
theorem
  for L being properly_defined Boolean well-complemented Lattice-like
  non empty ShefferOrthoLattStr holds L is satisfying_Sheffer_3
proof
  let L be properly_defined Boolean well-complemented Lattice-like non empty
  ShefferOrthoLattStr;
  let x, y, z be Element of L;
  x *' (y` + z`) = (y` *' x) + (z` *' x) by ROBBINS1:30;
  then (x` + (y | z)`)` = (y` *' x) + (z` *' x) by Def12;
  then (x | (y | z))` = (y` *' x) + (z` *' x) by Def12;
  then (x | (y | z)) | (x | (y | z)) = (y` *' x) + (z` *' x) by Def12
    .= (y` *' x``) + (z` *' x) by ROBBINS1:3
    .= (y` *' x``) + (z` *' x``) by ROBBINS1:3
    .= (y + x`)` + (z` *' x``) by Th1
    .= (y + x`)` + (z + x`)` by Th1
    .= (y + x`) | (z + x`) by Def12
    .= (y`` + x`) | (z + x`) by ROBBINS1:3
    .= (y`` + x`) | (z`` + x`) by ROBBINS1:3
    .= (y` | x) | (z`` + x`) by Def12
    .= (y` | x) | (z` | x) by Def12
    .= ((y | y) | x) | (z` | x) by Def12
    .= ((y | y) | x) | ((z | z) | x) by Def12;
  hence thesis;
end;
