let m, n be Nat; :: thesis: ( m > 1 & n > 0 implies m |^ n > 1 )
assume A1: ( m > 1 & n > 0 ) ; :: thesis: m |^ n > 1
defpred S₁[ Nat] means ( $1 > 0 implies m |^ $1 > 1 );
A2: S₁[ 0 ] ;
A3: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A4: S₁[n] ; :: thesis: S₁[n + 1]
A5: m |^ (n + 1) = (m |^ n) * (m |^ 1) by NEWTON:13
.= (m |^ n) * m by NEWTON:10 ;
S₁[n + 1]

proof

now

per cases ( n = 0 or n <> 0 ) ;

suppose n = 0 ; :: thesis: S₁[n + 1]

hence S₁[n + 1] by A1, NEWTON:10; :: thesis: verum

end;

suppose n <> 0 ; :: thesis: S₁[n + 1]

then (m |^ n) * m > 1 * m by A1, A4, XREAL_1:70;
hence S₁[n + 1] by A1, A5, XXREAL_0:2; :: thesis: verum

end;

end;

end;

hence S₁[n + 1] ; :: thesis: verum

end;

hence S₁[n + 1] ; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A2, A3);
hence m |^ n > 1 by A1; :: thesis: verum