
theorem Th35:
  for L being satisfying_Sheffer_1 satisfying_Sheffer_2
  satisfying_Sheffer_3 properly_defined non empty ShefferOrthoLattStr holds L
  is distributive
proof
  let L be satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3
  properly_defined non empty ShefferOrthoLattStr;
  let x, y, z be Element of L;
  set Y = y";
  set Z = z";
  x "/\" (y "\/" z) = x "/\" ((y | y) | (z | z)) by Def12
    .= (x | (Y | Z))" by Def12
    .= (Y" | x) | (Z" | x) by Def15
    .= (y | x) | (Z" | x) by Def13
    .= (y | x) | (z | x) by Def13
    .= (x | y) | (z | x) by Th31
    .= (x | y) | (x | z) by Th31
    .= ((x | y)")" | (x | z) by Def13
    .= ((x | y) | (x | y))" | ((x | z)")" by Def13
    .= ((x "/\" y) | ((x | y) | (x | y))) | (((x | z) | (x | z)) | ((x | z)
  | (x | z))) by Def12
    .= ((x "/\" y) | (x "/\" y)) | (((x | z) | (x | z)) | ((x | z) | (x | z)
  )) by Def12
    .= ((x "/\" y) | (x "/\" y)) | ((x "/\" z) | ((x | z) | (x | z))) by Def12
    .= ((x "/\" y) | (x "/\" y)) | ((x "/\" z) | (x "/\" z)) by Def12
    .= (x "/\" y) "\/" (x "/\" z) by Def12;
  hence thesis;
end;
