r)*|[-1,1]|) + r*
|[0,1]| by RLVECT_1:def 3
.= r*(G*(1,width G)) + (1-r)*(G*(1,width G))+(1-r)*|[-1,1]| + r*|[0,
1]| by RLVECT_1:def 3
.= (r+(1-r))*(G*(1,width G)) +(1-r)*|[-1,1]| + r*|[0,1]| by
RLVECT_1:def 6
.= G*(1,width G) +(1-r)*|[-1,1]| + r*|[0,1]| by RLVECT_1:def 8
.= G*(1,width G)+|[(1-r)*(-1),(1-r)*1]| + r*|[0,1]| by EUCLID:58
.= G*(1,width G)+|[r-1,1-r]| + |[r*0,r*1]| by EUCLID:58
.= |[r1+(r-1),s1+(1-r)]| + |[0,r]| by A7,EUCLID:56
.= |[r1+(r-1)+0,s1+(1-r)+r]| by EUCLID:56
.= |[r1-(1-r),s1+1]|;
hence p in Int cell(G,0,width G) by A4,A8;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem Th63:
LSeg(G*(len G,width G) + |[1,1]|,G*(len G,width G) + |[0,1]|) c=
Int cell(G,len G,width G) \/ { G*(len G,width G) + |[0,1]| }
proof
let x be object;
set r1 = G*(len G,1)`1, s1 = G*(1,width G)`2;
assume
A1: x in LSeg(G*(len G,width G)+|[1,1]|,G*(len G,width G) + |[0,1]|);
then reconsider p = x as Point of TOP-REAL 2;
consider r such that
A2: p = (1-r)*(G*(len G,width G)+|[1,1]|)+r*(G*(len G,width G) + |[0,1]| ) and
0<=r and
A3: r<=1 by A1;
now
per cases by A3,XXREAL_0:1;
case
r = 1;
then p = 0.TOP-REAL 2 + 1*(G*(len G,width G) + |[0,1]|) by A2,RLVECT_1:10
.= 1*(G*(len G,width G) + |[0,1]|) by RLVECT_1:4
.= G*(len G,width G) + |[0,1]| by RLVECT_1:def 8;
hence p in { G*(len G,width G) + |[0,1]| } by TARSKI:def 1;
end;
case
r < 1;
then 1 - r > 0 by XREAL_1:50;
then
A4: s1 < s1 +1 & r1 < r1+(1-r) by XREAL_1:29;
0 <> width G by MATRIX_0:def 10;
then
A5: 1 <= width G by NAT_1:14;
0 <> len G by MATRIX_0:def 10;
then
A6: 1 <= len G by NAT_1:14;
A7: G*(len G,width G) = |[G*(len G,width G)`1,G*(len G,width G)`2]| by
EUCLID:53
.= |[r1,G*(len G,width G)`2]| by A5,A6,GOBOARD5:2
.= |[r1,s1]| by A5,A6,GOBOARD5:1;
A8: Int cell(G,len G,width G) = { |[r9,s9]| : G*(len G,1)`1 < r9 & G*(1
,width G)`2 < s9 } by Th22;
p = (1-r)*(G*(len G,width G))+(1-r)*|[1,1]|+r*(G* (len G,width G) +
|[0,1]|) by A2,RLVECT_1:def 5
.= (1-r)*(G*(len G,width G))+(1-r)*|[1,1]|+(r*(G* (len G,width G)) +
r*|[0,1]|) by RLVECT_1:def 5
.= r*(G*(len G,width G))+((1-r)*(G* (len G,width G))+(1-r)*|[1,1]|)
+ r*|[0,1]| by RLVECT_1:def 3
.= r*(G*(len G,width G)) + (1-r)*(G* (len G,width G))+(1-r)*|[1,1]|
+ r*|[0,1]| by RLVECT_1:def 3
.= (r+(1-r))*(G*(len G,width G)) +(1-r)*|[1,1]| + r*|[0,1]| by
RLVECT_1:def 6
.= G*(len G,width G) +(1-r)*|[1,1]| + r*|[0,1]| by RLVECT_1:def 8
.= G*(len G,width G)+|[(1-r)*1,(1-r)*1]| + r*|[0,1]| by EUCLID:58
.= G*(len G,width G)+|[1-r,1-r]| + |[r*0,r*1]| by EUCLID:58
.= |[r1+(1-r),s1+(1-r)]| + |[0,r]| by A7,EUCLID:56
.= |[r1+(1-r)+0,s1+(1-r)+r]| by EUCLID:56
.= |[r1+(1-r),s1+1]|;
hence p in Int cell(G,len G,width G) by A4,A8;
end;
end;
hence thesis by XBOOLE_0:def 3;
end;
theorem
1 <= i & i < len G & 1 <= j & j+1 < width G implies LSeg(1/2*(G*(i,j)+
G*(i+1,j+1)),1/2*(G*(i,j+1)+G*(i+1,j+2))) c= Int cell(G,i,j) \/ Int cell(G,i,j+
1) \/ { 1/2*(G*(i,j+1)+G*(i+1,j+1)) }
proof
assume that
A1: 1 <= i and
A2: i < len G and
A3: 1 <= j and
A4: j+1 < width G;
set p1 = G*(i,j), p2 = G*(i,j+1), q2 = G*(i+1,j+1), q3 = G*(i+1,j+2), r = (
p2`2-p1`2)/(q3`2-p1`2);
A5: j+1 >= 1 by NAT_1:11;
set I1 = Int cell(G,i,j), I2 = Int cell(G,i,j+1);
j <= j+1 by NAT_1:11;
then
A6: j < width G by A4,XXREAL_0:2;
then
A7: LSeg(1/2*(p1+q2),1/2*(p2+q2)) c= I1 \/ { 1/2*(p2+q2) } by A1,A2,A3,Th41;
j < j+1 by XREAL_1:29;
then p1`2 < p2`2 by A1,A2,A3,A4,GOBOARD5:4;
then
A8: p2`2-p1`2 > 0 by XREAL_1:50;
A9: j+1+1 = j+(1+1);
then
A10: j+2 >= 1 by NAT_1:11;
A11: j+(1+1) <= width G by A4,A9,NAT_1:13;
A12: i+1 >= 1 & i+1 <= len G by A2,NAT_1:11,13;
then
A13: q2`1 = G*(i+1,1)`1 by A4,A5,GOBOARD5:2
.= q3`1 by A11,A10,A12,GOBOARD5:2;
A14: q2`2 = G*(1,j+1)`2 by A4,A5,A12,GOBOARD5:1
.= p2`2 by A1,A2,A4,A5,GOBOARD5:1;
j+1 < j+2 by XREAL_1:6;
then q2`2 < q3`2 by A5,A11,A12,GOBOARD5:4;
then
A15: p2`2-p1`2 < q3`2-p1`2 by A14,XREAL_1:9;
then
A16: r*(q3`2-p1`2) = p2`2-p1`2 by A8,XCMPLX_1:87;
p1`1 = G*(i,1)`1 by A1,A2,A3,A6,GOBOARD5:2
.= p2`1 by A1,A2,A4,A5,GOBOARD5:2;
then
A17: (p2+q2)`1 = (1-r)*(p1`1+q2`1)+r*(p2`1+q3`1) by A13,Lm1
.= (1-r)*(p1+q2)`1+r*(p2`1+q3`1) by Lm1
.= (1-r)*(p1+q2)`1+r*(p2+q3)`1 by Lm1
.= (1-r)*(p1+q2)`1+(r*(p2+q3))`1 by Lm3
.= ((1-r)*(p1+q2))`1+(r*(p2+q3))`1 by Lm3
.= ((1-r)*(p1+q2)+r*(p2+q3))`1 by Lm1;
(p2+q2)`2 = p2`2+(r+(1-r))*q2`2 by Lm1
.= (1-r)*(p1`2+q2`2)+r*(p2`2+q3`2) by A14,A16
.= (1-r)*(p1`2+q2`2)+r*(p2+q3)`2 by Lm1
.= (1-r)*(p1+q2)`2+r*(p2+q3)`2 by Lm1
.= (1-r)*(p1+q2)`2+(r*(p2+q3))`2 by Lm3
.= ((1-r)*(p1+q2))`2+(r*(p2+q3))`2 by Lm3
.= ((1-r)*(p1+q2)+r*(p2+q3))`2 by Lm1;
then (1-r)*(p1+q2)+r*(p2+q3) = |[(p2+q2)`1,(p2+q2)`2]| by A17,EUCLID:53
.= p2+q2 by EUCLID:53;
then
A18: 1/2*(p2+q2) = (1/2)*((1-r)*(p1+q2))+(1/2)*(r*(p2+q3)) by RLVECT_1:def 5
.= 1/2*(1-r)*(p1+q2)+1/2*(r*(p2+q3)) by RLVECT_1:def 7
.= (1-r)*(1/2*(p1+q2))+1/2*(r*(p2+q3)) by RLVECT_1:def 7
.= (1-r)*(1/2*(p1+q2))+1/2*r*(p2+q3) by RLVECT_1:def 7
.= (1-r)*((1/2)*(p1+q2))+r*((1/2)*(p2+q3)) by RLVECT_1:def 7;
r < 1 by A15,A8,XREAL_1:189;
then 1/2*(p2+q2) in LSeg(1/2*(p1+q2),1/2*(p2+q3)) by A15,A8,A18;
then
A19: LSeg(1/2*(p1+q2),1/2*(p2+q3)) = LSeg(1/2*(p1+q2),1/2*(p2+q2)) \/ LSeg(1
/2*(p2+q2),1/2*(p2+q3)) by TOPREAL1:5;
A20: I1 \/ I2 \/ { 1/2*(p2+q2) } = I1 \/ (I2 \/ ({ 1/2*(p2+q2) } \/ { 1/2*(
p2+q2) })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { 1/2*(p2+q2) } \/ { 1/2*(p2+q2) }) by XBOOLE_1:4
.= I1 \/ { 1/2*(p2+q2) } \/ (I2 \/ { 1/2*(p2+q2) }) by XBOOLE_1:4;
LSeg(1/2*(p2+q2),1/2*(p2+q3)) c= I2 \/ { 1/2*(p2+q2) } by A1,A2,A4,A5,A9,Th43
;
hence thesis by A19,A7,A20,XBOOLE_1:13;
end;
theorem
1 <= j & j < width G & 1 <= i & i+1 < len G implies LSeg(1/2*(G*(i,j)+
G*(i+1,j+1)),1/2*(G*(i+1,j)+G*(i+2,j+1))) c= Int cell(G,i,j) \/ Int cell(G,i+1,
j) \/ { 1/2*(G*(i+1,j)+G*(i+1,j+1)) }
proof
assume that
A1: 1 <= j and
A2: j < width G and
A3: 1 <= i and
A4: i+1 < len G;
set p1 = G*(i,j), p2 = G*(i+1,j), q2 = G*(i+1,j+1), q3 = G*(i+2,j+1), r = (
p2`1-p1`1)/(q3`1-p1`1);
A5: i+1 >= 1 by NAT_1:11;
set I1 = Int cell(G,i,j), I2 = Int cell(G,i+1,j);
i <= i+1 by NAT_1:11;
then
A6: i < len G by A4,XXREAL_0:2;
then
A7: LSeg(1/2*(p1+q2),1/2*(p2+q2)) c= I1 \/ { 1/2*(p2+q2) } by A1,A2,A3,Th42;
i < i+1 by XREAL_1:29;
then p1`1 < p2`1 by A1,A2,A3,A4,GOBOARD5:3;
then
A8: p2`1-p1`1 > 0 by XREAL_1:50;
A9: i+1+1 = i+(1+1);
then
A10: i+2 >= 1 by NAT_1:11;
A11: i+(1+1) <= len G by A4,A9,NAT_1:13;
A12: j+1 >= 1 & j+1 <= width G by A2,NAT_1:11,13;
then
A13: q2`2 = G*(1,j+1)`2 by A4,A5,GOBOARD5:1
.= q3`2 by A11,A10,A12,GOBOARD5:1;
A14: q2`1 = G*(i+1,1)`1 by A4,A5,A12,GOBOARD5:2
.= p2`1 by A1,A2,A4,A5,GOBOARD5:2;
i+1 < i+2 by XREAL_1:6;
then q2`1 < q3`1 by A5,A11,A12,GOBOARD5:3;
then
A15: p2`1-p1`1 < q3`1-p1`1 by A14,XREAL_1:9;
then
A16: r*(q3`1-p1`1) = p2`1-p1`1 by A8,XCMPLX_1:87;
p1`2 = G*(1,j)`2 by A1,A2,A3,A6,GOBOARD5:1
.= p2`2 by A1,A2,A4,A5,GOBOARD5:1;
then
A17: (p2+q2)`2 = (1-r)*(p1`2+q2`2)+r*(p2`2+q3`2) by A13,Lm1
.= (1-r)*(p1+q2)`2+r*(p2`2+q3`2) by Lm1
.= (1-r)*(p1+q2)`2+r*(p2+q3)`2 by Lm1
.= (1-r)*(p1+q2)`2+(r*(p2+q3))`2 by Lm3
.= ((1-r)*(p1+q2))`2+(r*(p2+q3))`2 by Lm3
.= ((1-r)*(p1+q2)+r*(p2+q3))`2 by Lm1;
(p2+q2)`1 = p2`1+(r+(1-r))*q2`1 by Lm1
.= (1-r)*(p1`1+q2`1)+r*(p2`1+q3`1) by A14,A16
.= (1-r)*(p1`1+q2`1)+r*(p2+q3)`1 by Lm1
.= (1-r)*(p1+q2)`1+r*(p2+q3)`1 by Lm1
.= (1-r)*(p1+q2)`1+(r*(p2+q3))`1 by Lm3
.= ((1-r)*(p1+q2))`1+(r*(p2+q3))`1 by Lm3
.= ((1-r)*(p1+q2)+r*(p2+q3))`1 by Lm1;
then (1-r)*(p1+q2)+r*(p2+q3) = |[(p2+q2)`1,(p2+q2)`2]| by A17,EUCLID:53
.= p2+q2 by EUCLID:53;
then
A18: 1/2*(p2+q2) = (1/2)*((1-r)*(p1+q2))+(1/2)*(r*(p2+q3)) by RLVECT_1:def 5
.= 1/2*(1-r)*(p1+q2)+1/2*(r*(p2+q3)) by RLVECT_1:def 7
.= (1-r)*(1/2*(p1+q2))+1/2*(r*(p2+q3)) by RLVECT_1:def 7
.= (1-r)*(1/2*(p1+q2))+1/2*r*(p2+q3) by RLVECT_1:def 7
.= (1-r)*((1/2)*(p1+q2))+r*((1/2)*(p2+q3)) by RLVECT_1:def 7;
r < 1 by A15,A8,XREAL_1:189;
then 1/2*(p2+q2) in LSeg(1/2*(p1+q2),1/2*(p2+q3)) by A15,A8,A18;
then
A19: LSeg(1/2*(p1+q2),1/2*(p2+q3)) = LSeg(1/2*(p1+q2),1/2*(p2+q2)) \/ LSeg(1
/2*(p2+q2),1/2*(p2+q3)) by TOPREAL1:5;
A20: I1 \/ I2 \/ { 1/2*(p2+q2) } = I1 \/ (I2 \/ ({ 1/2*(p2+q2) } \/ { 1/2*(
p2+q2) })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { 1/2*(p2+q2) } \/ { 1/2*(p2+q2) }) by XBOOLE_1:4
.= I1 \/ { 1/2*(p2+q2) } \/ (I2 \/ { 1/2*(p2+q2) }) by XBOOLE_1:4;
LSeg(1/2*(p2+q2),1/2*(p2+q3)) c= I2 \/ { 1/2*(p2+q2) } by A1,A2,A4,A5,A9,Th40
;
hence thesis by A19,A7,A20,XBOOLE_1:13;
end;
theorem
1 <= i & i < len G & 1 < width G implies LSeg(1/2*(G*(i,1)+G*(i+1,1))-
|[0,1]|,1/2*(G*(i,1)+G*(i+1,2))) c= Int cell(G,i,0) \/ Int cell(G,i,1) \/ { 1/2
*(G*(i,1)+G*(i+1,1)) }
proof
assume that
A1: 1 <= i and
A2: i < len G and
A3: 1 < width G;
set p1 = G*(i,1), q2 = G*(i+1,1), q3 = G*(i+1,2), r = 1/(1/2*(q3`2-p1`2)+1);
A4: i+1 >= 1 & i+1 <= len G by A2,NAT_1:11,13;
A5: 0+(1+1) <= width G by A3,NAT_1:13;
then
A6: q2`1 = q3`1 by A4,GOBOARD5:2;
A7: q2`2 = G*(1,0+1)`2 by A3,A4,GOBOARD5:1
.= p1`2 by A1,A2,A3,GOBOARD5:1;
then p1`2 < q3`2 by A5,A4,GOBOARD5:4;
then
A8: q3`2-p1`2 > 0 by XREAL_1:50;
then 1 < 1/2*(q3`2-p1`2)+1 by XREAL_1:29,129;
then
A9: r < 1 by XREAL_1:212;
set I1 = Int cell(G,i,0), I2 = Int cell(G,i,1);
A10: LSeg(1/2*(p1+q2)-|[0,1]|,1/2*(p1+q2)) c= I1 \/ { 1/2*(p1+q2) } by A1,A2
,Th46;
A11: I1 \/ I2 \/ { 1/2*(p1+q2) } = I1 \/ (I2 \/ ({ 1/2*(p1+q2) } \/ { 1/2*(
p1+q2) })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { 1/2*(p1+q2) } \/ { 1/2*(p1+q2) }) by XBOOLE_1:4
.= I1 \/ { 1/2*(p1+q2) } \/ (I2 \/ { 1/2*(p1+q2) }) by XBOOLE_1:4;
A12: ((1-r)*((1/2)*q2)+r*((1/2)*q3)-(1-r)*|[0,1]|)`1 = ((1-r)*((1/2)*q2)+r*(
(1/2)*q3))`1-((1-r)*|[0,1]|)`1 by Lm2
.= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`1-|[(1-r)*0,(1-r)*1]|`1 by EUCLID:58
.= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`1-0 by EUCLID:52
.= ((1-r)*((1/2)*q2))`1+(r*((1/2)*q3))`1 by Lm1
.= (1-r)*((1/2)*q2)`1+(r*((1/2)*q3))`1 by Lm3
.= (1-r)*((1/2)*q2)`1+r*((1/2)*q3)`1 by Lm3
.= (1-r)*((1/2)*q2`1)+r*((1/2)*q3)`1 by Lm3
.= (1-r)*((1/2)*q2`1)+r*((1/2)*q2`1) by A6,Lm3
.= ((1/2)*q2)`1 by Lm3;
A13: (1-r)*((1/2)*p1)+r*((1/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*q3) = (1-r)*((1
/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*p1)+r*((1/2)*q3) by RLVECT_1:def 3
.= (1-r)*((1/2)*p1)+(1-r)*((1/2)*q2)+(r*((1/2)*p1)+r*((1/2)*q3)) by
RLVECT_1:def 3
.= (1-r)*((1/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*p1+(1/2)*q3) by
RLVECT_1:def 5
.= (1-r)*((1/2)*p1+(1/2)*q2)+r*((1/2)*p1+(1/2)*q3) by RLVECT_1:def 5
.= (1-r)*((1/2)*p1+(1/2)*q2)+r*((1/2)*(p1+q3)) by RLVECT_1:def 5
.= (1-r)*((1/2)*(p1+q2))+r*((1/2)*(p1+q3)) by RLVECT_1:def 5;
A14: r*((1/2)*q3`2)-r*((1/2)*q2`2)+r = r*((1/2)*(q3`2-q2`2)+1)
.= 1 by A7,A8,XCMPLX_1:106;
((1-r)*((1/2)*q2)+r*((1/2)*q3)-(1-r)*|[0,1]|)`2 = ((1-r)*((1/2)*q2)+r*(
(1/2)*q3))`2-((1-r)*|[0,1]|)`2 by Lm2
.= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`2-|[(1-r)*0,(1-r)*1]|`2 by EUCLID:58
.= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`2-(1-r) by EUCLID:52
.= ((1-r)*((1/2)*q2))`2+(r*((1/2)*q3))`2-(1-r) by Lm1
.= (1-r)*((1/2)*q2)`2+(r*((1/2)*q3))`2-(1-r) by Lm3
.= (1-r)*((1/2)*q2)`2+r*((1/2)*q3)`2-(1-r) by Lm3
.= (1-r)*((1/2)*q2`2)+r*((1/2)*q3)`2-(1-r) by Lm3
.= (1-r)*((1/2)*q2`2)+r*((1/2)*q3`2)-(1-r) by Lm3
.= ((1/2)*q2)`2 by A14,Lm3;
then
A15: (1-r)*((1/2)*q2)+r*((1/2)*q3)-(1-r)*|[0,1]| = |[(1/2*q2)`1,(1/2* q2 )`2
]| by A12,EUCLID:53
.= 1/2*q2 by EUCLID:53;
1/2*(p1+q2) = 1/2*p1+1/2*q2 by RLVECT_1:def 5
.= (1-r+r)*((1/2)*p1)+1/2*q2 by RLVECT_1:def 8
.= (1-r)*((1/2)*p1)+r*((1/2)*p1)+1/2*q2 by RLVECT_1:def 6
.= (1-r)*((1/2)*p1)+r*((1/2)*p1)+ ((1-r)*((1/2)*q2)+r*((1/2)*q3))-(1-r)*
|[0,1]| by A15,RLVECT_1:def 3
.= (1-r)*((1/2)*p1)+r*((1/2)*p1)+ (1-r)*((1/2)*q2)+r*((1/2)*q3)-(1-r)*|[
0,1]| by RLVECT_1:def 3
.= (1-r)*((1/2)*(p1+q2))+(r*((1/2)*(p1+q3))-(1-r)*|[0,1]|) by A13,
RLVECT_1:def 3
.= (1-r)*((1/2)*(p1+q2))+-((1-r)*|[0,1]|-r*((1/2)*(p1+q3))) by RLVECT_1:33
.= (1-r)*((1/2)*(p1+q2))-((1-r)*|[0,1]|-r*((1/2)*(p1+q3)))
.= (1-r)*((1/2)*(p1+q2))-(1-r)*|[0,1]|+r*((1/2)*(p1+q3)) by RLVECT_1:29
.= (1-r)*((1/2)*(p1+q2)-|[0,1]|)+r*((1/2)*(p1+q3)) by RLVECT_1:34;
then 1/2*(p1+q2) in LSeg(1/2*(p1+q2)-|[0,1]|,1/2*(p1+q3)) by A8,A9;
then
A16: LSeg(1/2*(p1+q2)-|[0,1]|,1/2*(p1+q3)) = LSeg(1/2*(p1+q2)-|[0,1]|,1/2*(
p1+q2)) \/ LSeg(1/2*(p1+q2),1/2*(p1+q3)) by TOPREAL1:5;
0+1+1 = 0+(1+1);
then LSeg(1/2*(p1+q2),1/2*(p1+q3)) c= I2 \/ { 1/2*(p1+q2) } by A1,A2,A3,Th43;
hence thesis by A16,A10,A11,XBOOLE_1:13;
end;
theorem
1 <= i & i < len G & 1 < width G implies LSeg(1/2*(G*(i,width G)+G*(i+
1,width G))+|[0,1]|, 1/2*(G*(i,width G)+G*(i+1,width G -' 1))) c= Int cell(G,i,
width G -'1) \/ Int cell(G,i,width G) \/ { 1/2*(G*(i,width G)+G*(i+1,width G))
}
proof
assume that
A1: 1 <= i and
A2: i < len G and
A3: 1 < width G;
set I1 = Int cell(G,i,width G -'1), I2 = Int cell(G,i,width G);
set p1 = G*(i,width G), q2 = G*(i+1,width G), q3 = G*(i+1,width G -' 1), r =
1/(1/2*(p1`2-q3`2)+1);
A4: width G -'1 + 1 = width G by A3,XREAL_1:235;
then
A5: 1 <= width G -'1 by A3,NAT_1:13;
A6: width G -'1 < width G by A4,NAT_1:13;
then
G*(i,width G)+G*(i+1,width G -' 1) = G*(i,width G -' 1)+G*(i+1,width G)
by A1,A2,A4,A5,Th11;
then
A7: LSeg(1/2*(p1+q2),1/2*(p1+q3)) c= I1 \/ { 1/2*(p1+q2) } by A1,A2,A4,A5,A6
,Th41;
A8: i+1 >= 1 & i+1 <= len G by A2,NAT_1:11,13;
then
A9: q2`1 = G*(i+1,1)`1 by A3,GOBOARD5:2
.= q3`1 by A5,A6,A8,GOBOARD5:2;
A10: q2`2 = G*(1,width G)`2 by A3,A8,GOBOARD5:1
.= p1`2 by A1,A2,A3,GOBOARD5:1;
then q3`2 < p1`2 by A5,A6,A8,GOBOARD5:4;
then
A11: p1`2-q3`2 > 0 by XREAL_1:50;
then 1 < 1/2*(p1`2-q3`2)+1 by XREAL_1:29,129;
then
A12: r < 1 by XREAL_1:212;
A13: ((1-r)*((1/2)*q2)+r*((1/2)*q3)+(1-r)*|[0,1]|)`1 = ((1-r)*((1/2)*q2)+r*(
(1/2)*q3))`1+((1-r)*|[0,1]|)`1 by Lm1
.= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`1+|[(1-r)*0,(1-r)*1]|`1 by EUCLID:58
.= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`1+0 by EUCLID:52
.= ((1-r)*((1/2)*q2))`1+(r*((1/2)*q3))`1 by Lm1
.= (1-r)*((1/2)*q2)`1+(r*((1/2)*q3))`1 by Lm3
.= (1-r)*((1/2)*q2)`1+r*((1/2)*q3)`1 by Lm3
.= (1-r)*((1/2)*q2`1)+r*((1/2)*q3)`1 by Lm3
.= (1-r)*((1/2)*q2`1)+r*((1/2)*q2`1) by A9,Lm3
.= ((1/2)*q2)`1 by Lm3;
A14: I1 \/ I2 \/ { 1/2*(p1+q2) } = I1 \/ (I2 \/ ({ 1/2*(p1+q2) } \/ { 1/2*(
p1+q2) })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { 1/2*(p1+q2) } \/ { 1/2*(p1+q2) }) by XBOOLE_1:4
.= I1 \/ { 1/2*(p1+q2) } \/ (I2 \/ { 1/2*(p1+q2) }) by XBOOLE_1:4;
A15: (1-r)*((1/2)*p1)+r*((1/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*q3) = (1-r)*((1
/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*p1)+r*((1/2)*q3) by RLVECT_1:def 3
.= (1-r)*((1/2)*p1)+(1-r)*((1/2)*q2)+(r*((1/2)*p1)+r*((1/2)*q3)) by
RLVECT_1:def 3
.= (1-r)*((1/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*p1+(1/2)*q3) by
RLVECT_1:def 5
.= (1-r)*((1/2)*p1+(1/2)*q2)+r*((1/2)*p1+(1/2)*q3) by RLVECT_1:def 5
.= (1-r)*((1/2)*p1+(1/2)*q2)+r*((1/2)*(p1+q3)) by RLVECT_1:def 5
.= (1-r)*((1/2)*(p1+q2))+r*((1/2)*(p1+q3)) by RLVECT_1:def 5;
A16: r*((1/2)*q2`2)-r*((1/2)*q3`2)+r = r*((1/2)*(q2`2-q3`2)+1)
.= 1 by A10,A11,XCMPLX_1:106;
((1-r)*((1/2)*q2)+r*((1/2)*q3)+(1-r)*|[0,1]|)`2 = ((1-r)*((1/2)*q2)+r*(
(1/2)*q3))`2+((1-r)*|[0,1]|)`2 by Lm1
.= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`2+|[(1-r)*0,(1-r)*1]|`2 by EUCLID:58
.= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`2+(1-r) by EUCLID:52
.= ((1-r)*((1/2)*q2))`2+(r*((1/2)*q3))`2+(1-r) by Lm1
.= (1-r)*((1/2)*q2)`2+(r*((1/2)*q3))`2+(1-r) by Lm3
.= (1-r)*((1/2)*q2)`2+r*((1/2)*q3)`2+(1-r) by Lm3
.= (1-r)*((1/2)*q2`2)+r*((1/2)*q3)`2+(1-r) by Lm3
.= (1-r)*((1/2)*q2`2)+r*((1/2)*q3`2)+(1-r) by Lm3
.= ((1/2)*q2)`2 by A16,Lm3;
then
A17: (1-r)*((1/2)*q2)+r*((1/2)*q3)+(1-r)*|[0,1]| = |[(1/2*q2)`1,(1/2* q2 )`2
]| by A13,EUCLID:53
.= 1/2*q2 by EUCLID:53;
1/2*(p1+q2) = 1/2*p1+1/2*q2 by RLVECT_1:def 5
.= (1-r+r)*((1/2)*p1)+1/2*q2 by RLVECT_1:def 8
.= (1-r)*((1/2)*p1)+r*((1/2)*p1)+1/2*q2 by RLVECT_1:def 6
.= (1-r)*((1/2)*p1)+r*((1/2)*p1)+ ((1-r)*((1/2)*q2)+r*((1/2)*q3))+(1-r)*
|[0,1]| by A17,RLVECT_1:def 3
.= (1-r)*((1/2)*p1)+r*((1/2)*p1)+ (1-r)*((1/2)*q2)+r*((1/2)*q3)+(1-r)*|[
0,1]| by RLVECT_1:def 3
.= (1-r)*((1/2)*(p1+q2))+(1-r)*|[0,1]|+r*((1/2)*(p1+q3)) by A15,
RLVECT_1:def 3
.= (1-r)*((1/2)*(p1+q2)+|[0,1]|)+r*((1/2)*(p1+q3)) by RLVECT_1:def 5;
then 1/2*(p1+q2) in LSeg(1/2*(p1+q2)+|[0,1]|,1/2*(p1+q3)) by A11,A12;
then
A18: LSeg(1/2*(p1+q2)+|[0,1]|,1/2*(p1+q3)) = LSeg(1/2*(p1+q2)+|[0,1]|,1/2*(
p1+q2)) \/ LSeg(1/2*(p1+q2),1/2*(p1+q3)) by TOPREAL1:5;
LSeg(1/2*(p1+q2)+|[0,1]|,1/2*(p1+q2)) c= I2 \/ { 1/2*(p1+q2) } by A1,A2,Th47;
hence thesis by A18,A7,A14,XBOOLE_1:13;
end;
theorem
1 <= j & j < width G & 1 < len G implies LSeg(1/2*(G*(1,j)+G*(1,j+1))-
|[1,0]|,1/2*(G*(1,j)+G*(2,j+1))) c= Int cell(G,0,j) \/ Int cell(G,1,j) \/ { 1/2
*(G*(1,j)+G*(1,j+1)) }
proof
assume that
A1: 1 <= j and
A2: j < width G and
A3: 1 < len G;
set p1 = G*(1,j), q2 = G*(1,j+1), q3 = G*(2,j+1), r = 1/(1/2*(q3`1-p1`1)+1);
A4: j+1 >= 1 & j+1 <= width G by A2,NAT_1:11,13;
A5: 0+(1+1) <= len G by A3,NAT_1:13;
then
A6: q2`2 = q3`2 by A4,GOBOARD5:1;
A7: q2`1 = G*(1,1)`1 by A3,A4,GOBOARD5:2
.= p1`1 by A1,A2,A3,GOBOARD5:2;
then p1`1 < q3`1 by A5,A4,GOBOARD5:3;
then
A8: q3`1-p1`1 > 0 by XREAL_1:50;
then 1 < 1/2*(q3`1-p1`1)+1 by XREAL_1:29,129;
then
A9: r < 1 by XREAL_1:212;
set I1 = Int cell(G,0,j), I2 = Int cell(G,1,j);
A10: LSeg(1/2*(p1+q2)-|[1,0]|,1/2*(p1+q2)) c= I1 \/ { 1/2*(p1+q2) } by A1,A2
,Th44;
A11: I1 \/ I2 \/ { 1/2*(p1+q2) } = I1 \/ (I2 \/ ({ 1/2*(p1+q2) } \/ { 1/2*(
p1+q2) })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { 1/2*(p1+q2) } \/ { 1/2*(p1+q2) }) by XBOOLE_1:4
.= I1 \/ { 1/2*(p1+q2) } \/ (I2 \/ { 1/2*(p1+q2) }) by XBOOLE_1:4;
A12: ((1-r)*((1/2)*q2)+r*((1/2)*q3)-(1-r)*|[1,0]|)`2 = ((1-r)*((1/2)*q2)+r*(
(1/2)*q3))`2-((1-r)*|[1,0]|)`2 by Lm2
.= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`2-|[(1-r)*1,(1-r)*0]|`2 by EUCLID:58
.= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`2-0 by EUCLID:52
.= ((1-r)*((1/2)*q2))`2+(r*((1/2)*q3))`2 by Lm1
.= (1-r)*((1/2)*q2)`2+(r*((1/2)*q3))`2 by Lm3
.= (1-r)*((1/2)*q2)`2+r*((1/2)*q3)`2 by Lm3
.= (1-r)*((1/2)*q2`2)+r*((1/2)*q3)`2 by Lm3
.= (1-r)*((1/2)*q2`2)+r*((1/2)*q2`2) by A6,Lm3
.= ((1/2)*q2)`2 by Lm3;
A13: (1-r)*((1/2)*p1)+r*((1/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*q3) = (1-r)*((1
/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*p1)+r*((1/2)*q3) by RLVECT_1:def 3
.= (1-r)*((1/2)*p1)+(1-r)*((1/2)*q2)+(r*((1/2)*p1)+r*((1/2)*q3)) by
RLVECT_1:def 3
.= (1-r)*((1/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*p1+(1/2)*q3) by
RLVECT_1:def 5
.= (1-r)*((1/2)*p1+(1/2)*q2)+r*((1/2)*p1+(1/2)*q3) by RLVECT_1:def 5
.= (1-r)*((1/2)*p1+(1/2)*q2)+r*((1/2)*(p1+q3)) by RLVECT_1:def 5
.= (1-r)*((1/2)*(p1+q2))+r*((1/2)*(p1+q3)) by RLVECT_1:def 5;
A14: r*((1/2)*q3`1)-r*((1/2)*q2`1)+r = r*((1/2)*(q3`1-q2`1)+1)
.= 1 by A7,A8,XCMPLX_1:106;
((1-r)*((1/2)*q2)+r*((1/2)*q3)-(1-r)*|[1,0]|)`1 = ((1-r)*((1/2)*q2)+r*(
(1/2)*q3))`1-((1-r)*|[1,0]|)`1 by Lm2
.= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`1-|[(1-r)*1,(1-r)*0]|`1 by EUCLID:58
.= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`1-(1-r) by EUCLID:52
.= ((1-r)*((1/2)*q2))`1+(r*((1/2)*q3))`1-(1-r) by Lm1
.= (1-r)*((1/2)*q2)`1+(r*((1/2)*q3))`1-(1-r) by Lm3
.= (1-r)*((1/2)*q2)`1+r*((1/2)*q3)`1-(1-r) by Lm3
.= (1-r)*((1/2)*q2`1)+r*((1/2)*q3)`1-(1-r) by Lm3
.= (1-r)*((1/2)*q2`1)+r*((1/2)*q3`1)-(1-r) by Lm3
.= ((1/2)*q2)`1 by A14,Lm3;
then
A15: (1-r)*((1/2)*q2)+r*((1/2)*q3)-(1-r)*|[1,0]| = |[(1/2*q2)`1,(1/2* q2 )`2
]| by A12,EUCLID:53
.= 1/2*q2 by EUCLID:53;
1/2*(p1+q2) = 1/2*p1+1/2*q2 by RLVECT_1:def 5
.= (1-r+r)*((1/2)*p1)+1/2*q2 by RLVECT_1:def 8
.= (1-r)*((1/2)*p1)+r*((1/2)*p1)+1/2*q2 by RLVECT_1:def 6
.= (1-r)*((1/2)*p1)+r*((1/2)*p1)+ ((1-r)*((1/2)*q2)+r*((1/2)*q3))-(1-r)*
|[1,0]| by A15,RLVECT_1:def 3
.= (1-r)*((1/2)*p1)+r*((1/2)*p1)+ (1-r)*((1/2)*q2)+r*((1/2)*q3)-(1-r)*|[
1,0]| by RLVECT_1:def 3
.= (1-r)*((1/2)*(p1+q2))+(r*((1/2)*(p1+q3))-(1-r)*|[1,0]|) by A13,
RLVECT_1:def 3
.= (1-r)*((1/2)*(p1+q2))+-((1-r)*|[1,0]|-r*((1/2)*(p1+q3))) by RLVECT_1:33
.= (1-r)*((1/2)*(p1+q2))-((1-r)*|[1,0]|-r*((1/2)*(p1+q3)))
.= (1-r)*((1/2)*(p1+q2))-(1-r)*|[1,0]|+r*((1/2)*(p1+q3)) by RLVECT_1:29
.= (1-r)*((1/2)*(p1+q2)-|[1,0]|)+r*((1/2)*(p1+q3)) by RLVECT_1:34;
then 1/2*(p1+q2) in LSeg(1/2*(p1+q2)-|[1,0]|,1/2*(p1+q3)) by A8,A9;
then
A16: LSeg(1/2*(p1+q2)-|[1,0]|,1/2*(p1+q3)) = LSeg(1/2*(p1+q2)-|[1,0]|,1/2*(
p1+q2)) \/ LSeg(1/2*(p1+q2),1/2*(p1+q3)) by TOPREAL1:5;
0+1+1 = 0+(1+1);
then LSeg(1/2*(p1+q2),1/2*(p1+q3)) c= I2 \/ { 1/2*(p1+q2) } by A1,A2,A3,Th40;
hence thesis by A16,A10,A11,XBOOLE_1:13;
end;
theorem
1 <= j & j < width G & 1 < len G implies LSeg(1/2*(G*(len G,j)+G*(len
G,j+1))+|[1,0]|, 1/2*(G*(len G,j)+G*(len G -' 1,j+1))) c= Int cell(G,len G -' 1
,j) \/ Int cell(G,len G,j) \/ { 1/2*(G*(len G,j)+G*(len G,j+1)) }
proof
assume that
A1: 1 <= j and
A2: j < width G and
A3: 1 < len G;
set I1 = Int cell(G,len G -' 1,j), I2 = Int cell(G,len G,j);
set p1 = G*(len G,j), q2 = G*(len G,j+1), q3 = G*(len G -' 1,j+1), r = 1/(1/
2*(p1`1-q3`1)+1);
A4: len G -'1 + 1 = len G by A3,XREAL_1:235;
then
A5: 1 <= len G -'1 by A3,NAT_1:13;
A6: len G -'1 < len G by A4,NAT_1:13;
then
G*(len G -' 1,j)+G*(len G,j+1) = G*(len G,j)+G* (len G -' 1,j+1) by A1,A2,A4
,A5,Th11;
then
A7: LSeg(1/2*(p1+q2),1/2*(p1+q3)) c= I1 \/ { 1/2*(p1+q2) } by A1,A2,A4,A5,A6
,Th42;
A8: j+1 >= 1 & j+1 <= width G by A2,NAT_1:11,13;
then
A9: q2`2 = G*(1,j+1)`2 by A3,GOBOARD5:1
.= q3`2 by A5,A6,A8,GOBOARD5:1;
A10: q2`1 = G*(len G,1)`1 by A3,A8,GOBOARD5:2
.= p1`1 by A1,A2,A3,GOBOARD5:2;
then q3`1 < p1`1 by A5,A6,A8,GOBOARD5:3;
then
A11: p1`1-q3`1 > 0 by XREAL_1:50;
then 1 < 1/2*(p1`1-q3`1)+1 by XREAL_1:29,129;
then
A12: r < 1 by XREAL_1:212;
A13: ((1-r)*((1/2)*q2)+r*((1/2)*q3)+(1-r)*|[1,0]|)`2 = ((1-r)*((1/2)*q2)+r*(
(1/2)*q3))`2+((1-r)*|[1,0]|)`2 by Lm1
.= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`2+|[(1-r)*1,(1-r)*0]|`2 by EUCLID:58
.= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`2+0 by EUCLID:52
.= ((1-r)*((1/2)*q2))`2+(r*((1/2)*q3))`2 by Lm1
.= (1-r)*((1/2)*q2)`2+(r*((1/2)*q3))`2 by Lm3
.= (1-r)*((1/2)*q2)`2+r*((1/2)*q3)`2 by Lm3
.= (1-r)*((1/2)*q2`2)+r*((1/2)*q3)`2 by Lm3
.= (1-r)*((1/2)*q2`2)+r*((1/2)*q2`2) by A9,Lm3
.= ((1/2)*q2)`2 by Lm3;
A14: I1 \/ I2 \/ { 1/2*(p1+q2) } = I1 \/ (I2 \/ ({ 1/2*(p1+q2) } \/ { 1/2*(
p1+q2) })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { 1/2*(p1+q2) } \/ { 1/2*(p1+q2) }) by XBOOLE_1:4
.= I1 \/ { 1/2*(p1+q2) } \/ (I2 \/ { 1/2*(p1+q2) }) by XBOOLE_1:4;
A15: (1-r)*((1/2)*p1)+r*((1/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*q3) = (1-r)*((1
/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*p1)+r*((1/2)*q3) by RLVECT_1:def 3
.= (1-r)*((1/2)*p1)+(1-r)*((1/2)*q2)+(r*((1/2)*p1)+r*((1/2)*q3)) by
RLVECT_1:def 3
.= (1-r)*((1/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*p1+(1/2)*q3) by
RLVECT_1:def 5
.= (1-r)*((1/2)*p1+(1/2)*q2)+r*((1/2)*p1+(1/2)*q3) by RLVECT_1:def 5
.= (1-r)*((1/2)*p1+(1/2)*q2)+r*((1/2)*(p1+q3)) by RLVECT_1:def 5
.= (1-r)*((1/2)*(p1+q2))+r*((1/2)*(p1+q3)) by RLVECT_1:def 5;
A16: r*((1/2)*q2`1)-r*((1/2)*q3`1)+r = r*((1/2)*(q2`1-q3`1)+1)
.= 1 by A10,A11,XCMPLX_1:106;
((1-r)*((1/2)*q2)+r*((1/2)*q3)+(1-r)*|[1,0]|)`1 = ((1-r)*((1/2)*q2)+r*(
(1/2)*q3))`1+((1-r)*|[1,0]|)`1 by Lm1
.= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`1+|[(1-r)*1,(1-r)*0]|`1 by EUCLID:58
.= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`1+(1-r) by EUCLID:52
.= ((1-r)*((1/2)*q2))`1+(r*((1/2)*q3))`1+(1-r) by Lm1
.= (1-r)*((1/2)*q2)`1+(r*((1/2)*q3))`1+(1-r) by Lm3
.= (1-r)*((1/2)*q2)`1+r*((1/2)*q3)`1+(1-r) by Lm3
.= (1-r)*((1/2)*q2`1)+r*((1/2)*q3)`1+(1-r) by Lm3
.= (1-r)*((1/2)*q2`1)+r*((1/2)*q3`1)+(1-r) by Lm3
.= ((1/2)*q2)`1 by A16,Lm3;
then
A17: (1-r)*((1/2)*q2)+r*((1/2)*q3)+(1-r)*|[1,0]| = |[(1/2*q2)`1,(1/2* q2 )`2
]| by A13,EUCLID:53
.= 1/2*q2 by EUCLID:53;
1/2*(p1+q2) = 1/2*p1+1/2*q2 by RLVECT_1:def 5
.= (1-r+r)*((1/2)*p1)+1/2*q2 by RLVECT_1:def 8
.= (1-r)*((1/2)*p1)+r*((1/2)*p1)+1/2*q2 by RLVECT_1:def 6
.= (1-r)*((1/2)*p1)+r*((1/2)*p1)+ ((1-r)*((1/2)*q2)+r*((1/2)*q3))+(1-r)*
|[1,0]| by A17,RLVECT_1:def 3
.= (1-r)*((1/2)*p1)+r*((1/2)*p1)+ (1-r)*((1/2)*q2)+r*((1/2)*q3)+(1-r)*|[
1,0]| by RLVECT_1:def 3
.= (1-r)*((1/2)*(p1+q2))+(1-r)*|[1,0]|+r*((1/2)*(p1+q3)) by A15,
RLVECT_1:def 3
.= (1-r)*((1/2)*(p1+q2)+|[1,0]|)+r*((1/2)*(p1+q3)) by RLVECT_1:def 5;
then 1/2*(p1+q2) in LSeg(1/2*(p1+q2)+|[1,0]|,1/2*(p1+q3)) by A11,A12;
then
A18: LSeg(1/2*(p1+q2)+|[1,0]|,1/2*(p1+q3)) = LSeg(1/2*(p1+q2)+|[1,0]|,1/2*(
p1+q2)) \/ LSeg(1/2*(p1+q2),1/2*(p1+q3)) by TOPREAL1:5;
LSeg(1/2*(p1+q2)+|[1,0]|,1/2*(p1+q2)) c= I2 \/ { 1/2*(p1+q2) } by A1,A2,Th45;
hence thesis by A18,A7,A14,XBOOLE_1:13;
end;
Lm7: 1/2+1/2 = 1;
theorem
1 < len G & 1 <= j & j+1 < width G implies LSeg(1/2*(G*(1,j)+G*(1,j+1)
)-|[1,0]|,1/2*(G*(1,j+1)+G*(1,j+2))-|[1,0]|) c= Int cell(G,0,j) \/ Int cell(G,0
,j+1) \/ { G*(1,j+1)-|[1,0]| }
proof
assume that
A1: 1 < len G and
A2: 1 <= j and
A3: j+1 < width G;
set p1 = G*(1,j), p2 = G*(1,j+1), q3 = G*(1,j+2), r = (p2`2-p1`2)/(q3`2-p1`2
);
A4: j+1+1 = j+(1+1);
then
A5: j+2 >= 1 by NAT_1:11;
A6: j+(1+1) <= width G by A3,A4,NAT_1:13;
set I1 = Int cell(G,0,j), I2 = Int cell(G,0,j+1);
A7: I1 \/ I2 \/ { p2-|[1,0]| } = I1 \/ (I2 \/ ({ p2-|[1,0]| } \/ { p2-|[1,0
]| })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { p2-|[1,0]| } \/ { p2-|[1,0]| }) by XBOOLE_1:4
.= I1 \/ { p2-|[1,0]| } \/ (I2 \/ { p2-|[1,0]| }) by XBOOLE_1:4;
A8: LSeg(1/2*(p2+q3)-|[1,0]|,p2-|[1,0]|) c= I2 \/ { p2-|[1,0]| } by A3,A4,Th48,
NAT_1:11;
j < j+1 by XREAL_1:29;
then p1`2 < p2`2 by A1,A2,A3,GOBOARD5:4;
then
A9: p2`2-p1`2 > 0 by XREAL_1:50;
A10: j+1 >= 1 by NAT_1:11;
then
A11: p2`1 = G*(1,1)`1 by A1,A3,GOBOARD5:2
.= q3`1 by A1,A6,A5,GOBOARD5:2;
j <= j+1 by NAT_1:11;
then
A12: j < width G by A3,XXREAL_0:2;
then p1`1 = G*(1,1)`1 by A1,A2,GOBOARD5:2
.= p2`1 by A1,A3,A10,GOBOARD5:2;
then
A13: 1*p2`1 = (1-r)*p1`1+r*q3`1 by A11
.= ((1-r)*p1)`1+r*q3`1 by Lm3
.= ((1-r)*p1)`1+(r*q3)`1 by Lm3
.= ((1-r)*p1+r*q3)`1 by Lm1;
j+1 < j+2 by XREAL_1:6;
then p2`2 < q3`2 by A1,A10,A6,GOBOARD5:4;
then
A14: p2`2-p1`2 < q3`2-p1`2 by XREAL_1:9;
then r*(q3`2-p1`2) = p2`2-p1`2 by A9,XCMPLX_1:87;
then p2`2 = (1-r)*p1`2 +r*q3`2;
then 1*p2`2 = ((1-r)*p1)`2+r*q3`2 by Lm3
.= ((1-r)*p1)`2+(r*q3)`2 by Lm3
.= ((1-r)*p1+(r*q3))`2 by Lm1;
then
A15: (1-r)*p1+r*q3 = |[p2`1,p2`2]| by A13,EUCLID:53
.= p2 by EUCLID:53;
p2 = 1*p2 by RLVECT_1:def 8
.= 1/2*p2+1/2*p2 by Lm7,RLVECT_1:def 6
.= 1/2*(((1-r)+r)*p2) + 1/2*((1-r)*p1+r*q3) by A15,RLVECT_1:def 8
.= 1/2*((1-r)*p2+r*p2) + 1/2*((1-r)*p1+r*q3) by RLVECT_1:def 6
.= 1/2*((1-r)*p2)+1/2*(r*p2) + 1/2*((1-r)*p1+r*q3) by RLVECT_1:def 5
.= 1/2*((1-r)*p2)+1/2*(r*p2) + (1/2*((1-r)*p1)+1/2*(r*q3)) by
RLVECT_1:def 5
.= 1/2*((1-r)*p2)+(1/2*(r*p2) + (1/2*((1-r)*p1)+1/2*(r*q3))) by
RLVECT_1:def 3
.= 1/2*((1-r)*p2)+(1/2*((1-r)*p1)+(1/2*(r*p2)+1/2*(r*q3))) by
RLVECT_1:def 3
.= 1/2*((1-r)*p2)+1/2*((1-r)*p1)+(1/2*(r*p2)+1/2*(r*q3)) by RLVECT_1:def 3
.= (1/2*((1-r)*p2)+1/2*((1-r)*p1))+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 3
.= 1/2*((1-r)*p2+(1-r)*p1)+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 5
.= 1/2*((1-r)*(p1+p2))+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 5
.= 1/2*(1-r)*(p1+p2)+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 7
.= 1/2*(1-r)*(p1+p2)+(1/2*(r*p2)+1/2*(r*q3)) by RLVECT_1:def 3
.= 1/2*(1-r)*(p1+p2)+1/2*(r*p2+r*q3) by RLVECT_1:def 5
.= 1/2*(1-r)*(p1+p2)+1/2*(r*(p2+q3)) by RLVECT_1:def 5;
then
A16: p2 = (1-r)*(1/2*(p1+p2))+1/2*(r*(p2+q3)) by RLVECT_1:def 7
.= (1-r)*(1/2*(p1+p2))+1/2*r*(p2+q3) by RLVECT_1:def 7
.= (1-r)*((1/2)*(p1+p2))+r*((1/2)*(p2+q3)) by RLVECT_1:def 7;
A17: (1-r)*(1/2*(p1+p2)-|[1,0]|)+r*(1/2*(p2+q3)-|[1,0]|) = (1-r)*(1/2*(p1+p2
))-(1-r)*|[1,0]|+r*(1/2*(p2+q3)-|[1,0]|) by RLVECT_1:34
.= (1-r)*(1/2*(p1+p2))-(1-r)*|[1,0]|+(r*(1/2*(p2+q3))-r*|[1,0]|) by
RLVECT_1:34
.= r*(1/2*(p2+q3))+((1-r)*(1/2*(p1+p2))-(1-r)*|[1,0]|)-r*|[1,0]| by
RLVECT_1:def 3
.= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))-(1-r)*|[1,0]|-r*|[1,0]| by
RLVECT_1:def 3
.= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))-((1-r)*|[1,0]|+r*|[1,0]|) by
RLVECT_1:27
.= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))-((1-r)+r)*|[1,0]| by RLVECT_1:def 6
.= p2-|[1,0]| by A16,RLVECT_1:def 8;
r < 1 by A14,A9,XREAL_1:189;
then p2-|[1,0]| in LSeg(1/2*(p1+p2)-|[1,0]|,1/2*(p2+q3)-|[1,0]|) by A14,A9
,A17;
then
A18: LSeg(1/2*(p1+p2)-|[1,0]|,1/2*(p2+q3)-|[1,0]|) = LSeg(1/2*(p1+p2)-|[1,0
]|,p2-|[1,0]|) \/ LSeg(p2-|[1,0]|,1/2*(p2+q3)-|[1,0]|) by TOPREAL1:5;
LSeg(1/2*(p1+p2)-|[1,0]|,p2-|[1,0]|) c= I1 \/ { p2-|[1,0]| } by A2,A12,Th49;
hence thesis by A18,A8,A7,XBOOLE_1:13;
end;
theorem
1 < len G & 1 <= j & j+1 < width G implies LSeg(1/2*(G*(len G,j)+G*(
len G,j+1))+|[1,0]|, 1/2*(G*(len G,j+1)+G*(len G,j+2))+|[1,0]|) c= Int cell(G,
len G,j) \/ Int cell(G,len G,j+1) \/ { G*(len G,j+1)+|[1,0]| }
proof
assume that
A1: 1 < len G and
A2: 1 <= j and
A3: j+1 < width G;
set p1 = G*(len G,j), p2 = G*(len G,j+1), q3 = G*(len G,j+2), r = (p2`2-p1`2
)/(q3`2-p1`2);
A4: j+1+1 = j+(1+1);
then
A5: j+2 >= 1 by NAT_1:11;
A6: j+(1+1) <= width G by A3,A4,NAT_1:13;
set I1 = Int cell(G,len G,j), I2 = Int cell(G,len G,j+1);
A7: I1 \/ I2 \/ { p2+|[1,0]| } = I1 \/ (I2 \/ ({ p2+|[1,0]| } \/ { p2+|[1,0
]| })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { p2+|[1,0]| } \/ { p2+|[1,0]| }) by XBOOLE_1:4
.= I1 \/ { p2+|[1,0]| } \/ (I2 \/ { p2+|[1,0]| }) by XBOOLE_1:4;
A8: LSeg(1/2*(p2+q3)+|[1,0]|,p2+|[1,0]|) c= I2 \/ { p2+|[1,0]| } by A3,A4,Th50,
NAT_1:11;
j < j+1 by XREAL_1:29;
then p1`2 < p2`2 by A1,A2,A3,GOBOARD5:4;
then
A9: p2`2-p1`2 > 0 by XREAL_1:50;
A10: j+1 >= 1 by NAT_1:11;
then
A11: p2`1 = G*(len G,1)`1 by A1,A3,GOBOARD5:2
.= q3`1 by A1,A6,A5,GOBOARD5:2;
j <= j+1 by NAT_1:11;
then
A12: j < width G by A3,XXREAL_0:2;
then p1`1 = G*(len G,1)`1 by A1,A2,GOBOARD5:2
.= p2`1 by A1,A3,A10,GOBOARD5:2;
then
A13: 1*p2`1 = (1-r)*p1`1+r*q3`1 by A11
.= ((1-r)*p1)`1+r*q3`1 by Lm3
.= ((1-r)*p1)`1+(r*q3)`1 by Lm3
.= ((1-r)*p1+r*q3)`1 by Lm1;
j+1 < j+2 by XREAL_1:6;
then p2`2 < q3`2 by A1,A10,A6,GOBOARD5:4;
then
A14: p2`2-p1`2 < q3`2-p1`2 by XREAL_1:9;
then r*(q3`2-p1`2) = p2`2-p1`2 by A9,XCMPLX_1:87;
then p2`2 = (1-r)*p1`2 +r*q3`2;
then 1*p2`2 = ((1-r)*p1)`2+r*q3`2 by Lm3
.= ((1-r)*p1)`2+(r*q3)`2 by Lm3
.= ((1-r)*p1+(r*q3))`2 by Lm1;
then
A15: (1-r)*p1+r*q3 = |[p2`1,p2`2]| by A13,EUCLID:53
.= p2 by EUCLID:53;
p2 = 1*p2 by RLVECT_1:def 8
.= 1/2*p2+1/2*p2 by Lm7,RLVECT_1:def 6
.= 1/2*(((1-r)+r)*p2) + 1/2*((1-r)*p1+r*q3) by A15,RLVECT_1:def 8
.= 1/2*((1-r)*p2+r*p2) + 1/2*((1-r)*p1+r*q3) by RLVECT_1:def 6
.= 1/2*((1-r)*p2)+1/2*(r*p2) + 1/2*((1-r)*p1+r*q3) by RLVECT_1:def 5
.= 1/2*((1-r)*p2)+1/2*(r*p2) + (1/2*((1-r)*p1)+1/2*(r*q3)) by
RLVECT_1:def 5
.= 1/2*((1-r)*p2)+(1/2*(r*p2) + (1/2*((1-r)*p1)+1/2*(r*q3))) by
RLVECT_1:def 3
.= 1/2*((1-r)*p2)+(1/2*((1-r)*p1)+(1/2*(r*p2)+1/2*(r*q3))) by
RLVECT_1:def 3
.= 1/2*((1-r)*p2)+1/2*((1-r)*p1)+(1/2*(r*p2)+1/2*(r*q3)) by RLVECT_1:def 3
.= (1/2*((1-r)*p2)+1/2*((1-r)*p1))+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 3
.= 1/2*((1-r)*p2+(1-r)*p1)+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 5
.= 1/2*((1-r)*(p1+p2))+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 5
.= 1/2*(1-r)*(p1+p2)+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 7;
then
A16: p2 = 1/2*(1-r)*(p1+p2)+(1/2*(r*p2)+1/2*(r*q3)) by RLVECT_1:def 3
.= 1/2*(1-r)*(p1+p2)+1/2*(r*p2+r*q3) by RLVECT_1:def 5
.= 1/2*(1-r)*(p1+p2)+1/2*(r*(p2+q3)) by RLVECT_1:def 5
.= (1-r)*(1/2*(p1+p2))+1/2*(r*(p2+q3)) by RLVECT_1:def 7
.= (1-r)*(1/2*(p1+p2))+1/2*r*(p2+q3) by RLVECT_1:def 7
.= (1-r)*((1/2)*(p1+p2))+r*((1/2)*(p2+q3)) by RLVECT_1:def 7;
A17: (1-r)*(1/2*(p1+p2)+|[1,0]|)+r*(1/2*(p2+q3)+|[1,0]|) = (1-r)*(1/2*(p1+p2
))+(1-r)*|[1,0]|+r*(1/2*(p2+q3)+|[1,0]|) by RLVECT_1:def 5
.= (1-r)*(1/2*(p1+p2))+(1-r)*|[1,0]|+(r*(1/2*(p2+q3))+r*|[1,0]|) by
RLVECT_1:def 5
.= r*(1/2*(p2+q3))+((1-r)*(1/2*(p1+p2))+(1-r)*|[1,0]|)+r*|[1,0]| by
RLVECT_1:def 3
.= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))+(1-r)*|[1,0]|+r*|[1,0]| by
RLVECT_1:def 3
.= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))+((1-r)*|[1,0]|+r*|[1,0]|) by
RLVECT_1:def 3
.= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))+((1-r)+r)*|[1,0]| by RLVECT_1:def 6
.= p2+|[1,0]| by A16,RLVECT_1:def 8;
r < 1 by A14,A9,XREAL_1:189;
then p2+|[1,0]| in LSeg(1/2*(p1+p2)+|[1,0]|,1/2*(p2+q3)+|[1,0]|) by A14,A9
,A17;
then
A18: LSeg(1/2*(p1+p2)+|[1,0]|,1/2*(p2+q3)+|[1,0]|) = LSeg(1/2*(p1+p2)+|[1,0
]|,p2+|[1,0]|) \/ LSeg(p2+|[1,0]|,1/2*(p2+q3)+|[1,0]|) by TOPREAL1:5;
LSeg(1/2*(p1+p2)+|[1,0]|,p2+|[1,0]|) c= I1 \/ { p2+|[1,0]| } by A2,A12,Th51;
hence thesis by A18,A8,A7,XBOOLE_1:13;
end;
theorem
1 < width G & 1 <= i & i+1 < len G implies LSeg(1/2*(G*(i,1)+G*(i+1,1)
)-|[0,1]|,1/2*(G*(i+1,1)+G*(i+2,1))-|[0,1]|) c= Int cell(G,i,0) \/ Int cell(G,i
+1,0) \/ { G*(i+1,1)-|[0,1]| }
proof
assume that
A1: 1 < width G and
A2: 1 <= i and
A3: i+1 < len G;
set p1 = G*(i,1), p2 = G*(i+1,1), q3 = G*(i+2,1), r = (p2`1-p1`1)/(q3`1-p1`1
);
A4: i+1+1 = i+(1+1);
then
A5: i+2 >= 1 by NAT_1:11;
A6: i+(1+1) <= len G by A3,A4,NAT_1:13;
set I1 = Int cell(G,i,0), I2 = Int cell(G,i+1,0);
A7: I1 \/ I2 \/ { p2-|[0,1]| } = I1 \/ (I2 \/ ({ p2-|[0,1]| } \/ { p2-|[0,1
]| })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { p2-|[0,1]| } \/ { p2-|[0,1]| }) by XBOOLE_1:4
.= I1 \/ { p2-|[0,1]| } \/ (I2 \/ { p2-|[0,1]| }) by XBOOLE_1:4;
A8: LSeg(1/2*(p2+q3)-|[0,1]|,p2-|[0,1]|) c= I2 \/ { p2-|[0,1]| } by A3,A4,Th52,
NAT_1:11;
i < i+1 by XREAL_1:29;
then p1`1 < p2`1 by A1,A2,A3,GOBOARD5:3;
then
A9: p2`1-p1`1 > 0 by XREAL_1:50;
A10: i+1 >= 1 by NAT_1:11;
then
A11: p2`2 = G*(1,1)`2 by A1,A3,GOBOARD5:1
.= q3`2 by A1,A6,A5,GOBOARD5:1;
i <= i+1 by NAT_1:11;
then
A12: i < len G by A3,XXREAL_0:2;
then p1`2 = G*(1,1)`2 by A1,A2,GOBOARD5:1
.= p2`2 by A1,A3,A10,GOBOARD5:1;
then
A13: 1*p2`2 = (1-r)*p1`2+r*q3`2 by A11
.= ((1-r)*p1)`2+r*q3`2 by Lm3
.= ((1-r)*p1)`2+(r*q3)`2 by Lm3
.= ((1-r)*p1+r*q3)`2 by Lm1;
i+1 < i+2 by XREAL_1:6;
then p2`1 < q3`1 by A1,A10,A6,GOBOARD5:3;
then
A14: p2`1-p1`1 < q3`1-p1`1 by XREAL_1:9;
then r*(q3`1-p1`1) = p2`1-p1`1 by A9,XCMPLX_1:87;
then p2`1 = (1-r)*p1`1 +r*q3`1;
then 1*p2`1 = ((1-r)*p1)`1+r*q3`1 by Lm3
.= ((1-r)*p1)`1+(r*q3)`1 by Lm3
.= ((1-r)*p1+(r*q3))`1 by Lm1;
then
A15: (1-r)*p1+r*q3 = |[p2`1,p2`2]| by A13,EUCLID:53
.= p2 by EUCLID:53;
p2 = 1*p2 by RLVECT_1:def 8
.= 1/2*p2+1/2*p2 by Lm7,RLVECT_1:def 6
.= 1/2*(((1-r)+r)*p2) + 1/2*((1-r)*p1+r*q3) by A15,RLVECT_1:def 8
.= 1/2*((1-r)*p2+r*p2) + 1/2*((1-r)*p1+r*q3) by RLVECT_1:def 6
.= 1/2*((1-r)*p2)+1/2*(r*p2) + 1/2*((1-r)*p1+r*q3) by RLVECT_1:def 5
.= 1/2*((1-r)*p2)+1/2*(r*p2) + (1/2*((1-r)*p1)+1/2*(r*q3)) by
RLVECT_1:def 5
.= 1/2*((1-r)*p2)+(1/2*(r*p2) + (1/2*((1-r)*p1)+1/2*(r*q3))) by
RLVECT_1:def 3
.= 1/2*((1-r)*p2)+(1/2*((1-r)*p1)+(1/2*(r*p2)+1/2*(r*q3))) by
RLVECT_1:def 3
.= 1/2*((1-r)*p2)+1/2*((1-r)*p1)+(1/2*(r*p2)+1/2*(r*q3)) by RLVECT_1:def 3
.= (1/2*((1-r)*p2)+1/2*((1-r)*p1))+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 3
.= 1/2*((1-r)*p2+(1-r)*p1)+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 5
.= 1/2*((1-r)*(p1+p2))+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 5
.= 1/2*(1-r)*(p1+p2)+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 7
.= 1/2*(1-r)*(p1+p2)+(1/2*(r*p2)+1/2*(r*q3)) by RLVECT_1:def 3
.= 1/2*(1-r)*(p1+p2)+1/2*(r*p2+r*q3) by RLVECT_1:def 5;
then
A16: p2 = 1/2*(1-r)*(p1+p2)+1/2*(r*(p2+q3)) by RLVECT_1:def 5
.= (1-r)*(1/2*(p1+p2))+1/2*(r*(p2+q3)) by RLVECT_1:def 7
.= (1-r)*(1/2*(p1+p2))+1/2*r*(p2+q3) by RLVECT_1:def 7
.= (1-r)*((1/2)*(p1+p2))+r*((1/2)*(p2+q3)) by RLVECT_1:def 7;
A17: (1-r)*(1/2*(p1+p2)-|[0,1]|)+r*(1/2*(p2+q3)-|[0,1]|) = (1-r)*(1/2*(p1+p2
))-(1-r)*|[0,1]|+r*(1/2*(p2+q3)-|[0,1]|) by RLVECT_1:34
.= (1-r)*(1/2*(p1+p2))-(1-r)*|[0,1]|+(r*(1/2*(p2+q3))-r*|[0,1]|) by
RLVECT_1:34
.= r*(1/2*(p2+q3))+((1-r)*(1/2*(p1+p2))-(1-r)*|[0,1]|)-r*|[0,1]| by
RLVECT_1:def 3
.= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))-(1-r)*|[0,1]|-r*|[0,1]| by
RLVECT_1:def 3
.= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))-((1-r)*|[0,1]|+r*|[0,1]|) by
RLVECT_1:27
.= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))-((1-r)+r)*|[0,1]| by RLVECT_1:def 6
.= p2-|[0,1]| by A16,RLVECT_1:def 8;
r < 1 by A14,A9,XREAL_1:189;
then p2-|[0,1]| in LSeg(1/2*(p1+p2)-|[0,1]|,1/2*(p2+q3)-|[0,1]|) by A14,A9
,A17;
then
A18: LSeg(1/2*(p1+p2)-|[0,1]|,1/2*(p2+q3)-|[0,1]|) = LSeg(1/2*(p1+p2)-|[0,1
]|,p2-|[0,1]|) \/ LSeg(p2-|[0,1]|,1/2*(p2+q3)-|[0,1]|) by TOPREAL1:5;
LSeg(1/2*(p1+p2)-|[0,1]|,p2-|[0,1]|) c= I1 \/ { p2-|[0,1]| } by A2,A12,Th53;
hence thesis by A18,A8,A7,XBOOLE_1:13;
end;
theorem
1 < width G & 1 <= i & i+1 < len G implies LSeg(1/2*(G*(i,width G)+G*(
i+1,width G))+|[0,1]|, 1/2*(G*(i+1,width G)+G*(i+2,width G))+|[0,1]|) c= Int
cell(G,i,width G) \/ Int cell(G,i+1,width G) \/ { G*(i+1,width G)+|[0,1]| }
proof
assume that
A1: 1 < width G and
A2: 1 <= i and
A3: i+1 < len G;
set p1 = G*(i,width G), p2 = G*(i+1,width G), q3 = G*(i+2,width G), r = (p2
`1-p1`1)/(q3`1-p1`1);
A4: i+1+1 = i+(1+1);
then
A5: i+2 >= 1 by NAT_1:11;
A6: i+(1+1) <= len G by A3,A4,NAT_1:13;
set I1 = Int cell(G,i,width G), I2 = Int cell(G,i+1,width G);
A7: I1 \/ I2 \/ { p2+|[0,1]| } = I1 \/ (I2 \/ ({ p2+|[0,1]| } \/ { p2+|[0,1
]| })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { p2+|[0,1]| } \/ { p2+|[0,1]| }) by XBOOLE_1:4
.= I1 \/ { p2+|[0,1]| } \/ (I2 \/ { p2+|[0,1]| }) by XBOOLE_1:4;
A8: LSeg(1/2*(p2+q3)+|[0,1]|,p2+|[0,1]|) c= I2 \/ { p2+|[0,1]| } by A3,A4,Th54,
NAT_1:11;
i < i+1 by XREAL_1:29;
then p1`1 < p2`1 by A1,A2,A3,GOBOARD5:3;
then
A9: p2`1-p1`1 > 0 by XREAL_1:50;
A10: i+1 >= 1 by NAT_1:11;
then
A11: p2`2 = G*(1,width G)`2 by A1,A3,GOBOARD5:1
.= q3`2 by A1,A6,A5,GOBOARD5:1;
i <= i+1 by NAT_1:11;
then
A12: i < len G by A3,XXREAL_0:2;
then p1`2 = G*(1,width G)`2 by A1,A2,GOBOARD5:1
.= p2`2 by A1,A3,A10,GOBOARD5:1;
then
A13: 1*p2`2 = (1-r)*p1`2+r*q3`2 by A11
.= ((1-r)*p1)`2+r*q3`2 by Lm3
.= ((1-r)*p1)`2+(r*q3)`2 by Lm3
.= ((1-r)*p1+r*q3)`2 by Lm1;
i+1 < i+2 by XREAL_1:6;
then p2`1 < q3`1 by A1,A10,A6,GOBOARD5:3;
then
A14: p2`1-p1`1 < q3`1-p1`1 by XREAL_1:9;
then r*(q3`1-p1`1) = p2`1-p1`1 by A9,XCMPLX_1:87;
then p2`1 = (1-r)*p1`1 +r*q3`1;
then 1*p2`1 = ((1-r)*p1)`1+r*q3`1 by Lm3
.= ((1-r)*p1)`1+(r*q3)`1 by Lm3
.= ((1-r)*p1+(r*q3))`1 by Lm1;
then
A15: (1-r)*p1+r*q3 = |[p2`1,p2`2]| by A13,EUCLID:53
.= p2 by EUCLID:53;
p2 = 1*p2 by RLVECT_1:def 8
.= 1/2*p2+1/2*p2 by Lm7,RLVECT_1:def 6
.= 1/2*(((1-r)+r)*p2) + 1/2*((1-r)*p1+r*q3) by A15,RLVECT_1:def 8
.= 1/2*((1-r)*p2+r*p2) + 1/2*((1-r)*p1+r*q3) by RLVECT_1:def 6
.= 1/2*((1-r)*p2)+1/2*(r*p2) + 1/2*((1-r)*p1+r*q3) by RLVECT_1:def 5
.= 1/2*((1-r)*p2)+1/2*(r*p2) + (1/2*((1-r)*p1)+1/2*(r*q3)) by
RLVECT_1:def 5
.= 1/2*((1-r)*p2)+(1/2*(r*p2) + (1/2*((1-r)*p1)+1/2*(r*q3))) by
RLVECT_1:def 3
.= 1/2*((1-r)*p2)+(1/2*((1-r)*p1)+(1/2*(r*p2)+1/2*(r*q3))) by
RLVECT_1:def 3
.= 1/2*((1-r)*p2)+1/2*((1-r)*p1)+(1/2*(r*p2)+1/2*(r*q3)) by RLVECT_1:def 3
.= (1/2*((1-r)*p2)+1/2*((1-r)*p1))+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 3
.= 1/2*((1-r)*p2+(1-r)*p1)+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 5
.= 1/2*((1-r)*(p1+p2))+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 5
.= 1/2*(1-r)*(p1+p2)+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 7;
then
A16: p2 = 1/2*(1-r)*(p1+p2)+(1/2*(r*p2)+1/2*(r*q3)) by RLVECT_1:def 3
.= 1/2*(1-r)*(p1+p2)+1/2*(r*p2+r*q3) by RLVECT_1:def 5
.= 1/2*(1-r)*(p1+p2)+1/2*(r*(p2+q3)) by RLVECT_1:def 5
.= (1-r)*(1/2*(p1+p2))+1/2*(r*(p2+q3)) by RLVECT_1:def 7
.= (1-r)*(1/2*(p1+p2))+1/2*r*(p2+q3) by RLVECT_1:def 7
.= (1-r)*((1/2)*(p1+p2))+r*((1/2)*(p2+q3)) by RLVECT_1:def 7;
A17: (1-r)*(1/2*(p1+p2)+|[0,1]|)+r*(1/2*(p2+q3)+|[0,1]|) = (1-r)*(1/2*(p1+p2
))+(1-r)*|[0,1]|+r*(1/2*(p2+q3)+|[0,1]|) by RLVECT_1:def 5
.= (1-r)*(1/2*(p1+p2))+(1-r)*|[0,1]|+(r*(1/2*(p2+q3))+r*|[0,1]|) by
RLVECT_1:def 5
.= r*(1/2*(p2+q3))+((1-r)*(1/2*(p1+p2))+(1-r)*|[0,1]|)+r*|[0,1]| by
RLVECT_1:def 3
.= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))+(1-r)*|[0,1]|+r*|[0,1]| by
RLVECT_1:def 3
.= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))+((1-r)*|[0,1]|+r*|[0,1]|) by
RLVECT_1:def 3
.= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))+((1-r)+r)*|[0,1]| by RLVECT_1:def 6
.= p2+|[0,1]| by A16,RLVECT_1:def 8;
r < 1 by A14,A9,XREAL_1:189;
then p2+|[0,1]| in LSeg(1/2*(p1+p2)+|[0,1]|,1/2*(p2+q3)+|[0,1]|) by A14,A9
,A17;
then
A18: LSeg(1/2*(p1+p2)+|[0,1]|,1/2*(p2+q3)+|[0,1]|) = LSeg(1/2*(p1+p2)+|[0,1
]|,p2+|[0,1]|) \/ LSeg(p2+|[0,1]|,1/2*(p2+q3)+|[0,1]|) by TOPREAL1:5;
LSeg(1/2*(p1+p2)+|[0,1]|,p2+|[0,1]|) c= I1 \/ { p2+|[0,1]| } by A2,A12,Th55;
hence thesis by A18,A8,A7,XBOOLE_1:13;
end;
theorem
1 < len G & 1 < width G implies LSeg(G*(1,1)-|[1,1]|,1/2*(G*(1,1)+G*(1
,2))-|[1,0]|) c= Int cell(G,0,0) \/ Int cell(G,0,1) \/ { G*(1,1)-|[1,0]| }
proof
assume that
A1: 1 < len G and
A2: 1 < width G;
set q2 = G*(1,1), q3 = G*(1,2), r = 1/(1/2*(q3`2-q2`2)+1);
A3: 0+(1+1) <= width G by A2,NAT_1:13;
then
A4: q2`1 = q3`1 by A1,GOBOARD5:2;
q2`2 < q3`2 by A1,A3,GOBOARD5:4;
then
A5: q3`2-q2`2 > 0 by XREAL_1:50;
then 1 < 1/2*(q3`2-q2`2)+1 by XREAL_1:29,129;
then
A6: r < 1 by XREAL_1:212;
A7: ((1-r)*(q2-|[1,1]|)+r*(1/2*(q2+q3)-|[1,0]|))`1 = ((1-r)*(q2-|[1,1]|))`1
+(r*(1/2*(q2+q3)-|[1,0]|))`1 by Lm1
.= (1-r)*(q2-|[1,1]|)`1+(r*(1/2*(q2+q3)-|[1,0]|))`1 by Lm3
.= (1-r)*(q2-|[1,1]|)`1+r*(1/2*(q2+q3)-|[1,0]|)`1 by Lm3
.= (1-r)*(q2`1-|[1,1]|`1)+r*(1/2*(q2+q3)-|[1,0]|)`1 by Lm2
.= (1-r)*(q2`1-|[1,1]|`1)+r*((1/2*(q2+q3))`1-|[1,0]|`1) by Lm2
.= (1-r)*(q2`1-1)+r*((1/2*(q2+q3))`1-|[1,0]|`1) by EUCLID:52
.= (1-r)*q2`1-(1-r)*1+r*((1/2*(q2+q3))`1-1) by EUCLID:52
.= (1-r)*q2`1+r*(1/2*(q2+q3))`1-((1-r)+r)
.= (1-r)*q2`1+r*(1/2*(q2+q3)`1)-1 by Lm3
.= (1-r)*q2`1+r*(1/2*(q2`1+q2`1))-1 by A4,Lm1
.= q2`1-|[1,0]|`1 by EUCLID:52
.= (q2-|[1,0]|)`1 by Lm2;
A8: r*((1/2)*q3`2)-r*((1/2)*q2`2)+r = r*((1/2)*(q3`2-q2`2)+1)
.= 1 by A5,XCMPLX_1:106;
((1-r)*(q2-|[1,1]|)+r*(1/2*(q2+q3)-|[1,0]|))`2 = ((1-r)*(q2-|[1,1]|))`2
+(r*(1/2*(q2+q3)-|[1,0]|))`2 by Lm1
.= ((1-r)*q2-(1-r)*|[1,1]|)`2+(r*(1/2*(q2+q3)-|[1,0]|))`2 by RLVECT_1:34
.= ((1-r)*q2)`2-((1-r)*|[1,1]|)`2+(r*(1/2*(q2+q3)-|[1,0]|))`2 by Lm2
.= ((1-r)*q2)`2-(1-r)*|[1,1]|`2+(r*(1/2*(q2+q3)-|[1,0]|))`2 by Lm3
.= ((1-r)*q2)`2-(1-r)*1+(r*(1/2*(q2+q3)-|[1,0]|))`2 by EUCLID:52
.= (1-r)*q2`2-(1-r)*1+(r*(1/2*(q2+q3)-|[1,0]|))`2 by Lm3
.= (1-r)*q2`2-(1-r)+r*(1/2*(q2+q3)-|[1,0]|)`2 by Lm3
.= (1-r)*q2`2-(1-r)+r*((1/2*(q2+q3))`2-|[1,0]|`2) by Lm2
.= (1-r)*q2`2-(1-r)+r*((1/2*(q2+q3))`2-0) by EUCLID:52
.= (1-r)*q2`2-(1-r)+r*(1/2*(q2+q3)`2) by Lm3
.= (1-r)*q2`2-(1-r)+r*(1/2*(q2`2+q3`2)) by Lm1
.= q2`2-0 by A8
.= q2`2-|[1,0]|`2 by EUCLID:52
.= (q2-|[1,0]|)`2 by Lm2;
then
(1-r)*(q2-|[1,1]|)+r*(1/2*(q2+q3)-|[1,0]|) = |[(q2-|[1,0]|)`1,(q2-|[1,0
]|)`2]| by A7,EUCLID:53
.= q2-|[1,0]| by EUCLID:53;
then q2-|[1,0]| in LSeg(q2-|[1,1]|,1/2*(q2+q3)-|[1,0]|) by A5,A6;
then
A9: LSeg(q2-|[1,1]|,1/2*(q2+q3)-|[1,0]|) = LSeg(q2-|[1,1]|,q2-|[1,0]|) \/
LSeg(q2-|[1,0]|,1/2*(q2+q3)-|[1,0]|) by TOPREAL1:5;
set I1 = Int cell(G,0,0), I2 = Int cell(G,0,1);
0+1+1 = 0+(1+1);
then
A10: LSeg(q2-|[1,0]|,1/2*(q2+q3)-|[1,0]|) c= I2 \/ { q2-|[1,0]| } by A2,Th48;
A11: I1 \/ I2 \/ { q2-|[1,0]| } = I1 \/ (I2 \/ ({ q2-|[1,0]| } \/ { q2-|[1,0
]| })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { q2-|[1,0]| } \/ { q2-|[1,0]| }) by XBOOLE_1:4
.= I1 \/ { q2-|[1,0]| } \/ (I2 \/ { q2-|[1,0]| }) by XBOOLE_1:4;
LSeg(q2-|[1,1]|,q2-|[1,0]|) c= I1 \/ { q2-|[1,0]| } by Th56;
hence thesis by A9,A10,A11,XBOOLE_1:13;
end;
theorem
1 < len G & 1 < width G implies LSeg(G*(len G,1)+|[1,-1]|,1/2*(G*(len
G,1)+G*(len G,2))+|[1,0]|) c= Int cell(G,len G,0) \/ Int cell(G,len G,1) \/ { G
*(len G,1)+|[1,0]| }
proof
assume that
A1: 1 < len G and
A2: 1 < width G;
set q2 = G*(len G,1), q3 = G*(len G,2), r = 1/(1/2*(q3`2-q2`2)+1);
A3: 0+(1+1) <= width G by A2,NAT_1:13;
then
A4: q2`1 = q3`1 by A1,GOBOARD5:2;
q2`2 < q3`2 by A1,A3,GOBOARD5:4;
then
A5: q3`2-q2`2 > 0 by XREAL_1:50;
then 1 < 1/2*(q3`2-q2`2)+1 by XREAL_1:29,129;
then
A6: r < 1 by XREAL_1:212;
A7: ((1-r)*(q2+|[1,-1]|)+r*(1/2*(q2+q3)+|[1,0]|))`1 = ((1-r)*(q2+|[1,-1]|))
`1+(r*(1/2*(q2+q3)+|[1,0]|))`1 by Lm1
.= (1-r)*(q2+|[1,-1]|)`1+(r*(1/2*(q2+q3)+|[1,0]|))`1 by Lm3
.= (1-r)*(q2+|[1,-1]|)`1+r*(1/2*(q2+q3)+|[1,0]|)`1 by Lm3
.= (1-r)*(q2`1+|[1,-1]|`1)+r*(1/2*(q2+q3)+|[1,0]|)`1 by Lm1
.= (1-r)*(q2`1+|[1,-1]|`1)+r*((1/2*(q2+q3))`1+|[1,0]|`1) by Lm1
.= (1-r)*(q2`1+1)+r*((1/2*(q2+q3))`1+|[1,0]|`1) by EUCLID:52
.= (1-r)*q2`1+(1-r)*1+r*((1/2*(q2+q3))`1+1) by EUCLID:52
.= (1-r)*q2`1+r*(1/2*(q2+q3))`1+((1-r)+r)
.= (1-r)*q2`1+r*(1/2*(q2+q3)`1)+1 by Lm3
.= (1-r)*q2`1+r*(1/2*(q2`1+q2`1))+1 by A4,Lm1
.= q2`1+|[1,0]|`1 by EUCLID:52
.= (q2+|[1,0]|)`1 by Lm1;
A8: r*((1/2)*q3`2)-r*((1/2)*q2`2)+r = r*((1/2)*(q3`2-q2`2)+1)
.= 1 by A5,XCMPLX_1:106;
((1-r)*(q2+|[1,-1]|)+r*(1/2*(q2+q3)+|[1,0]|))`2 = ((1-r)*(q2+|[1,-1]|))
`2+(r*(1/2*(q2+q3)+|[1,0]|))`2 by Lm1
.= ((1-r)*q2+(1-r)*|[1,-1]|)`2+(r*(1/2*(q2+q3)+|[1,0]|))`2 by
RLVECT_1:def 5
.= ((1-r)*q2)`2+((1-r)*|[1,-1]|)`2+(r*(1/2*(q2+q3)+|[1,0]|))`2 by Lm1
.= ((1-r)*q2)`2+(1-r)*|[1,-1]|`2+(r*(1/2*(q2+q3)+|[1,0]|))`2 by Lm3
.= ((1-r)*q2)`2+(1-r)*(-1)+(r*(1/2*(q2+q3)+|[1,0]|))`2 by EUCLID:52
.= (1-r)*q2`2+-(1-r)+(r*(1/2*(q2+q3)+|[1,0]|))`2 by Lm3
.= (1-r)*q2`2-(1-r)+r*(1/2*(q2+q3)+|[1,0]|)`2 by Lm3
.= (1-r)*q2`2-(1-r)+r*((1/2*(q2+q3))`2+|[1,0]|`2) by Lm1
.= (1-r)*q2`2-(1-r)+r*((1/2*(q2+q3))`2+0) by EUCLID:52
.= (1-r)*q2`2-(1-r)+r*(1/2*(q2+q3)`2) by Lm3
.= (1-r)*q2`2-(1-r)+r*(1/2*(q2`2+q3`2)) by Lm1
.= q2`2+0 by A8
.= q2`2+|[1,0]|`2 by EUCLID:52
.= (q2+|[1,0]|)`2 by Lm1;
then
(1-r)*(q2+|[1,-1]|)+r*(1/2*(q2+q3)+|[1,0]|) = |[(q2+|[1,0]|)`1,(q2+|[1,
0]|)`2]| by A7,EUCLID:53
.= q2+|[1,0]| by EUCLID:53;
then q2+|[1,0]| in LSeg(q2+|[1,-1]|,1/2*(q2+q3)+|[1,0]|) by A5,A6;
then
A9: LSeg(q2+|[1,-1]|,1/2*(q2+q3)+|[1,0]|) = LSeg(q2+|[1,-1]|,q2+|[1,0]|) \/
LSeg(q2+|[1,0]|,1/2*(q2+q3)+|[1,0]|) by TOPREAL1:5;
set I1 = Int cell(G,len G,0), I2 = Int cell(G,len G,1);
0+1+1 = 0+(1+1);
then
A10: LSeg(q2+|[1,0]|,1/2*(q2+q3)+|[1,0]|) c= I2 \/ { q2+|[1,0]| } by A2,Th50;
A11: I1 \/ I2 \/ { q2+|[1,0]| } = I1 \/ (I2 \/ ({ q2+|[1,0]| } \/ { q2+|[1,0
]| })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { q2+|[1,0]| } \/ { q2+|[1,0]| }) by XBOOLE_1:4
.= I1 \/ { q2+|[1,0]| } \/ (I2 \/ { q2+|[1,0]| }) by XBOOLE_1:4;
LSeg(q2+|[1,-1]|,q2+|[1,0]|) c= I1 \/ { q2+|[1,0]| } by Th57;
hence thesis by A9,A10,A11,XBOOLE_1:13;
end;
theorem
1 < len G & 1 < width G implies LSeg(G*(1,width G)+|[-1,1]|,1/2*(G*(1,
width G)+G* (1,width G -' 1))-|[1,0]|) c= Int cell(G,0,width G) \/ Int cell(G,0
,width G -' 1) \/ { G*(1,width G)-|[1,0]| }
proof
assume that
A1: 1 < len G and
A2: 1 < width G;
set q2 = G*(1,width G), q3 = G*(1,width G -' 1), r = 1/(1/2*(q2`2-q3`2)+1);
A3: width G -' 1 + 1 = width G by A2,XREAL_1:235;
then
A4: width G -' 1 >= 1 by A2,NAT_1:13;
A5: width G -'1 < width G by A3,NAT_1:13;
then q3`2 < q2`2 by A1,A4,GOBOARD5:4;
then
A6: q2`2-q3`2 > 0 by XREAL_1:50;
then 1 < 1/2*(q2`2-q3`2)+1 by XREAL_1:29,129;
then
A7: r < 1 by XREAL_1:212;
A8: q2`1 = G*(1,1)`1 by A1,A2,GOBOARD5:2
.= q3`1 by A1,A4,A5,GOBOARD5:2;
A9: ((1-r)*(q2+|[-1,1]|)+r*(1/2*(q2+q3)-|[1,0]|))`1 = ((1-r)*(q2+|[-1,1]|))
`1+(r*(1/2*(q2+q3)-|[1,0]|))`1 by Lm1
.= (1-r)*(q2+|[-1,1]|)`1+(r*(1/2*(q2+q3)-|[1,0]|))`1 by Lm3
.= (1-r)*(q2+|[-1,1]|)`1+r*(1/2*(q2+q3)-|[1,0]|)`1 by Lm3
.= (1-r)*(q2`1+|[-1,1]|`1)+r*(1/2*(q2+q3)-|[1,0]|)`1 by Lm1
.= (1-r)*(q2`1+|[-1,1]|`1)+r*((1/2*(q2+q3))`1-|[1,0]|`1) by Lm2
.= (1-r)*(q2`1+-1)+r*((1/2*(q2+q3))`1-|[1,0]|`1) by EUCLID:52
.= (1-r)*(q2`1-1)+r*((1/2*(q2+q3))`1-1) by EUCLID:52
.= (1-r)*q2`1+r*(1/2*(q2+q3))`1-1
.= (1-r)*q2`1+r*(1/2*(q2+q3)`1)-1 by Lm3
.= (1-r)*q2`1+r*(1/2*(q2`1+q2`1))-1 by A8,Lm1
.= q2`1-|[1,0]|`1 by EUCLID:52
.= (q2-|[1,0]|)`1 by Lm2;
A10: r*((1/2)*q2`2)-r*((1/2)*q3`2)+r = r*((1/2)*(q2`2-q3`2)+1)
.= 1 by A6,XCMPLX_1:106;
((1-r)*(q2+|[-1,1]|)+r*(1/2*(q2+q3)-|[1,0]|))`2 = ((1-r)*(q2+|[-1,1]|))
`2+(r*(1/2*(q2+q3)-|[1,0]|))`2 by Lm1
.= ((1-r)*q2+(1-r)*|[-1,1]|)`2+(r*(1/2*(q2+q3)-|[1,0]|))`2 by
RLVECT_1:def 5
.= ((1-r)*q2)`2+((1-r)*|[-1,1]|)`2+(r*(1/2*(q2+q3)-|[1,0]|))`2 by Lm1
.= ((1-r)*q2)`2+(1-r)*|[-1,1]|`2+(r*(1/2*(q2+q3)-|[1,0]|))`2 by Lm3
.= ((1-r)*q2)`2+(1-r)*1+(r*(1/2*(q2+q3)-|[1,0]|))`2 by EUCLID:52
.= (1-r)*q2`2+(1-r)*1+(r*(1/2*(q2+q3)-|[1,0]|))`2 by Lm3
.= (1-r)*q2`2+(1-r)+r*(1/2*(q2+q3)-|[1,0]|)`2 by Lm3
.= (1-r)*q2`2+(1-r)+r*((1/2*(q2+q3))`2-|[1,0]|`2) by Lm2
.= (1-r)*q2`2+(1-r)+r*((1/2*(q2+q3))`2-0) by EUCLID:52
.= (1-r)*q2`2+(1-r)+r*(1/2*(q2+q3)`2) by Lm3
.= (1-r)*q2`2+(1-r)+r*(1/2*(q2`2+q3`2)) by Lm1
.= q2`2-0 by A10
.= q2`2-|[1,0]|`2 by EUCLID:52
.= (q2-|[1,0]|)`2 by Lm2;
then
(1-r)*(q2+|[-1,1]|)+r*(1/2*(q2+q3)-|[1,0]|) = |[(q2-|[1,0]|)`1,(q2-|[1,
0]|)`2]| by A9,EUCLID:53
.= q2-|[1,0]| by EUCLID:53;
then q2-|[1,0]| in LSeg(q2+|[-1,1]|,1/2*(q2+q3)-|[1,0]|) by A6,A7;
then
A11: LSeg(q2+|[-1,1]|,1/2*(q2+q3)-|[1,0]|) = LSeg(q2+|[-1,1]|,q2-|[1,0]|) \/
LSeg(q2-|[1,0]|,1/2*(q2+q3)-|[1,0]|) by TOPREAL1:5;
set I1 = Int cell(G,0,width G), I2 = Int cell(G,0,width G -' 1);
A12: I1 \/ I2 \/ { q2-|[1,0]| } = I1 \/ (I2 \/ ({ q2-|[1,0]| } \/ { q2-|[1,0
]| })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { q2-|[1,0]| } \/ { q2-|[1,0]| }) by XBOOLE_1:4
.= I1 \/ { q2-|[1,0]| } \/ (I2 \/ { q2-|[1,0]| }) by XBOOLE_1:4;
A13: LSeg(q2+|[-1,1]|,q2-|[1,0]|) c= I1 \/ { q2-|[1,0]| } by Th58;
LSeg(q2-|[1,0]|,1/2*(q2+q3)-|[1,0]|) c= I2 \/ { q2-|[1,0]| } by A3,A4,A5,Th49
;
hence thesis by A11,A13,A12,XBOOLE_1:13;
end;
theorem
1 < len G & 1 < width G implies LSeg(G*(len G,width G)+|[1,1]|, 1/2*(G
*(len G,width G)+G*(len G,width G -' 1))+|[1,0]|) c= Int cell(G,len G,width G)
\/ Int cell(G,len G,width G -' 1) \/ { G*(len G,width G)+|[1,0]| }
proof
assume that
A1: 1 < len G and
A2: 1 < width G;
set q2 = G*(len G,width G), q3 = G*(len G,width G -' 1), r = 1/(1/2*(q2`2-q3
`2)+1);
A3: width G -' 1 + 1 = width G by A2,XREAL_1:235;
then
A4: width G -' 1 >= 1 by A2,NAT_1:13;
A5: width G -' 1 < width G by A3,NAT_1:13;
then q3`2 < q2`2 by A1,A4,GOBOARD5:4;
then
A6: q2`2-q3`2 > 0 by XREAL_1:50;
then 1 < 1/2*(q2`2-q3`2)+1 by XREAL_1:29,129;
then
A7: r < 1 by XREAL_1:212;
A8: q2`1 = G*(len G,1)`1 by A1,A2,GOBOARD5:2
.= q3`1 by A1,A4,A5,GOBOARD5:2;
A9: ((1-r)*(q2+|[1,1]|)+r*(1/2*(q2+q3)+|[1,0]|))`1 = ((1-r)*(q2+|[1,1]|))`1
+(r*(1/2*(q2+q3)+|[1,0]|))`1 by Lm1
.= (1-r)*(q2+|[1,1]|)`1+(r*(1/2*(q2+q3)+|[1,0]|))`1 by Lm3
.= (1-r)*(q2+|[1,1]|)`1+r*(1/2*(q2+q3)+|[1,0]|)`1 by Lm3
.= (1-r)*(q2`1+|[1,1]|`1)+r*(1/2*(q2+q3)+|[1,0]|)`1 by Lm1
.= (1-r)*(q2`1+|[1,1]|`1)+r*((1/2*(q2+q3))`1+|[1,0]|`1) by Lm1
.= (1-r)*(q2`1+1)+r*((1/2*(q2+q3))`1+|[1,0]|`1) by EUCLID:52
.= (1-r)*q2`1+(1-r)*1+r*((1/2*(q2+q3))`1+1) by EUCLID:52
.= (1-r)*q2`1+r*(1/2*(q2+q3))`1+((1-r)+r)
.= (1-r)*q2`1+r*(1/2*(q2+q3)`1)+1 by Lm3
.= (1-r)*q2`1+r*(1/2*(q2`1+q2`1))+1 by A8,Lm1
.= q2`1+|[1,0]|`1 by EUCLID:52
.= (q2+|[1,0]|)`1 by Lm1;
A10: r*((1/2)*q2`2)-r*((1/2)*q3`2)+r = r*((1/2)*(q2`2-q3`2)+1)
.= 1 by A6,XCMPLX_1:106;
((1-r)*(q2+|[1,1]|)+r*(1/2*(q2+q3)+|[1,0]|))`2 = ((1-r)*(q2+|[1,1]|))`2
+(r*(1/2*(q2+q3)+|[1,0]|))`2 by Lm1
.= ((1-r)*q2+(1-r)*|[1,1]|)`2+(r*(1/2*(q2+q3)+|[1,0]|))`2 by RLVECT_1:def 5
.= ((1-r)*q2)`2+((1-r)*|[1,1]|)`2+(r*(1/2*(q2+q3)+|[1,0]|))`2 by Lm1
.= ((1-r)*q2)`2+(1-r)*|[1,1]|`2+(r*(1/2*(q2+q3)+|[1,0]|))`2 by Lm3
.= ((1-r)*q2)`2+(1-r)*1+(r*(1/2*(q2+q3)+|[1,0]|))`2 by EUCLID:52
.= (1-r)*q2`2+(1-r)+(r*(1/2*(q2+q3)+|[1,0]|))`2 by Lm3
.= (1-r)*q2`2+(1-r)+r*(1/2*(q2+q3)+|[1,0]|)`2 by Lm3
.= (1-r)*q2`2+(1-r)+r*((1/2*(q2+q3))`2+|[1,0]|`2) by Lm1
.= (1-r)*q2`2+(1-r)+r*((1/2*(q2+q3))`2+0) by EUCLID:52
.= (1-r)*q2`2+(1-r)+r*(1/2*(q2+q3)`2) by Lm3
.= (1-r)*q2`2+(1-r)+r*(1/2*(q2`2+q3`2)) by Lm1
.= q2`2+0 by A10
.= q2`2+|[1,0]|`2 by EUCLID:52
.= (q2+|[1,0]|)`2 by Lm1;
then
(1-r)*(q2+|[1,1]|)+r*(1/2*(q2+q3)+|[1,0]|) = |[(q2+|[1,0]|)`1,(q2+|[1,0
]|)`2]| by A9,EUCLID:53
.= q2+|[1,0]| by EUCLID:53;
then q2+|[1,0]| in LSeg(q2+|[1,1]|,1/2*(q2+q3)+|[1,0]|) by A6,A7;
then
A11: LSeg(q2+|[1,1]|,1/2*(q2+q3)+|[1,0]|) = LSeg(q2+|[1,1]|,q2+|[1,0]|) \/
LSeg(q2+|[1,0]|,1/2*(q2+q3)+|[1,0]|) by TOPREAL1:5;
set I1 = Int cell(G,len G,width G), I2 = Int cell(G,len G,width G -' 1);
A12: I1 \/ I2 \/ { q2+|[1,0]| } = I1 \/ (I2 \/ ({ q2+|[1,0]| } \/ { q2+|[1,0
]| })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { q2+|[1,0]| } \/ { q2+|[1,0]| }) by XBOOLE_1:4
.= I1 \/ { q2+|[1,0]| } \/ (I2 \/ { q2+|[1,0]| }) by XBOOLE_1:4;
A13: LSeg(q2+|[1,1]|,q2+|[1,0]|) c= I1 \/ { q2+|[1,0]| } by Th59;
LSeg(q2+|[1,0]|,1/2*(q2+q3)+|[1,0]|) c= I2 \/ { q2+|[1,0]| } by A3,A4,A5,Th51
;
hence thesis by A11,A13,A12,XBOOLE_1:13;
end;
theorem
1 < width G & 1 < len G implies LSeg(G*(1,1)-|[1,1]|,1/2*(G*(1,1)+G*(2
,1))-|[0,1]|) c= Int cell(G,0,0) \/ Int cell(G,1,0) \/ { G*(1,1)-|[0,1]| }
proof
assume that
A1: 1 < width G and
A2: 1 < len G;
set q2 = G*(1,1), q3 = G*(2,1), r = 1/(1/2*(q3`1-q2`1)+1);
A3: 0+(1+1) <= len G by A2,NAT_1:13;
then
A4: q2`2 = q3`2 by A1,GOBOARD5:1;
q2`1 < q3`1 by A1,A3,GOBOARD5:3;
then
A5: q3`1-q2`1 > 0 by XREAL_1:50;
then 1 < 1/2*(q3`1-q2`1)+1 by XREAL_1:29,129;
then
A6: r < 1 by XREAL_1:212;
A7: ((1-r)*(q2-|[1,1]|)+r*(1/2*(q2+q3)-|[0,1]|))`2 = ((1-r)*(q2-|[1,1]|))`2
+(r*(1/2*(q2+q3)-|[0,1]|))`2 by Lm1
.= (1-r)*(q2-|[1,1]|)`2+(r*(1/2*(q2+q3)-|[0,1]|))`2 by Lm3
.= (1-r)*(q2-|[1,1]|)`2+r*(1/2*(q2+q3)-|[0,1]|)`2 by Lm3
.= (1-r)*(q2`2-|[1,1]|`2)+r*(1/2*(q2+q3)-|[0,1]|)`2 by Lm2
.= (1-r)*(q2`2-|[1,1]|`2)+r*((1/2*(q2+q3))`2-|[0,1]|`2) by Lm2
.= (1-r)*(q2`2-1)+r*((1/2*(q2+q3))`2-|[0,1]|`2) by EUCLID:52
.= (1-r)*q2`2-(1-r)*1+r*((1/2*(q2+q3))`2-1) by EUCLID:52
.= (1-r)*q2`2+r*(1/2*(q2+q3))`2-((1-r)+r)
.= (1-r)*q2`2+r*(1/2*(q2+q3)`2)-1 by Lm3
.= (1-r)*q2`2+r*(1/2*(q2`2+q2`2))-1 by A4,Lm1
.= q2`2-|[0,1]|`2 by EUCLID:52
.= (q2-|[0,1]|)`2 by Lm2;
A8: r*((1/2)*q3`1)-r*((1/2)*q2`1)+r = r*((1/2)*(q3`1-q2`1)+1)
.= 1 by A5,XCMPLX_1:106;
((1-r)*(q2-|[1,1]|)+r*(1/2*(q2+q3)-|[0,1]|))`1 = ((1-r)*(q2-|[1,1]|))`1
+(r*(1/2*(q2+q3)-|[0,1]|))`1 by Lm1
.= ((1-r)*q2-(1-r)*|[1,1]|)`1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by RLVECT_1:34
.= ((1-r)*q2)`1-((1-r)*|[1,1]|)`1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by Lm2
.= ((1-r)*q2)`1-(1-r)*|[1,1]|`1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by Lm3
.= ((1-r)*q2)`1-(1-r)*1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by EUCLID:52
.= (1-r)*q2`1-(1-r)*1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by Lm3
.= (1-r)*q2`1-(1-r)+r*(1/2*(q2+q3)-|[0,1]|)`1 by Lm3
.= (1-r)*q2`1-(1-r)+r*((1/2*(q2+q3))`1-|[0,1]|`1) by Lm2
.= (1-r)*q2`1-(1-r)+r*((1/2*(q2+q3))`1-0) by EUCLID:52
.= (1-r)*q2`1-(1-r)+r*(1/2*(q2+q3)`1) by Lm3
.= (1-r)*q2`1-(1-r)+r*(1/2*(q2`1+q3`1)) by Lm1
.= q2`1-0 by A8
.= q2`1-|[0,1]|`1 by EUCLID:52
.= (q2-|[0,1]|)`1 by Lm2;
then
(1-r)*(q2-|[1,1]|)+r*(1/2*(q2+q3)-|[0,1]|) = |[(q2-|[0,1]|)`1,(q2-|[0,1
]|)`2]| by A7,EUCLID:53
.= q2-|[0,1]| by EUCLID:53;
then q2-|[0,1]| in LSeg(q2-|[1,1]|,1/2*(q2+q3)-|[0,1]|) by A5,A6;
then
A9: LSeg(q2-|[1,1]|,1/2*(q2+q3)-|[0,1]|) = LSeg(q2-|[1,1]|,q2-|[0,1]|) \/
LSeg(q2-|[0,1]|,1/2*(q2+q3)-|[0,1]|) by TOPREAL1:5;
set I1 = Int cell(G,0,0), I2 = Int cell(G,1,0);
0+1+1 = 0+(1+1);
then
A10: LSeg(q2-|[0,1]|,1/2*(q2+q3)-|[0,1]|) c= I2 \/ { q2-|[0,1]| } by A2,Th52;
A11: I1 \/ I2 \/ { q2-|[0,1]| } = I1 \/ (I2 \/ ({ q2-|[0,1]| } \/ { q2-|[0,1
]| })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { q2-|[0,1]| } \/ { q2-|[0,1]| }) by XBOOLE_1:4
.= I1 \/ { q2-|[0,1]| } \/ (I2 \/ { q2-|[0,1]| }) by XBOOLE_1:4;
LSeg(q2-|[1,1]|,q2-|[0,1]|) c= I1 \/ { q2-|[0,1]| } by Th60;
hence thesis by A9,A10,A11,XBOOLE_1:13;
end;
theorem
1 < width G & 1 < len G implies LSeg(G*(1,width G)+|[-1,1]|,1/2*(G*(1,
width G)+G*(2,width G))+|[0,1]|) c= Int cell(G,0,width G) \/ Int cell(G,1,width
G) \/ { G*(1,width G)+|[0,1]| }
proof
assume that
A1: 1 < width G and
A2: 1 < len G;
set q2 = G*(1,width G), q3 = G*(2,width G), r = 1/(1/2*(q3`1-q2`1)+1);
A3: 0+(1+1) <= len G by A2,NAT_1:13;
then
A4: q2`2 = q3`2 by A1,GOBOARD5:1;
q2`1 < q3`1 by A1,A3,GOBOARD5:3;
then
A5: q3`1-q2`1 > 0 by XREAL_1:50;
then 1 < 1/2*(q3`1-q2`1)+1 by XREAL_1:29,129;
then
A6: r < 1 by XREAL_1:212;
A7: ((1-r)*(q2+|[-1,1]|)+r*(1/2*(q2+q3)+|[0,1]|))`2 = ((1-r)*(q2+|[-1,1]|))
`2+(r*(1/2*(q2+q3)+|[0,1]|))`2 by Lm1
.= (1-r)*(q2+|[-1,1]|)`2+(r*(1/2*(q2+q3)+|[0,1]|))`2 by Lm3
.= (1-r)*(q2+|[-1,1]|)`2+r*(1/2*(q2+q3)+|[0,1]|)`2 by Lm3
.= (1-r)*(q2`2+|[-1,1]|`2)+r*(1/2*(q2+q3)+|[0,1]|)`2 by Lm1
.= (1-r)*(q2`2+|[-1,1]|`2)+r*((1/2*(q2+q3))`2+|[0,1]|`2) by Lm1
.= (1-r)*(q2`2+1)+r*((1/2*(q2+q3))`2+|[0,1]|`2) by EUCLID:52
.= (1-r)*q2`2+(1-r)*1+r*((1/2*(q2+q3))`2+1) by EUCLID:52
.= (1-r)*q2`2+r*(1/2*(q2+q3))`2+((1-r)+r)
.= (1-r)*q2`2+r*(1/2*(q2+q3)`2)+1 by Lm3
.= (1-r)*q2`2+r*(1/2*(q2`2+q2`2))+1 by A4,Lm1
.= q2`2+|[0,1]|`2 by EUCLID:52
.= (q2+|[0,1]|)`2 by Lm1;
A8: r*((1/2)*q3`1)-r*((1/2)*q2`1)+r = r*((1/2)*(q3`1-q2`1)+1)
.= 1 by A5,XCMPLX_1:106;
((1-r)*(q2+|[-1,1]|)+r*(1/2*(q2+q3)+|[0,1]|))`1 = ((1-r)*(q2+|[-1,1]|))
`1+(r*(1/2*(q2+q3)+|[0,1]|))`1 by Lm1
.= ((1-r)*q2+(1-r)*|[-1,1]|)`1+(r*(1/2*(q2+q3)+|[0,1]|))`1 by
RLVECT_1:def 5
.= ((1-r)*q2)`1+((1-r)*|[-1,1]|)`1+(r*(1/2*(q2+q3)+|[0,1]|))`1 by Lm1
.= ((1-r)*q2)`1+(1-r)*|[-1,1]|`1+(r*(1/2*(q2+q3)+|[0,1]|))`1 by Lm3
.= ((1-r)*q2)`1+(1-r)*(-1)+(r*(1/2*(q2+q3)+|[0,1]|))`1 by EUCLID:52
.= ((1-r)*q2)`1-(1-r)+(r*(1/2*(q2+q3)+|[0,1]|))`1
.= (1-r)*q2`1-(1-r)+(r*(1/2*(q2+q3)+|[0,1]|))`1 by Lm3
.= (1-r)*q2`1-(1-r)+r*(1/2*(q2+q3)+|[0,1]|)`1 by Lm3
.= (1-r)*q2`1-(1-r)+r*((1/2*(q2+q3))`1+|[0,1]|`1) by Lm1
.= (1-r)*q2`1-(1-r)+r*((1/2*(q2+q3))`1+0) by EUCLID:52
.= (1-r)*q2`1-(1-r)+r*(1/2*(q2+q3)`1) by Lm3
.= (1-r)*q2`1-(1-r)+r*(1/2*(q2`1+q3`1)) by Lm1
.= q2`1+0 by A8
.= q2`1+|[0,1]|`1 by EUCLID:52
.= (q2+|[0,1]|)`1 by Lm1;
then
(1-r)*(q2+|[-1,1]|)+r*(1/2*(q2+q3)+|[0,1]|) = |[(q2+|[0,1]|)`1,(q2+|[0,
1]|)`2]| by A7,EUCLID:53
.= q2+|[0,1]| by EUCLID:53;
then q2+|[0,1]| in LSeg(q2+|[-1,1]|,1/2*(q2+q3)+|[0,1]|) by A5,A6;
then
A9: LSeg(q2+|[-1,1]|,1/2*(q2+q3)+|[0,1]|) = LSeg(q2+|[-1,1]|,q2+|[0,1]|) \/
LSeg(q2+|[0,1]|,1/2*(q2+q3)+|[0,1]|) by TOPREAL1:5;
set I1 = Int cell(G,0,width G), I2 = Int cell(G,1,width G);
0+1+1 = 0+(1+1);
then
A10: LSeg(q2+|[0,1]|,1/2*(q2+q3)+|[0,1]|) c= I2 \/ { q2+|[0,1]| } by A2,Th54;
A11: I1 \/ I2 \/ { q2+|[0,1]| } = I1 \/ (I2 \/ ({ q2+|[0,1]| } \/ { q2+|[0,1
]| })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { q2+|[0,1]| } \/ { q2+|[0,1]| }) by XBOOLE_1:4
.= I1 \/ { q2+|[0,1]| } \/ (I2 \/ { q2+|[0,1]| }) by XBOOLE_1:4;
LSeg(q2+|[-1,1]|,q2+|[0,1]|) c= I1 \/ { q2+|[0,1]| } by Th62;
hence thesis by A9,A10,A11,XBOOLE_1:13;
end;
theorem
1 < width G & 1 < len G implies LSeg(G*(len G,1)+|[1,-1]|,1/2*(G*(len
G,1)+G*(len G -' 1,1))-|[0,1]|) c= Int cell(G,len G,0) \/ Int cell(G,len G -' 1
,0) \/ { G*(len G,1)-|[0,1]| }
proof
assume that
A1: 1 < width G and
A2: 1 < len G;
set q2 = G*(len G,1), q3 = G*(len G -' 1,1), r = 1/(1/2*(q2`1-q3`1)+1);
A3: len G -' 1 + 1 = len G by A2,XREAL_1:235;
then
A4: len G -' 1 >= 1 by A2,NAT_1:13;
A5: len G -'1 < len G by A3,NAT_1:13;
then q3`1 < q2`1 by A1,A4,GOBOARD5:3;
then
A6: q2`1-q3`1 > 0 by XREAL_1:50;
then 1 < 1/2*(q2`1-q3`1)+1 by XREAL_1:29,129;
then
A7: r < 1 by XREAL_1:212;
A8: q2`2 = G*(1,1)`2 by A1,A2,GOBOARD5:1
.= q3`2 by A1,A4,A5,GOBOARD5:1;
A9: ((1-r)*(q2+|[1,-1]|)+r*(1/2*(q2+q3)-|[0,1]|))`2 = ((1-r)*(q2+|[1,-1]|))
`2+(r*(1/2*(q2+q3)-|[0,1]|))`2 by Lm1
.= (1-r)*(q2+|[1,-1]|)`2+(r*(1/2*(q2+q3)-|[0,1]|))`2 by Lm3
.= (1-r)*(q2+|[1,-1]|)`2+r*(1/2*(q2+q3)-|[0,1]|)`2 by Lm3
.= (1-r)*(q2`2+|[1,-1]|`2)+r*(1/2*(q2+q3)-|[0,1]|)`2 by Lm1
.= (1-r)*(q2`2+|[1,-1]|`2)+r*((1/2*(q2+q3))`2-|[0,1]|`2) by Lm2
.= (1-r)*(q2`2+-1)+r*((1/2*(q2+q3))`2-|[0,1]|`2) by EUCLID:52
.= (1-r)*(q2`2-1)+r*((1/2*(q2+q3))`2-1) by EUCLID:52
.= (1-r)*q2`2+r*(1/2*(q2+q3))`2-1
.= (1-r)*q2`2+r*(1/2*(q2+q3)`2)-1 by Lm3
.= (1-r)*q2`2+r*(1/2*(q2`2+q2`2))-1 by A8,Lm1
.= q2`2-|[0,1]|`2 by EUCLID:52
.= (q2-|[0,1]|)`2 by Lm2;
A10: r*((1/2)*q2`1)-r*((1/2)*q3`1)+r = r*((1/2)*(q2`1-q3`1)+1)
.= 1 by A6,XCMPLX_1:106;
((1-r)*(q2+|[1,-1]|)+r*(1/2*(q2+q3)-|[0,1]|))`1 = ((1-r)*(q2+|[1,-1]|))
`1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by Lm1
.= ((1-r)*q2+(1-r)*|[1,-1]|)`1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by
RLVECT_1:def 5
.= ((1-r)*q2)`1+((1-r)*|[1,-1]|)`1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by Lm1
.= ((1-r)*q2)`1+(1-r)*|[1,-1]|`1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by Lm3
.= ((1-r)*q2)`1+(1-r)*1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by EUCLID:52
.= (1-r)*q2`1+(1-r)*1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by Lm3
.= (1-r)*q2`1+(1-r)+r*(1/2*(q2+q3)-|[0,1]|)`1 by Lm3
.= (1-r)*q2`1+(1-r)+r*((1/2*(q2+q3))`1-|[0,1]|`1) by Lm2
.= (1-r)*q2`1+(1-r)+r*((1/2*(q2+q3))`1-0) by EUCLID:52
.= (1-r)*q2`1+(1-r)+r*(1/2*(q2+q3)`1) by Lm3
.= (1-r)*q2`1+(1-r)+r*(1/2*(q2`1+q3`1)) by Lm1
.= q2`1-0 by A10
.= q2`1-|[0,1]|`1 by EUCLID:52
.= (q2-|[0,1]|)`1 by Lm2;
then
(1-r)*(q2+|[1,-1]|)+r*(1/2*(q2+q3)-|[0,1]|) = |[(q2-|[0,1]|)`1,(q2-|[0,
1]|)`2]| by A9,EUCLID:53
.= q2-|[0,1]| by EUCLID:53;
then q2-|[0,1]| in LSeg(q2+|[1,-1]|,1/2*(q2+q3)-|[0,1]|) by A6,A7;
then
A11: LSeg(q2+|[1,-1]|,1/2*(q2+q3)-|[0,1]|) = LSeg(q2+|[1,-1]|,q2-|[0,1]|) \/
LSeg(q2-|[0,1]|,1/2*(q2+q3)-|[0,1]|) by TOPREAL1:5;
set I1 = Int cell(G,len G,0), I2 = Int cell(G,len G -' 1,0);
A12: I1 \/ I2 \/ { q2-|[0,1]| } = I1 \/ (I2 \/ ({ q2-|[0,1]| } \/ { q2-|[0,1
]| })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { q2-|[0,1]| } \/ { q2-|[0,1]| }) by XBOOLE_1:4
.= I1 \/ { q2-|[0,1]| } \/ (I2 \/ { q2-|[0,1]| }) by XBOOLE_1:4;
A13: LSeg(q2+|[1,-1]|,q2-|[0,1]|) c= I1 \/ { q2-|[0,1]| } by Th61;
LSeg(q2-|[0,1]|,1/2*(q2+q3)-|[0,1]|) c= I2 \/ { q2-|[0,1]| } by A3,A4,A5,Th53
;
hence thesis by A11,A13,A12,XBOOLE_1:13;
end;
theorem
1 < width G & 1 < len G implies LSeg(G*(len G,width G)+|[1,1]|, 1/2*(G
*(len G,width G)+G*(len G -' 1,width G))+|[0,1]|) c= Int cell(G,len G,width G)
\/ Int cell(G,len G -' 1,width G) \/ { G*(len G,width G)+|[0,1]| }
proof
assume that
A1: 1 < width G and
A2: 1 < len G;
set q2 = G*(len G,width G), q3 = G*(len G -' 1,width G), r = 1/(1/2*(q2`1-q3
`1)+1);
A3: len G -' 1 + 1 = len G by A2,XREAL_1:235;
then
A4: len G -' 1 >= 1 by A2,NAT_1:13;
A5: len G -' 1 < len G by A3,NAT_1:13;
then q3`1 < q2`1 by A1,A4,GOBOARD5:3;
then
A6: q2`1-q3`1 > 0 by XREAL_1:50;
then 1 < 1/2*(q2`1-q3`1)+1 by XREAL_1:29,129;
then
A7: r < 1 by XREAL_1:212;
A8: q2`2 = G*(1,width G)`2 by A1,A2,GOBOARD5:1
.= q3`2 by A1,A4,A5,GOBOARD5:1;
A9: ((1-r)*(q2+|[1,1]|)+r*(1/2*(q2+q3)+|[0,1]|))`2 = ((1-r)*(q2+|[1,1]|))`2
+(r*(1/2*(q2+q3)+|[0,1]|))`2 by Lm1
.= (1-r)*(q2+|[1,1]|)`2+(r*(1/2*(q2+q3)+|[0,1]|))`2 by Lm3
.= (1-r)*(q2+|[1,1]|)`2+r*(1/2*(q2+q3)+|[0,1]|)`2 by Lm3
.= (1-r)*(q2`2+|[1,1]|`2)+r*(1/2*(q2+q3)+|[0,1]|)`2 by Lm1
.= (1-r)*(q2`2+|[1,1]|`2)+r*((1/2*(q2+q3))`2+|[0,1]|`2) by Lm1
.= (1-r)*(q2`2+1)+r*((1/2*(q2+q3))`2+|[0,1]|`2) by EUCLID:52
.= (1-r)*q2`2+(1-r)*1+r*((1/2*(q2+q3))`2+1) by EUCLID:52
.= (1-r)*q2`2+r*(1/2*(q2+q3))`2+((1-r)+r)
.= (1-r)*q2`2+r*(1/2*(q2+q3)`2)+1 by Lm3
.= (1-r)*q2`2+r*(1/2*(q2`2+q2`2))+1 by A8,Lm1
.= q2`2+|[0,1]|`2 by EUCLID:52
.= (q2+|[0,1]|)`2 by Lm1;
A10: r*((1/2)*q2`1)-r*((1/2)*q3`1)+r = r*((1/2)*(q2`1-q3`1)+1)
.= 1 by A6,XCMPLX_1:106;
((1-r)*(q2+|[1,1]|)+r*(1/2*(q2+q3)+|[0,1]|))`1 = ((1-r)*(q2+|[1,1]|))`1
+(r*(1/2*(q2+q3)+|[0,1]|))`1 by Lm1
.= ((1-r)*q2+(1-r)*|[1,1]|)`1+(r*(1/2*(q2+q3)+|[0,1]|))`1 by RLVECT_1:def 5
.= ((1-r)*q2)`1+((1-r)*|[1,1]|)`1+(r*(1/2*(q2+q3)+|[0,1]|))`1 by Lm1
.= ((1-r)*q2)`1+(1-r)*|[1,1]|`1+(r*(1/2*(q2+q3)+|[0,1]|))`1 by Lm3
.= ((1-r)*q2)`1+(1-r)*1+(r*(1/2*(q2+q3)+|[0,1]|))`1 by EUCLID:52
.= (1-r)*q2`1+(1-r)+(r*(1/2*(q2+q3)+|[0,1]|))`1 by Lm3
.= (1-r)*q2`1+(1-r)+r*(1/2*(q2+q3)+|[0,1]|)`1 by Lm3
.= (1-r)*q2`1+(1-r)+r*((1/2*(q2+q3))`1+|[0,1]|`1) by Lm1
.= (1-r)*q2`1+(1-r)+r*((1/2*(q2+q3))`1+0) by EUCLID:52
.= (1-r)*q2`1+(1-r)+r*(1/2*(q2+q3)`1) by Lm3
.= (1-r)*q2`1+(1-r)+r*(1/2*(q2`1+q3`1)) by Lm1
.= q2`1+0 by A10
.= q2`1+|[0,1]|`1 by EUCLID:52
.= (q2+|[0,1]|)`1 by Lm1;
then
(1-r)*(q2+|[1,1]|)+r*(1/2*(q2+q3)+|[0,1]|) = |[(q2+|[0,1]|)`1,(q2+|[0,1
]|)`2]| by A9,EUCLID:53
.= q2+|[0,1]| by EUCLID:53;
then q2+|[0,1]| in LSeg(q2+|[1,1]|,1/2*(q2+q3)+|[0,1]|) by A6,A7;
then
A11: LSeg(q2+|[1,1]|,1/2*(q2+q3)+|[0,1]|) = LSeg(q2+|[1,1]|,q2+|[0,1]|) \/
LSeg(q2+|[0,1]|,1/2*(q2+q3)+|[0,1]|) by TOPREAL1:5;
set I1 = Int cell(G,len G,width G), I2 = Int cell(G,len G -' 1,width G);
A12: I1 \/ I2 \/ { q2+|[0,1]| } = I1 \/ (I2 \/ ({ q2+|[0,1]| } \/ { q2+|[0,1
]| })) by XBOOLE_1:4
.= I1 \/ (I2 \/ { q2+|[0,1]| } \/ { q2+|[0,1]| }) by XBOOLE_1:4
.= I1 \/ { q2+|[0,1]| } \/ (I2 \/ { q2+|[0,1]| }) by XBOOLE_1:4;
A13: LSeg(q2+|[1,1]|,q2+|[0,1]|) c= I1 \/ { q2+|[0,1]| } by Th63;
LSeg(q2+|[0,1]|,1/2*(q2+q3)+|[0,1]|) c= I2 \/ { q2+|[0,1]| } by A3,A4,A5,Th55
;
hence thesis by A11,A13,A12,XBOOLE_1:13;
end;
theorem
1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G implies LSeg(1/2*(G*(i
,j)+G*(i+1,j+1)),p) meets Int cell(G,i,j)
proof
assume
A1: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G;
now
take a = 1/2*(G*(i,j)+G*(i+1,j+1));
thus a in LSeg(1/2*(G*(i,j)+G*(i+1,j+1)),p) by RLTOPSP1:68;
thus a in Int cell(G,i,j) by A1,Th31;
end;
hence thesis by XBOOLE_0:3;
end;
theorem
1 <= i & i+1 <= len G implies LSeg(p,1/2*(G*(i,width G)+G*(i+1,width G
))+|[0,1]|) meets Int cell(G,i,width G)
proof
assume
A1: 1 <= i & i+1 <= len G;
now
take a = 1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]|;
thus a in LSeg(p,1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]|) by
RLTOPSP1:68;
thus a in Int cell(G,i,width G) by A1,Th32;
end;
hence thesis by XBOOLE_0:3;
end;
theorem
1 <= i & i+1 <= len G implies LSeg(1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|,p)
meets Int cell(G,i,0)
proof
assume
A1: 1 <= i & i+1 <= len G;
now
take a = 1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|;
thus a in LSeg(1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|,p) by RLTOPSP1:68;
thus a in Int cell(G,i,0) by A1,Th33;
end;
hence thesis by XBOOLE_0:3;
end;
theorem
1 <= j & j+1 <= width G implies LSeg(1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|,p
) meets Int cell(G,0,j)
proof
assume
A1: 1 <= j & j+1 <= width G;
now
take a = 1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|;
thus a in LSeg(1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|,p) by RLTOPSP1:68;
thus a in Int cell(G,0,j) by A1,Th35;
end;
hence thesis by XBOOLE_0:3;
end;
theorem
1 <= j & j+1 <= width G implies LSeg(p,1/2*(G*(len G,j)+G*(len G,j+1))
+|[1,0]|) meets Int cell(G,len G,j)
proof
assume
A1: 1 <= j & j+1 <= width G;
now
take a = 1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|;
thus a in LSeg(p,1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|) by RLTOPSP1:68;
thus a in Int cell(G,len G,j) by A1,Th34;
end;
hence thesis by XBOOLE_0:3;
end;
theorem
LSeg(p,G*(1,1)-|[1,1]|) meets Int cell(G,0,0)
proof
now
take a = G*(1,1)-|[1,1]|;
thus a in LSeg(p,G*(1,1)-|[1,1]|) by RLTOPSP1:68;
thus a in Int cell(G,0,0) by Th36;
end;
hence thesis by XBOOLE_0:3;
end;
theorem
LSeg(p,G*(len G,width G)+|[1,1]|) meets Int cell(G,len G,width G)
proof
now
take a = G*(len G,width G)+|[1,1]|;
thus a in LSeg(p,G*(len G,width G)+|[1,1]|) by RLTOPSP1:68;
thus a in Int cell(G,len G,width G) by Th37;
end;
hence thesis by XBOOLE_0:3;
end;
theorem
LSeg(p,G*(1,width G)+|[-1,1]|) meets Int cell(G,0,width G)
proof
now
take a = G*(1,width G)+|[-1,1]|;
thus a in LSeg(p,G*(1,width G)+|[-1,1]|) by RLTOPSP1:68;
thus a in Int cell(G,0,width G) by Th38;
end;
hence thesis by XBOOLE_0:3;
end;
theorem
LSeg(p,G*(len G,1)+|[1,-1]|) meets Int cell(G,len G,0)
proof
now
take a = G*(len G,1)+|[1,-1]|;
thus a in LSeg(p,G*(len G,1)+|[1,-1]|) by RLTOPSP1:68;
thus a in Int cell(G,len G,0) by Th39;
end;
hence thesis by XBOOLE_0:3;
end;
theorem Th91:
for M being non empty MetrSpace, p being Point of M, q being
Point of TopSpaceMetr M, r being Real st p = q & r > 0 holds Ball (p, r)
is a_neighborhood of q
proof
let M be non empty MetrSpace, p be Point of M, q be Point of TopSpaceMetr M,
r be Real;
reconsider A = Ball (p, r) as Subset of TopSpaceMetr M by TOPMETR:12;
assume p = q & r > 0;
then q in A by Th1;
hence thesis by CONNSP_2:3,TOPMETR:14;
end;
theorem
for M being non empty MetrSpace, A being Subset of TopSpaceMetr M, p
being Point of M holds p in Cl A iff for r being Real st r > 0 holds
Ball (p, r) meets A
proof
let M be non empty MetrSpace, A be Subset of TopSpaceMetr M, p be Point of M;
reconsider p9 = p as Point of TopSpaceMetr M by TOPMETR:12;
hereby
assume
A1: p in Cl A;
let r be Real;
reconsider B = Ball (p, r) as Subset of TopSpaceMetr M by TOPMETR:12;
assume r > 0;
then B is a_neighborhood of p9 by Th91;
hence Ball (p, r) meets A by A1,CONNSP_2:27;
end;
assume
A2: for r being Real st r > 0 holds Ball (p, r) meets A;
for G being a_neighborhood of p9 holds G meets A
proof
let G be a_neighborhood of p9;
p in Int G by CONNSP_2:def 1;
then ex r being Real st r > 0 & Ball (p, r) c= G by Th4;
hence thesis by A2,XBOOLE_1:63;
end;
hence thesis by CONNSP_2:27;
end;
:: Moved from JORDAN19, AG 20.01.2006
theorem
for A be Subset of TOP-REAL n for p be Point of TOP-REAL n for p9 be
Point of Euclid n st p = p9 for s be Real st s > 0 holds p in Cl A iff
for r be Real st 0 < r & r < s holds Ball (p9,r) meets A
proof
let A be Subset of TOP-REAL n;
let p be Point of TOP-REAL n;
let p9 be Point of Euclid n;
assume
A1: p = p9;
let s be Real;
assume
A2: s > 0;
hereby
assume
A3: p in Cl A;
let r be Real;
assume that
A4: 0 < r and
r < s;
reconsider B = Ball (p9,r) as Subset of TOP-REAL n by TOPREAL3:8;
B is a_neighborhood of p by A1,A4,Th2;
hence Ball (p9, r) meets A by A3,CONNSP_2:27;
end;
assume
A5: for r be Real st 0 < r & r < s holds Ball (p9,r) meets A;
for G be a_neighborhood of p holds G meets A
proof
let G be a_neighborhood of p;
p in Int G by CONNSP_2:def 1;
then consider r9 be Real such that
A6: r9 > 0 and
A7: Ball (p9,r9) c= G by A1,Th5;
set r = min(r9,s/2);
Ball (p9,r) c= Ball (p9,r9) by PCOMPS_1:1,XXREAL_0:17;
then
A8: Ball (p9,r) c= G by A7;
s/2 < s & r <= s/2 by A2,XREAL_1:216,XXREAL_0:17;
then
A9: r < s by XXREAL_0:2;
s/2 > 0 by A2,XREAL_1:215;
then r > 0 by A6,XXREAL_0:15;
hence thesis by A5,A8,A9,XBOOLE_1:63;
end;
hence thesis by CONNSP_2:27;
end;