let P1, P2 be Subset of (TOP-REAL 2); :: 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 & 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 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
A21: 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,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
A22: 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,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 A3;

suppose A23: ( i1 = i2 & j1 + 1 = j2 ) ; :: thesis: P1 = P2

A24: j2 <= j2 + 1 by NAT_1:11;
A25: j1 < j2 by A23, XREAL_1:29;
hence P1 = cell (G,i1,j1) by A1, A2, A21, A24
.= P2 by A1, A2, A22, A25, A24 ;
:: thesis: verum

end;

suppose A26: ( i1 + 1 = i2 & j1 = j2 ) ; :: thesis: P1 = P2

A27: i2 <= i2 + 1 by NAT_1:11;
A28: i1 < i2 by A26, XREAL_1:29;
hence P1 = cell (G,i1,(j1 -' 1)) by A1, A2, A21, A27
.= P2 by A1, A2, A22, A28, A27 ;
:: thesis: verum

end;

suppose A29: ( i1 = i2 + 1 & j1 = j2 ) ; :: thesis: P1 = P2

A30: i1 <= i1 + 1 by NAT_1:11;
A31: i2 < i1 by A29, XREAL_1:29;
hence P1 = cell (G,i2,j2) by A1, A2, A21, A30
.= P2 by A1, A2, A22, A31, A30 ;
:: thesis: verum

end;

suppose A32: ( i1 = i2 & j1 = j2 + 1 ) ; :: thesis: P1 = P2

A33: j1 <= j1 + 1 by NAT_1:11;
A34: j2 < j1 by A32, XREAL_1:29;
hence P1 = cell (G,(i1 -' 1),j2) by A1, A2, A21, A33
.= P2 by A1, A2, A22, A34, A33 ;
:: thesis: verum

end;

end;