
theorem
  for L being with_suprema Poset, X being Subset of L, x being Element
  of L st ex_inf_of {x} "\/" X,L & ex_inf_of X,L holds x "\/" inf X <= inf ({x}
  "\/" X)
proof
  let L be with_suprema Poset, X be Subset of L, x be Element of L such that
A1: ex_inf_of {x} "\/" X,L and
A2: ex_inf_of X,L;
A3: {x} "\/" X = {x "\/" y where y is Element of L : y in X} by Th15;
  {x} "\/" X is_>=_than x "\/" inf X
  proof
    let q be Element of L;
    assume q in {x} "\/" X;
    then consider y being Element of L such that
A4: q = x "\/" y and
A5: y in X by A3;
    x <= x & y >= inf X by A2,A5,Th2;
    hence thesis by A4,YELLOW_3:3;
  end;
  hence thesis by A1,YELLOW_0:def 10;
end;
