let seq be ExtREAL_sequence; :: thesis: ( seq is convergent_to_-infty implies ( not seq is convergent_to_+infty & not seq is convergent_to_finite_number ) )
assume A1: seq is convergent_to_-infty ; :: thesis: ( not seq is convergent_to_+infty & not seq is convergent_to_finite_number )

hereby :: thesis: not seq is convergent_to_finite_number

assume seq is convergent_to_+infty ; :: thesis: contradiction
then consider n1 being Nat such that
A2: for m being Nat st n1 <= m holds
R_EAL 1 <= seq . m by Def9;
consider n2 being Nat such that
A3: for m being Nat st n2 <= m holds
seq . m <= R_EAL (- 1) by A1, Def10;
reconsider n1 = n1, n2 = n2 as Element of NAT by ORDINAL1:def 13;
set m = max (n1,n2);
seq . (max (n1,n2)) <= R_EAL (- 1) by A3, XXREAL_0:25;
hence contradiction by A2, XXREAL_0:25; :: thesis: verum

end;

assume seq is convergent_to_finite_number ; :: thesis: contradiction
then consider g being real number such that
A4: for p being real number st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.((seq . m) - (R_EAL g)).| < p by Def8;
reconsider g1 = - g as Real by XREAL_0:def 1;

per cases ( g > 0 or g = 0 or g < 0 ) ;

suppose A5: g > 0 ; :: thesis: contradiction

then consider n1 being Nat such that
A6: for m being Nat st n1 <= m holds
|.((seq . m) - (R_EAL g)).| < R_EAL g by A4;

A7: now

let m be Element of NAT ; :: thesis: ( n1 <= m implies 0 < seq . m )
assume n1 <= m ; :: thesis: 0 < seq . m
then |.((seq . m) - (R_EAL g)).| < R_EAL g by A6;
then - (R_EAL g) < (seq . m) - (R_EAL g) by EXTREAL2:58;
then (- g) + g < seq . m by XXREAL_3:64;
hence 0 < seq . m ; :: thesis: verum

end;

consider n2 being Nat such that
A8: for m being Nat st n2 <= m holds
seq . m <= R_EAL g1 by A1, A5, Def10;
reconsider n1 = n1, n2 = n2 as Element of NAT by ORDINAL1:def 13;
set m = max (n1,n2);
0 < seq . (max (n1,n2)) by A7, XXREAL_0:25;
hence contradiction by A5, A8, XXREAL_0:25; :: thesis: verum

end;

suppose A9: g = 0 ; :: thesis: contradiction

consider n1 being Nat such that
A10: for m being Nat st n1 <= m holds
|.((seq . m) - (R_EAL g)).| < R_EAL 1 by A4;
consider n2 being Nat such that
A11: for m being Nat st n2 <= m holds
seq . m <= R_EAL (- 1) by A1, Def10;
reconsider n1 = n1, n2 = n2 as Element of NAT by ORDINAL1:def 13;
set m = max (n1,n2);
|.((seq . (max (n1,n2))) - (R_EAL g)).| < R_EAL 1 by A10, XXREAL_0:25;
then - (R_EAL 1) < (seq . (max (n1,n2))) - (R_EAL g) by EXTREAL2:58;
then (- (R_EAL 1)) + (R_EAL g) < seq . (max (n1,n2)) by XXREAL_3:64;
then - (R_EAL 1) < seq . (max (n1,n2)) by A9, XXREAL_3:4;
then R_EAL (- 1) < seq . (max (n1,n2)) by SUPINF_2:3;
hence contradiction by A11, XXREAL_0:25; :: thesis: verum

end;

suppose A12: g < 0 ; :: thesis: contradiction

then consider n1 being Nat such that
A13: for m being Nat st n1 <= m holds
|.((seq . m) - (R_EAL g)).| < R_EAL g1 by A4;

A14: now

let m be Element of NAT ; :: thesis: ( n1 <= m implies R_EAL (2 * g) < seq . m )
assume n1 <= m ; :: thesis: R_EAL (2 * g) < seq . m
then |.((seq . m) - (R_EAL g)).| < R_EAL g1 by A13;
then - (R_EAL g1) < (seq . m) - (R_EAL g) by EXTREAL2:58;
then (- (R_EAL g1)) + (R_EAL g) < seq . m by XXREAL_3:64;
then R_EAL ((- g1) + g) < seq . m by SUPINF_2:1;
hence R_EAL (2 * g) < seq . m ; :: thesis: verum

end;

consider n2 being Nat such that
A15: for m being Nat st n2 <= m holds
seq . m <= R_EAL (2 * g) by A1, A12, Def10;
reconsider n1 = n1, n2 = n2 as Element of NAT by ORDINAL1:def 13;
set m = max (n1,n2);
seq . (max (n1,n2)) <= R_EAL (2 * g) by A15, XXREAL_0:25;
hence contradiction by A14, XXREAL_0:25; :: thesis: verum

end;

end;