reserve r,t for Real;
reserve i for Integer;
reserve k,n for Nat;
reserve p for Polynomial of F_Real;
reserve e for Element of F_Real;
reserve L for non empty ZeroStr;
reserve z,z0,z1,z2 for Element of L;

theorem Th48:
  for i being Nat, r being Element of F_Real
  holds power(F_Real).(r,i) = r |^ i
  proof
    let i be Nat;
    let r be Element of F_Real;
    defpred P[Nat] means power(F_Real).(r,$1) = r |^ $1;
    power(F_Real).(r,0) = 1_F_Real by GROUP_1:def 7
    .= r |^ 0 by NEWTON:4;
    then A1: P[0];
A2: now
      let n be Nat;
      assume A3: P[n];
      power(F_Real).(r,n+1) = power(F_Real).(r,n)*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 power(F_Real).(r,i) = r |^ i;
  end;
