MML Identifier: TOPREAL7.

(1) For all real numbers a, b such that max(a,b)≤ a holds max(a,b) = a. (2) For all real numbers a, b, c, d holds max(a+ c,b+d)≤ max(a,b)+max(c,d). (3) For all real numbers a, b, c, d, e, f such that a ≤ b + c and d ≤ e + f holds max(a,d) ≤ max(b,e)+max(c, f ). (4) For all finite sequences f , g holds domg ⊆ dom( f a g). (5) For all finite sequences f , g such that len f < i and i≤ len f + leng holds i− len f ∈ domg. (6) For all finite sequences f , g, h, k such that f a g = h a k and len f = lenh and leng = lenk holds f = h and g = k. (7) If len f = leng or dom f = domg, then len( f +g) = len f and dom( f +g) = dom f . (8) If len f = leng or dom f = domg, then len( f −g) = len f and dom( f −g) = dom f . (9) len f = len2 f and dom f = dom2 f . (10) len f = len| f | and dom f = dom| f |. (11) 2( f a g) = (2 f )a (2g). (12) | f a g|= | f |a |g|. (13) If len f = lenh and leng = lenk, then 2( f a g+h a k) = (2( f +h))a (2(g+ k)). (14) If len f = lenh and leng = lenk, then | f a g+h a k|= | f +h|a |g+ k|. (15) If len f = lenh and leng = lenk, then 2( f a g−h a k) = (2( f −h))a (2(g− k)).