let L be non empty join-commutative meet-commutative distributive join-idempotent upper-bounded' lower-bounded' distributive' complemented' LattStr ; :: thesis: for x, y being Element of L holds
( x is_a_complement'_of y iff x is_a_complement_of y )
let x, y be Element of L; :: thesis: ( x is_a_complement'_of y iff x is_a_complement_of y )

hereby :: thesis: ( x is_a_complement_of y implies x is_a_complement'_of y )

assume A1: x is_a_complement'_of y ; :: thesis: x is_a_complement_of y
then x "/\" y = Bot' L by Def6;
then A2: x "/\" y = Bottom L by Th20;
x "\/" y = Top' L by A1, Def6;
then x "\/" y = Top L by Th19;
hence x is_a_complement_of y by A2, LATTICES:def 18; :: thesis: verum

end;

assume A3: x is_a_complement_of y ; :: thesis: x is_a_complement'_of y
then x "/\" y = Bottom L by LATTICES:def 18;
then A4: x "/\" y = Bot' L by Th20;
x "\/" y = Top L by A3, LATTICES:def 18;
then x "\/" y = Top' L by Th19;
hence x is_a_complement'_of y by A4, Def6; :: thesis: verum