reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem
  for f being real-valued Function, r1,r2 being Real st f <= r1 <= r2
  holds f <= r2
  proof
    let f be real-valued Function;
    let r1,r2 be Real such that
A1: f <= r1 and
A2: r1 <= r2;
    let x be object;
    assume x in dom f;
    then f.x <= r1 by A1;
    hence f.x <= r2 by A2,XXREAL_0:2;
  end;
