let i, j, k be Nat; :: thesis: for f being FinSequence of (TOP-REAL 2)

for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds

front_left_cell (f,k,G) = cell (G,i,(j -' 1))

let f be FinSequence of (TOP-REAL 2); :: thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds

front_left_cell (f,k,G) = cell (G,i,(j -' 1))

let G be Go-board; :: thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) implies front_left_cell (f,k,G) = cell (G,i,(j -' 1)) )

A1: ( j < j + 1 & j + 1 <= (j + 1) + 1 ) by XREAL_1:29;

assume ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) ) ; :: thesis: front_left_cell (f,k,G) = cell (G,i,(j -' 1))

hence front_left_cell (f,k,G) = cell (G,i,(j -' 1)) by A1, Def4; :: thesis: verum

for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds

front_left_cell (f,k,G) = cell (G,i,(j -' 1))

let f be FinSequence of (TOP-REAL 2); :: thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds

front_left_cell (f,k,G) = cell (G,i,(j -' 1))

let G be Go-board; :: thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) implies front_left_cell (f,k,G) = cell (G,i,(j -' 1)) )

A1: ( j < j + 1 & j + 1 <= (j + 1) + 1 ) by XREAL_1:29;

assume ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) ) ; :: thesis: front_left_cell (f,k,G) = cell (G,i,(j -' 1))

hence front_left_cell (f,k,G) = cell (G,i,(j -' 1)) by A1, Def4; :: thesis: verum