let n be natural number ; :: thesis: for x being real number holds (exp_R x) #R n = exp_R (n * x)
let x be real number ; :: thesis: (exp_R x) #R n = exp_R (n * x)
reconsider x = x as Real by XREAL_0:def 1;
defpred S₁[ natural number ] means (exp_R x) #R $1 = exp_R ($1 * x);
A1: S₁[ 0 ] by PREPOWER:85, SIN_COS:56, SIN_COS:60;
A2: for k being natural number st S₁[k] holds
S₁[k + 1]

proof

let k be natural number ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
reconsider k1 = k as Element of NAT by ORDINAL1:def 13;
thus (exp_R x) #R (k + 1) = ((exp_R x) #R k) * ((exp_R x) #R 1) by PREPOWER:89, SIN_COS:60
.= ((exp_R x) #R k) * (exp_R x) by PREPOWER:86, SIN_COS:60
.= exp_R ((k1 * x) + x) by A3, SIN_COS:55
.= exp_R ((k + 1) * x) ; :: thesis: verum

end;

for n being natural number holds S₁[n] from NAT_1:sch 2(A1, A2);
hence (exp_R x) #R n = exp_R (n * x) ; :: thesis: verum