
theorem Th94:
  for f be Function of [:NAT,NAT:],ExtREAL st
    f is P-convergent_to_-infty holds
    not f is P-convergent_to_+infty &
    not f is P-convergent_to_finite_number
proof
   let f be Function of [:NAT,NAT:],ExtREAL;
   assume A1: f is P-convergent_to_-infty;
   hereby assume f is P-convergent_to_+infty; then
    consider N1 be Nat such that
A3:  for n,m be Nat st n>=N1 & m>=N1 holds f.(n,m) >= 1;
    consider N2 be Nat such that
A4:  for n,m be Nat st n>=N2 & m>=N2 holds -1 >= f.(n,m) by A1;
    reconsider N1,N2 as Element of NAT by ORDINAL1:def 12;
    set N = max(N1,N2);
A5: N>=N1 & N>=N2 by XXREAL_0:25; then
    f.(N,N) >= 1 by A3;
    hence contradiction by A4,A5;
   end;
   assume f is P-convergent_to_finite_number; then
   consider p be Real such that
A6: for e be Real st 0<e
     ex N be Nat st
      for n,m be Nat st n>=N & m>=N holds
       |. f.(n,m) - p qua ExtReal .| < e;
   reconsider p1=p as ExtReal;
   per cases;
   suppose A9: p > 0;
    then consider N1 be Nat such that
A7: for n,m be Nat st n>=N1 & m>=N1 holds |.f.(n,m)- p1.| < p by A6;
A8: now
     let n,m be Nat;
     assume n>=N1 & m>=N1;
     then |.f.(n,m)- p qua ExtReal.| <  p by A7;
     then -p1 < f.(n,m) -  p by EXTREAL1:21;
     then -p1 + p < f.(n,m) by XXREAL_3:53;
     hence 0 < f.(n,m) by XXREAL_3:7;
    end;
    consider N2 be Nat such that
A10: for n,m be Nat st n>=N2 & m>=N2 holds -(2*p) >= f.(n,m) by A1,A9;
    reconsider N1,N2 as Element of NAT by ORDINAL1:def 12;
    set N = max(N1,N2);
A11:N>=N1 & N>=N2 by XXREAL_0:25; then
    0 < f.(N,N) by A8;
    hence contradiction by A9,A11,A10;
   end;
   suppose A12: p = 0;
    consider N1 be Nat such that
A13: for n,m be Nat st n>=N1 & m>=N1 holds |. f.(n,m)- p .| <  1 by A6;
    consider N2 be Nat such that
A14: for n,m be Nat st n>=N2 & m>=N2 holds -1 >= f.(n,m) by A1;
    reconsider N1,N2 as Element of NAT by ORDINAL1:def 12;
    reconsider jj =1 as ExtReal;
    set N = max(N1,N2);
A15:N>=N1 & N>=N2 by XXREAL_0:25; then
    |. f.(N,N)- p1 .| <  jj by A13;
    then -jj < f.(N,N) -  p1 by EXTREAL1:21;
    then -jj + p < f.(N,N) by XXREAL_3:53;
    then -jj < f.(N,N) by A12,XXREAL_3:4;
    then -1 < f.(N,N) by XXREAL_3:def 3;
    hence contradiction by A14,A15;
   end;
   suppose
A16:p < 0; then
    consider N1 be Nat such that
A17: for n,m be Nat st n>=N1 & m>=N1 holds |.f.(n,m) - p .| < -p by A6;
A18:now let n,m be Nat;
     assume n>=N1 & m>=N1; then
     |.f.(n,m) - p qua ExtReal .| < -p by A17; then
     -(-p1) < f.(n,m) - p1 by EXTREAL1:21; then
     p1 + p1 < f.(n,m) by XXREAL_3:53;
     then 2*p1 < f.(n,m) by XXREAL_3:94;
     hence (2*p) < f.(n,m) by XXREAL_3:def 5;
    end;
    consider N2 be Nat such that
A19: for n,m be Nat st n>=N2 & m>=N2 holds f.(n,m) <= 2*p by A1,A16;
    reconsider N1,N2 as Element of NAT by ORDINAL1:def 12;
    set N = max(N1,N2);
A20:N>=N1 & N>=N2 by XXREAL_0:25; then
    (2*p) < f.(N,N) by A18;
    hence contradiction by A19,A20;
   end;
end;
