
theorem Th12:
  for x,y be Element of BOOLEAN holds (x 'or' y = TRUE iff x =
  TRUE or y = TRUE) & (x 'or' y = FALSE iff x = FALSE & y = FALSE)
proof
  let x,y be Element of BOOLEAN;
  thus x 'or' y = TRUE implies x = TRUE or y = TRUE
  proof
    assume x 'or' y = TRUE;
    then 'not' x = FALSE or 'not' y = FALSE by MARGREL1:12;
    hence thesis;
  end;
  thus x = TRUE or y = TRUE implies x 'or' y = TRUE;
  thus x 'or' y = FALSE implies x = FALSE & y = FALSE
  proof
    assume
A1: x 'or' y = FALSE;
    then 'not' x = TRUE by MARGREL1:12;
    hence thesis by A1;
  end;
  thus thesis;
end;
