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