
theorem Th31:
  for L being satisfying_Sheffer_1 satisfying_Sheffer_2
  satisfying_Sheffer_3 properly_defined non empty ShefferOrthoLattStr, x, y
  being Element of L holds x | y = y | x
proof
  let L be satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3
  properly_defined non empty ShefferOrthoLattStr;
  let x, y be Element of L;
  x | y = ((x | y)")" by Def13
    .= ((x | (y")")")" by Def13
    .= (((y")" |x)")" by Def15
    .= ((y | x)")" by Def13
    .= y | x by Def13;
  hence thesis;
end;
