let P1, P2 be Subset of (TOP-REAL 2); :: thesis: ( ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (GoB f) & [i2,j2] in Indices (GoB f) & f /. k = (GoB f) * i1,j1 & f /. (k + 1) = (GoB f) * i2,j2 & not ( i1 = i2 & j1 + 1 = j2 & P1 = cell (GoB f),i1,j1 ) & not ( i1 + 1 = i2 & j1 = j2 & P1 = cell (GoB f),i1,(j1 -' 1) ) & not ( i1 = i2 + 1 & j1 = j2 & P1 = cell (GoB f),i2,j2 ) holds
( i1 = i2 & j1 = j2 + 1 & P1 = cell (GoB f),(i1 -' 1),j2 ) ) & ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (GoB f) & [i2,j2] in Indices (GoB f) & f /. k = (GoB f) * i1,j1 & f /. (k + 1) = (GoB f) * i2,j2 & not ( i1 = i2 & j1 + 1 = j2 & P2 = cell (GoB f),i1,j1 ) & not ( i1 + 1 = i2 & j1 = j2 & P2 = cell (GoB f),i1,(j1 -' 1) ) & not ( i1 = i2 + 1 & j1 = j2 & P2 = cell (GoB f),i2,j2 ) holds
( i1 = i2 & j1 = j2 + 1 & P2 = cell (GoB f),(i1 -' 1),j2 ) ) implies P1 = P2 )
assume that
A16: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (GoB f) & [i2,j2] in Indices (GoB f) & f /. k = (GoB f) * i1,j1 & f /. (k + 1) = (GoB f) * i2,j2 & not ( i1 = i2 & j1 + 1 = j2 & P1 = cell (GoB f),i1,j1 ) & not ( i1 + 1 = i2 & j1 = j2 & P1 = cell (GoB f),i1,(j1 -' 1) ) & not ( i1 = i2 + 1 & j1 = j2 & P1 = cell (GoB f),i2,j2 ) holds
( i1 = i2 & j1 = j2 + 1 & P1 = cell (GoB f),(i1 -' 1),j2 ) and
A17: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (GoB f) & [i2,j2] in Indices (GoB f) & f /. k = (GoB f) * i1,j1 & f /. (k + 1) = (GoB f) * i2,j2 & not ( i1 = i2 & j1 + 1 = j2 & P2 = cell (GoB f),i1,j1 ) & not ( i1 + 1 = i2 & j1 = j2 & P2 = cell (GoB f),i1,(j1 -' 1) ) & not ( i1 = i2 + 1 & j1 = j2 & P2 = cell (GoB f),i2,j2 ) holds
( i1 = i2 & j1 = j2 + 1 & P2 = cell (GoB f),(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 A7;

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

then A18: j1 < j2 by XREAL_1:31;
A19: j2 <= j2 + 1 by NAT_1:11;
hence P1 = cell (GoB f),i1,j1 by A3, A5, A16, A18
.= P2 by A3, A5, A17, A18, A19 ;
:: thesis: verum

end;

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

then A20: i1 < i2 by XREAL_1:31;
A21: i2 <= i2 + 1 by NAT_1:11;
hence P1 = cell (GoB f),i1,(j1 -' 1) by A3, A5, A16, A20
.= P2 by A3, A5, A17, A20, A21 ;
:: thesis: verum

end;

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

then A22: i2 < i1 by XREAL_1:31;
A23: i1 <= i1 + 1 by NAT_1:11;
hence P1 = cell (GoB f),i2,j2 by A3, A5, A16, A22
.= P2 by A3, A5, A17, A22, A23 ;
:: thesis: verum

end;

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

then A24: j2 < j1 by XREAL_1:31;
A25: j1 <= j1 + 1 by NAT_1:11;
hence P1 = cell (GoB f),(i1 -' 1),j2 by A3, A5, A16, A24
.= P2 by A3, A5, A17, A24, A25 ;
:: thesis: verum

end;

end;