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 A2;

suppose A25: ( i1 = i2 & j1 + 1 = j2 ) ; :: thesis: ex b₁ being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * i1,j1 & f /. (k + 1) = G * i2,j2 & not ( i1 = i2 & j1 + 1 = j2 & b₁ = cell G,(i2 -' 1),j2 ) & not ( i1 + 1 = i2 & j1 = j2 & b₁ = cell G,i2,j2 ) & not ( i1 = i2 + 1 & j1 = j2 & b₁ = cell G,(i2 -' 1),(j2 -' 1) ) holds
( i1 = i2 & j1 = j2 + 1 & b₁ = cell G,i2,(j2 -' 1) )

take cell G,(i2 -' 1),j2 ; :: thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * i1,j1 & f /. (k + 1) = G * i2,j2 & not ( i1 = i2 & j1 + 1 = j2 & cell G,(i2 -' 1),j2 = cell G,(i2 -' 1),j2 ) & not ( i1 + 1 = i2 & j1 = j2 & cell G,(i2 -' 1),j2 = cell G,i2,j2 ) & not ( i1 = i2 + 1 & j1 = j2 & cell G,(i2 -' 1),j2 = cell G,(i2 -' 1),(j2 -' 1) ) holds
( i1 = i2 & j1 = j2 + 1 & cell G,(i2 -' 1),j2 = cell G,i2,(j2 -' 1) )
let i19, j19, i29, j29 be Element of NAT ; :: thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * i19,j19 & f /. (k + 1) = G * i29,j29 & not ( i19 = i29 & j19 + 1 = j29 & cell G,(i2 -' 1),j2 = cell G,(i29 -' 1),j29 ) & not ( i19 + 1 = i29 & j19 = j29 & cell G,(i2 -' 1),j2 = cell G,i29,j29 ) & not ( i19 = i29 + 1 & j19 = j29 & cell G,(i2 -' 1),j2 = cell G,(i29 -' 1),(j29 -' 1) ) implies ( i19 = i29 & j19 = j29 + 1 & cell G,(i2 -' 1),j2 = cell G,i29,(j29 -' 1) ) )
assume A26: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * i19,j19 & f /. (k + 1) = G * i29,j29 ) ; :: thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell G,(i2 -' 1),j2 = cell G,(i29 -' 1),j29 ) or ( i19 + 1 = i29 & j19 = j29 & cell G,(i2 -' 1),j2 = cell G,i29,j29 ) or ( i19 = i29 + 1 & j19 = j29 & cell G,(i2 -' 1),j2 = cell G,(i29 -' 1),(j29 -' 1) ) or ( i19 = i29 & j19 = j29 + 1 & cell G,(i2 -' 1),j2 = cell G,i29,(j29 -' 1) ) )
then ( i2 = i29 & j1 = j19 ) by A1, GOBOARD1:21;
hence ( ( i19 = i29 & j19 + 1 = j29 & cell G,(i2 -' 1),j2 = cell G,(i29 -' 1),j29 ) or ( i19 + 1 = i29 & j19 = j29 & cell G,(i2 -' 1),j2 = cell G,i29,j29 ) or ( i19 = i29 + 1 & j19 = j29 & cell G,(i2 -' 1),j2 = cell G,(i29 -' 1),(j29 -' 1) ) or ( i19 = i29 & j19 = j29 + 1 & cell G,(i2 -' 1),j2 = cell G,i29,(j29 -' 1) ) ) by A1, A25, A26, GOBOARD1:21; :: thesis: verum

end;

suppose A27: ( i1 + 1 = i2 & j1 = j2 ) ; :: thesis: ex b₁ being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * i1,j1 & f /. (k + 1) = G * i2,j2 & not ( i1 = i2 & j1 + 1 = j2 & b₁ = cell G,(i2 -' 1),j2 ) & not ( i1 + 1 = i2 & j1 = j2 & b₁ = cell G,i2,j2 ) & not ( i1 = i2 + 1 & j1 = j2 & b₁ = cell G,(i2 -' 1),(j2 -' 1) ) holds
( i1 = i2 & j1 = j2 + 1 & b₁ = cell G,i2,(j2 -' 1) )

take cell G,i2,j2 ; :: thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * i1,j1 & f /. (k + 1) = G * i2,j2 & not ( i1 = i2 & j1 + 1 = j2 & cell G,i2,j2 = cell G,(i2 -' 1),j2 ) & not ( i1 + 1 = i2 & j1 = j2 & cell G,i2,j2 = cell G,i2,j2 ) & not ( i1 = i2 + 1 & j1 = j2 & cell G,i2,j2 = cell G,(i2 -' 1),(j2 -' 1) ) holds
( i1 = i2 & j1 = j2 + 1 & cell G,i2,j2 = cell G,i2,(j2 -' 1) )
let i19, j19, i29, j29 be Element of NAT ; :: thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * i19,j19 & f /. (k + 1) = G * i29,j29 & not ( i19 = i29 & j19 + 1 = j29 & cell G,i2,j2 = cell G,(i29 -' 1),j29 ) & not ( i19 + 1 = i29 & j19 = j29 & cell G,i2,j2 = cell G,i29,j29 ) & not ( i19 = i29 + 1 & j19 = j29 & cell G,i2,j2 = cell G,(i29 -' 1),(j29 -' 1) ) implies ( i19 = i29 & j19 = j29 + 1 & cell G,i2,j2 = cell G,i29,(j29 -' 1) ) )
assume A28: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * i19,j19 & f /. (k + 1) = G * i29,j29 ) ; :: thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell G,i2,j2 = cell G,(i29 -' 1),j29 ) or ( i19 + 1 = i29 & j19 = j29 & cell G,i2,j2 = cell G,i29,j29 ) or ( i19 = i29 + 1 & j19 = j29 & cell G,i2,j2 = cell G,(i29 -' 1),(j29 -' 1) ) or ( i19 = i29 & j19 = j29 + 1 & cell G,i2,j2 = cell G,i29,(j29 -' 1) ) )
then ( i2 = i29 & j1 = j19 ) by A1, GOBOARD1:21;
hence ( ( i19 = i29 & j19 + 1 = j29 & cell G,i2,j2 = cell G,(i29 -' 1),j29 ) or ( i19 + 1 = i29 & j19 = j29 & cell G,i2,j2 = cell G,i29,j29 ) or ( i19 = i29 + 1 & j19 = j29 & cell G,i2,j2 = cell G,(i29 -' 1),(j29 -' 1) ) or ( i19 = i29 & j19 = j29 + 1 & cell G,i2,j2 = cell G,i29,(j29 -' 1) ) ) by A1, A27, A28, GOBOARD1:21; :: thesis: verum

end;

suppose A29: ( i1 = i2 + 1 & j1 = j2 ) ; :: thesis: ex b₁ being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * i1,j1 & f /. (k + 1) = G * i2,j2 & not ( i1 = i2 & j1 + 1 = j2 & b₁ = cell G,(i2 -' 1),j2 ) & not ( i1 + 1 = i2 & j1 = j2 & b₁ = cell G,i2,j2 ) & not ( i1 = i2 + 1 & j1 = j2 & b₁ = cell G,(i2 -' 1),(j2 -' 1) ) holds
( i1 = i2 & j1 = j2 + 1 & b₁ = cell G,i2,(j2 -' 1) )

take cell G,(i2 -' 1),(j2 -' 1) ; :: thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * i1,j1 & f /. (k + 1) = G * i2,j2 & not ( i1 = i2 & j1 + 1 = j2 & cell G,(i2 -' 1),(j2 -' 1) = cell G,(i2 -' 1),j2 ) & not ( i1 + 1 = i2 & j1 = j2 & cell G,(i2 -' 1),(j2 -' 1) = cell G,i2,j2 ) & not ( i1 = i2 + 1 & j1 = j2 & cell G,(i2 -' 1),(j2 -' 1) = cell G,(i2 -' 1),(j2 -' 1) ) holds
( i1 = i2 & j1 = j2 + 1 & cell G,(i2 -' 1),(j2 -' 1) = cell G,i2,(j2 -' 1) )
let i19, j19, i29, j29 be Element of NAT ; :: thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * i19,j19 & f /. (k + 1) = G * i29,j29 & not ( i19 = i29 & j19 + 1 = j29 & cell G,(i2 -' 1),(j2 -' 1) = cell G,(i29 -' 1),j29 ) & not ( i19 + 1 = i29 & j19 = j29 & cell G,(i2 -' 1),(j2 -' 1) = cell G,i29,j29 ) & not ( i19 = i29 + 1 & j19 = j29 & cell G,(i2 -' 1),(j2 -' 1) = cell G,(i29 -' 1),(j29 -' 1) ) implies ( i19 = i29 & j19 = j29 + 1 & cell G,(i2 -' 1),(j2 -' 1) = cell G,i29,(j29 -' 1) ) )
assume A30: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * i19,j19 & f /. (k + 1) = G * i29,j29 ) ; :: thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell G,(i2 -' 1),(j2 -' 1) = cell G,(i29 -' 1),j29 ) or ( i19 + 1 = i29 & j19 = j29 & cell G,(i2 -' 1),(j2 -' 1) = cell G,i29,j29 ) or ( i19 = i29 + 1 & j19 = j29 & cell G,(i2 -' 1),(j2 -' 1) = cell G,(i29 -' 1),(j29 -' 1) ) or ( i19 = i29 & j19 = j29 + 1 & cell G,(i2 -' 1),(j2 -' 1) = cell G,i29,(j29 -' 1) ) )
then ( i2 = i29 & j1 = j19 ) by A1, GOBOARD1:21;
hence ( ( i19 = i29 & j19 + 1 = j29 & cell G,(i2 -' 1),(j2 -' 1) = cell G,(i29 -' 1),j29 ) or ( i19 + 1 = i29 & j19 = j29 & cell G,(i2 -' 1),(j2 -' 1) = cell G,i29,j29 ) or ( i19 = i29 + 1 & j19 = j29 & cell G,(i2 -' 1),(j2 -' 1) = cell G,(i29 -' 1),(j29 -' 1) ) or ( i19 = i29 & j19 = j29 + 1 & cell G,(i2 -' 1),(j2 -' 1) = cell G,i29,(j29 -' 1) ) ) by A1, A29, A30, GOBOARD1:21; :: thesis: verum

end;

suppose A31: ( i1 = i2 & j1 = j2 + 1 ) ; :: thesis: ex b₁ being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * i1,j1 & f /. (k + 1) = G * i2,j2 & not ( i1 = i2 & j1 + 1 = j2 & b₁ = cell G,(i2 -' 1),j2 ) & not ( i1 + 1 = i2 & j1 = j2 & b₁ = cell G,i2,j2 ) & not ( i1 = i2 + 1 & j1 = j2 & b₁ = cell G,(i2 -' 1),(j2 -' 1) ) holds
( i1 = i2 & j1 = j2 + 1 & b₁ = cell G,i2,(j2 -' 1) )

take cell G,i2,(j2 -' 1) ; :: thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * i1,j1 & f /. (k + 1) = G * i2,j2 & not ( i1 = i2 & j1 + 1 = j2 & cell G,i2,(j2 -' 1) = cell G,(i2 -' 1),j2 ) & not ( i1 + 1 = i2 & j1 = j2 & cell G,i2,(j2 -' 1) = cell G,i2,j2 ) & not ( i1 = i2 + 1 & j1 = j2 & cell G,i2,(j2 -' 1) = cell G,(i2 -' 1),(j2 -' 1) ) holds
( i1 = i2 & j1 = j2 + 1 & cell G,i2,(j2 -' 1) = cell G,i2,(j2 -' 1) )
let i19, j19, i29, j29 be Element of NAT ; :: thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * i19,j19 & f /. (k + 1) = G * i29,j29 & not ( i19 = i29 & j19 + 1 = j29 & cell G,i2,(j2 -' 1) = cell G,(i29 -' 1),j29 ) & not ( i19 + 1 = i29 & j19 = j29 & cell G,i2,(j2 -' 1) = cell G,i29,j29 ) & not ( i19 = i29 + 1 & j19 = j29 & cell G,i2,(j2 -' 1) = cell G,(i29 -' 1),(j29 -' 1) ) implies ( i19 = i29 & j19 = j29 + 1 & cell G,i2,(j2 -' 1) = cell G,i29,(j29 -' 1) ) )
assume A32: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * i19,j19 & f /. (k + 1) = G * i29,j29 ) ; :: thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell G,i2,(j2 -' 1) = cell G,(i29 -' 1),j29 ) or ( i19 + 1 = i29 & j19 = j29 & cell G,i2,(j2 -' 1) = cell G,i29,j29 ) or ( i19 = i29 + 1 & j19 = j29 & cell G,i2,(j2 -' 1) = cell G,(i29 -' 1),(j29 -' 1) ) or ( i19 = i29 & j19 = j29 + 1 & cell G,i2,(j2 -' 1) = cell G,i29,(j29 -' 1) ) )
then ( i2 = i29 & j1 = j19 ) by A1, GOBOARD1:21;
hence ( ( i19 = i29 & j19 + 1 = j29 & cell G,i2,(j2 -' 1) = cell G,(i29 -' 1),j29 ) or ( i19 + 1 = i29 & j19 = j29 & cell G,i2,(j2 -' 1) = cell G,i29,j29 ) or ( i19 = i29 + 1 & j19 = j29 & cell G,i2,(j2 -' 1) = cell G,(i29 -' 1),(j29 -' 1) ) or ( i19 = i29 & j19 = j29 + 1 & cell G,i2,(j2 -' 1) = cell G,i29,(j29 -' 1) ) ) by A1, A31, A32, GOBOARD1:21; :: thesis: verum

end;

end;