theorem
  f c= g & h c= h1 implies max(f,h) c= max(g,h1)
proof
  assume
A1: f c= g & h c= h1;
  let x be Element of C;
  f.x <= g.x & h.x <= h1.x by A1;
  then max(f.x,h.x) <= max(g.x,h1.x) by XXREAL_0:26;
  then max(f,h).x <= max(g.x,h1.x) by Def4;
  hence thesis by Def4;
end;
