let G be Go-board; :: thesis: for f being FinSequence of (TOP-REAL 2)
for k being Nat st 1 <= k & k + 1 <= len f & f is_sequence_on G holds
left_cell (f,k,G) is closed
let f be FinSequence of (TOP-REAL 2); :: thesis: for k being Nat st 1 <= k & k + 1 <= len f & f is_sequence_on G holds
left_cell (f,k,G) is closed
let k be Nat; :: thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G implies left_cell (f,k,G) is closed )
assume that
A1: 1 <= k and
A2: k + 1 <= len f and
A3: f is_sequence_on G ; :: thesis: left_cell (f,k,G) is closed
consider i1, j1, i2, j2 being Nat such that
A4: [i1,j1] in Indices G and
A5: f /. k = G * (i1,j1) and
A6: [i2,j2] in Indices G and
A7: f /. (k + 1) = G * (i2,j2) and
A8: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A1, A2, A3, JORDAN8:3;
( ( i1 = i2 & j1 + 1 = j2 & left_cell (f,k,G) = cell (G,(i1 -' 1),j1) ) or ( i1 + 1 = i2 & j1 = j2 & left_cell (f,k,G) = cell (G,i1,j1) ) or ( i1 = i2 + 1 & j1 = j2 & left_cell (f,k,G) = cell (G,i2,(j2 -' 1)) ) or ( i1 = i2 & j1 = j2 + 1 & left_cell (f,k,G) = cell (G,i1,j2) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, GOBRD13:def 3;
hence left_cell (f,k,G) is closed by GOBRD11:33; :: thesis: verum