let i, j, k be Element of NAT ; :: thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg ((GoB f) * i,j),((GoB f) * (i + 1),j) = LSeg f,k & LSeg ((GoB f) * (i + 1),j),((GoB f) * (i + 1),(j + 1)) = LSeg f,(k + 1) holds
( f /. k = (GoB f) * i,j & f /. (k + 1) = (GoB f) * (i + 1),j & f /. (k + 2) = (GoB f) * (i + 1),(j + 1) )
let f be non constant standard special_circular_sequence; :: thesis: ( 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg ((GoB f) * i,j),((GoB f) * (i + 1),j) = LSeg f,k & LSeg ((GoB f) * (i + 1),j),((GoB f) * (i + 1),(j + 1)) = LSeg f,(k + 1) implies ( f /. k = (GoB f) * i,j & f /. (k + 1) = (GoB f) * (i + 1),j & f /. (k + 2) = (GoB f) * (i + 1),(j + 1) ) )
assume that
A1: 1 <= i and
A2: i + 1 <= len (GoB f) and
A3: 1 <= j and
A4: j + 1 <= width (GoB f) and
A5: 1 <= k and
A6: k + 1 < len f and
A7: LSeg ((GoB f) * i,j),((GoB f) * (i + 1),j) = LSeg f,k and
A8: LSeg ((GoB f) * (i + 1),j),((GoB f) * (i + 1),(j + 1)) = LSeg f,(k + 1) ; :: thesis: ( f /. k = (GoB f) * i,j & f /. (k + 1) = (GoB f) * (i + 1),j & f /. (k + 2) = (GoB f) * (i + 1),(j + 1) )
A9: i < len (GoB f) by A2, NAT_1:13;
A10: i < i + 1 by NAT_1:13;
A11: 1 <= j + 1 by NAT_1:11;
j < width (GoB f) by A4, NAT_1:13;
then ((GoB f) * i,j) `1 = ((GoB f) * i,1) `1 by A1, A3, A9, GOBOARD5:3
.= ((GoB f) * i,(j + 1)) `1 by A1, A4, A11, A9, GOBOARD5:3 ;
then A12: (GoB f) * i,j <> (GoB f) * (i + 1),(j + 1) by A1, A2, A4, A11, A10, GOBOARD5:4;
A13: 1 <= k + 1 by NAT_1:11;
A14: k + (1 + 1) = (k + 1) + 1 ;
then k + 2 <= len f by A6, NAT_1:13;
then A15: LSeg ((GoB f) * (i + 1),j),((GoB f) * (i + 1),(j + 1)) = LSeg (f /. (k + 1)),(f /. (k + 2)) by A8, A14, A13, TOPREAL1:def 5;
then A16: ( ( (GoB f) * (i + 1),(j + 1) = f /. (k + 1) & (GoB f) * (i + 1),j = f /. (k + 2) ) or ( (GoB f) * (i + 1),(j + 1) = f /. (k + 2) & (GoB f) * (i + 1),j = f /. (k + 1) ) ) by SPPOL_1:25;
A17: LSeg ((GoB f) * i,j),((GoB f) * (i + 1),j) = LSeg (f /. k),(f /. (k + 1)) by A5, A6, A7, TOPREAL1:def 5;
then ( ( (GoB f) * i,j = f /. k & (GoB f) * (i + 1),j = f /. (k + 1) ) or ( (GoB f) * i,j = f /. (k + 1) & (GoB f) * (i + 1),j = f /. k ) ) by SPPOL_1:25;
hence f /. k = (GoB f) * i,j by A15, A12, SPPOL_1:25; :: thesis: ( f /. (k + 1) = (GoB f) * (i + 1),j & f /. (k + 2) = (GoB f) * (i + 1),(j + 1) )
thus f /. (k + 1) = (GoB f) * (i + 1),j by A17, A16, A12, SPPOL_1:25; :: thesis: f /. (k + 2) = (GoB f) * (i + 1),(j + 1)
thus f /. (k + 2) = (GoB f) * (i + 1),(j + 1) by A17, A16, A12, SPPOL_1:25; :: thesis: verum