let a be real number ; :: thesis: for n being natural number st 0 < a & a <= 1 & 1 <= n holds
a |^ n <= a
let n be natural number ; :: thesis: ( 0 < a & a <= 1 & 1 <= n implies a |^ n <= a )
assume A1: ( 0 < a & a <= 1 & 1 <= n ) ; :: thesis: a |^ n <= a
then consider m being Nat such that
A2: n = 1 + m by NAT_1:10;
A3: m in NAT by ORDINAL1:def 13;
A4: a |^ (1 + 0 ) = (a GeoSeq ) . (0 + 1) by Def1
.= ((a GeoSeq ) . 0 ) * a by Th4
.= 1 * a by Th4
.= a ;
A5: a * a <= a * 1 by A1, XREAL_1:66;
defpred S₁[ Element of NAT ] means a |^ (1 + $1) <= a;
A6: S₁[ 0 ] by A4;
A7: for m1 being Element of NAT st S₁[m1] holds
S₁[m1 + 1]

proof

let m1 be Element of NAT ; :: thesis: ( S₁[m1] implies S₁[m1 + 1] )
assume a |^ (1 + m1) <= a ; :: thesis: S₁[m1 + 1]
then (a |^ (1 + m1)) * a <= a * a by A1, XREAL_1:66;
then a |^ (1 + (m1 + 1)) <= a * a by NEWTON:11;
hence S₁[m1 + 1] by A5, XXREAL_0:2; :: thesis: verum

end;

for m1 being Element of NAT holds S₁[m1] from NAT_1:sch 1(A6, A7);
hence a |^ n <= a by A2, A3; :: thesis: verum