theorem
  f c= g implies min(f,h) c= min(g,h)
proof
  assume
A1: f c= g;
  let x be Element of C;
  f.x <= g.x by A1;
  then min(f.x,h.x) <= min(g.x,h.x) by XXREAL_0:18;
  then min(f,h).x <= min(g.x,h.x) by Def3;
  hence thesis by Def3;
end;
