reserve            x for object,
               X,Y,Z for set,
         i,j,k,l,m,n for Nat,
                 r,s for Real,
                  no for Element of OrderedNAT,
                   A for Subset of [:NAT,NAT:];
reserve X,Y,X1,X2 for non empty set,
          cA1,cB1 for filter_base of X1,
          cA2,cB2 for filter_base of X2,
              cF1 for Filter of X1,
              cF2 for Filter of X2,
             cBa1 for basis of cF1,
             cBa2 for basis of cF2;
reserve T for non empty TopSpace,
        s for Function of [:NAT,NAT:], the carrier of T,
        M for Subset of the carrier of T;
reserve cF3,cF4 for Filter of the carrier of T;

theorem Th53:
  x in s.:(square-uparrow n) iff ex i,j st n <= i & n <= j & x = s.(i,j)
  proof
    hereby
      assume x in s.:(square-uparrow n);
      then consider y be object such that
A1:   y in dom s and
A2:   y in square-uparrow n and
A3:   x = s.y by FUNCT_1:def 6;
      reconsider z = y as Element of [:NAT,NAT:] by A1;
      consider i,j such that
A4:   z`1 = i and
A5:   z`2 = j and
A6:   n <= i and
A7:   n <= j by A2,Def3;
      consider m1,m2 be object such that
      m1 in NAT and
      m2 in NAT and
A8:   z = [m1,m2] by ZFMISC_1:def 2;
      x = s.(i,j) by A4,A5,A8,A3,BINOP_1:def 1;
      hence ex i,j st n <= i & n <= j & x = s.(i,j) by A6,A7;
    end;
    assume ex i,j st n <= i & n <= j & x = s.(i,j);
    then consider i,j such that
A9: n <= i and
A10: n <= j and
A11: x = s.(i,j);
A12: dom s = [:NAT,NAT:] by FUNCT_2:def 1;
A13: i in NAT & j in NAT by ORDINAL1:def 12;
A14: x = s.([i,j]) by A11,BINOP_1:def 1;
    [i,j]`1 = i & [i,j]`2 = j & [i,j] in [:NAT,NAT:]
      by A13,ZFMISC_1:def 2;
    then [i,j] in square-uparrow n by A9,A10,Def3;
    hence x in s.:(square-uparrow n) by A12,A14,FUNCT_1:def 6;
  end;
