
theorem
  for L being with_infima with_suprema naturally_sup-generated
naturally_inf-generated Lattice-like OrderInvolutive PartialOrdered non empty
  OrthoLattRelStr holds L is de_Morgan
proof
  let L be with_infima with_suprema naturally_sup-generated
naturally_inf-generated Lattice-like OrderInvolutive PartialOrdered non empty
  OrthoLattRelStr;
A1: for x,y being Element of L holds x` "|_|" y` <= ( x "|^|" y )`
  proof
    let a,b be Element of L;
    set i = a "|^|" b;
    set s = a` "|_|" b`;
    i <= a by YELLOW_0:23;
    then a` <= i` by Th7;
    then
A2: s <= i` "|_|" b` by WAYBEL_1:2;
    i <= b by YELLOW_0:23;
    then b` <= i` by Th7;
    then i` = b` "|_|" i` by Th24;
    hence thesis by A2,LATTICE3:13;
  end;
A3: for x,y being Element of L holds ( x "|_|" y )` <= x` "|^|" y`
  proof
    let a,b be Element of L;
    set i = a` "|^|" b`;
    set s = a "|_|" b;
    a <= s by YELLOW_0:22;
    then s` <= a` by Th7;
    then
A4: s` "|^|" b` <= i by WAYBEL_1:1;
    b <= s by YELLOW_0:22;
    then s` <= b` by Th7;
    hence thesis by A4,Th24;
  end;
  for x,y being Element of L holds (x` |_| y`)` = x |^| y
  proof
    let a,b be Element of L;
    set s = a` "|_|" b`;
    set i = a "|^|" b;
    a`` = a & b`` = b by Th6;
    then
A5: s` <= i by A3;
    i`` <= s` by A1,Th7;
    then
A6: i <= s` by Th6;
    i = a |^| b & s = a` |_| b` by Th25,Th26;
    hence thesis by A5,A6,ORDERS_2:2;
  end;
  hence thesis by ROBBINS1:def 23;
end;
