
theorem Th5:
  for L being LATTICE holds L is distributive iff for x,y,z being
  Element of L holds x "\/" (y "/\" z) = (x "\/" y) "/\" (x "\/" z)
proof
  let L be LATTICE;
  hereby
    assume
A1: L is distributive;
    let x,y,z be Element of L;
    thus x"\/"(y"/\"z) = (x"\/"(z"/\"x))"\/"(y"/\"z) by LATTICE3:17
      .= x"\/"((z"/\"x)"\/"(z"/\"y)) by LATTICE3:14
      .= x"\/"((x"\/"y)"/\"z) by A1
      .= ((x"\/"y)"/\"x)"\/"((x"\/"y)"/\"z) by LATTICE3:18
      .= (x"\/"y)"/\"(x"\/"z) by A1;
  end;
  assume
A2: for x,y,z being Element of L holds x "\/" (y "/\" z) = (x "\/" y)
  "/\" (x "\/" z);
  let x,y,z be Element of L;
  thus x"/\"(y"\/"z) = (x"/\"(x"\/"z))"/\"(y"\/"z) by LATTICE3:18
    .= x"/\"((z"\/"x)"/\"(y"\/"z)) by LATTICE3:16
    .= x"/\"(z"\/"(x"/\"y)) by A2
    .= ((y"/\"x)"\/"x)"/\"((x"/\"y)"\/"z) by LATTICE3:17
    .= (x"/\"y)"\/"(x"/\"z) by A2;
end;
