let L be Real; :: thesis: ( 0 < L & L < 1 implies for k being Nat holds
( L to_power k <= 1 & 0 < L to_power k ) )
assume that
A1: 0 < L and
A2: L < 1 ; :: thesis: for k being Nat holds
( L to_power k <= 1 & 0 < L to_power k )
defpred S₁[ Nat] means ( L to_power $1 <= 1 & 0 < L to_power $1 );
A3: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume that
A4: L to_power k <= 1 and
A5: 0 < L to_power k ; :: thesis: S₁[k + 1]
A6: L to_power (k + 1) = (L to_power k) * (L to_power 1) by A1, POWER:27
.= (L to_power k) * L by POWER:25 ;
(L to_power k) * L <= L to_power k by A2, A5, XREAL_1:153;
hence S₁[k + 1] by A1, A4, A5, A6, XREAL_1:129, XXREAL_0:2; :: thesis: verum

end;

A7: S₁[ 0 ] by POWER:24;
thus for k being Nat holds S₁[k] from NAT_1:sch 2(A7, A3); :: thesis: verum