let k be Element of NAT ; :: thesis: for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k + 1 <= len f holds
( Int (left_cell f,k,G) is connected & Int (right_cell f,k,G) is connected )
let G be Go-board; :: thesis: for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k + 1 <= len f holds
( Int (left_cell f,k,G) is connected & Int (right_cell f,k,G) is connected )
let f be FinSequence of (TOP-REAL 2); :: thesis: ( f is_sequence_on G & 1 <= k & k + 1 <= len f implies ( Int (left_cell f,k,G) is connected & Int (right_cell f,k,G) is connected ) )
assume A1: ( f is_sequence_on G & 1 <= k & k + 1 <= len f ) ; :: thesis: ( Int (left_cell f,k,G) is connected & Int (right_cell f,k,G) is connected )
then consider i1, j1, i2, j2 being Element of NAT such that
A2: [i1,j1] in Indices G and
A3: f /. k = G * i1,j1 and
A4: [i2,j2] in Indices G and
A5: f /. (k + 1) = G * i2,j2 and
A6: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by JORDAN8:6;
A7: i2 <= len G by A4, MATRIX_1:39;
A8: i1 <= len G by A2, MATRIX_1:39;
then A9: i1 -' 1 <= len G by NAT_D:44;
A10: j2 <= width G by A4, MATRIX_1:39;
then A11: j2 -' 1 <= width G by NAT_D:44;
A12: ( j1 + 1 > j1 & j2 + 1 > j2 ) by NAT_1:13;
A13: j1 <= width G by A2, MATRIX_1:39;
then A14: j1 -' 1 <= width G by NAT_D:44;
A15: ( i1 + 1 > i1 & i2 + 1 > i2 ) by NAT_1:13;

per cases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A6;

suppose ( i1 = i2 & j1 + 1 = j2 ) ; :: thesis: ( Int (left_cell f,k,G) is connected & Int (right_cell f,k,G) is connected )

then ( right_cell f,k,G = cell G,i1,j1 & left_cell f,k,G = cell G,(i1 -' 1),j1 ) by A1, A2, A3, A4, A5, A12, GOBRD13:def 2, GOBRD13:def 3;
hence ( Int (left_cell f,k,G) is connected & Int (right_cell f,k,G) is connected ) by A8, A13, A9, GOBOARD9:21; :: thesis: verum

end;

suppose ( i1 + 1 = i2 & j1 = j2 ) ; :: thesis: ( Int (left_cell f,k,G) is connected & Int (right_cell f,k,G) is connected )

then ( right_cell f,k,G = cell G,i1,(j1 -' 1) & left_cell f,k,G = cell G,i1,j1 ) by A1, A2, A3, A4, A5, A15, GOBRD13:def 2, GOBRD13:def 3;
hence ( Int (left_cell f,k,G) is connected & Int (right_cell f,k,G) is connected ) by A8, A13, A14, GOBOARD9:21; :: thesis: verum

end;

suppose ( i1 = i2 + 1 & j1 = j2 ) ; :: thesis: ( Int (left_cell f,k,G) is connected & Int (right_cell f,k,G) is connected )

then ( right_cell f,k,G = cell G,i2,j2 & left_cell f,k,G = cell G,i2,(j2 -' 1) ) by A1, A2, A3, A4, A5, A15, GOBRD13:def 2, GOBRD13:def 3;
hence ( Int (left_cell f,k,G) is connected & Int (right_cell f,k,G) is connected ) by A7, A10, A11, GOBOARD9:21; :: thesis: verum

end;

suppose ( i1 = i2 & j1 = j2 + 1 ) ; :: thesis: ( Int (left_cell f,k,G) is connected & Int (right_cell f,k,G) is connected )

then ( right_cell f,k,G = cell G,(i1 -' 1),j2 & left_cell f,k,G = cell G,i1,j2 ) by A1, A2, A3, A4, A5, A12, GOBRD13:def 2, GOBRD13:def 3;
hence ( Int (left_cell f,k,G) is connected & Int (right_cell f,k,G) is connected ) by A8, A10, A9, GOBOARD9:21; :: thesis: verum

end;

end;