Article SHEFFER2, MML version 4.99.1005

:: SHEFFER2:attrnot 1 => SHEFFER2:attr 1
definition
  let a1 be non empty ShefferStr;
  attr a1 is satisfying_Sh_1 means
    for b1, b2, b3 being Element of the carrier of a1 holds
    (b1 | ((b2 | b1) | b1)) | (b2 | (b3 | b1)) = b2;
end;

:: SHEFFER2:dfs 1
definiens
  let a1 be non empty ShefferStr;
To prove
     a1 is satisfying_Sh_1
it is sufficient to prove
  thus for b1, b2, b3 being Element of the carrier of a1 holds
    (b1 | ((b2 | b1) | b1)) | (b2 | (b3 | b1)) = b2;

:: SHEFFER2:def 1
theorem
for b1 being non empty ShefferStr holds
      b1 is satisfying_Sh_1
   iff
      for b2, b3, b4 being Element of the carrier of b1 holds
      (b2 | ((b3 | b2) | b2)) | (b3 | (b4 | b2)) = b3;

:: SHEFFER2:condreg 1
registration
  cluster non empty trivial -> satisfying_Sh_1 (ShefferStr);
end;

:: SHEFFER2:exreg 1
registration
  cluster non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 satisfying_Sh_1 ShefferStr;
end;

:: SHEFFER2:th 1
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
((b2 | (b3 | b4)) | (b2 | (b2 | (b3 | b4)))) | ((b4 | ((b2 | b4) | b4)) | (b5 | (b2 | (b3 | b4)))) = b4 | ((b2 | b4) | b4);

:: SHEFFER2:th 2
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 | b3) | (((b3 | ((b4 | b3) | b3)) | (b2 | b3)) | (b2 | b3))) | b4 = b3 | ((b4 | b3) | b3);

:: SHEFFER2:th 3
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | ((b3 | b2) | b2)) | (b3 | (b4 | ((b2 | b4) | b4))) = b3;

:: SHEFFER2:th 4
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 | ((b2 | ((b2 | b2) | b2)) | (b3 | (b2 | ((b2 | b2) | b2)))) = b2 | ((b2 | b2) | b2);

:: SHEFFER2:th 5
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2 being Element of the carrier of b1 holds
   b2 | ((b2 | b2) | b2) = b2 | b2;

:: SHEFFER2:th 6
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2 being Element of the carrier of b1 holds
   (b2 | ((b2 | b2) | b2)) | (b2 | b2) = b2;

:: SHEFFER2:th 7
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b2 | b2) | (b2 | (b3 | b2)) = b2;

:: SHEFFER2:th 8
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b2 | (((b3 | b3) | b2) | b2)) | b3 = b3 | b3;

:: SHEFFER2:th 9
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
((b2 | b3) | (((b2 | b3) | (b2 | b3)) | (b2 | b3))) | ((b2 | b3) | (b2 | b3)) = b3 | ((((b2 | b3) | (b2 | b3)) | b3) | b3);

:: SHEFFER2:th 10
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 | ((((b3 | b2) | (b3 | b2)) | b2) | b2) = b3 | b2;

:: SHEFFER2:th 11
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b2 | b2) | (b3 | b2) = b2;

:: SHEFFER2:th 12
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 | (b3 | (b2 | b2)) = b2 | b2;

:: SHEFFER2:th 13
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
((b2 | b3) | (b2 | b3)) | b3 = b2 | b3;

:: SHEFFER2:th 14
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 | ((b3 | b2) | b2) = b3 | b2;

:: SHEFFER2:th 15
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | b3) | (b2 | (b4 | b3)) = b2;

:: SHEFFER2:th 16
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | (b3 | b4)) | (b2 | b4) = b2;

:: SHEFFER2:th 17
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | ((b2 | b3) | (b4 | b3)) = b2 | b3;

:: SHEFFER2:th 18
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 | (b3 | b4)) | b4) | b2 = b2 | (b3 | b4);

:: SHEFFER2:th 19
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 | ((b3 | b2) | b2) = b2 | b3;

:: SHEFFER2:th 20
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 | b3 = b3 | b2;

:: SHEFFER2:th 21
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b2 | b3) | (b2 | b2) = b2;

:: SHEFFER2:th 22
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | b3) | (b3 | (b4 | b2)) = b3;

:: SHEFFER2:th 23
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | (b3 | b4)) | (b4 | b2) = b2;

:: SHEFFER2:th 24
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | b3) | (b3 | (b2 | b4)) = b3;

:: SHEFFER2:th 25
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | (b3 | b4)) | (b3 | b2) = b2;

:: SHEFFER2:th 26
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 | b3) | (b2 | b4)) | b4 = b2 | b4;

:: SHEFFER2:th 27
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | (b3 | (b2 | (b3 | b4))) = b2 | (b3 | b4);

:: SHEFFER2:th 28
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | (b3 | (b2 | b4))) | b3 = b3 | (b2 | b4);

:: SHEFFER2:th 29
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(b2 | (b3 | b4)) | (b2 | (b5 | (b3 | b2))) = (b2 | (b3 | b4)) | (b3 | b2);

:: SHEFFER2:th 30
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | (b3 | (b2 | b4))) | b3 = b3 | (b4 | b2);

:: SHEFFER2:th 31
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(b2 | (b3 | b4)) | (b2 | (b5 | (b3 | b2))) = b2;

:: SHEFFER2:th 32
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 | (b3 | (b2 | b3)) = b2 | b2;

:: SHEFFER2:th 33
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | (b3 | b4) = b2 | (b4 | b3);

:: SHEFFER2:th 34
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | (b3 | (b2 | (b4 | (b3 | b2)))) = b2 | b2;

:: SHEFFER2:th 35
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | (b3 | b4)) | ((b3 | b2) | b2) = (b2 | (b3 | b4)) | (b2 | (b3 | b4));

:: SHEFFER2:th 36
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b2 | (b3 | b2)) | b3 = b3 | b3;

:: SHEFFER2:th 37
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | b3) | b4 = b4 | (b3 | b2);

:: SHEFFER2:th 38
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | (b3 | (b4 | (b2 | b3))) = b2 | (b3 | b3);

:: SHEFFER2:th 39
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 | b3) | b3) | (b3 | (b4 | b2)) = (b3 | (b4 | b2)) | (b3 | (b4 | b2));

:: SHEFFER2:th 40
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(b2 | b3) | (b4 | b5) = (b5 | b4) | (b3 | b2);

:: SHEFFER2:th 41
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | (b3 | ((b3 | b2) | b4)) = b2 | (b3 | b3);

:: SHEFFER2:th 42
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 | (b3 | b2) = b2 | (b3 | b3);

:: SHEFFER2:th 43
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b2 | b3) | b3 = b3 | (b2 | b2);

:: SHEFFER2:th 44
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 | (b3 | b3) = b2 | (b2 | b3);

:: SHEFFER2:th 45
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | (b3 | b3)) | (b2 | (b4 | b3)) = (b2 | (b4 | b3)) | (b2 | (b4 | b3));

:: SHEFFER2:th 46
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | (b3 | b4)) | (b2 | (b3 | b3)) = (b2 | (b3 | b4)) | (b2 | (b3 | b4));

:: SHEFFER2:th 47
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | ((b3 | b3) | (b4 | (b2 | (b2 | b3)))) = b2 | ((b3 | b3) | (b3 | b3));

:: SHEFFER2:th 48
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 | (b3 | b4)) | (b2 | (b3 | b4))) | (b3 | b3) = b2 | (b3 | b3);

:: SHEFFER2:th 49
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | ((b3 | b3) | (b4 | (b2 | (b2 | b3)))) = b2 | b3;

:: SHEFFER2:th 50
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(((b2 | b3) | (b2 | b3)) | ((b4 | ((b2 | b3) | b4)) | (b2 | b3))) | (b2 | b2) = (b4 | ((b2 | b3) | b4)) | (b2 | b2);

:: SHEFFER2:th 51
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | ((b3 | b4) | b2)) | (b3 | b3) = (b3 | b4) | (b3 | b3);

:: SHEFFER2:th 52
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | ((b3 | b4) | b2)) | (b3 | b3) = b3;

:: SHEFFER2:th 53
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | ((b3 | ((b2 | b4) | b3)) | b2) = b3 | ((b2 | b4) | b3);

:: SHEFFER2:th 54
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | ((b3 | (b3 | (b4 | b2))) | b2) = b3 | ((b2 | (b3 | (b2 | b4))) | b3);

:: SHEFFER2:th 55
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | ((b3 | (b3 | (b4 | b2))) | b2) = b3 | (b3 | (b4 | b2));

:: SHEFFER2:th 56
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
b2 | (b3 | (b4 | (b4 | (b5 | (b3 | b2))))) = b2 | (b3 | b3);

:: SHEFFER2:th 57
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | (b3 | (b3 | (b4 | (b2 | b3)))) = b2 | (b3 | (b2 | b2));

:: SHEFFER2:th 58
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | (b3 | (b3 | (b4 | (b2 | b3)))) = b2 | b2;

:: SHEFFER2:th 59
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 | (b3 | (b3 | b3)) = b2 | b2;

:: SHEFFER2:th 60
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | (((b3 | (b4 | b2)) | (b3 | (b4 | b2))) | (b4 | b4)) = b2 | (b3 | (b4 | b2));

:: SHEFFER2:th 61
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | (b3 | (b4 | b4)) = b2 | (b3 | (b4 | b2));

:: SHEFFER2:th 62
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | (b3 | ((b4 | b4) | b2)) = b2 | (b3 | b4);

:: SHEFFER2:th 63
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | (b3 | b3)) | (b2 | (b4 | ((b3 | b3) | b2))) = (b2 | (b4 | b3)) | (b2 | (b4 | b3));

:: SHEFFER2:th 64
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | (b3 | b3)) | (b2 | (b4 | (b2 | (b3 | b3)))) = (b2 | (b4 | b3)) | (b2 | (b4 | b3));

:: SHEFFER2:th 65
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | (b3 | b3)) | (b2 | (b4 | b4)) = (b2 | (b4 | b3)) | (b2 | (b4 | b3));

:: SHEFFER2:th 66
theorem
for b1 being non empty satisfying_Sh_1 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 | b2) | b3) | ((b4 | b4) | b3) = (b3 | (b2 | b4)) | (b3 | (b2 | b4));

:: SHEFFER2:th 67
theorem
for b1 being non empty ShefferStr
      st b1 is satisfying_Sh_1
   holds b1 is satisfying_Sheffer_1;

:: SHEFFER2:th 68
theorem
for b1 being non empty ShefferStr
      st b1 is satisfying_Sh_1
   holds b1 is satisfying_Sheffer_2;

:: SHEFFER2:th 69
theorem
for b1 being non empty ShefferStr
      st b1 is satisfying_Sh_1
   holds b1 is satisfying_Sheffer_3;

:: SHEFFER2:exreg 2
registration
  cluster non empty Lattice-like Boolean well-complemented de_Morgan properly_defined satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 satisfying_Sh_1 ShefferOrthoLattStr;
end;

:: SHEFFER2:condreg 2
registration
  cluster non empty properly_defined satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 -> Lattice-like Boolean (ShefferOrthoLattStr);
end;

:: SHEFFER2:condreg 3
registration
  cluster non empty Lattice-like Boolean well-complemented properly_defined -> satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 (ShefferOrthoLattStr);
end;

:: SHEFFER2:th 70
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b3 | ((b2 | b2) | b2) = b3 | b3;

:: SHEFFER2:th 71
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b3 = (b3 | b3) | (b2 | (b2 | b2));

:: SHEFFER2:th 72
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b3) | (b3 | (b2 | (b2 | b2))) = b3;

:: SHEFFER2:th 73
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
((b5 | (b4 | (b4 | b4))) | b3) | ((b2 | b2) | b3) = (b3 | (b5 | b2)) | (b3 | (b5 | b2));

:: SHEFFER2:th 74
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b4 | b3) | ((b2 | b2) | b3) = (b3 | ((b4 | b4) | b2)) | (b3 | ((b4 | b4) | b2));

:: SHEFFER2:th 75
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
((b2 | b2) | b3) | ((b5 | (b4 | (b4 | b4))) | b3) = (b3 | (b2 | b5)) | (b3 | (b2 | b5));

:: SHEFFER2:th 76
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b3 = (b3 | b3) | ((b2 | b2) | b2);

:: SHEFFER2:th 77
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b3) | (b3 | ((b2 | b2) | b2)) = b3;

:: SHEFFER2:th 78
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | ((b2 | b2) | b2)) | (b3 | b3) = b3;

:: SHEFFER2:th 79
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b4 | ((b3 | b3) | b3)) | (b2 | (b2 | b2)) = b4;

:: SHEFFER2:th 80
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b4 | (b3 | b3)) | (b4 | (b3 | b3))) | (b2 | (b2 | b2)) = (b3 | b3) | b4;

:: SHEFFER2:th 81
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b3 | (b2 | b2) = (b2 | b2) | b3;

:: SHEFFER2:th 82
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b3 | b2 = ((b2 | b2) | (b2 | b2)) | b3;

:: SHEFFER2:th 83
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b3 | b2 = b2 | b3;

:: SHEFFER2:th 84
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b2) | (b3 | ((b2 | b2) | (b2 | b2))) = (((b2 | b2) | (b2 | b2)) | b3) | (((b2 | b2) | (b2 | b2)) | b3);

:: SHEFFER2:th 85
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b2) | (b3 | b2) = (((b2 | b2) | (b2 | b2)) | b3) | (((b2 | b2) | (b2 | b2)) | b3);

:: SHEFFER2:th 86
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b2) | (b3 | b2) = (b2 | b3) | (((b2 | b2) | (b2 | b2)) | b3);

:: SHEFFER2:th 87
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b2) | (b3 | b2) = (b2 | b3) | (b2 | b3);

:: SHEFFER2:th 88
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b4 | b3) | ((b2 | b2) | b2)) | ((b4 | b3) | (b4 | b3)) = ((b4 | b4) | (b4 | b3)) | ((b3 | b3) | (b4 | b3));

:: SHEFFER2:th 89
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b3 | b2 = ((b3 | b3) | (b3 | b2)) | ((b2 | b2) | (b3 | b2));

:: SHEFFER2:th 90
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b3) | (b3 | ((b2 | b2) | b2)) = (((b2 | b2) | (b2 | b2)) | b3) | ((b2 | b2) | b3);

:: SHEFFER2:th 91
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 = (((b3 | b3) | (b3 | b3)) | b2) | ((b3 | b3) | b2);

:: SHEFFER2:th 92
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 = (b3 | b2) | ((b3 | b3) | b2);

:: SHEFFER2:th 93
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(((b4 | b4) | b3) | ((b2 | b2) | b3)) | ((b3 | (b4 | b2)) | (b3 | (b4 | b2))) = ((b3 | b3) | (b3 | (b4 | b2))) | (((b4 | b2) | (b4 | b2)) | (b3 | (b4 | b2)));

:: SHEFFER2:th 94
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(((b4 | b4) | b3) | ((b2 | b2) | b3)) | ((b3 | (b4 | b2)) | (b3 | (b4 | b2))) = b3 | (b4 | b2);

:: SHEFFER2:th 95
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b3) | (b3 | (b2 | (b2 | b2))) = ((b2 | b2) | b3) | (((b2 | b2) | (b2 | b2)) | b3);

:: SHEFFER2:th 96
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 = ((b3 | b3) | b2) | (((b3 | b3) | (b3 | b3)) | b2);

:: SHEFFER2:th 97
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 = ((b3 | b3) | b2) | (b3 | b2);

:: SHEFFER2:th 98
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(((b4 | b4) | b3) | ((b2 | b2) | b3)) | ((b3 | (b4 | b2)) | (b3 | (b4 | b2))) = (((b2 | b2) | (b2 | b2)) | ((b4 | b4) | b3)) | ((b3 | b3) | ((b4 | b4) | b3));

:: SHEFFER2:th 99
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | (b4 | b3) = (((b3 | b3) | (b3 | b3)) | ((b4 | b4) | b2)) | ((b2 | b2) | ((b4 | b4) | b2));

:: SHEFFER2:th 100
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | (b4 | b3) = (b3 | ((b4 | b4) | b2)) | ((b2 | b2) | ((b4 | b4) | b2));

:: SHEFFER2:th 101
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 | b2 = ((b3 | b3) | b3) | b2;

:: SHEFFER2:th 102
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b2) | ((b2 | b2) | b3) = b3;

:: SHEFFER2:th 103
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b3) | ((b3 | b3) | ((b2 | b2) | b2)) = (b2 | b2) | b2;

:: SHEFFER2:th 104
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b3) | b3 = (b2 | b2) | b2;

:: SHEFFER2:th 105
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b2) | (b2 | (b3 | b3)) = b2;

:: SHEFFER2:th 106
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | (b2 | b2)) | (b2 | b3) = b3;

:: SHEFFER2:th 107
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b2) | (b3 | (b2 | b2)) = b3;

:: SHEFFER2:th 108
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b3 | (((b3 | (b2 | b2)) | (b3 | (b2 | b2))) | (b2 | b3)) = b2 | b3;

:: SHEFFER2:th 109
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | (b2 | b2)) | (b3 | b2) = b3;

:: SHEFFER2:th 110
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(((b5 | b5) | b5) | b4) | ((b3 | b3) | b4) = (b4 | (((b2 | (b2 | b2)) | (b2 | (b2 | b2))) | b3)) | (b4 | (((b2 | (b2 | b2)) | (b2 | (b2 | b2))) | b3));

:: SHEFFER2:th 111
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | b2) | ((b3 | b3) | b2) = (b2 | (((b4 | (b4 | b4)) | (b4 | (b4 | b4))) | b3)) | (b2 | (((b4 | (b4 | b4)) | (b4 | (b4 | b4))) | b3));

:: SHEFFER2:th 112
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | b4) | (b2 | (b4 | (b3 | (b3 | b3)))) = b2;

:: SHEFFER2:th 113
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | (b4 | (b3 | (b3 | b3)))) | (b2 | b4) = b2;

:: SHEFFER2:th 114
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(((b4 | b4) | b4) | b3) | ((b2 | b2) | b3) = (b3 | ((b3 | b3) | b2)) | (b3 | ((b3 | b3) | b2));

:: SHEFFER2:th 115
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 | ((b4 | b4) | b3)) | (b2 | ((b4 | b4) | b3))) | ((b4 | b2) | ((b3 | b3) | b2)) = (((b3 | b3) | (b3 | b3)) | (b4 | b2)) | ((b2 | b2) | (b4 | b2));

:: SHEFFER2:th 116
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 | ((b4 | b4) | b3)) | (b2 | ((b4 | b4) | b3))) | ((b4 | b2) | ((b3 | b3) | b2)) = (b3 | (b4 | b2)) | ((b2 | b2) | (b4 | b2));

:: SHEFFER2:th 117
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 | b2) | ((b4 | b4) | b3)) | ((b3 | ((b3 | b3) | b4)) | (b3 | ((b3 | b3) | b4))) = (((b4 | b4) | b3) | (b2 | b3)) | (((b4 | b4) | b3) | (b2 | b3));

:: SHEFFER2:th 118
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b3 | (b4 | b3)) | (b3 | (b4 | b3))) | (b2 | (b2 | b2)) = (b4 | b4) | b3;

:: SHEFFER2:th 119
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 | (b3 | b2) = (b3 | b3) | b2;

:: SHEFFER2:th 120
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b2) | ((b3 | b3) | b2) = (b2 | b2) | (b3 | b2);

:: SHEFFER2:th 121
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 = (b2 | b2) | (b3 | b2);

:: SHEFFER2:th 122
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b2) | b2 = ((b2 | (b3 | b3)) | (b2 | (b3 | b3))) | (b3 | b2);

:: SHEFFER2:th 123
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 | ((b4 | b4) | b3)) | (b2 | ((b4 | b4) | b3))) | ((b4 | b2) | ((b3 | b3) | b2)) = (b3 | (b4 | b2)) | b2;

:: SHEFFER2:th 124
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | (((b4 | (b4 | b4)) | (b4 | (b4 | b4))) | b3)) | (b2 | (((b4 | (b4 | b4)) | (b4 | (b4 | b4))) | b3)) = b2;

:: SHEFFER2:th 125
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | ((b4 | b4) | b3)) | b3 = b3 | (b4 | b2);

:: SHEFFER2:th 126
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
b2 | ((b3 | b2) | b2) = b3 | b2;

:: SHEFFER2:th 127
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b3 = (((b4 | b4) | b4) | b3) | ((b2 | b2) | b3);

:: SHEFFER2:th 128
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | ((b3 | b3) | b2)) | (b3 | ((b3 | b3) | b2)) = b3;

:: SHEFFER2:th 129
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(((b4 | b4) | b3) | (b2 | b3)) | (((b4 | b4) | b3) | (b2 | b3)) = ((b2 | b2) | ((b4 | b4) | b3)) | b3;

:: SHEFFER2:th 130
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(((b4 | b4) | b3) | (b2 | b3)) | (((b4 | b4) | b3) | (b2 | b3)) = b3 | (b4 | (b2 | b2));

:: SHEFFER2:th 131
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
((b3 | b2) | (b3 | b2)) | b3 = b3 | b2;

:: SHEFFER2:th 132
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b3) | (b3 | b2) = b3;

:: SHEFFER2:th 133
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b2) | (b2 | b2) = b2;

:: SHEFFER2:th 134
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b4 | (b3 | (b3 | b3))) | (b4 | b2) = b4;

:: SHEFFER2:th 135
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3 being Element of the carrier of b1 holds
(b3 | b2) | (b3 | b3) = b3;

:: SHEFFER2:th 136
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b4 | b3) | (b4 | (b2 | (b2 | b2))) = b4;

:: SHEFFER2:th 137
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1 holds
(((b6 | (b5 | (b5 | b5))) | b4) | ((b3 | b3) | b4)) | ((b4 | (b6 | b3)) | (b4 | (b6 | b3))) = ((b4 | (b2 | (b2 | b2))) | (b4 | (b6 | b3))) | (((b6 | b3) | (b6 | b3)) | (b4 | (b6 | b3)));

:: SHEFFER2:th 138
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(((b5 | (b4 | (b4 | b4))) | b3) | ((b2 | b2) | b3)) | ((b3 | (b5 | b2)) | (b3 | (b5 | b2))) = b3 | (((b5 | b2) | (b5 | b2)) | (b3 | (b5 | b2)));

:: SHEFFER2:th 139
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(((b5 | (b4 | (b4 | b4))) | b3) | ((b2 | b2) | b3)) | ((b3 | (b5 | b2)) | (b3 | (b5 | b2))) = b3 | (b5 | b2);

:: SHEFFER2:th 140
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1 holds
(((b6 | (b5 | (b5 | b5))) | b4) | ((b2 | b2) | b4)) | ((b4 | (b6 | b2)) | (b4 | (b6 | b2))) = (((b2 | b2) | (b3 | (b3 | b3))) | ((b6 | (b5 | (b5 | b5))) | b4)) | ((b4 | b4) | ((b6 | (b5 | (b5 | b5))) | b4));

:: SHEFFER2:th 141
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1 holds
b4 | (b6 | b2) = (((b2 | b2) | (b3 | (b3 | b3))) | ((b6 | (b5 | (b5 | b5))) | b4)) | ((b4 | b4) | ((b6 | (b5 | (b5 | b5))) | b4));

:: SHEFFER2:th 142
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
b3 | (b5 | b2) = (b2 | ((b5 | (b4 | (b4 | b4))) | b3)) | ((b3 | b3) | ((b5 | (b4 | (b4 | b4))) | b3));

:: SHEFFER2:th 143
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
b3 | (b5 | b2) = (b2 | ((b5 | (b4 | (b4 | b4))) | b3)) | b3;

:: SHEFFER2:th 144
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1 holds
(b3 | (b4 | (b6 | b5))) | (((b6 | (b2 | (b2 | b2))) | b4) | ((b5 | b5) | b4)) = b4 | (b6 | b5);

:: SHEFFER2:th 145
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 | ((b4 | b4) | b3)) | (b2 | ((b4 | b4) | b3))) | ((b4 | b2) | ((b3 | b3) | b2)) = b2 | ((b4 | b4) | b3);

:: SHEFFER2:th 146
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | (b4 | b3)) | b3 = b3 | ((b4 | b4) | b2);

:: SHEFFER2:th 147
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(((b2 | b2) | b3) | ((b5 | (b4 | (b4 | b4))) | b3)) | b3 = b3 | (b2 | b5);

:: SHEFFER2:th 148
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b3 | (b2 | ((b4 | b4) | b3)) = b3 | (b4 | b2);

:: SHEFFER2:th 149
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
((b3 | (b5 | b2)) | (b3 | (b5 | b2))) | (((b5 | (b4 | (b4 | b4))) | b3) | ((b2 | b2) | b3)) = (((b2 | b2) | b3) | ((b5 | (b4 | (b4 | b4))) | b3)) | (((b2 | b2) | b3) | ((b5 | (b4 | (b4 | b4))) | b3));

:: SHEFFER2:th 150
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
b3 | (b5 | b2) = (((b2 | b2) | b3) | ((b5 | (b4 | (b4 | b4))) | b3)) | (((b2 | b2) | b3) | ((b5 | (b4 | (b4 | b4))) | b3));

:: SHEFFER2:th 151
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
b2 | (b5 | b3) = b2 | (b3 | ((b5 | (b4 | (b4 | b4))) | (b5 | (b4 | (b4 | b4)))));

:: SHEFFER2:th 152
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | (b4 | b3) = b2 | (b3 | b4);

:: SHEFFER2:th 153
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b4 | b3) | b2 = b2 | (b3 | b4);

:: SHEFFER2:th 154
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b4 | b3) | b2) | b4 = b4 | ((b3 | b3) | b2);

:: SHEFFER2:th 155
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
b3 | b5 = b3 | ((b5 | b2) | (((b5 | (b4 | (b4 | b4))) | (b5 | (b4 | (b4 | b4)))) | b3));

:: SHEFFER2:th 156
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 | b4 = b2 | ((b4 | b3) | (b4 | b2));

:: SHEFFER2:th 157
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(b3 | b5) | (((b3 | (b5 | (b4 | (b4 | b4)))) | b2) | b3) = (b3 | b5) | (b3 | (b5 | (b4 | (b4 | b4))));

:: SHEFFER2:th 158
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(b2 | b5) | (b2 | (((b5 | (b4 | (b4 | b4))) | (b5 | (b4 | (b4 | b4)))) | b3)) = (b2 | b5) | (b2 | (b5 | (b4 | (b4 | b4))));

:: SHEFFER2:th 159
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(b3 | b5) | (b3 | (b5 | b4)) = (b3 | b5) | (b3 | (b5 | (b2 | (b2 | b2))));

:: SHEFFER2:th 160
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 | b4) | (b2 | (b4 | b3)) = b2;

:: SHEFFER2:th 161
theorem
for b1 being non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 ShefferStr holds
   b1 is satisfying_Sh_1;

:: SHEFFER2:condreg 4
registration
  cluster non empty satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 -> satisfying_Sh_1 (ShefferStr);
end;

:: SHEFFER2:condreg 5
registration
  cluster non empty satisfying_Sh_1 -> satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3 (ShefferStr);
end;

:: SHEFFER2:condreg 6
registration
  cluster non empty properly_defined satisfying_Sh_1 -> Lattice-like Boolean (ShefferOrthoLattStr);
end;

:: SHEFFER2:condreg 7
registration
  cluster non empty Lattice-like Boolean well-complemented properly_defined -> satisfying_Sh_1 (ShefferOrthoLattStr);
end;