 reserve x for Real,
    p,k,l,m,n,s,h,i,j,k1,t,t1 for Nat,
    X for Subset of REAL;

theorem
  0 < m implies for n ex k,t st n = (m*k)+t & t < m
proof
  defpred P[Nat] means ex k,t st $1 = (m*k)+t & t < m;
  assume
A1: 0 < m;
A2: for n st P[n] holds P[n+1]
  proof
    let n;
    given k1,t1 such that
A3: n = (m*k1)+t1 and
A4: t1 < m;
A5: t1+1 < m implies ex k,t st n+1 = (m*k)+t & t < m
    proof
      assume
A6:   t1+1 < m;
      take k = k1, t = t1+1;
      thus n+1 = m*k+t by A3;
      thus thesis by A6;
    end;
A7: t1+1 = m implies ex k,t st n+1 = (m*k)+t & t < m
    proof
      assume
A8:   t1+1 = m;
      take k = k1+1, t = 0;
      thus n+1 = m*k+t by A3,A8;
      thus thesis by A1;
    end;
    t1+1 <= m by A4,Th13;
    hence thesis by A5,A7,XXREAL_0:1;
  end;
  0 = m*0+0;
  then
A9: P[0] by A1;
  thus for n holds P[n] from NatInd(A9,A2);
end;
