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

theorem
  for k,n being natural Number holds k <= n implies n - k is Element of NAT
proof
  let k,n be natural Number;
  assume
A1: k <= n;
  per cases by A1,XXREAL_0:1;
  suppose
    k < n;
    then k + 1 <= n by Th13;
    then consider j being Nat such that
A2: n = k + 1 + j by Th10;
    reconsider j as Element of NAT by ORDINAL1:def 12;
    n - k = 1 + j by A2;
    hence thesis;
  end;
  suppose
    k = n;
    then n - k = 0;
    hence thesis;
  end;
end;
