defpred S₁[ R_eal] means ( ex g being Real st
( $1 = g & ( for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.((seq . m) - $1).| < p ) & seq is convergent_to_finite_number ) or ( $1 = +infty & seq is convergent_to_+infty ) or ( $1 = -infty & seq is convergent_to_-infty ) );
given g1, g2 being R_eal such that A4: S₁[g1] and
A5: S₁[g2] and
A6: g1 <> g2 ; :: thesis: contradiction

per cases ( seq is convergent_to_finite_number or seq is convergent_to_+infty or seq is convergent_to_-infty ) by A1;

suppose A7: seq is convergent_to_finite_number ; :: thesis: contradiction

then consider g being Real such that
A8: g1 = g and
A9: for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.((seq . m) - g1).| < p and
seq is convergent_to_finite_number by A4, Th50, Th51;
consider h being Real such that
A10: g2 = h and
A11: for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.((seq . m) - g2).| < p and
seq is convergent_to_finite_number by A5, A7, Th50, Th51;
reconsider g = g, h = h as Complex ;
g - h <> 0 by A6, A8, A10;
then A12: |.(g - h).| > 0 ;
then consider n1 being Nat such that
A13: for m being Nat st n1 <= m holds
|.((seq . m) - g1).| < |.(g - h).| / 2 by A9;
consider n2 being Nat such that
A14: for m being Nat st n2 <= m holds
|.((seq . m) - g2).| < |.(g - h).| / 2 by A11, A12;
reconsider n1 = n1, n2 = n2 as Element of NAT by ORDINAL1:def 12;
set m = max (n1,n2);
A15: |.((seq . (max (n1,n2))) - g1).| < |.(g - h).| / 2 by A13, XXREAL_0:25;
A16: |.((seq . (max (n1,n2))) - g2).| < |.(g - h).| / 2 by A14, XXREAL_0:25;
reconsider g = g, h = h as Complex ;
A17: (seq . (max (n1,n2))) - g2 < |.(g - h).| / 2 by A16, EXTREAL1:21;
A18: - (|.(g - h).| / 2) < (seq . (max (n1,n2))) - g2 by A16, EXTREAL1:21;
then reconsider w = (seq . (max (n1,n2))) - g2 as Element of REAL by A17, XXREAL_0:48;
A19: (seq . (max (n1,n2))) - g2 in REAL by A18, A17, XXREAL_0:48;
then A20: seq . (max (n1,n2)) <> +infty by A10;
A21: (- (seq . (max (n1,n2)))) + g1 = - ((seq . (max (n1,n2))) - g1) by XXREAL_3:26;
then A22: |.((- (seq . (max (n1,n2)))) + g1).| < |.(g - h).| / 2 by A15, EXTREAL1:29;
then A23: (- (seq . (max (n1,n2)))) + g1 < |.(g - h).| / 2 by EXTREAL1:21;
- (|.(g - h).| / 2) < (- (seq . (max (n1,n2)))) + g1 by A22, EXTREAL1:21;
then A24: (- (seq . (max (n1,n2)))) + g1 in REAL by A23, XXREAL_0:48;
A25: seq . (max (n1,n2)) <> -infty by A10, A19;
|.(g1 - g2).| = |.((g1 + 0.) - g2).| by XXREAL_3:4
.= |.((g1 + ((seq . (max (n1,n2))) + (- (seq . (max (n1,n2)))))) - g2).| by XXREAL_3:7
.= |.((((- (seq . (max (n1,n2)))) + g1) + (seq . (max (n1,n2)))) - g2).| by A8, A20, A25, XXREAL_3:29
.= |.(((- (seq . (max (n1,n2)))) + g1) + ((seq . (max (n1,n2))) - g2)).| by A10, A24, XXREAL_3:30 ;
then |.(g1 - g2).| <= |.((- (seq . (max (n1,n2)))) + g1).| + |.((seq . (max (n1,n2))) - g2).| by EXTREAL1:24;
then A26: |.(g1 - g2).| <= |.((seq . (max (n1,n2))) - g1).| + |.((seq . (max (n1,n2))) - g2).| by A21, EXTREAL1:29;
|.w.| in REAL by XREAL_0:def 1;
then |.((seq . (max (n1,n2))) - g2).| in REAL ;
then A27: |.((seq . (max (n1,n2))) - g1).| + |.((seq . (max (n1,n2))) - g2).| < (|.(g - h).| / 2) + |.((seq . (max (n1,n2))) - g2).| by A15, XXREAL_3:43;
|.(g - h).| / 2 in REAL by XREAL_0:def 1;
then |.(g - h).| / 2 in REAL ;
then (|.(g - h).| / 2) + |.((seq . (max (n1,n2))) - g2).| < (|.(g - h).| / 2) + (|.(g - h).| / 2) by A16, XXREAL_3:43;
then A28: |.((seq . (max (n1,n2))) - g1).| + |.((seq . (max (n1,n2))) - g2).| < (|.(g - h).| / 2) + (|.(g - h).| / 2) by A27, XXREAL_0:2;
g - h = g1 - g2 by A8, A10, SUPINF_2:3;
then |.(g - h).| = |.(g1 - g2).| by EXTREAL1:12;
then |.(g - h).| < (|.(g - h).| / 2) + (|.(g - h).| / 2) by A28, A26;
hence contradiction ; :: thesis: verum

end;

suppose ( seq is convergent_to_+infty or seq is convergent_to_-infty ) ; :: thesis: contradiction

hence contradiction by A4, A5, A6, Th50, Th51; :: thesis: verum

end;

end;