let f be non constant standard special_circular_sequence; :: thesis: for k being Element of NAT st 1 <= k & k + 1 <= len f holds
( left_cell f,k,(GoB f) is convex & right_cell f,k,(GoB f) is convex )
let k be Element of NAT ; :: thesis: ( 1 <= k & k + 1 <= len f implies ( left_cell f,k,(GoB f) is convex & right_cell f,k,(GoB f) is convex ) )
assume A1: ( 1 <= k & k + 1 <= len f ) ; :: thesis: ( left_cell f,k,(GoB f) is convex & right_cell f,k,(GoB f) is convex )
left_cell f,k = left_cell f,k,(GoB f) by A1, JORDAN1H:27;
hence left_cell f,k,(GoB f) is convex by A1, Th22; :: thesis: right_cell f,k,(GoB f) is convex
A2: len f = len (Rev f) by FINSEQ_5:def 3;
k <= len f by A1, NAT_1:13;
then A3: ((len f) -' k) + k = len f by XREAL_1:237;
then A4: ((len f) -' k) + 1 <= len f by A1, XREAL_1:8;
A5: (len f) -' k >= 1 by A1, A3, XREAL_1:8;
then A6: left_cell (Rev f),((len f) -' k) is convex by A2, A4, Th22;
right_cell f,k = left_cell (Rev f),((len f) -' k) by A1, A3, A5, GOBOARD9:13;
hence right_cell f,k,(GoB f) is convex by A1, A6, JORDAN1H:29; :: thesis: verum