reserve Y for non empty set,
  a,b,c,d for Function of Y,BOOLEAN;
reserve Y for non empty set,
  a,b,c for Function of Y,BOOLEAN;

theorem
  O_el(Y) 'nand' O_el(Y) = I_el(Y) & O_el(Y) 'nand' I_el(Y) = I_el(Y) &
  I_el(Y) 'nand' I_el(Y) = O_el(Y)
proof
  thus O_el(Y) 'nand' O_el(Y) = I_el(Y) by Th4;
  thus O_el(Y) 'nand' I_el(Y) = I_el(Y) by Th4;
  thus I_el(Y) 'nand' I_el(Y) = 'not' I_el(Y) by Th3
    .= O_el(Y) by BVFUNC_1:2;
end;
