
theorem Th35:
  for L being bounded LATTICE, x,y being Element of L holds y
  is_a_complement_of x iff y~ is_a_complement_of x~
proof
  let L be bounded LATTICE, x,y be Element of L;
  hereby
    assume
A1: y is_a_complement_of x;
    then
A2: (x~)"\/"(y~) = (Bottom L)~ by Th21
      .= Top (L opp) by Th12,YELLOW_0:42;
    (x~)"/\"(y~) = (Top L)~ by Th23,A1
      .= Bottom (L opp) by Th13,YELLOW_0:43;
    hence y~ is_a_complement_of x~ by A2;
  end;
  assume that
A3: (x~)"\/"(y~) = Top (L opp) and
A4: (x~)"/\"(y~) = Bottom (L opp);
  thus x "\/" y = (x~)"/\"(y~) by Th23
    .= Top L by A4,Th13,YELLOW_0:43;
  thus x "/\" y = (x~)"\/"(y~) by Th21
    .= Bottom L by A3,Th12,YELLOW_0:42;
end;
