reserve x,y,y1,y2 for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,h,g,h1 for Membership_Func of C;

theorem
  f c= g & h c= h1 implies min(f,h) c= min(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 min(f.x,h.x) <= min(g.x,h1.x) by XXREAL_0:18;
  then min(f,h).x <= min(g.x,h1.x) by Def3;
  hence thesis by Def3;
end;
