reserve a,b,c,k,m,n for Nat;
reserve i,j,x,y for Integer;
reserve p,q for Prime;
reserve r,s for Real;

theorem Th1:
  for i,j being natural Number holds
  i < j implies ex k being positive Nat st j = i + k
  proof
    let i,j be natural Number;
    assume
A1: i < j;
    then consider k being Nat such that
A2: j = i + k by NAT_1:10;
    now
      assume k <= 0;
      then k = 0;
      hence contradiction by A1,A2;
    end;
    then reconsider k as positive Nat;
    take k;
    thus thesis by A2;
  end;
