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
  a '&' (a 'nand' b) = a '&' 'not' b
proof
  thus a '&' (a 'nand' b) =(a '&' 'not' a) 'or' (a '&' 'not' b) by Th50
    .=O_el(Y) 'or' (a '&' 'not' b) by BVFUNC_4:5
    .=a '&' 'not' b by BVFUNC_1:9;
end;
