let i, j be Element of NAT ; :: thesis:
let f be non constant standard special_circular_sequence; :: thesis:
set mi = (1 / 2) * (((GoB f) * i,j) + ((GoB f) * (i + 1),j));
assume that
A1: ( 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j <= width (GoB f) ) and
A2: (1 / 2) * (((GoB f) * i,j) + ((GoB f) * (i + 1),j)) in L~ f ; :: thesis:
L~ f = union { (LSeg f,k) where k is Element of NAT : ( 1 <= k & k + 1 <= len f ) } by TOPREAL1:def 6;
then consider x being set such that
A3: (1 / 2) * (((GoB f) * i,j) + ((GoB f) * (i + 1),j)) in x and
A4: x in { (LSeg f,k) where k is Element of NAT : ( 1 <= k & k + 1 <= len f ) } by A2, TARSKI:def 4;
consider k being Element of NAT such that
A5: x = LSeg f,k and
A6: 1 <= k and
A7: k + 1 <= len f by A4;
A8: f is_sequence_on GoB f by GOBOARD5:def 5;
A9: (1 / 2) * (((GoB f) * i,j) + ((GoB f) * (i + 1),j)) in LSeg (f /. k),(f /. (k + 1)) by A3, A5, A6, A7, TOPREAL1:def 5;
k <= k + 1 by NAT_1:11;
then k <= len f by A7, XXREAL_0:2;
then A10: k in dom f by A6, FINSEQ_3:27;
then consider i1, j1 being Element of NAT such that
A11: [i1,j1] in Indices (GoB f) and
A12: f /. k = (GoB f) * i1,j1 by A8, GOBOARD1:def 11;
A13: 1 <= j1 by A11, MATRIX_1:39;
take k ; :: thesis:
thus ( 1 <= k & k + 1 <= len f ) by A6, A7; :: thesis:
1 <= k + 1 by NAT_1:11;
then A14: k + 1 in dom f by A7, FINSEQ_3:27;
then consider i2, j2 being Element of NAT such that
A15: [i2,j2] in Indices (GoB f) and
A16: f /. (k + 1) = (GoB f) * i2,j2 by A8, GOBOARD1:def 11;
A17: 1 <= j2 by A15, MATRIX_1:39;
A18: i2 <= len (GoB f) by A15, MATRIX_1:39;
(abs (j1 - j2)) + (abs (i1 - i2)) = 1 by A8, A10, A11, A12, A14, A15, A16, GOBOARD1:def 11;
then A19: ( ( abs (j1 - j2) = 1 & i1 = i2 ) or ( abs (i1 - i2) = 1 & j1 = j2 ) ) by GOBOARD1:2;
A20: j1 <= width (GoB f) by A11, MATRIX_1:39;
A21: i1 <= len (GoB f) by A11, MATRIX_1:39;
A22: 1 <= i1 by A11, MATRIX_1:39;
A23: j2 <= width (GoB f) by A15, MATRIX_1:39;
A24: 1 <= i2 by A15, MATRIX_1:39;

per cases by A19, GOBOARD1:1;

suppose A25: ( i1 = i2 & j1 = j2 + 1 ) ; :: thesis:

then (1 / 2) * (((GoB f) * i,j) + ((GoB f) * (i + 1),j)) in LSeg ((GoB f) * i2,j2),((GoB f) * i2,(j2 + 1)) by A3, A5, A6, A7, A12, A16, TOPREAL1:def 5;
hence LSeg ((GoB f) * i,j),((GoB f) * (i + 1),j) = LSeg f,k by A1, A20, A17, A24, A18, A25, Th29; :: thesis:

end;

suppose A26: ( i1 = i2 & j1 + 1 = j2 ) ; :: thesis:

then (1 / 2) * (((GoB f) * i,j) + ((GoB f) * (i + 1),j)) in LSeg ((GoB f) * i1,j1),((GoB f) * i1,(j1 + 1)) by A3, A5, A6, A7, A12, A16, TOPREAL1:def 5;
hence LSeg ((GoB f) * i,j),((GoB f) * (i + 1),j) = LSeg f,k by A1, A13, A22, A21, A23, A26, Th29; :: thesis:

end;

suppose A27: ( i1 = i2 + 1 & j1 = j2 ) ; :: thesis:

then ( j = j2 & i = i2 ) by A1, A12, A16, A13, A20, A21, A24, A9, Th28;
hence LSeg ((GoB f) * i,j),((GoB f) * (i + 1),j) = LSeg f,k by A6, A7, A12, A16, A27, TOPREAL1:def 5; :: thesis:

end;

suppose A28: ( i1 + 1 = i2 & j1 = j2 ) ; :: thesis:

then ( j = j1 & i = i1 ) by A1, A12, A16, A13, A20, A22, A18, A9, Th28;
hence LSeg ((GoB f) * i,j),((GoB f) * (i + 1),j) = LSeg f,k by A6, A7, A12, A16, A28, TOPREAL1:def 5; :: thesis:

end;