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

for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & i in dom f & i + 1 in dom f & n in Seg (width G) & f /. i in rng (Col (G,n)) & not f /. (i + 1) in rng (Col (G,n)) holds

for k being Nat st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds

|.(n - k).| = 1

let G be Go-board; :: thesis: for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & i in dom f & i + 1 in dom f & n in Seg (width G) & f /. i in rng (Col (G,n)) & not f /. (i + 1) in rng (Col (G,n)) holds

for k being Nat st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds

|.(n - k).| = 1

let f be FinSequence of (TOP-REAL 2); :: thesis: ( f is_sequence_on G & i in dom f & i + 1 in dom f & n in Seg (width G) & f /. i in rng (Col (G,n)) & not f /. (i + 1) in rng (Col (G,n)) implies for k being Nat st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds

|.(n - k).| = 1 )

assume that

A1: f is_sequence_on G and

A2: i in dom f and

A3: i + 1 in dom f and

A4: ( n in Seg (width G) & f /. i in rng (Col (G,n)) ) ; :: thesis: ( f /. (i + 1) in rng (Col (G,n)) or for k being Nat st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds

|.(n - k).| = 1 )

consider j1, j2 being Nat such that

A5: [j1,j2] in Indices G and

A6: f /. (i + 1) = G * (j1,j2) by A1, A3;

A7: Indices G = [:(dom G),(Seg (width G)):] by MATRIX_0:def 4;

then A8: j1 in dom G by A5, ZFMISC_1:87;

A9: j2 in Seg (width G) by A5, A7, ZFMISC_1:87;

len (Col (G,j2)) = len G by MATRIX_0:def 8;

then A10: j1 in dom (Col (G,j2)) by A8, FINSEQ_3:29;

consider i1, i2 being Nat such that

A11: [i1,i2] in Indices G and

A12: f /. i = G * (i1,i2) by A1, A2;

A13: i1 in dom G by A11, A7, ZFMISC_1:87;

len (Col (G,i2)) = len G by MATRIX_0:def 8;

then A14: i1 in dom (Col (G,i2)) by A13, FINSEQ_3:29;

(Col (G,i2)) . i1 = f /. i by A12, A13, MATRIX_0:def 8;

then A15: f /. i in rng (Col (G,i2)) by A14, FUNCT_1:def 3;

i2 in Seg (width G) by A11, A7, ZFMISC_1:87;

then i2 = n by A4, A15, Th3;

then A16: |.(i1 - j1).| + |.(n - j2).| = 1 by A1, A2, A3, A11, A12, A5, A6;

A17: (Col (G,j2)) . j1 = f /. (i + 1) by A6, A8, MATRIX_0:def 8;

then A18: f /. (i + 1) in rng (Col (G,j2)) by A10, FUNCT_1:def 3;

|.(n - k).| = 1 ) ; :: thesis: verum

for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & i in dom f & i + 1 in dom f & n in Seg (width G) & f /. i in rng (Col (G,n)) & not f /. (i + 1) in rng (Col (G,n)) holds

for k being Nat st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds

|.(n - k).| = 1

let G be Go-board; :: thesis: for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & i in dom f & i + 1 in dom f & n in Seg (width G) & f /. i in rng (Col (G,n)) & not f /. (i + 1) in rng (Col (G,n)) holds

for k being Nat st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds

|.(n - k).| = 1

let f be FinSequence of (TOP-REAL 2); :: thesis: ( f is_sequence_on G & i in dom f & i + 1 in dom f & n in Seg (width G) & f /. i in rng (Col (G,n)) & not f /. (i + 1) in rng (Col (G,n)) implies for k being Nat st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds

|.(n - k).| = 1 )

assume that

A1: f is_sequence_on G and

A2: i in dom f and

A3: i + 1 in dom f and

A4: ( n in Seg (width G) & f /. i in rng (Col (G,n)) ) ; :: thesis: ( f /. (i + 1) in rng (Col (G,n)) or for k being Nat st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds

|.(n - k).| = 1 )

consider j1, j2 being Nat such that

A5: [j1,j2] in Indices G and

A6: f /. (i + 1) = G * (j1,j2) by A1, A3;

A7: Indices G = [:(dom G),(Seg (width G)):] by MATRIX_0:def 4;

then A8: j1 in dom G by A5, ZFMISC_1:87;

A9: j2 in Seg (width G) by A5, A7, ZFMISC_1:87;

len (Col (G,j2)) = len G by MATRIX_0:def 8;

then A10: j1 in dom (Col (G,j2)) by A8, FINSEQ_3:29;

consider i1, i2 being Nat such that

A11: [i1,i2] in Indices G and

A12: f /. i = G * (i1,i2) by A1, A2;

A13: i1 in dom G by A11, A7, ZFMISC_1:87;

len (Col (G,i2)) = len G by MATRIX_0:def 8;

then A14: i1 in dom (Col (G,i2)) by A13, FINSEQ_3:29;

(Col (G,i2)) . i1 = f /. i by A12, A13, MATRIX_0:def 8;

then A15: f /. i in rng (Col (G,i2)) by A14, FUNCT_1:def 3;

i2 in Seg (width G) by A11, A7, ZFMISC_1:87;

then i2 = n by A4, A15, Th3;

then A16: |.(i1 - j1).| + |.(n - j2).| = 1 by A1, A2, A3, A11, A12, A5, A6;

A17: (Col (G,j2)) . j1 = f /. (i + 1) by A6, A8, MATRIX_0:def 8;

then A18: f /. (i + 1) in rng (Col (G,j2)) by A10, FUNCT_1:def 3;

now :: thesis: ( f /. (i + 1) in rng (Col (G,n)) or for k being Nat st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds

|.(n - k).| = 1 )end;

hence
( f /. (i + 1) in rng (Col (G,n)) or for k being Nat st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds |.(n - k).| = 1 )

per cases
( ( |.(i1 - j1).| = 1 & n = j2 ) or ( |.(n - j2).| = 1 & i1 = j1 ) )
by A16, SEQM_3:42;

end;

suppose
( |.(i1 - j1).| = 1 & n = j2 )
; :: thesis: ( f /. (i + 1) in rng (Col (G,n)) or for k being Nat st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds

|.(n - k).| = 1 )

|.(n - k).| = 1 )

hence
( f /. (i + 1) in rng (Col (G,n)) or for k being Nat st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds

|.(n - k).| = 1 ) by A17, A10, FUNCT_1:def 3; :: thesis: verum

end;|.(n - k).| = 1 ) by A17, A10, FUNCT_1:def 3; :: thesis: verum

suppose
( |.(n - j2).| = 1 & i1 = j1 )
; :: thesis: ( f /. (i + 1) in rng (Col (G,n)) or for k being Nat st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds

|.(n - k).| = 1 )

|.(n - k).| = 1 )

hence
( f /. (i + 1) in rng (Col (G,n)) or for k being Nat st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds

|.(n - k).| = 1 ) by A9, A18, Th3; :: thesis: verum

end;|.(n - k).| = 1 ) by A9, A18, Th3; :: thesis: verum

|.(n - k).| = 1 ) ; :: thesis: verum