reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;

theorem
  k - n in Seg k iff n < k
proof
  thus k - n in Seg k implies n < k
  proof
    assume
A1: k - n in Seg k;
    then reconsider x = k - n as Element of NAT;
    assume not n < k;
    then k - n <= n - n by XREAL_1:9;
    then x = 0;
    hence contradiction by A1,FINSEQ_1:1;
  end;
  thus thesis by Th12,FINSEQ_1:3;
end;
