per cases by A10;

suppose A49: ( i1 = i2 & j1 + 1 = j2 ) ; :: thesis:

take cell ((GoB f),(i1 -' 1),j1) ; :: thesis:
let i19, j19, i29, j29 be Nat; :: thesis:
assume that
A50: [i19,j19] in Indices (GoB f) and
A51: [i29,j29] in Indices (GoB f) and
A52: f /. k = (GoB f) * (i19,j19) and
A53: f /. (k + 1) = (GoB f) * (i29,j29) ; :: thesis:
A54: i1 = i19 by A4, A5, A50, A52, GOBOARD1:5;
j1 = j19 by A4, A5, A50, A52, GOBOARD1:5;
hence ( ( i19 = i29 & j19 + 1 = j29 & cell ((GoB f),(i1 -' 1),j1) = cell ((GoB f),(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell ((GoB f),(i1 -' 1),j1) = cell ((GoB f),i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell ((GoB f),(i1 -' 1),j1) = cell ((GoB f),i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell ((GoB f),(i1 -' 1),j1) = cell ((GoB f),i19,j29) ) ) by A7, A8, A49, A51, A53, A54, GOBOARD1:5; :: thesis:

end;

suppose A55: ( i1 + 1 = i2 & j1 = j2 ) ; :: thesis:

take cell ((GoB f),i1,j1) ; :: thesis:
let i19, j19, i29, j29 be Nat; :: thesis:
assume that
A56: [i19,j19] in Indices (GoB f) and
A57: [i29,j29] in Indices (GoB f) and
A58: f /. k = (GoB f) * (i19,j19) and
A59: f /. (k + 1) = (GoB f) * (i29,j29) ; :: thesis:
A60: i1 = i19 by A4, A5, A56, A58, GOBOARD1:5;
j1 = j19 by A4, A5, A56, A58, GOBOARD1:5;
hence ( ( i19 = i29 & j19 + 1 = j29 & cell ((GoB f),i1,j1) = cell ((GoB f),(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell ((GoB f),i1,j1) = cell ((GoB f),i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell ((GoB f),i1,j1) = cell ((GoB f),i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell ((GoB f),i1,j1) = cell ((GoB f),i19,j29) ) ) by A7, A8, A55, A57, A59, A60, GOBOARD1:5; :: thesis:

end;

suppose A61: ( i1 = i2 + 1 & j1 = j2 ) ; :: thesis:

take cell ((GoB f),i2,(j2 -' 1)) ; :: thesis:
let i19, j19, i29, j29 be Nat; :: thesis:
assume that
A62: [i19,j19] in Indices (GoB f) and
A63: [i29,j29] in Indices (GoB f) and
A64: f /. k = (GoB f) * (i19,j19) and
A65: f /. (k + 1) = (GoB f) * (i29,j29) ; :: thesis:
A66: i2 = i29 by A7, A8, A63, A65, GOBOARD1:5;
j1 = j19 by A4, A5, A62, A64, GOBOARD1:5;
hence ( ( i19 = i29 & j19 + 1 = j29 & cell ((GoB f),i2,(j2 -' 1)) = cell ((GoB f),(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell ((GoB f),i2,(j2 -' 1)) = cell ((GoB f),i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell ((GoB f),i2,(j2 -' 1)) = cell ((GoB f),i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell ((GoB f),i2,(j2 -' 1)) = cell ((GoB f),i19,j29) ) ) by A4, A5, A7, A8, A61, A62, A63, A64, A65, A66, GOBOARD1:5; :: thesis:

end;

suppose A67: ( i1 = i2 & j1 = j2 + 1 ) ; :: thesis:

take cell ((GoB f),i1,j2) ; :: thesis:
let i19, j19, i29, j29 be Nat; :: thesis:
assume that
A68: [i19,j19] in Indices (GoB f) and
A69: [i29,j29] in Indices (GoB f) and
A70: f /. k = (GoB f) * (i19,j19) and
A71: f /. (k + 1) = (GoB f) * (i29,j29) ; :: thesis:
A72: i1 = i19 by A4, A5, A68, A70, GOBOARD1:5;
j1 = j19 by A4, A5, A68, A70, GOBOARD1:5;
hence ( ( i19 = i29 & j19 + 1 = j29 & cell ((GoB f),i1,j2) = cell ((GoB f),(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell ((GoB f),i1,j2) = cell ((GoB f),i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell ((GoB f),i1,j2) = cell ((GoB f),i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell ((GoB f),i1,j2) = cell ((GoB f),i19,j29) ) ) by A7, A8, A67, A69, A71, A72, GOBOARD1:5; :: thesis:

end;

end;