let a be Real; :: thesis: ( 1 < a implies seq_a^ (a,1,0) is increasing )
assume AS: 1 < a ; :: thesis: seq_a^ (a,1,0) is increasing
C1: for n being Element of NAT holds (seq_a^ (a,1,0)) . n < (seq_a^ (a,1,0)) . (n + 1)

proof

let n be Element of NAT ; :: thesis: (seq_a^ (a,1,0)) . n < (seq_a^ (a,1,0)) . (n + 1)
L2: (seq_a^ (a,1,0)) . n = a to_power ((1 * n) + 0) by ASYMPT_1:def 1
.= a to_power n ;
(seq_a^ (a,1,0)) . (n + 1) = a to_power ((1 * (n + 1)) + 0) by ASYMPT_1:def 1
.= a to_power (n + 1) ;
hence (seq_a^ (a,1,0)) . n < (seq_a^ (a,1,0)) . (n + 1) by L2, LC5aa, AS; :: thesis: verum

end;

reconsider S = seq_a^ (a,1,0) as Real_Sequence ;
for n being Nat holds S . n < S . (n + 1)

proof

let n be Nat; :: thesis: S . n < S . (n + 1)
reconsider n = n as Element of NAT by ORDINAL1:def 12;
(seq_a^ (a,1,0)) . n < (seq_a^ (a,1,0)) . (n + 1) by C1;
hence S . n < S . (n + 1) ; :: thesis: verum

end;

hence seq_a^ (a,1,0) is increasing by SEQM_3:def 6; :: thesis: verum