let m, i be Nat; :: thesis: for G being Go-board

for f being FinSequence of (TOP-REAL 2) st 1 <= len f & f /. (len f) in rng (Line (G,(len G))) & f is_sequence_on G & i in dom G & i + 1 in dom G & m in dom f & f /. m in rng (Line (G,i)) & ( for k being Nat st k in dom f & f /. k in rng (Line (G,i)) holds

k <= m ) holds

( m + 1 in dom f & f /. (m + 1) in rng (Line (G,(i + 1))) )

let G be Go-board; :: thesis: for f being FinSequence of (TOP-REAL 2) st 1 <= len f & f /. (len f) in rng (Line (G,(len G))) & f is_sequence_on G & i in dom G & i + 1 in dom G & m in dom f & f /. m in rng (Line (G,i)) & ( for k being Nat st k in dom f & f /. k in rng (Line (G,i)) holds

k <= m ) holds

( m + 1 in dom f & f /. (m + 1) in rng (Line (G,(i + 1))) )

let f be FinSequence of (TOP-REAL 2); :: thesis: ( 1 <= len f & f /. (len f) in rng (Line (G,(len G))) & f is_sequence_on G & i in dom G & i + 1 in dom G & m in dom f & f /. m in rng (Line (G,i)) & ( for k being Nat st k in dom f & f /. k in rng (Line (G,i)) holds

k <= m ) implies ( m + 1 in dom f & f /. (m + 1) in rng (Line (G,(i + 1))) ) )

assume that

A1: 1 <= len f and

A2: f /. (len f) in rng (Line (G,(len G))) and

A3: f is_sequence_on G and

A4: i in dom G and

A5: i + 1 in dom G and

A6: m in dom f and

A7: f /. m in rng (Line (G,i)) and

A8: for k being Nat st k in dom f & f /. k in rng (Line (G,i)) holds

k <= m ; :: thesis: ( m + 1 in dom f & f /. (m + 1) in rng (Line (G,(i + 1))) )

reconsider p = f /. (len f), q = f /. m as Point of (TOP-REAL 2) ;

defpred S_{1}[ Nat, set ] means ( $2 in dom G & ( for k being Nat st k = $2 holds

f /. $1 in rng (Line (G,k)) ) );

A9: for n being Nat st n in dom f holds

ex k being Nat st

( k in dom G & f /. n in rng (Line (G,k)) )

ex r being Element of REAL st S_{1}[n,r]

A21: dom v = Seg (len f) and

A22: for n being Nat st n in Seg (len f) holds

S_{1}[n,v . n]
from FINSEQ_1:sch 5(A18);

A23: dom f = Seg (len f) by FINSEQ_1:def 3;

A24: for k being Nat st k in dom v & v . k = i holds

k <= m

A29: for k being Nat st 1 <= k & k <= (len v) - 1 holds

for r, s being Real st r = v . k & s = v . (k + 1) & not |.(r - s).| = 1 holds

r = s

then 1 <= len f by XXREAL_0:2;

then A52: len f in dom f by FINSEQ_3:25;

A53: v . (len v) = len G

hence m + 1 in dom f by A4, A5, A6, A21, A23, A27, A53, A29, A49, A24, SEQM_3:45; :: thesis: f /. (m + 1) in rng (Line (G,(i + 1)))

( m + 1 in dom v & v . (m + 1) = i + 1 ) by A4, A5, A6, A21, A23, A57, A27, A53, A29, A49, A24, SEQM_3:45;

hence f /. (m + 1) in rng (Line (G,(i + 1))) by A21, A22; :: thesis: verum

for f being FinSequence of (TOP-REAL 2) st 1 <= len f & f /. (len f) in rng (Line (G,(len G))) & f is_sequence_on G & i in dom G & i + 1 in dom G & m in dom f & f /. m in rng (Line (G,i)) & ( for k being Nat st k in dom f & f /. k in rng (Line (G,i)) holds

k <= m ) holds

( m + 1 in dom f & f /. (m + 1) in rng (Line (G,(i + 1))) )

let G be Go-board; :: thesis: for f being FinSequence of (TOP-REAL 2) st 1 <= len f & f /. (len f) in rng (Line (G,(len G))) & f is_sequence_on G & i in dom G & i + 1 in dom G & m in dom f & f /. m in rng (Line (G,i)) & ( for k being Nat st k in dom f & f /. k in rng (Line (G,i)) holds

k <= m ) holds

( m + 1 in dom f & f /. (m + 1) in rng (Line (G,(i + 1))) )

let f be FinSequence of (TOP-REAL 2); :: thesis: ( 1 <= len f & f /. (len f) in rng (Line (G,(len G))) & f is_sequence_on G & i in dom G & i + 1 in dom G & m in dom f & f /. m in rng (Line (G,i)) & ( for k being Nat st k in dom f & f /. k in rng (Line (G,i)) holds

k <= m ) implies ( m + 1 in dom f & f /. (m + 1) in rng (Line (G,(i + 1))) ) )

assume that

A1: 1 <= len f and

A2: f /. (len f) in rng (Line (G,(len G))) and

A3: f is_sequence_on G and

A4: i in dom G and

A5: i + 1 in dom G and

A6: m in dom f and

A7: f /. m in rng (Line (G,i)) and

A8: for k being Nat st k in dom f & f /. k in rng (Line (G,i)) holds

k <= m ; :: thesis: ( m + 1 in dom f & f /. (m + 1) in rng (Line (G,(i + 1))) )

reconsider p = f /. (len f), q = f /. m as Point of (TOP-REAL 2) ;

defpred S

f /. $1 in rng (Line (G,k)) ) );

A9: for n being Nat st n in dom f holds

ex k being Nat st

( k in dom G & f /. n in rng (Line (G,k)) )

proof

A18:
for n being Nat st n in Seg (len f) holds
assume
ex n being Nat st

( n in dom f & ( for k being Nat holds

( not k in dom G or not f /. n in rng (Line (G,k)) ) ) ) ; :: thesis: contradiction

then consider n being Nat such that

A10: n in dom f and

A11: for k being Nat st k in dom G holds

not f /. n in rng (Line (G,k)) ;

consider i, j being Nat such that

A12: [i,j] in Indices G and

A13: f /. n = G * (i,j) by A3, A10;

A14: [i,j] in [:(dom G),(Seg (width G)):] by A12, MATRIX_0:def 4;

then i in dom G by ZFMISC_1:87;

then A15: not f /. n in rng (Line (G,i)) by A11;

A16: j in Seg (width G) by A14, ZFMISC_1:87;

then j in Seg (len (Line (G,i))) by MATRIX_0:def 7;

then A17: j in dom (Line (G,i)) by FINSEQ_1:def 3;

(Line (G,i)) . j = G * (i,j) by A16, MATRIX_0:def 7;

hence contradiction by A13, A15, A17, FUNCT_1:def 3; :: thesis: verum

end;( n in dom f & ( for k being Nat holds

( not k in dom G or not f /. n in rng (Line (G,k)) ) ) ) ; :: thesis: contradiction

then consider n being Nat such that

A10: n in dom f and

A11: for k being Nat st k in dom G holds

not f /. n in rng (Line (G,k)) ;

consider i, j being Nat such that

A12: [i,j] in Indices G and

A13: f /. n = G * (i,j) by A3, A10;

A14: [i,j] in [:(dom G),(Seg (width G)):] by A12, MATRIX_0:def 4;

then i in dom G by ZFMISC_1:87;

then A15: not f /. n in rng (Line (G,i)) by A11;

A16: j in Seg (width G) by A14, ZFMISC_1:87;

then j in Seg (len (Line (G,i))) by MATRIX_0:def 7;

then A17: j in dom (Line (G,i)) by FINSEQ_1:def 3;

(Line (G,i)) . j = G * (i,j) by A16, MATRIX_0:def 7;

hence contradiction by A13, A15, A17, FUNCT_1:def 3; :: thesis: verum

ex r being Element of REAL st S

proof

consider v being FinSequence of REAL such that
let n be Nat; :: thesis: ( n in Seg (len f) implies ex r being Element of REAL st S_{1}[n,r] )

assume n in Seg (len f) ; :: thesis: ex r being Element of REAL st S_{1}[n,r]

then n in dom f by FINSEQ_1:def 3;

then consider k being Nat such that

A19: k in dom G and

A20: f /. n in rng (Line (G,k)) by A9;

reconsider r = k as Element of REAL by XREAL_0:def 1;

take r ; :: thesis: S_{1}[n,r]

thus r in dom G by A19; :: thesis: for k being Nat st k = r holds

f /. n in rng (Line (G,k))

let m be Nat; :: thesis: ( m = r implies f /. n in rng (Line (G,m)) )

assume m = r ; :: thesis: f /. n in rng (Line (G,m))

hence f /. n in rng (Line (G,m)) by A20; :: thesis: verum

end;assume n in Seg (len f) ; :: thesis: ex r being Element of REAL st S

then n in dom f by FINSEQ_1:def 3;

then consider k being Nat such that

A19: k in dom G and

A20: f /. n in rng (Line (G,k)) by A9;

reconsider r = k as Element of REAL by XREAL_0:def 1;

take r ; :: thesis: S

thus r in dom G by A19; :: thesis: for k being Nat st k = r holds

f /. n in rng (Line (G,k))

let m be Nat; :: thesis: ( m = r implies f /. n in rng (Line (G,m)) )

assume m = r ; :: thesis: f /. n in rng (Line (G,m))

hence f /. n in rng (Line (G,m)) by A20; :: thesis: verum

A21: dom v = Seg (len f) and

A22: for n being Nat st n in Seg (len f) holds

S

A23: dom f = Seg (len f) by FINSEQ_1:def 3;

A24: for k being Nat st k in dom v & v . k = i holds

k <= m

proof

A27:
rng v c= dom G
let k be Nat; :: thesis: ( k in dom v & v . k = i implies k <= m )

assume that

A25: k in dom v and

A26: v . k = i ; :: thesis: k <= m

f /. k in rng (Line (G,i)) by A21, A22, A25, A26;

hence k <= m by A8, A21, A23, A25; :: thesis: verum

end;assume that

A25: k in dom v and

A26: v . k = i ; :: thesis: k <= m

f /. k in rng (Line (G,i)) by A21, A22, A25, A26;

hence k <= m by A8, A21, A23, A25; :: thesis: verum

proof

A28:
len v = len f
by A21, FINSEQ_1:def 3;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng v or x in dom G )

assume x in rng v ; :: thesis: x in dom G

then ex y being Nat st

( y in dom v & v . y = x ) by FINSEQ_2:10;

hence x in dom G by A21, A22; :: thesis: verum

end;assume x in rng v ; :: thesis: x in dom G

then ex y being Nat st

( y in dom v & v . y = x ) by FINSEQ_2:10;

hence x in dom G by A21, A22; :: thesis: verum

A29: for k being Nat st 1 <= k & k <= (len v) - 1 holds

for r, s being Real st r = v . k & s = v . (k + 1) & not |.(r - s).| = 1 holds

r = s

proof

A49:
v . m = i
let k be Nat; :: thesis: ( 1 <= k & k <= (len v) - 1 implies for r, s being Real st r = v . k & s = v . (k + 1) & not |.(r - s).| = 1 holds

r = s )

assume that

A30: 1 <= k and

A31: k <= (len v) - 1 ; :: thesis: for r, s being Real st r = v . k & s = v . (k + 1) & not |.(r - s).| = 1 holds

r = s

A32: k + 1 <= len v by A31, XREAL_1:19;

let r, s be Real; :: thesis: ( r = v . k & s = v . (k + 1) & not |.(r - s).| = 1 implies r = s )

assume that

A33: r = v . k and

A34: s = v . (k + 1) ; :: thesis: ( |.(r - s).| = 1 or r = s )

1 <= k + 1 by NAT_1:11;

then A35: k + 1 in dom f by A28, A32, FINSEQ_3:25;

then A36: s in rng v by A21, A23, A34, FUNCT_1:def 3;

then A37: s in dom G by A27;

k <= k + 1 by NAT_1:11;

then k <= len f by A28, A32, XXREAL_0:2;

then A38: k in dom f by A30, FINSEQ_3:25;

then A39: r in rng v by A21, A23, A33, FUNCT_1:def 3;

then r in dom G by A27;

then reconsider n1 = r, n2 = s as Element of NAT by A37;

set L1 = Line (G,n1);

set L2 = Line (G,n2);

f /. k in rng (Line (G,n1)) by A22, A23, A38, A33;

then consider x being Nat such that

A40: x in dom (Line (G,n1)) and

A41: (Line (G,n1)) . x = f /. k by FINSEQ_2:10;

A42: ( dom (Line (G,n1)) = Seg (len (Line (G,n1))) & len (Line (G,n1)) = width G ) by FINSEQ_1:def 3, MATRIX_0:def 7;

then A43: f /. k = G * (n1,x) by A40, A41, MATRIX_0:def 7;

f /. (k + 1) in rng (Line (G,n2)) by A22, A23, A35, A34;

then consider y being Nat such that

A44: y in dom (Line (G,n2)) and

A45: (Line (G,n2)) . y = f /. (k + 1) by FINSEQ_2:10;

reconsider x = x, y = y as Element of NAT by ORDINAL1:def 12;

[n1,x] in [:(dom G),(Seg (width G)):] by A27, A39, A40, A42, ZFMISC_1:87;

then A46: [n1,x] in Indices G by MATRIX_0:def 4;

A47: ( Seg (len (Line (G,n2))) = dom (Line (G,n2)) & len (Line (G,n2)) = width G ) by FINSEQ_1:def 3, MATRIX_0:def 7;

then [n2,y] in [:(dom G),(Seg (width G)):] by A27, A36, A44, ZFMISC_1:87;

then A48: [n2,y] in Indices G by MATRIX_0:def 4;

f /. (k + 1) = G * (n2,y) by A47, A44, A45, MATRIX_0:def 7;

then |.(n1 - n2).| + |.(x - y).| = 1 by A3, A38, A35, A43, A46, A48;

hence ( |.(r - s).| = 1 or r = s ) by SEQM_3:42; :: thesis: verum

end;r = s )

assume that

A30: 1 <= k and

A31: k <= (len v) - 1 ; :: thesis: for r, s being Real st r = v . k & s = v . (k + 1) & not |.(r - s).| = 1 holds

r = s

A32: k + 1 <= len v by A31, XREAL_1:19;

let r, s be Real; :: thesis: ( r = v . k & s = v . (k + 1) & not |.(r - s).| = 1 implies r = s )

assume that

A33: r = v . k and

A34: s = v . (k + 1) ; :: thesis: ( |.(r - s).| = 1 or r = s )

1 <= k + 1 by NAT_1:11;

then A35: k + 1 in dom f by A28, A32, FINSEQ_3:25;

then A36: s in rng v by A21, A23, A34, FUNCT_1:def 3;

then A37: s in dom G by A27;

k <= k + 1 by NAT_1:11;

then k <= len f by A28, A32, XXREAL_0:2;

then A38: k in dom f by A30, FINSEQ_3:25;

then A39: r in rng v by A21, A23, A33, FUNCT_1:def 3;

then r in dom G by A27;

then reconsider n1 = r, n2 = s as Element of NAT by A37;

set L1 = Line (G,n1);

set L2 = Line (G,n2);

f /. k in rng (Line (G,n1)) by A22, A23, A38, A33;

then consider x being Nat such that

A40: x in dom (Line (G,n1)) and

A41: (Line (G,n1)) . x = f /. k by FINSEQ_2:10;

A42: ( dom (Line (G,n1)) = Seg (len (Line (G,n1))) & len (Line (G,n1)) = width G ) by FINSEQ_1:def 3, MATRIX_0:def 7;

then A43: f /. k = G * (n1,x) by A40, A41, MATRIX_0:def 7;

f /. (k + 1) in rng (Line (G,n2)) by A22, A23, A35, A34;

then consider y being Nat such that

A44: y in dom (Line (G,n2)) and

A45: (Line (G,n2)) . y = f /. (k + 1) by FINSEQ_2:10;

reconsider x = x, y = y as Element of NAT by ORDINAL1:def 12;

[n1,x] in [:(dom G),(Seg (width G)):] by A27, A39, A40, A42, ZFMISC_1:87;

then A46: [n1,x] in Indices G by MATRIX_0:def 4;

A47: ( Seg (len (Line (G,n2))) = dom (Line (G,n2)) & len (Line (G,n2)) = width G ) by FINSEQ_1:def 3, MATRIX_0:def 7;

then [n2,y] in [:(dom G),(Seg (width G)):] by A27, A36, A44, ZFMISC_1:87;

then A48: [n2,y] in Indices G by MATRIX_0:def 4;

f /. (k + 1) = G * (n2,y) by A47, A44, A45, MATRIX_0:def 7;

then |.(n1 - n2).| + |.(x - y).| = 1 by A3, A38, A35, A43, A46, A48;

hence ( |.(r - s).| = 1 or r = s ) by SEQM_3:42; :: thesis: verum

proof

( 1 <= m & m <= len f )
by A6, FINSEQ_3:25;
A50:
v . m in dom G
by A6, A22, A23;

then reconsider k = v . m as Element of NAT ;

assume A51: v . m <> i ; :: thesis: contradiction

q in rng (Line (G,k)) by A6, A22, A23;

hence contradiction by A4, A7, A51, A50, Th2; :: thesis: verum

end;then reconsider k = v . m as Element of NAT ;

assume A51: v . m <> i ; :: thesis: contradiction

q in rng (Line (G,k)) by A6, A22, A23;

hence contradiction by A4, A7, A51, A50, Th2; :: thesis: verum

then 1 <= len f by XXREAL_0:2;

then A52: len f in dom f by FINSEQ_3:25;

A53: v . (len v) = len G

proof

A57:
( dom G = Seg (len G) & v <> {} )
by A1, A21, FINSEQ_1:def 3;
0 < len G
by MATRIX_0:44;

then 0 + 1 <= len G by NAT_1:13;

then A54: len G in dom G by FINSEQ_3:25;

A55: v . (len v) in dom G by A22, A28, A23, A52;

then reconsider k = v . (len v) as Element of NAT ;

assume A56: v . (len v) <> len G ; :: thesis: contradiction

p in rng (Line (G,k)) by A22, A28, A23, A52;

hence contradiction by A2, A56, A55, A54, Th2; :: thesis: verum

end;then 0 + 1 <= len G by NAT_1:13;

then A54: len G in dom G by FINSEQ_3:25;

A55: v . (len v) in dom G by A22, A28, A23, A52;

then reconsider k = v . (len v) as Element of NAT ;

assume A56: v . (len v) <> len G ; :: thesis: contradiction

p in rng (Line (G,k)) by A22, A28, A23, A52;

hence contradiction by A2, A56, A55, A54, Th2; :: thesis: verum

hence m + 1 in dom f by A4, A5, A6, A21, A23, A27, A53, A29, A49, A24, SEQM_3:45; :: thesis: f /. (m + 1) in rng (Line (G,(i + 1)))

( m + 1 in dom v & v . (m + 1) = i + 1 ) by A4, A5, A6, A21, A23, A57, A27, A53, A29, A49, A24, SEQM_3:45;

hence f /. (m + 1) in rng (Line (G,(i + 1))) by A21, A22; :: thesis: verum