let P1, P2 be Subset of (TOP-REAL 2); ( ( 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 & P1 = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & P1 = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & P1 = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & P1 = cell (G,(i2 -' 1),(j2 -' 1)) ) ) & ( 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 & P2 = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & P2 = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & P2 = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & P2 = cell (G,(i2 -' 1),(j2 -' 1)) ) ) implies P1 = P2 )
assume that
A11:
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 & P1 = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & P1 = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & P1 = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & P1 = cell (G,(i2 -' 1),(j2 -' 1)) )
and
A12:
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 & P2 = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & P2 = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & P2 = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & P2 = cell (G,(i2 -' 1),(j2 -' 1)) )
; P1 = P2
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;
end;