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

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

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

assume that
A1: 1 <= len f and
A2: f /. (len f) in rng (Col (G,())) and
A3: f is_sequence_on G and
A4: i in Seg () and
A5: i + 1 in Seg () and
A6: m in dom f and
A7: f /. m in rng (Col (G,i)) and
A8: for k being Nat st k in dom f & f /. k in rng (Col (G,i)) holds
k <= m ; :: thesis: ( m + 1 in dom f & f /. (m + 1) in rng (Col (G,(i + 1))) )
defpred S1[ Nat, set ] means ( \$2 in Seg () & ( for k being Nat st k = \$2 holds
f /. \$1 in rng (Col (G,k)) ) );
A9: dom G = Seg (len G) by FINSEQ_1:def 3;
A10: for n being Nat st n in dom f holds
ex k being Nat st
( k in Seg () & f /. n in rng (Col (G,k)) )
proof
assume ex n being Nat st
( n in dom f & ( for k being Nat holds
( not k in Seg () or not f /. n in rng (Col (G,k)) ) ) ) ; :: thesis: contradiction
then consider n being Nat such that
A11: n in dom f and
A12: for k being Nat st k in Seg () holds
not f /. n in rng (Col (G,k)) ;
consider i, j being Nat such that
A13: [i,j] in Indices G and
A14: f /. n = G * (i,j) by ;
A15: [i,j] in [:(dom G),(Seg ()):] by ;
then j in Seg () by ZFMISC_1:87;
then A16: not f /. n in rng (Col (G,j)) by A12;
A17: i in dom G by ;
then i in Seg (len (Col (G,j))) by ;
then A18: i in dom (Col (G,j)) by FINSEQ_1:def 3;
(Col (G,j)) . i = G * (i,j) by ;
hence contradiction by A14, A16, A18, FUNCT_1:def 3; :: thesis: verum
end;
A19: for n being Nat st n in Seg (len f) holds
ex r being Element of REAL st S1[n,r]
proof
let n be Nat; :: thesis: ( n in Seg (len f) implies ex r being Element of REAL st S1[n,r] )
assume n in Seg (len f) ; :: thesis: ex r being Element of REAL st S1[n,r]
then n in dom f by FINSEQ_1:def 3;
then consider k being Nat such that
A20: k in Seg () and
A21: f /. n in rng (Col (G,k)) by A10;
reconsider r = k as Element of REAL by XREAL_0:def 1;
take r ; :: thesis: S1[n,r]
thus r in Seg () by A20; :: thesis: for k being Nat st k = r holds
f /. n in rng (Col (G,k))

let m be Nat; :: thesis: ( m = r implies f /. n in rng (Col (G,m)) )
assume m = r ; :: thesis: f /. n in rng (Col (G,m))
hence f /. n in rng (Col (G,m)) by A21; :: thesis: verum
end;
consider v being FinSequence of REAL such that
A22: dom v = Seg (len f) and
A23: for n being Nat st n in Seg (len f) holds
S1[n,v . n] from A24: dom f = Seg (len f) by FINSEQ_1:def 3;
A25: for k being Nat st k in dom v & v . k = i holds
k <= m
proof
let k be Nat; :: thesis: ( k in dom v & v . k = i implies k <= m )
assume that
A26: k in dom v and
A27: v . k = i ; :: thesis: k <= m
f /. k in rng (Col (G,i)) by A22, A23, A26, A27;
hence k <= m by A8, A22, A24, A26; :: thesis: verum
end;
A28: rng v c= Seg ()
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng v or x in Seg () )
assume x in rng v ; :: thesis: x in Seg ()
then ex y being Nat st
( y in dom v & v . y = x ) by FINSEQ_2:10;
hence x in Seg () by ; :: thesis: verum
end;
A29: len v = len f by ;
A30: 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
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
A31: 1 <= k and
A32: 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

A33: k + 1 <= len v by ;
let r, s be Real; :: thesis: ( r = v . k & s = v . (k + 1) & not |.(r - s).| = 1 implies r = s )
assume that
A34: r = v . k and
A35: s = v . (k + 1) ; :: thesis: ( |.(r - s).| = 1 or r = s )
1 <= k + 1 by NAT_1:11;
then A36: k + 1 in dom f by ;
then A37: s in rng v by ;
then A38: s in Seg () by A28;
k <= k + 1 by NAT_1:11;
then k <= len f by ;
then A39: k in dom f by ;
then A40: r in rng v by ;
then r in Seg () by A28;
then reconsider n1 = r, n2 = s as Element of NAT by A38;
set L1 = Col (G,n1);
set L2 = Col (G,n2);
f /. k in rng (Col (G,n1)) by A23, A24, A39, A34;
then consider x being Nat such that
A41: x in dom (Col (G,n1)) and
A42: (Col (G,n1)) . x = f /. k by FINSEQ_2:10;
A43: ( dom (Col (G,n1)) = Seg (len (Col (G,n1))) & len (Col (G,n1)) = len G ) by ;
then A44: f /. k = G * (x,n1) by ;
f /. (k + 1) in rng (Col (G,n2)) by A23, A24, A36, A35;
then consider y being Nat such that
A45: y in dom (Col (G,n2)) and
A46: (Col (G,n2)) . y = f /. (k + 1) by FINSEQ_2:10;
reconsider x = x, y = y as Element of NAT by ORDINAL1:def 12;
[x,n1] in [:(dom G),(Seg ()):] by ;
then A47: [x,n1] in Indices G by MATRIX_0:def 4;
A48: ( dom (Col (G,n2)) = Seg (len (Col (G,n2))) & len (Col (G,n2)) = len G ) by ;
then [y,n2] in [:(dom G),(Seg ()):] by ;
then A49: [y,n2] in Indices G by MATRIX_0:def 4;
f /. (k + 1) = G * (y,n2) by ;
then |.(x - y).| + |.(n1 - n2).| = 1 by A3, A39, A36, A44, A47, A49;
hence ( |.(r - s).| = 1 or r = s ) by SEQM_3:42; :: thesis: verum
end;
A50: v . m = i
proof
A51: v . m in Seg () by A6, A23, A24;
then reconsider k = v . m as Element of NAT ;
assume A52: v . m <> i ; :: thesis: contradiction
f /. m in rng (Col (G,k)) by A6, A23, A24;
hence contradiction by A4, A7, A52, A51, Th3; :: thesis: verum
end;
( 1 <= m & m <= len f ) by ;
then 1 <= len f by XXREAL_0:2;
then A53: len f in dom f by FINSEQ_3:25;
A54: v . (len v) = width G
proof
0 < width G by MATRIX_0:44;
then 0 + 1 <= width G by NAT_1:13;
then A55: width G in Seg () by FINSEQ_1:1;
A56: v . (len v) in Seg () by A23, A29, A24, A53;
then reconsider k = v . (len v) as Element of NAT ;
assume A57: v . (len v) <> width G ; :: thesis: contradiction
f /. (len f) in rng (Col (G,k)) by A23, A29, A24, A53;
hence contradiction by A2, A57, A56, A55, Th3; :: thesis: verum
end;
A58: v <> {} by ;
hence m + 1 in dom f by A4, A5, A6, A22, A24, A28, A54, A30, A50, A25, SEQM_3:45; :: thesis: f /. (m + 1) in rng (Col (G,(i + 1)))
( m + 1 in dom v & v . (m + 1) = i + 1 ) by A4, A5, A6, A22, A24, A58, A28, A54, A30, A50, A25, SEQM_3:45;
hence f /. (m + 1) in rng (Col (G,(i + 1))) by ; :: thesis: verum