let P1, P2 be Subset of (TOP-REAL 2); :: 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 & P1 = cell G,i1,j1 ) & not ( i1 + 1 = i2 & j1 = j2 & P1 = cell G,i1,(j1 -' 1) ) & not ( i1 = i2 + 1 & j1 = j2 & P1 = cell G,i2,j2 ) holds
( i1 = i2 & j1 = j2 + 1 & P1 = cell G,(i1 -' 1),j2 ) ) & ( 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 & P2 = cell G,i1,j1 ) & not ( i1 + 1 = i2 & j1 = j2 & P2 = cell G,i1,(j1 -' 1) ) & not ( i1 = i2 + 1 & j1 = j2 & P2 = cell G,i2,j2 ) holds
( i1 = i2 & j1 = j2 + 1 & P2 = cell G,(i1 -' 1),j2 ) ) implies P1 = P2 )
assume that
A10: 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 & P1 = cell G,i1,j1 ) & not ( i1 + 1 = i2 & j1 = j2 & P1 = cell G,i1,(j1 -' 1) ) & not ( i1 = i2 + 1 & j1 = j2 & P1 = cell G,i2,j2 ) holds
( i1 = i2 & j1 = j2 + 1 & P1 = cell G,(i1 -' 1),j2 ) and
A11: 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 & P2 = cell G,i1,j1 ) & not ( i1 + 1 = i2 & j1 = j2 & P2 = cell G,i1,(j1 -' 1) ) & not ( i1 = i2 + 1 & j1 = j2 & P2 = cell G,i2,j2 ) holds
( i1 = i2 & j1 = j2 + 1 & P2 = cell G,(i1 -' 1),j2 ) ; :: thesis: 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 A5;

suppose ( i1 = i2 & j1 + 1 = j2 ) ; :: thesis: P1 = P2

then A12: j1 < j2 by XREAL_1:31;
A13: j2 <= j2 + 1 by NAT_1:11;
hence P1 = cell G,i1,j1 by A3, A4, A10, A12
.= P2 by A3, A4, A11, A12, A13 ;
:: thesis: verum

end;

suppose ( i1 + 1 = i2 & j1 = j2 ) ; :: thesis: P1 = P2

then A14: i1 < i2 by XREAL_1:31;
A15: i2 <= i2 + 1 by NAT_1:11;
hence P1 = cell G,i1,(j1 -' 1) by A3, A4, A10, A14
.= P2 by A3, A4, A11, A14, A15 ;
:: thesis: verum

end;

suppose ( i1 = i2 + 1 & j1 = j2 ) ; :: thesis: P1 = P2

then A16: i2 < i1 by XREAL_1:31;
A17: i1 <= i1 + 1 by NAT_1:11;
hence P1 = cell G,i2,j2 by A3, A4, A10, A16
.= P2 by A3, A4, A11, A16, A17 ;
:: thesis: verum

end;

suppose ( i1 = i2 & j1 = j2 + 1 ) ; :: thesis: P1 = P2

then A18: j2 < j1 by XREAL_1:31;
A19: j1 <= j1 + 1 by NAT_1:11;
hence P1 = cell G,(i1 -' 1),j2 by A3, A4, A10, A18
.= P2 by A3, A4, A11, A18, A19 ;
:: thesis: verum

end;

end;