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;

theorem
  square-uparrow n is infinite Subset of [:NAT,NAT:]
  proof
    assume not square-uparrow n is infinite Subset of [:NAT,NAT:];
    then reconsider A = square-uparrow n as finite Subset of [:NAT,NAT:];
    consider i,j be Nat such that
A1: A c= [:Segm i,Segm j:] by Th16;
    consider k,l be Nat such that
    k = [n,n]`1 and l = [n,n]`2 and
A2: n <= k and
A3: n <= l;
    k <= k + i & l <= l + j by NAT_1:11; then
A4: n <= k + i & n <= l + j by XXREAL_0:2,A2,A3;
    k + i in NAT & l + j in NAT by ORDINAL1:def 12;
    then reconsider y = [k + i,l + j] as Element of [:NAT,NAT:]
      by ZFMISC_1:def 2;
    y`1 = k + i & y`2 = l + j;
    then y in square-uparrow n by Def3,A4;
    then ex a,b be object st a in Segm i & b in Segm j & y = [a,b]
      by A1,ZFMISC_1:def 2;
    then k + i in Segm i & l + j in Segm j by XTUPLE_0:1;
    then k + i - i < i - i & l + j - j < j - j by NAT_1:44,XREAL_1:14;
    then k < 0 & l < 0;
    hence thesis;
  end;
