let a be Real; :: thesis: for n being Nat st 0 < a & a < 1 & 2 <= n holds

a |^ n < a

let n be Nat; :: thesis: ( 0 < a & a < 1 & 2 <= n implies a |^ n < a )

assume that

A1: 0 < a and

A2: a < 1 and

A3: 2 <= n ; :: thesis: a |^ n < a

consider m being Nat such that

A4: n = 2 + m by A3, NAT_1:10;

defpred S_{1}[ Nat] means a |^ (2 + $1) < a;

A5: a * a < a * 1 by A1, A2, XREAL_1:68;

A6: for m1 being Nat st S_{1}[m1] holds

S_{1}[m1 + 1]

.= ((a GeoSeq) . (0 + 1)) * a by Th3

.= (((a GeoSeq) . 0) * a) * a by Th3

.= (1 * a) * a by Th3

.= a * a ;

then A7: S_{1}[ 0 ]
by A5;

A8: for m1 being Nat holds S_{1}[m1]
from NAT_1:sch 2(A7, A6);

thus a |^ n < a by A4, A8; :: thesis: verum

a |^ n < a

let n be Nat; :: thesis: ( 0 < a & a < 1 & 2 <= n implies a |^ n < a )

assume that

A1: 0 < a and

A2: a < 1 and

A3: 2 <= n ; :: thesis: a |^ n < a

consider m being Nat such that

A4: n = 2 + m by A3, NAT_1:10;

defpred S

A5: a * a < a * 1 by A1, A2, XREAL_1:68;

A6: for m1 being Nat st S

S

proof

a |^ (2 + 0) =
(a GeoSeq) . (1 + 1)
by Def1
let m1 be Nat; :: thesis: ( S_{1}[m1] implies S_{1}[m1 + 1] )

assume a |^ (2 + m1) < a ; :: thesis: S_{1}[m1 + 1]

then (a |^ (2 + m1)) * a < a * a by A1, XREAL_1:68;

then a |^ ((2 + m1) + 1) < a * a by NEWTON:6;

hence S_{1}[m1 + 1]
by A5, XXREAL_0:2; :: thesis: verum

end;assume a |^ (2 + m1) < a ; :: thesis: S

then (a |^ (2 + m1)) * a < a * a by A1, XREAL_1:68;

then a |^ ((2 + m1) + 1) < a * a by NEWTON:6;

hence S

.= ((a GeoSeq) . (0 + 1)) * a by Th3

.= (((a GeoSeq) . 0) * a) * a by Th3

.= (1 * a) * a by Th3

.= a * a ;

then A7: S

A8: for m1 being Nat holds S

thus a |^ n < a by A4, A8; :: thesis: verum