reserve X for set,
  x,y,z for Element of BooleLatt X,
  s for set;
reserve y for Element of BooleLatt X;
reserve L for Lattice,
  p,q for Element of L;
reserve A for RelStr,
  a,b,c for Element of A;

theorem
  for L being Lattice holds LattPOSet L is with_suprema with_infima
proof
  let L;
  thus LattPOSet L is with_suprema
  proof
    let x,y be Element of LattPOSet L;
    take z = (%x"\/"%y)%;
A1: %x [= %x"\/"%y by LATTICES:5;
A2: %y [= %y"\/"%x by LATTICES:5;
A3: (%x)% = %x;
A4: (%y)% = %y;
    hence x <= z & y <= z by A1,A2,A3,Th7;
    let z9 be Element of LattPOSet L;
A5: (%z9)% = %z9;
    assume that
A6: x <= z9 and
A7: y <= z9;
A8: %x [= %z9 by A3,A5,A6,Th7;
    %y [= %z9 by A4,A5,A7,Th7;
    then %x"\/"%y [= %z9 by A8,FILTER_0:6;
    hence thesis by A5,Th7;
  end;
  let x,y be Element of LattPOSet L;
  take z = (%x"/\"%y)%;
A9: %x"/\"%y [= %x by LATTICES:6;
A10: %y"/\"%x [= %y by LATTICES:6;
A11: (%x)% = %x;
A12: (%y)% = %y;
  hence z <= x & z <= y by A9,A10,A11,Th7;
  let z9 be Element of LattPOSet L;
A13: (%z9)% = %z9;
  assume that
A14: z9 <= x and
A15: z9 <= y;
A16: %z9 [= %x by A11,A13,A14,Th7;
  %z9 [= %y by A12,A13,A15,Th7;
  then %z9 [= %x"/\"%y by A16,FILTER_0:7;
  hence thesis by A13,Th7;
end;
