
theorem Th36:
  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 X = (x | x);
  x "\/" (y "/\" z) = x "\/" ((y | z) | (y | z)) by Def12
    .= X | ((y | z)")" by Def12
    .= X | (y | z) by Def13
    .= ((X |(y | z))")" by Def13
    .= ((y" | X) | (z" | X))" by Def15
    .= ((X | y") | (z" | X))" by Th31
    .= ((X | (y | y)) | (X | z"))" by Th31
    .= (((x "\/" y) | (X | (z | z))) | ((X | (y | y)) | (X | (z | z)))) by
Def12
    .= (((x "\/" y) | (x "\/" z)) | ((X | (y | y)) | (X | (z | z)))) by Def12
    .= (((x "\/" y) | (x "\/" z)) | ((x "\/" y) | (X | (z | z)))) by Def12
    .= (((x "\/" y) | (x "\/" z)) | ((x "\/" y) | (x "\/" z))) by Def12
    .= (x "\/" y) "/\" (x "\/" z) by Def12;
  hence thesis;
end;
