reserve c for Complex;
reserve r for Real;
reserve m,n for Nat;
reserve f for complex-valued Function;
reserve f,g for differentiable Function of REAL,REAL;
reserve L for non empty ZeroStr;
reserve x for Element of L;
reserve p,q for Polynomial of F_Real;

theorem Th39:
  for r being Element of F_Real holds power(r,n) = r|^n
  proof
    let r be Element of F;
    defpred P[Nat] means power(r,$1) = r|^$1;
    power(r,0) = 1_F by GROUP_1:def 7;
    then
A1: P[0] by NEWTON:4;
A2: now
      let n be Nat;
      reconsider n1 = n as Element of NAT by ORDINAL1:def 12;
      assume
A3:   P[n];
      power(r,n+1) = power(r,n1)*r by GROUP_1:def 7
      .= r|^(n+1) by A3,NEWTON:6;
      hence P[n+1];
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
