reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;
reserve n for non zero Nat;
reserve n for non zero Nat;
reserve n for Nat,
        X for set,
        S for Subset-Family of X;
reserve n for Nat,
        S for Subset-Family of REAL;
reserve n       for Nat,
        a,b,c,d for Element of REAL n;

theorem
  a < b & b < c implies a < c
  proof
    assume that
A1: a < b and
A2: b < c;
    for i be Nat st i in Seg n holds a.i < c.i
    proof
      let i be Nat;
      assume i in Seg n;
      then a.i < b.i & b.i < c.i by A1,A2;
      hence thesis by XXREAL_0:2;
    end;
    hence thesis;
  end;
