let Z be open Subset of REAL; :: thesis: for n being Nat holds (diff (exp_R,Z)) . n = exp_R | Z
let n be Nat; :: thesis: (diff (exp_R,Z)) . n = exp_R | Z
defpred S₁[ Nat] means (diff (exp_R,Z)) . $1 = exp_R | Z;
A1: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A2: S₁[k] ; :: thesis: S₁[k + 1]
A3: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
(diff (exp_R,Z)) . (k + 1) = ((diff (exp_R,Z)) . k) `| Z by TAYLOR_1:def 5
.= exp_R `| Z by A2, A3, FDIFF_2:16
.= exp_R | Z by Th5 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A4: S₁[ 0 ] by TAYLOR_1:def 5;
for n being Nat holds S₁[n] from NAT_1:sch 2(A4, A1);
hence (diff (exp_R,Z)) . n = exp_R | Z ; :: thesis: verum