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 A3;

suppose A35: ( i1 = i2 & j1 + 1 = j2 ) ; :: thesis: ex b₁ being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being 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,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b₁ = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & b₁ = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b₁ = cell (G,i1,j2) )

take cell (G,(i1 -' 1),j1) ; :: thesis: for i1, j1, i2, j2 being 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,(i1 -' 1),j1) = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,(i1 -' 1),j1) = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,(i1 -' 1),j1) = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,(i1 -' 1),j1) = cell (G,i1,j2) )
let i19, j19, i29, j29 be 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,(i1 -' 1),j1) = cell (G,(i19 -' 1),j19) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,(i1 -' 1),j1) = cell (G,i19,j19) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,(i1 -' 1),j1) = cell (G,i29,(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,(i1 -' 1),j1) = cell (G,i19,j29) ) )
assume that
A36: [i19,j19] in Indices G and
A37: [i29,j29] in Indices G and
A38: f /. k = G * (i19,j19) and
A39: f /. (k + 1) = G * (i29,j29) ; :: thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,(i1 -' 1),j1) = cell (G,(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,(i1 -' 1),j1) = cell (G,i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,(i1 -' 1),j1) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,(i1 -' 1),j1) = cell (G,i19,j29) ) )
( i1 = i19 & j1 = j19 ) by A1, A36, A38, GOBOARD1:5;
hence ( ( i19 = i29 & j19 + 1 = j29 & cell (G,(i1 -' 1),j1) = cell (G,(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,(i1 -' 1),j1) = cell (G,i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,(i1 -' 1),j1) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,(i1 -' 1),j1) = cell (G,i19,j29) ) ) by A2, A35, A37, A39, GOBOARD1:5; :: thesis: verum

end;

suppose A40: ( i1 + 1 = i2 & j1 = j2 ) ; :: thesis: ex b₁ being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being 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,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b₁ = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & b₁ = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b₁ = cell (G,i1,j2) )

take cell (G,i1,j1) ; :: thesis: for i1, j1, i2, j2 being 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,i1,j1) = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,i1,j1) = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,i1,j1) = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,i1,j1) = cell (G,i1,j2) )
let i19, j19, i29, j29 be 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,i1,j1) = cell (G,(i19 -' 1),j19) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,j1) = cell (G,i19,j19) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,j1) = cell (G,i29,(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,j1) = cell (G,i19,j29) ) )
assume that
A41: [i19,j19] in Indices G and
A42: [i29,j29] in Indices G and
A43: f /. k = G * (i19,j19) and
A44: f /. (k + 1) = G * (i29,j29) ; :: thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i1,j1) = cell (G,(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,j1) = cell (G,i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,j1) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,j1) = cell (G,i19,j29) ) )
( i1 = i19 & j1 = j19 ) by A1, A41, A43, GOBOARD1:5;
hence ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i1,j1) = cell (G,(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,j1) = cell (G,i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,j1) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,j1) = cell (G,i19,j29) ) ) by A2, A40, A42, A44, GOBOARD1:5; :: thesis: verum

end;

suppose A45: ( i1 = i2 + 1 & j1 = j2 ) ; :: thesis: ex b₁ being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being 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,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b₁ = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & b₁ = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b₁ = cell (G,i1,j2) )

take cell (G,i2,(j2 -' 1)) ; :: thesis: for i1, j1, i2, j2 being 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,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,i2,(j2 -' 1)) = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,i2,(j2 -' 1)) = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,i2,(j2 -' 1)) = cell (G,i1,j2) )
let i19, j19, i29, j29 be 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,(i19 -' 1),j19) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i19,j19) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,(j2 -' 1)) = cell (G,i19,j29) ) )
assume A46: ( [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,(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,(j2 -' 1)) = cell (G,i19,j29) ) )
then ( i2 = i29 & j1 = j19 ) by A1, A2, GOBOARD1:5;
hence ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,(j2 -' 1)) = cell (G,i19,j29) ) ) by A1, A2, A45, A46, GOBOARD1:5; :: thesis: verum

end;

suppose A47: ( i1 = i2 & j1 = j2 + 1 ) ; :: thesis: ex b₁ being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being 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,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b₁ = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & b₁ = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b₁ = cell (G,i1,j2) )

take cell (G,i1,j2) ; :: thesis: for i1, j1, i2, j2 being 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,i1,j2) = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,i1,j2) = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,i1,j2) = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,i1,j2) = cell (G,i1,j2) )
let i19, j19, i29, j29 be 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,i1,j2) = cell (G,(i19 -' 1),j19) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,j2) = cell (G,i19,j19) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,j2) = cell (G,i29,(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,j2) = cell (G,i19,j29) ) )
assume that
A48: [i19,j19] in Indices G and
A49: [i29,j29] in Indices G and
A50: f /. k = G * (i19,j19) and
A51: f /. (k + 1) = G * (i29,j29) ; :: thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i1,j2) = cell (G,(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,j2) = cell (G,i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,j2) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,j2) = cell (G,i19,j29) ) )
( i1 = i19 & j1 = j19 ) by A1, A48, A50, GOBOARD1:5;
hence ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i1,j2) = cell (G,(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,j2) = cell (G,i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,j2) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,j2) = cell (G,i19,j29) ) ) by A2, A47, A49, A51, GOBOARD1:5; :: thesis: verum

end;

end;