LATQUASI

:: Formalization of Quasilattices
::  by Dominik Kulesza and Adam Grabowski
:: 
:: Received May 31, 2020
:: Copyright (c) 2020-2026 Association of Mizar Users
::           (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.

environ

 vocabularies LATQUASI, EQREL_1, LATTICES, STRUCT_0, PBOOLE, CARD_1, FUNCOP_1,
      FUNCT_1, XXREAL_0, MEMBERED, XREAL_0, ZFMISC_1, SHEFFER1, XBOOLE_0,
      SUBSET_1, ROBBINS1, BINOP_1, FUNCT_7;
 notations TARSKI, ENUMSET1, XXREAL_0, SUBSET_1, ORDINAL1, CARD_1, XBOOLE_0,
      STRUCT_0, MEMBERED, LATTICES, BINOP_1, FUNCOP_1, FUNCT_2, XREAL_0,
      ROBBINS1, SHEFFER1;
 constructors BINOP_1, ENUMSET1, LATTICES, STRUCT_0, ROBBINS1, CARD_1,
      FUNCOP_1, ORDINAL1, SUBSET_1, XXREAL_0, XBOOLE_0, XREAL_0, MEMBERED,
      FUNCT_1, ZFMISC_1, FUNCT_2, RELSET_1, SHEFFER1;
 registrations RELSET_1, STRUCT_0, LATTICES, XXREAL_0, XREAL_0, ORDINAL1,
      MEMBERED, SUBSET_1, LATTAD_1;
 requirements SUBSET, NUMERALS, BOOLE, REAL;
 definitions BINOP_1, LATTICES;
 equalities LATTICES;
 theorems LATTICES, XXREAL_0, ENUMSET1, FUNCOP_1, SUBSET_1, BINOP_1, STRUCT_0,
      SHEFFER1, ROBBINS1;
 schemes BINOP_1;

begin :: Preliminaries

reserve L for non empty LattStr;
reserve v3,v101,v100,v102,v103,v2,v1,v0 for Element of L;

definition let L be non empty LattStr;
  attr L is satisfying_QLT1 means
    for v0,v2,v1 being Element of L holds
      (v0"/\"(v1"\/"v2)) "\/" (v0"/\"v1) = v0 "/\" (v1"\/"v2);
  attr L is satisfying_QLT2 means
    for v0,v2,v1 being Element of L holds
      (v0"\/"(v1"/\"v2)) "/\" (v0"\/"v1) = v0 "\/" (v1"/\"v2);
  attr L is QLT-distributive means
    for v1,v2,v0 being Element of L holds
      v0 "/\" (v1"\/"(v0"/\"v2)) = v0 "/\" (v1"\/"v2);
end;

registration
  cluster trivial -> satisfying_QLT1 satisfying_QLT2
     QLT-distributive for non empty LattStr;
  coherence by STRUCT_0:def 10;
end;

registration
  cluster trivial -> join-idempotent meet-idempotent
        for non empty LattStr;
  coherence
  proof
    let L be non empty LattStr;
    assume
A1: L is trivial; then
    for x being Element of L holds x "\/" x = x by STRUCT_0:def 10;
    hence L is join-idempotent by ROBBINS1:def 7;
    for x being Element of L holds x "/\" x = x by A1,STRUCT_0:def 10;
    hence thesis by SHEFFER1:def 9;
  end;
end;

registration
  cluster join-commutative join-associative join-idempotent
          meet-commutative meet-associative meet-idempotent
          satisfying_QLT1 satisfying_QLT2 for non empty LattStr;
  existence
  proof
    take the trivial Lattice;
    thus thesis;
  end;
end;

definition let L be join-commutative non empty LattStr;
  redefine attr L is satisfying_QLT1 means  :: redefines satisfying_QLT1 (LATQUASI, definition)
    for v0,v1,v2 being Element of L holds
      v0 "/\" v1 [= v0 "/\" (v1 "\/" v2);
  compatibility
  proof
    thus L is satisfying_QLT1 implies
    for v0,v1,v2 being Element of L holds
      v0 "/\" v1 [= v0 "/\" (v1 "\/" v2)
    proof
      assume
A1:   L is satisfying_QLT1;
      let v0,v1,v2 be Element of L;
      (v0"/\"(v1"\/"v2)) "\/" (v0"/\"v1) = v0 "/\" (v1"\/"v2)
        by A1;
      hence thesis by LATTICES:def 3;
    end;
    assume
B1: for v0,v1,v2 being Element of L holds
      v0 "/\" v1 [= v0 "/\" (v1 "\/" v2);
    let v0,v2,v1 be Element of L;
    (v0"/\"(v1"\/"v2)) "\/" (v0"/\"v1) = v0 "/\" (v1"\/"v2)
      by B1,LATTICES:def 3;
    hence thesis;
  end;
end;

registration
  cluster {0,1,2} -> real-membered;
  coherence;
end;

registration
  cluster -> real for Element of {0,1,2};
  coherence;
end;

T2: for x,y being Element of {0,1,2} holds min (x,y) in {0,1,2}
  proof
    let x,y be Element of {0,1,2};
    min (x,y) = x or min (x,y) = y by XXREAL_0:15;
    hence thesis;
  end;

definition let x,y be Element of {0,1,2};
  func OpEx2 (x,y) -> Element of {0,1,2} equals :OpEx2Def:
  1 if x = 1 or y = 1,
  min (x,y) if x <> 1 & y <> 1;
  coherence by T2;
  consistency;
end;

definition
  func QLT_Ex1 -> BinOp of {0,1,2} means :QLTEx1Def:
    for x,y being Element of {0,1,2} holds
      (x = y implies it.(x,y) = x) &
      (x <> y implies it.(x,y) = 0);
  existence
  proof
    deffunc X() = {0,1,2};
    deffunc F(object,object) = In (IFEQ ($1,$2,$1,0),X());
    ex f being BinOp of X() st
    for x being Element of X()
    for y being Element of X() holds f.(x,y) = F(x,y) from BINOP_1:sch 4; then
    consider f being BinOp of X() such that
A1: for x being Element of X()
    for y being Element of X() holds f.(x,y) = F(x,y);
    take f;
    let x,y be Element of {0,1,2};
A4: 0 in X() by ENUMSET1:def 1;
    hereby assume
A2:   x = y;
      thus f.(x,y) = F(x,y) by A1
          .= In (x,X()) by A2,FUNCOP_1:def 8
          .= x;
    end;
    assume
A3: x <> y;
    thus f.(x,y) = F(x,y) by A1
          .= In (0,X()) by A3,FUNCOP_1:def 8
          .= 0 by SUBSET_1:def 8,A4;
  end;
  uniqueness
  proof
    let f1,f2 be BinOp of {0,1,2} such that
A1: for x,y being Element of {0,1,2} holds
      (x = y implies f1.(x,y) = x) &
      (x <> y implies f1.(x,y) = 0) and
A2: for x,y being Element of {0,1,2} holds
      (x = y implies f2.(x,y) = x) &
      (x <> y implies f2.(x,y) = 0);
    for x,y being Element of {0,1,2} holds
      f1.(x,y) = f2.(x,y)
    proof
      let x,y be Element of {0,1,2};
      per cases;
      suppose
A3:     x = y; then
        f1.(x,y) = x by A1 .= f2.(x,y) by A2,A3;
        hence thesis;
      end;
      suppose
A3:     x <> y; then
        f1.(x,y) = 0 by A1 .= f2.(x,y) by A2,A3;
        hence thesis;
      end;
    end;
    hence thesis by BINOP_1:2;
  end;
  func QLT_Ex2 -> BinOp of {0,1,2} means :QLTEx2Def:
    for x,y being Element of {0,1,2} holds
      (x = 1 or y = 1 implies it.(x,y) = 1) &
      (x <> 1 & y <> 1 implies it.(x,y) = min(x,y));
  existence
  proof
    deffunc X() = {0,1,2};
    deffunc F(Element of X(), Element of X()) = OpEx2 ($1,$2);
    ex f being BinOp of X() st
    for x,y being Element of X() holds f.(x,y) = F(x,y)
      from BINOP_1:sch 4; then
    consider f being BinOp of X() such that
A1: for x,y being Element of X() holds f.(x,y) = F(x,y);
    take f;
    let x,y be Element of {0,1,2};
    hereby assume
A2:   x = 1 or y = 1;
      f.(x,y) = OpEx2 (x,y) by A1;
      hence f.(x,y) = 1 by A2,OpEx2Def;
    end;
    assume
A2: x <> 1 & y <> 1;
    f.(x,y) = OpEx2 (x,y) by A1;
    hence thesis by OpEx2Def,A2;
  end;
  uniqueness
  proof
    let f1,f2 be BinOp of {0,1,2} such that
A1: for x,y being Element of {0,1,2} holds
      (x = 1 or y = 1 implies f1.(x,y) = 1) &
      (x <> 1 & y <> 1 implies f1.(x,y) = min(x,y)) and
A2: for x,y being Element of {0,1,2} holds
      (x = 1 or y = 1 implies f2.(x,y) = 1) &
      (x <> 1 & y <> 1 implies f2.(x,y) = min(x,y));
    for x,y being Element of {0,1,2} holds
      f1.(x,y) = f2.(x,y)
    proof
      let x,y be Element of {0,1,2};
      per cases;
      suppose
A3:     x = 1 or y = 1; then
        f1.(x,y) = 1 by A1 .= f2.(x,y) by A2,A3;
        hence thesis;
      end;
      suppose
A3:     x <> 1 & y <> 1; then
        f1.(x,y) = min(x,y) by A1 .= f2.(x,y) by A2,A3;
        hence thesis;
      end;
    end;
    hence thesis by BINOP_1:2;
  end;
end;

theorem WazneQLT7:
  QLT_Ex1 <> QLT_Ex2
  proof
A0: 0 in {0,1,2} & 1 in {0,1,2} by ENUMSET1:def 1; then
    QLT_Ex1.(0,1) = 0 by QLTEx1Def;
    hence thesis by A0,QLTEx2Def;
  end;

::: Construction of quasi-lattices needed to prove QLT-7

::  x <= y iff x "\/" y = y
::: if absorption is true, then x <= y iff x "/\" y = x

definition
  func QLTLattice1 -> strict non empty LattStr equals
    LattStr (# {0,1,2}, QLT_Ex1, QLT_Ex1 #);
  coherence;
  func QLTLattice2 -> strict non empty LattStr equals
    LattStr (# {0,1,2}, QLT_Ex1, QLT_Ex2 #);
  coherence;
end;

registration
  cluster QLT_Ex1 -> commutative associative idempotent;
  coherence
  proof
    set X = {0,1,2};
XX: 0 in X by ENUMSET1:def 1;
    set f = QLT_Ex1;
    thus f is commutative
    proof
      let a,b be Element of X;
      per cases;
      suppose
        a = b;
        hence thesis;
      end;
      suppose
A2:     a <> b; then
        f.(a,b) = 0 by QLTEx1Def
               .= f.(b,a) by A2,QLTEx1Def;
        hence thesis;
      end;
    end;
    thus f is associative
    proof
      let a,b,c be Element of X;
      per cases;
      suppose
A1:     a = b & b = c; then
        f.(a,b) = a by QLTEx1Def;
        hence thesis by A1;
      end;
      suppose
A1:     a <> b & b = c; then
A2:     f.(a,b) = 0 by QLTEx1Def;
D2:     0 in X by ENUMSET1:def 1;
        f.(f.(a,b),c) = f.(0,c) by A1,QLTEx1Def
                     .= 0 by QLTEx1Def,D2;
        hence thesis by A2,QLTEx1Def,A1;
      end;
      suppose
A1:     a = b & b <> c; then
A2:     f.(a,b) = a by QLTEx1Def;
A3:     f.(b,c) = 0 by QLTEx1Def,A1;
        per cases;
        suppose
          a = 0;
          hence thesis by A3,A1,A2;
        end;
        suppose
          a <> 0;
          hence thesis by QLTEx1Def,A3,A1,A2;
        end;
      end;
      suppose
A1:     a <> b & b <> c; then
A3:     f.(b,c) = 0 by QLTEx1Def;
        per cases;
        suppose
B1:       a = 0;
          f.(f.(a,b),c) = f.(0,c) by A1,QLTEx1Def
                       .= 0 by B1,QLTEx1Def;
          hence thesis by B1,QLTEx1Def,A3;
        end;
        suppose
b1:       a <> 0;
          f.(f.(a,b),c) = f.(0,c) by A1,QLTEx1Def
                       .= 0 by QLTEx1Def,XX;
          hence thesis by b1,QLTEx1Def,A3;
        end;
      end;
    end;
    let a be Element of X;
    thus thesis by QLTEx1Def;
  end;
end;

registration
  cluster QLT_Ex2 -> commutative associative idempotent;
  coherence
  proof
    set X = {0,1,2};
    set f = QLT_Ex2;
    thus f is commutative
    proof
      let a,b be Element of X;
      per cases;
      suppose
A1:     a = 1 or b = 1; then
        f.(a,b) = 1 by QLTEx2Def
               .= f.(b,a) by A1,QLTEx2Def;
        hence thesis;
      end;
      suppose
A2:     a <> 1 & b <> 1; then
        f.(a,b) = min(a,b) by QLTEx2Def
               .= f.(b,a) by A2,QLTEx2Def;
        hence thesis;
      end;
    end;
    thus f is associative
    proof
      let a,b,c be Element of X;
      per cases;
      suppose
A1:     a = 1 & (b = 1 or c = 1); then
A2:     f.(a,b) = 1 by QLTEx2Def;
A3:     f.(b,c) = 1 by QLTEx2Def,A1;
        f.(f.(a,b),c) = 1 by QLTEx2Def,A2;
        hence thesis by QLTEx2Def,A3;
      end;
      suppose
A1:     (a = 1 or b = 1) & c = 1; then
A5:     f.(f.(a,b),c) = 1 by QLTEx2Def;
        f.(b,c) = 1 by QLTEx2Def,A1;
        hence thesis by A5,QLTEx2Def;
      end;
      suppose
A1:     a <> 1 & b = 1 & c <> 1; then
        f.(a,b) = 1 by QLTEx2Def; then
A5:     f.(f.(a,b),c) = 1 by QLTEx2Def;
        f.(b,c) = 1 by QLTEx2Def,A1;
        hence thesis by A5,QLTEx2Def;
      end;
      suppose
A1:     a = 1 & b <> 1 & c = 1; then
        f.(b,c) = 1 by QLTEx2Def;
        hence thesis by QLTEx2Def,A1;
      end;
      suppose
A1:     a = 1 & b <> 1 & c <> 1; then
A2:     f.(a,b) = 1 by QLTEx2Def;
A3:     f.(b,c) = min(b,c) by QLTEx2Def,A1;
          per cases by XXREAL_0:15;
          suppose
            min(b,c) = b;
            hence f.(f.(a,b),c) = f.(a,f.(b,c)) by A2,A3,QLTEx2Def;
          end;
          suppose
            min(b,c) = c;
            hence f.(f.(a,b),c) = f.(a,f.(b,c)) by A1,QLTEx2Def,A2;
          end;
      end;
      suppose
A1:     a <> 1 & b = 1 & c <> 1; then
A2:     f.(a,b) = 1 by QLTEx2Def;
A3:     f.(b,c) = 1 by QLTEx2Def,A1;
        f.(f.(a,b),c) = 1 by QLTEx2Def,A2;
        hence thesis by QLTEx2Def,A3;
      end;
      suppose
A1:     a <> 1 & b <> 1 & c = 1; then
A2:     f.(a,b) = min(a,b) by QLTEx2Def;
A3:     f.(b,c) = 1 by QLTEx2Def,A1;
        per cases by XXREAL_0:15;
        suppose
          min(a,b) = a;
          f.(a,f.(b,c)) = f.(a,1) by QLTEx2Def,A1 .= 1 by QLTEx2Def,A1;
          hence f.(f.(a,b),c) = f.(a,f.(b,c)) by QLTEx2Def,A1;
        end;
        suppose
          min(a,b) = b;
          hence f.(f.(a,b),c) = f.(a,f.(b,c)) by QLTEx2Def,A3,A2;
        end;
      end;
      suppose
A1:     a <> 1 & b <> 1 & c <> 1; then
A2:     f.(a,b) = min(a,b) by QLTEx2Def;
A3:     f.(b,c) = min(b,c) by QLTEx2Def,A1;
          per cases by XXREAL_0:15;
          suppose
C1:       min(a,c) = a;
          per cases by XXREAL_0:15;
            suppose
              min(a,b) = a & min(b,c) = b;
              hence f.(f.(a,b),c) = f.(a,f.(b,c)) by A2,A3,C1,QLTEx2Def,A1;
            end;
            suppose
              min(a,b) = a & min(b,c) = c;
              hence f.(f.(a,b),c) = f.(a,f.(b,c)) by QLTEx2Def,A1,A2;
            end;
            suppose
              min(a,b) = b & min(b,c) = b;
              hence f.(f.(a,b),c) = f.(a,f.(b,c)) by A3,A2;
            end;
            suppose
B1:           min(a,b) = b & min(b,c) = c; then
              a <= c & b <= a & c <= b by C1,XXREAL_0:def 9; then
              a <= c & c <= a by XXREAL_0:2; then
              a = c by XXREAL_0:1;
              hence f.(f.(a,b),c) = f.(a,f.(b,c)) by B1,A3;
            end;
          end;
          suppose
C1:       min(a,c) = c;
          per cases by XXREAL_0:15;
            suppose
B1:           min(a,b) = a & min(b,c) = b; then
              c <= a & a <= b & b <= c by C1,XXREAL_0:def 9; then
              a <= c & c <= a by XXREAL_0:2; then
              a = c by XXREAL_0:1;
              hence f.(f.(a,b),c) = f.(a,f.(b,c)) by A2,B1;
            end;
            suppose
              min(a,b) = a & min(b,c) = c;
              hence f.(f.(a,b),c) = f.(a,f.(b,c)) by QLTEx2Def,A1,A2;
            end;
            suppose
              min(a,b) = b & min(b,c) = b;
              hence f.(f.(a,b),c) = f.(a,f.(b,c)) by A2,A3;
            end;
            suppose
              min(a,b) = b & min(b,c) = c;
              hence f.(f.(a,b),c) = f.(a,f.(b,c)) by C1,QLTEx2Def,A1,A3,A2;
            end;
          end;
      end;
     end;
     let a be Element of X;
     per cases;
     suppose
       a = 1;
       hence thesis by QLTEx2Def;
     end;
     suppose
       a <> 1; then
       f.(a,a) = min(a,a) by QLTEx2Def
              .= a;
       hence thesis;
     end;
  end;
end;

registration
  cluster QLTLattice1 -> join-commutative join-associative join-idempotent;
  coherence
  proof
    set L = QLTLattice1;
    set f = the L_join of L;
Y1: for x,y being Element of L holds x "\/" y = y "\/" x by BINOP_1:def 2;
S1: for x,y,z being Element of L holds
      x "\/" (y "\/" z) = (x "\/" y) "\/" z by BINOP_1:def 3;
    for x being Element of L holds x "\/" x = x by BINOP_1:def 4;
    hence thesis by Y1,LATTICES:def 4,def 5,S1,ROBBINS1:def 7;
  end;
end;

registration
  cluster QLTLattice1 -> meet-commutative meet-associative meet-idempotent;
  coherence
  proof
    set L = QLTLattice1;
    set f = the L_meet of L;
Y1: for x,y being Element of L holds x "/\" y = y "/\" x by BINOP_1:def 2;
S1: for x,y,z being Element of L holds
      x "/\" (y "/\" z) = (x "/\" y) "/\" z by BINOP_1:def 3;
    for x being Element of L holds x "/\" x = x by BINOP_1:def 4;
    hence thesis by Y1,LATTICES:def 6,def 7,S1,SHEFFER1:def 9;
  end;
end;

theorem Lemacik00:
  for v0,v1 being Element of QLTLattice1 st v1 = 0 holds
    v0 "/\" v1 = v1
  proof
    let v0,v1 be Element of QLTLattice1;
    assume
A1: v1 = 0;
    per cases;
    suppose v0 = v1;
      hence thesis by QLTEx1Def;
    end;
    suppose v0 <> v1;
      hence thesis by A1,QLTEx1Def;
    end;
  end;

theorem Lemacik001:
  for v0,v1 being Element of QLTLattice1 st v1 = 0 holds
    v0 "\/" v1 = v1
  proof
    let v0,v1 be Element of QLTLattice1;
    assume
A1: v1 = 0;
    per cases;
    suppose v0 = v1;
      hence thesis by QLTEx1Def;
    end;
    suppose v0 <> v1;
      hence thesis by A1,QLTEx1Def;
    end;
  end;

registration
  cluster QLTLattice1 -> satisfying_QLT1;
  coherence
  proof
    set L = QLTLattice1;
    for v0,v2,v1 being Element of L holds
      (v0"/\"(v1"\/"v2)) "\/" (v0"/\"v1) = v0 "/\" (v1"\/"v2)
    proof
      let v0,v2,v1 be Element of L;
      reconsider z = 0 as Element of L by ENUMSET1:def 1;
      per cases;
      suppose v0 = v1 & v1 = v2; then
        v1 "\/" v2 = v1 & v0 "/\" v1 = v0 by QLTEx1Def;
        hence thesis by ROBBINS1:def 7;
      end;
      suppose v0 = v1 & v1 <> v2; then
A1:    v1 "\/" v2 = 0 & v0 "/\" v1 = v0 by QLTEx1Def; then
       (v0"/\"(v1"\/"v2)) "\/" (v0"/\"v1) = z "\/" v0 by Lemacik00
           .= z by Lemacik001
           .= v0 "/\" (v1 "\/" v2) by A1,Lemacik00;
        hence thesis;
      end;
      suppose v0 <> v1 & v1 <> v2; then
A1:     v1 "\/" v2 = 0 & v0 "/\" v1 = 0 by QLTEx1Def; then
        (v0"/\"(v1"\/"v2)) "\/" (v0"/\"v1) = z by Lemacik001
           .= v0 "/\" (v1 "\/" v2) by A1,Lemacik00;
        hence thesis;
      end;
      suppose v0 <> v1 & v1 = v2; then
        v1 "\/" v2 = v1 & v0 "/\" v1 = 0 by QLTEx1Def;
        hence thesis by Lemacik001;
      end;
    end;
    hence thesis;
  end;
end;

registration
  cluster QLTLattice1 -> satisfying_QLT2;
  coherence
  proof
    set L = QLTLattice1;
    let v0,v2,v1 be Element of L;
    reconsider z = 0 as Element of L by ENUMSET1:def 1;
    per cases;
    suppose v0 = v1 & v1 = v2; then
      v1 "/\" v2 = v1 & v0 "\/" v1 = v0 by QLTEx1Def;
      hence thesis by SHEFFER1:def 9;
    end;
    suppose v0 = v1 & v1 <> v2; then
A1:     v1 "/\" v2 = 0 & v0 "\/" v1 = v0 by QLTEx1Def;
      z "\/" v0 = v0 "\/" z; then
      (v0"\/"(v1"/\"v2)) "/\" (v0"\/"v1) = z "/\" v0 by Lemacik00,A1
         .= z by Lemacik00
         .= v0 "\/" (v1 "/\" v2) by A1,Lemacik001;
      hence thesis;
    end;
    suppose v0 <> v1 & v1 <> v2; then
A1:    v1 "/\" v2 = 0 & v0 "\/" v1 = 0 by QLTEx1Def; then
      (v0"\/"(v1"/\"v2)) "/\" (v0"\/"v1) = z by Lemacik00
         .= v0 "\/" (v1 "/\" v2) by A1,Lemacik001;
      hence thesis;
    end;
    suppose v0 <> v1 & v1 = v2; then
      v1 "/\" v2 = v1 & v0 "\/" v1 = 0 by QLTEx1Def;
      hence thesis by Lemacik00;
    end;
  end;
end;

:: Every QLTLattice is QuasiLattice

registration
  cluster -> real for Element of QLTLattice2;
  coherence;
end;

registration
  cluster QLTLattice2 -> join-commutative join-associative join-idempotent;
  coherence
  proof
    set L = QLTLattice2;
Y1: for x,y being Element of L holds x "\/" y = y "\/" x by BINOP_1:def 2;
S1: for x,y,z being Element of L holds
      x "\/" (y "\/" z) = (x "\/" y) "\/" z by BINOP_1:def 3;
    for x being Element of L holds x "\/" x = x by BINOP_1:def 4;
    hence thesis by Y1,LATTICES:def 4,def 5,S1,ROBBINS1:def 7;
  end;
end;

registration
  cluster QLTLattice2 -> meet-commutative meet-associative meet-idempotent;
  coherence
  proof
    set L = QLTLattice2;
Y1: for x,y being Element of L holds x "/\" y = y "/\" x by BINOP_1:def 2;
S1: for x,y,z being Element of L holds
      x "/\" (y "/\" z) = (x "/\" y) "/\" z by BINOP_1:def 3;
    for x being Element of L holds x "/\" x = x by BINOP_1:def 4;
    hence thesis by Y1,LATTICES:def 6,def 7,S1,SHEFFER1:def 9;
  end;
end;

registration
  cluster QLTLattice2 -> satisfying_QLT1;
  coherence
  proof
    set L = QLTLattice2;
    for v0,v2,v1 being Element of L holds
      (v0"/\"(v1"\/"v2)) "\/" (v0"/\"v1) = v0 "/\" (v1"\/"v2)
    proof
      let v0,v2,v1 be Element of L;
      reconsider z = 0 as Element of L by ENUMSET1:def 1;
p2:   z <= v0 by ENUMSET1:def 1;
      reconsider o = 1 as Element of L by ENUMSET1:def 1;
      reconsider dwa = 2 as Element of L by ENUMSET1:def 1;
      per cases by ENUMSET1:def 1;
      suppose v0 = v1 & v1 = v2; then
        v1 "\/" v2 = v1 & v0 "/\" v1 = v1 by BINOP_1:def 4;
        hence thesis by ROBBINS1:def 7;
      end;
      suppose
Z1:     v0 = v1 & v1 <> v2 & v0 = 1; then
        v1 "\/" v2 = 0 & v0 "/\" v1 = 1 by QLTEx1Def,BINOP_1:def 4; then
        (v0"/\"(v1"\/"v2)) "\/" (v0"/\"v1) = o "\/" v0 by QLTEx2Def,Z1
           .= o by QLTEx1Def,Z1
           .= v0 "/\" (v1 "\/" v2) by Z1,QLTEx2Def;
        hence thesis;
      end;
      suppose
A0:     v0 = v1 & v1 <> v2 & v0 = 0; then
        v1 "\/" v2 = 0 & v0 "/\" v1 = min(v0,v1) by QLTEx1Def,QLTEx2Def;
        hence thesis by QLTEx1Def,A0;
      end;
      suppose
A0:     v0 = v1 & v1 <> v2 & v0 = 2; then
A1:     v1 "\/" v2 = 0 & v0 "/\" v1 = min(v0,v1) & v0 <> 1
          by QLTEx1Def,QLTEx2Def;
        v0 "/\" z = min (v0,z) by A0,QLTEx2Def
          .= z by XXREAL_0:def 9,p2;
        hence thesis by A1,QLTEx1Def,A0;
      end;
      suppose
A0:     v0 <> v1 & v1 <> v2 & v0 = 1; then
A1:     v1 "\/" v2 = 0 & v0 "/\" v1 = 1 by QLTEx1Def,QLTEx2Def;
        v0 "/\" z = o by A0,QLTEx2Def;
        hence thesis by QLTEx1Def,A1;
      end;
      suppose
A0:     v0 <> v1 & v1 <> v2 & v0 = 2;
        per cases;
        suppose
          v1 = 1; then
A1:       v1 "\/" v2 = 0 & v0 "/\" v1 = 1 by QLTEx1Def,QLTEx2Def,A0;
          v0 "/\" z = min (v0,z) by A0,QLTEx2Def
                   .= z by XXREAL_0:def 9,p2;
          hence thesis by QLTEx1Def,A1;
        end;
        suppose
B:        v1 <> 1; then
A1:       v1 "\/" v2 = 0 & v0 "/\" v1 = min(v0,v1)
            by QLTEx1Def,QLTEx2Def,A0;
          v1 = 0 or v1 = 2 by ENUMSET1:def 1,B;
          hence thesis by QLTEx1Def,A0,A1;
        end;
      end;
      suppose
A0:     v0 <> v1 & v1 <> v2 & v0 = 0;
        per cases;
        suppose
          v1 = 1; then
A1:       v1 "\/" v2 = 0 & v0 "/\" v1 = 1 by QLTEx1Def,QLTEx2Def,A0;
          v0 "/\" z = min (v0,z) by A0,QLTEx2Def
                   .= z by XXREAL_0:def 9,p2;
          hence thesis by A1,QLTEx1Def;
        end;
        suppose
          v1 <> 1; then
A1:       v1 "\/" v2 = 0 & v0 "/\" v1 = min(v0,v1)
            by QLTEx1Def,QLTEx2Def,A0;
p2:       v0 <= v1 by A0,ENUMSET1:def 1;
P4:       v0 "/\" z = min (v0,z) by A0,QLTEx2Def
            .= z by A0;
          (v0"/\"(v1"\/"v2)) "\/" (v0"/\"v1) = (v0"/\"z) "\/" v0
            by p2,A1,XXREAL_0:def 9
           .= v0 "/\" (v1 "\/" v2) by A1,P4,QLTEx1Def,A0;
          hence thesis;
        end;
      end;
      suppose
        v0 <> v1 & v1 = v2 & v0 = 1; then
        v1 "\/" v2 = v1 & v0 "/\" v1 = 1 by QLTEx1Def,QLTEx2Def;
        hence thesis by QLTEx1Def;
      end;
      suppose
A0:     v0 <> v1 & v1 = v2 & v0 = 0;
        per cases;
        suppose
          v1 = 1; then
          v1 "\/" v2 = v1 & v0 "/\" v1 = 1 by QLTEx1Def,QLTEx2Def,A0;
          hence thesis by ROBBINS1:def 7;
        end;
        suppose
          v1 <> 1; then
          v1 "\/" v2 = v1 & v0 "/\" v1 = min(v0,v1)
            by QLTEx1Def,QLTEx2Def,A0;
          hence thesis by ROBBINS1:def 7;
        end;
      end;
      suppose
A0:     v0 <> v1 & v1 = v2 & v0 = 2;
        per cases;
        suppose
          v1 = 1; then
          v1 "\/" v2 = v1 & v0 "/\" v1 = 1 by QLTEx1Def,QLTEx2Def,A0;
          hence thesis by ROBBINS1:def 7;
        end;
        suppose
          v1 <> 1; then
          v1 "\/" v2 = v1 & v0 "/\" v1 = min(v0,v1)
            by QLTEx1Def,QLTEx2Def,A0;
          hence thesis by ROBBINS1:def 7;
        end;
      end;
    end;
    hence thesis;
  end;
end;

registration
  cluster QLTLattice2 -> satisfying_QLT2;
  coherence
  proof
    set L = QLTLattice2;
    for v0,v2,v1 being Element of L holds
      (v0"\/"(v1"/\"v2)) "/\" (v0"\/"v1) = v0 "\/" (v1"/\"v2)
    proof
      let v0,v2,v1 be Element of L;
      reconsider z = 0 as Element of L by ENUMSET1:def 1;
      reconsider o = 1 as Element of L by ENUMSET1:def 1;
      reconsider dwa = 2 as Element of L by ENUMSET1:def 1;
      per cases by ENUMSET1:def 1;
      suppose v0 = v1 & v1 = v2; then
        v1 "/\" v2 = v1 & v0 "\/" v1 = v1 by BINOP_1:def 4;
        hence thesis by SHEFFER1:def 9;
      end;
      suppose
Z1:     v0 = v1 & v1 <> v2 & v0 = 1; then
        v1 "/\" v2 = 1 & v0 "\/" v1 = v1 by QLTEx1Def,QLTEx2Def;
        hence thesis by SHEFFER1:def 9,Z1;
      end;
      suppose
A0:     v0 = v1 & v1 <> v2 & v0 = 0;
        per cases;
        suppose
          v2 = 1; then
A1:       v1 "/\" v2 = 1 & v0 "\/" v1 = v0 by QLTEx1Def,QLTEx2Def,A0;
P4:       v0 "/\" z = min(v0,z) by A0,QLTEx2Def
                   .= z by A0;
          (v0"\/"(v1"/\"v2)) "/\" (v0"\/"v1)
              = z by P4,QLTEx1Def,A0,A1;
          hence thesis by QLTEx1Def,A0,A1;
        end;
        suppose
          v2 <> 1; then
A1:       v1 "/\" v2 = min(v1,v2) & v0 "\/" v1 = v0 by QLTEx1Def,QLTEx2Def,A0;
P3:       v1 <= v2 by A0,ENUMSET1:def 1;
          v1 "/\" v2 = v1 by XXREAL_0:def 9,P3,A1;
          hence thesis by SHEFFER1:def 9;
        end;
      end;
      suppose
A0:     v0 = v1 & v1 <> v2 & v0 = 2;
P4:     v0 "/\" z = min(v0,z) by A0,QLTEx2Def
           .= z by XXREAL_0:def 9,A0;
        per cases;
        suppose
          v2 = 1; then
A1:       v1 "/\" v2 = 1 & v0 "\/" v1 = v0 by QLTEx1Def,QLTEx2Def,A0; then
          (v0"\/"(v1"/\"v2)) "/\" (v0"\/"v1) = z by P4,QLTEx1Def,A0;
          hence thesis by QLTEx1Def,A0,A1;
        end;
        suppose
B:        v2 <> 1; then
A1:       v1 "/\" v2 = min(v1,v2) & v0 "\/" v1 = v0
            by QLTEx1Def,QLTEx2Def,A0;
p4:       v2 = 0 or v2 = 2 by ENUMSET1:def 1,B; then
P4A:      v1 "/\" v2 = v2 by XXREAL_0:def 9,A1,A0;
          per cases by ENUMSET1:def 1,B;
          suppose
            v2 = 0;
            (v0"\/"(v1"/\"v2)) "/\" (v0"\/"v1) = z by QLTEx1Def,A0,A1,p4,P4;
            hence thesis by QLTEx1Def,A0,P4A;
          end;
          suppose
            v2 = 2;
            hence thesis by A0;
           end;
         end;
      end;
      suppose
A0:     v0 <> v1 & v1 <> v2 & v0 = 1;
        per cases;
        suppose
          v2 = 1; then
A1:       v1 "/\" v2 = 1 & v0 "\/" v1 = 0 by QLTEx1Def,QLTEx2Def,A0; then
          (v0"\/"(v1"/\"v2)) "/\" (v0"\/"v1) = v0 "/\" z by A0,QLTEx1Def
             .= 1 by QLTEx2Def,A0;
          hence thesis by A0,QLTEx1Def,A1;
        end;
        suppose
B:        v2 <> 1; then
A1:       v1 "/\" v2 = min(v1,v2) & v0 "\/" v1 = 0
            by QLTEx1Def,QLTEx2Def,A0;
          per cases by XXREAL_0:15;
          suppose
C1:         min(v1,v2) = v1;
            (v0"\/"(v1"/\"v2)) "/\" (v0"\/"v1) = (v0"\/"v1) "/\" (v0"\/"v1)
              by B,C1,QLTEx2Def,A0
               .= (v0"\/"v1) by SHEFFER1:def 9;
            hence thesis by B,C1,QLTEx2Def,A0;
          end;
          suppose
D1:         min(v1,v2) = v2; then
            (v0"\/"(v1"/\"v2)) "/\" (v0"\/"v1) = z "/\" z by A0,B,QLTEx1Def,A1
                .= z by BINOP_1:def 4;
            hence thesis by A0,B,QLTEx1Def,A1,D1;
          end;
        end;
      end;
      suppose
A0:     v0 <> v1 & v1 <> v2 & v0 = 2;
        per cases;
        suppose
          v2 = 1 or v1 = 1; then
A1:       v1 "/\" v2 = 1 & v0 "\/" v1 = 0 by QLTEx1Def,QLTEx2Def,A0; then
          (v0"\/"(v1"/\"v2)) "/\" (v0"\/"v1) = z "/\" z by A0,QLTEx1Def
             .= z by BINOP_1:def 4;
          hence thesis by A0,QLTEx1Def,A1;
        end;
        suppose
B:        v2 <> 1 & v1 <> 1; then
A1:       v1 "/\" v2 = min(v1,v2) & v0 "\/" v1 = 0
            by QLTEx1Def,QLTEx2Def,A0;
          per cases by XXREAL_0:15;
          suppose
            min(v1,v2) = v1;
            hence thesis by SHEFFER1:def 9,A1;
          end;
          suppose
D1:         min(v1,v2) = v2;
            (v1 = 1 & v2 <> 1) or (v1 = 0 & v2 <> 0) by A0,ENUMSET1:def 1; then
            (v1 = 1 & (v2 = 0 or v2 = 2)) or (v1 = 0 & (v2 = 1 or v2 = 2))
              by ENUMSET1:def 1;
            hence thesis by XXREAL_0:def 9,D1,B;
          end;
        end;
      end;
      suppose
A0:     v0 <> v1 & v1 <> v2 & v0 = 0;
        per cases;
        suppose
          v2 = 1 or v1 = 1; then
A1:       v1 "/\" v2 = 1 & v0 "\/" v1 = 0 by QLTEx1Def,QLTEx2Def,A0;
          v0 "\/" o = z by QLTEx1Def,A0;
          hence thesis by BINOP_1:def 4,A1;
        end;
        suppose
          v2 <> 1 & v1 <> 1; then
A1:       v1 "/\" v2 = min(v1,v2) & v0 "\/" v1 = 0
            by QLTEx1Def,QLTEx2Def,A0;
          per cases by XXREAL_0:15;
          suppose
            min(v1,v2) = v1;
            hence thesis by SHEFFER1:def 9,A1;
          end;
          suppose
D1:         min(v1,v2) = v2;
            per cases;
            suppose
D2:           v2 = v0;
              v0 "/\" z = min (v0,z) by QLTEx2Def,A0
                 .= v0 by A0; then
              (v0"\/"(v1"/\"v2)) "/\" (v0"\/"v1) = v0
                by A1,D2,D1,QLTEx1Def;
              hence thesis by D2,QLTEx1Def,A1,D1;
            end;
            suppose
D2:           v2 <> v0; then
              (v0"\/"(v1"/\"v2)) "/\" (v0"\/"v1) = z "/\" z
                by QLTEx1Def,A1,D1
                  .= z by BINOP_1:def 4;
              hence thesis by D2,QLTEx1Def,A1,D1;
            end;
          end;
        end;
      end;
      suppose v0 <> v1 & v1 = v2; then
        v1 "/\" v2 = v1 & v0 "\/" v1 = 0 by QLTEx1Def,BINOP_1:def 4;
        hence thesis by SHEFFER1:def 9;
      end;
    end;
    hence thesis;
  end;
end;

registration
  cluster QLTLattice2 -> non join-absorbing;
  coherence
  proof
    set L = QLTLattice2;
    reconsider z = 0 as Element of L by ENUMSET1:def 1;
    reconsider o = 1 as Element of L by ENUMSET1:def 1;
    reconsider dwa = 2 as Element of L by ENUMSET1:def 1;
    dwa "/\" (dwa "\/" o) = dwa "/\" z by QLTEx1Def
      .= min (dwa,z) by QLTEx2Def
      .= z by XXREAL_0:def 9;
    hence thesis by LATTICES:def 9;
  end;
  cluster QLTLattice2 -> non meet-absorbing;
  coherence
  proof
    set L = QLTLattice2;
    reconsider z = 0 as Element of L by ENUMSET1:def 1;
    reconsider o = 1 as Element of L by ENUMSET1:def 1;
    reconsider dwa = 2 as Element of L by ENUMSET1:def 1;
    (dwa "/\" o) "\/" dwa = o "\/" dwa by QLTEx2Def
      .= z by QLTEx1Def;
    hence thesis by LATTICES:def 8;
  end;
end;

registration
  cluster QLTLattice1 -> non join-absorbing;
  coherence
  proof
    set L = QLTLattice1;
    reconsider z = 0 as Element of L by ENUMSET1:def 1;
    reconsider o = 1 as Element of L by ENUMSET1:def 1;
    reconsider dwa = 2 as Element of L by ENUMSET1:def 1;
    dwa "/\" (dwa "\/" o) = dwa "/\" z by QLTEx1Def
      .= z by QLTEx1Def;
    hence thesis by LATTICES:def 9;
  end;
  cluster QLTLattice1 -> non meet-absorbing;
  coherence
  proof
    set L = QLTLattice1;
    reconsider z = 0 as Element of L by ENUMSET1:def 1;
    reconsider o = 1 as Element of L by ENUMSET1:def 1;
    reconsider dwa = 2 as Element of L by ENUMSET1:def 1;
    (dwa "/\" o) "\/" dwa = z "\/" dwa by QLTEx1Def
      .= z by QLTEx1Def;
    hence thesis by LATTICES:def 8;
  end;
end;

definition
  mode QuasiLattice is join-commutative join-associative
          meet-commutative meet-associative
          join-idempotent meet-idempotent
          satisfying_QLT1 satisfying_QLT2 non empty LattStr;
end;

begin :: Properties of QuasiLattices: QLT-1

:: In QLT-1 we don't need two QLT axioms:
:: for v0 holds v0"/\"v0 = v0 & :: /\-idempotent
::   (for v2,v1,v0 holds
::      (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2)) & :: /\-associative

theorem Lemma1: :: QLT-1
   (for v1,v0 holds v0"/\"v1 = v1"/\"v0) & :: /\-commutative
   (for v0,v2,v1 holds ::: link law
      (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2)) &
   (for v0 holds v0"\/"v0 = v0) & :: \/-idempotent
   (for v2,v1,v0 holds
      (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) & :: \/-associative
   (for v1,v0 holds v0"\/"v1 = v1"\/"v0) & :: \/-commutative
   (for v0,v2,v1 holds  :: link law
      (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
   (for v1,v2,v0 holds :: QLT-distributivity
       v0"/\"(v1"\/"(v0"/\"v2)) = v0"/\"(v1"\/"v2)) implies
   for v1,v2,v3 holds :: distributivity
       (v1"/\"v2)"\/"(v1"/\"v3) = v1"/\"(v2"\/"v3)
proof
assume A1: for v1,v0 holds v0"/\"v1 = v1"/\"v0;
assume A2: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2);
assume A3: for v0 holds v0"\/"v0 = v0;
assume A4: for v2,v1,v0 holds
  (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2);
assume A5: for v1,v0 holds v0"\/"v1 = v1"\/"v0;
assume A6: for v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2);

A8: for v0,v2,v1 holds
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) =
    (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) by A1;
  hence thesis by A6;
end;

assume A9: for v1,v2,v0 holds
  v0"/\"(v1"\/"(v0"/\"v2)) = v0"/\"(v1"\/"v2);

A12: for v0,v2,v1 holds
  (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) =
    (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) by A5;
  hence thesis by A2;
end;

A15: for v102,v100 holds (v100"\/"v102)"\/"v102 = v100"\/"v102
proof let v102,v100;
  v102"\/"v102 = v102 by A3;
  hence thesis by A4;
end;

A18: for v1,v0 holds v1"\/"(v0"\/"v1) = v0"\/"v1
proof let v1,v0;
  (v0"\/"v1)"\/"v1 = v1"\/"(v0"\/"v1) by A5;
  hence thesis by A15;
end;

A21: for v2,v0,v1 holds (v1"\/"v0)"\/"v2 = v0"\/"(v1"\/"v2)
proof let v2,v0,v1;
  v0"\/"v1 = v1"\/"v0 by A5;
  hence thesis by A4;
end;

A24: for v0,v2,v1 holds
  v0"\/"(v1"\/"v2) = v1"\/"(v0"\/"v2)
proof let v0,v2,v1;
  (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2) by A4;
  hence thesis by A21;
end;

A27: for v102,v101 holds
 v101"/\"(v101"\/"(v101"/\"v102)) = v101"\/"(v101"/\"v102)
proof let v102,v101;
  v101"\/"v101 = v101 by A3;
  hence thesis by A8;
end;

A30: for v1,v0 holds v0"/\"(v0"\/"v1) = v0"\/"(v0"/\"v1)
proof let v1,v0;
  v0"/\"(v0"\/"(v0"/\"v1)) = v0"/\"(v0"\/"v1) by A9;
  hence thesis by A27;
end;

A33: for v1,v0,v2 holds
  v0"/\"(v1"\/"(v2"/\"v0)) = v0"/\"(v1"\/"v2)
proof let v1,v0,v2;
  v0"/\"v2 = v2"/\"v0 by A1;
  hence thesis by A9;
end;

A35: for v0,v1,v2 holds
  (v0"/\"v1)"\/"(v0"/\"(v2"\/"v1)) = v0"/\"(v1"\/"v2)
proof let v0,v1,v2;
  v1"\/"v2 = v2"\/"v1 by A5;
  hence thesis by A12;
end;

A37: for v0,v1 holds v0"\/"(v1"/\"v0) = v0"/\"(v0"\/"v1)
proof let v0,v1;
  v0"/\"v1 = v1"/\"v0 by A1;
  hence thesis by A30;
end;

A40: for v1,v2,v101 holds
 (v1"\/"(v101"/\"v2))"\/"(v101"/\"(v1"\/"v2)) =
    (v1"\/"(v101"/\"v2))"/\"((v1"\/"(v101"/\"v2))"\/"v101)
proof let v1,v2,v101;
  v101"/\"(v1"\/"(v101"/\"v2)) = v101"/\"(v1"\/"v2) by A9;
  hence thesis by A37;
end;

A43: for v1,v2,v0 holds
 (v1"/\"(v0"\/"v2))"\/"(v0"\/"(v1"/\"v2)) =
   (v0"\/"(v1"/\"v2))"/\"((v0"\/"(v1"/\"v2))"\/"v1)
proof let v1,v2,v0;
  (v0"\/"(v1"/\"v2))"\/"(v1"/\"(v0"\/"v2)) =
    (v1"/\"(v0"\/"v2))"\/"(v0"\/"(v1"/\"v2)) by A5;
  hence thesis by A40;
end;

A46: for v0,v2,v1 holds
 v1"\/"((v0"/\"(v1"\/"v2))"\/"(v0"/\"v2)) =
  (v1"\/"(v0"/\"v2))"/\"((v1"\/"(v0"/\"v2))"\/"v0)
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"(v1"\/"(v0"/\"v2)) =
    v1"\/"((v0"/\"(v1"\/"v2))"\/"(v0"/\"v2)) by A24;
  hence thesis by A43;
end;

A49: for v1,v2,v0 holds
 v0"\/"((v1"/\"v2)"\/"(v1"/\"(v0"\/"v2))) =
   (v0"\/"(v1"/\"v2))"/\"((v0"\/"(v1"/\"v2))"\/"v1)
proof let v1,v2,v0;
  (v1"/\"(v0"\/"v2))"\/"(v1"/\"v2) =
    (v1"/\"v2)"\/"(v1"/\"(v0"\/"v2)) by A5;
  hence thesis by A46;
end;

A51: for v1,v0,v2 holds
 v0"\/"(v1"/\"(v2"\/"v0)) =
  (v0"\/"(v1"/\"v2))"/\"((v0"\/"(v1"/\"v2))"\/"v1)
proof let v1,v0,v2;
  (v1"/\"v2)"\/"(v1"/\"(v0"\/"v2)) = v1"/\"(v2"\/"v0) by A35;
  hence thesis by A49;
end;

A53: for v1,v0,v2 holds
 v0"\/"(v1"/\"(v2"\/"v0)) = (v0"\/"(v1"/\"v2))"/\"(v1"\/"(v0"\/"(v1"/\"v2)))
proof let v1,v0,v2;
  (v0"\/"(v1"/\"v2))"\/"v1 = v1"\/"(v0"\/"(v1"/\"v2)) by A5;
  hence thesis by A51;
end;

A56: for v101,v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"(v101"\/"(v0"\/"(v1"/\"v2))) =
   (v0"\/"(v1"/\"v2))"/\"(v101"\/"(v0"\/"v1))
proof let v101,v0,v2,v1;
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2) by A8;
  hence thesis by A33;
end;

A59: for v1,v0,v2 holds
 v0"\/"(v1"/\"(v2"\/"v0)) = (v0"\/"(v1"/\"v2))"/\"(v1"\/"(v0"\/"v1))
proof let v1,v0,v2;
  (v0"\/"(v1"/\"v2))"/\"(v1"\/"(v0"\/"(v1"/\"v2))) =
     (v0"\/"(v1"/\"v2))"/\"(v1"\/"(v0"\/"v1)) by A56;
  hence thesis by A53;
end;

A61: for v1,v0,v2 holds
 v0"\/"(v1"/\"(v2"\/"v0)) = (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1)
proof let v1,v0,v2;
  v1"\/"(v0"\/"v1) = v0"\/"v1 by A18;
  hence thesis by A59;
end;

A63: for v1,v0,v2 holds
  v0"\/"(v1"/\"(v2"\/"v0)) = (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2))
proof let v1,v0,v2;
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) =
    (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) by A1;
  hence thesis by A61;
end;

A65: for v1,v0,v2 holds
  v0"\/"(v1"/\"(v2"\/"v0)) = v0"\/"(v1"/\"v2)
proof let v1,v0,v2;
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2) by A8;
  hence thesis by A63;
end;

A68: for v101,v2,v102 holds
 (v101"/\"v2)"\/"(v101"/\"(v102"\/"v2)) =
    (v101"/\"v2)"\/"(v101"/\"v102)
proof let v101,v2,v102;
  v101"/\"(v102"\/"(v101"/\"v2)) = v101"/\"(v102"\/"v2) by A9;
  hence thesis by A65;
end;

for v0,v2,v1 holds
  v0"/\"(v1"\/"v2) = (v0"/\"v1)"\/"(v0"/\"v2)
proof let v0,v2,v1;
  (v0"/\"v1)"\/"(v0"/\"(v2"\/"v1)) = v0"/\"(v1"\/"v2) by A35;
  hence thesis by A68;
end;
hence thesis;
end;

theorem Cluster1:
  L is meet-commutative satisfying_QLT1 join-idempotent join-associative
       join-commutative satisfying_QLT2 QLT-distributive implies
    L is distributive
proof
  assume that
A1: L is meet-commutative and
A2: L is satisfying_QLT1 and
A3: L is join-idempotent and
A4: L is join-associative and
A5: L is join-commutative and
A6: L is satisfying_QLT2 and
A7: L is QLT-distributive;
S:  for v1,v0 holds v0"/\"v1 = v1"/\"v0 by LATTICES:def 6,A1;
S2: for v0 holds v0"\/"v0 = v0 by A3,ROBBINS1:def 7;
S3: for v2,v1,v0 holds
       (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2) by A4,LATTICES:def 5;
S4: for v1,v0 holds v0"\/"v1 = v1"\/"v0 by A5,LATTICES:def 4;
  let v1,v2,v3 be Element of L;
  thus thesis by Lemma1,S,A2,A6,S2,S3,S4,A7;
end;

registration
  cluster meet-commutative satisfying_QLT1 join-idempotent
     join-associative join-commutative satisfying_QLT2 QLT-distributive ->
        distributive for non empty LattStr;
  coherence by Cluster1;
end;

begin :: QLT-2

theorem ThQLT2:
(for v0 holds v0"/\"v0 = v0) &
(for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2)) &
(for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
(for v0 holds v0"\/"v0 = v0) &
(for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
(for v0,v2,v1 holds (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
(for v0,v2,v1 holds v0"/\"(v1"\/"v2) = (v0"/\"v1)"\/"(v0"/\"v2)) implies
  for v1,v2,v3 holds v1"\/"(v2"/\"v3) = (v1"\/"v2)"/\"(v1"\/"v3)
proof
assume A2: for v0 holds v0"/\"v0 = v0;
assume A3: for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2);
assume A4: for v1,v0 holds v0"/\"v1 = v1"/\"v0;
assume A5: for v0 holds v0"\/"v0 = v0;
assume A6: for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2);
assume A7: for v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2);

A9: for v0,v2,v1 holds
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) =
    (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) by A4;
  hence thesis by A7;
end;

assume A10: for v0,v2,v1 holds
  v0"/\"(v1"\/"v2) = (v0"/\"v1)"\/"(v0"/\"v2);

A15: for v102,v101 holds v101"/\"v102 = v101"/\"(v101"/\"v102)
proof let v102,v101;
  v101"/\"v101 = v101 by A2;
  hence thesis by A3;
end;

A20: for v102,v101 holds
  v101"\/"v102 = v101"\/"(v101"\/"v102)
proof let v102,v101;
  v101"\/"v101 = v101 by A5;
  hence thesis by A6;
end;

A24: for v2,v0,v1 holds
  (v1"/\"v0)"\/"(v0"/\"v2) = v0"/\"(v1"\/"v2)
proof let v2,v0,v1;
  v0"/\"v1 = v1"/\"v0 by A4;
  hence thesis by A10;
end;

A28: for v102,v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"v102 = (v0"/\"v1)"\/"((v0"/\"v2)"\/"v102)
proof let v102,v0,v2,v1;
  (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v1"\/"v2) by A10;
  hence thesis by A6;
end;

A32: for v102,v1,v100 holds
 (v100"/\"v1)"\/"(v100"/\"v102) = v100"/\"((v100"/\"v1)"\/"v102)
proof let v102,v1,v100;
  v100"/\"(v100"/\"v1) = v100"/\"v1 by A15;
  hence thesis by A10;
end;

A35: for v0,v2,v1 holds v0"/\"(v1"\/"v2) = v0"/\"((v0"/\"v1)"\/"v2)
proof let v0,v2,v1;
  (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v1"\/"v2) by A10;
  hence thesis by A32;
end;

A39: for v1,v101,v100 holds
 (v100"/\"v101)"\/"(v100"/\"v1) = v100"/\"(v101"\/"(v100"/\"v1))
proof let v1,v101,v100;
  v100"/\"(v100"/\"v1) = v100"/\"v1 by A15;
  hence thesis by A10;
end;

A42: for v0,v2,v1 holds v0"/\"(v1"\/"v2) = v0"/\"(v1"\/"(v0"/\"v2))
proof let v0,v2,v1;
  (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v1"\/"v2) by A10;
  hence thesis by A39;
end;

A46: for v102,v1,v100 holds
 (v100"\/"v1)"/\"(v100"\/"((v100"\/"v1)"/\"v102)) =
    v100"\/"((v100"\/"v1)"/\"v102)
proof let v102,v1,v100;
  v100"\/"(v100"\/"v1) = v100"\/"v1 by A20;
  hence thesis by A9;
end;

A49: for v2,v1,v0 holds
  (v0"\/"v1)"/\"(v0"\/"v2) = v0"\/"((v0"\/"v1)"/\"v2)
proof let v2,v1,v0;
  (v0"\/"v1)"/\"(v0"\/"((v0"\/"v1)"/\"v2)) =
    (v0"\/"v1)"/\"(v0"\/"v2) by A42;
  hence thesis by A46;
end;

A53: for v2,v1,v101 holds
 (v101"\/"v1)"/\"(((v101"\/"v1)"/\"v101)"\/"(v1"/\"v2)) =
    v101"\/"(v1"/\"v2)
proof let v2,v1,v101;
  (v101"\/"v1)"/\"(v101"\/"(v1"/\"v2)) = v101"\/"(v1"/\"v2) by A9;
  hence thesis by A35;
end;

A56: for v2,v1,v0 holds
  (v0"\/"v1)"/\"((v0"/\"(v0"\/"v1))"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  (v0"\/"v1)"/\"v0 = v0"/\"(v0"\/"v1) by A4;
  hence thesis by A53;
end;

A58: for v2,v1,v0 holds
  (v0"\/"v1)"/\"((v0"/\"v0)"\/"((v0"/\"v1)"\/"(v1"/\"v2))) =
    v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"\/"(v1"/\"v2) =
    (v0"/\"v0)"\/"((v0"/\"v1)"\/"(v1"/\"v2)) by A28;
  hence thesis by A56;
end;

A60: for v2,v1,v0 holds
  (v0"\/"v1)"/\"(v0"\/"((v0"/\"v1)"\/"(v1"/\"v2))) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  v0"/\"v0 = v0 by A2;
  hence thesis by A58;
end;

A62: for v1,v2,v0 holds
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"(v0"\/"v2))) = v0"\/"(v1"/\"v2)
proof let v1,v2,v0;
  (v0"/\"v1)"\/"(v1"/\"v2) = v1"/\"(v0"\/"v2) by A24;
  hence thesis by A60;
end;

A64: for v1,v2,v0 holds
  v0"\/"(v1"/\"(v0"\/"v2)) = v0"\/"(v1"/\"v2)
proof let v1,v2,v0;
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"(v0"\/"v2))) =
    v0"\/"(v1"/\"(v0"\/"v2)) by A9;
  hence thesis by A62;
end;

A66: for v1,v2,v0 holds
  v0"\/"((v0"\/"v2)"/\"v1) = v0"\/"(v1"/\"v2)
proof let v1,v2,v0;
  v1"/\"(v0"\/"v2) = (v0"\/"v2)"/\"v1 by A4;
  hence thesis by A64;
end;

A69: for v2,v1,v0 holds
  (v0"\/"v1)"/\"(v0"\/"v2) = v0"\/"(v2"/\"v1)
proof let v2,v1,v0;
  v0"\/"((v0"\/"v1)"/\"v2) = (v0"\/"v1)"/\"(v0"\/"v2) by A49;
  hence thesis by A66;
end;

  let v1,v2,v3;
  v1"\/"(v2"/\"v3) = (v1"\/"v3)"/\"(v1"\/"v2) by A69;
  hence thesis by A4;
end;

theorem Cluster2:
  L is meet-idempotent meet-associative meet-commutative
       join-idempotent join-associative satisfying_QLT2 distributive implies
    L is distributive'
  proof
    assume L is meet-idempotent meet-associative meet-commutative
       join-idempotent join-associative satisfying_QLT2 distributive; then
    (for v0 holds v0"/\"v0 = v0) &
    (for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2)) &
    (for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
    (for v0 holds v0"\/"v0 = v0) &
    (for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
    (for v0,v2,v1 holds (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
    (for v0,v2,v1 holds v0"/\"(v1"\/"v2) = (v0"/\"v1)"\/"(v0"/\"v2))
      by LATTICES:def 5, def 6, def 7, def 11,
      SHEFFER1:def 9, ROBBINS1:def 7; then
    for v1,v2,v3 holds v1"\/"(v2"/\"v3) = (v1"\/"v2)"/\"(v1"\/"v3) by ThQLT2;
    hence thesis by SHEFFER1:def 5;
  end;

registration
  cluster meet-idempotent meet-associative meet-commutative
       join-idempotent join-associative satisfying_QLT2 distributive ->
    distributive' for non empty LattStr;
  coherence by Cluster2;
end;

begin :: QLT-3

definition let L;
  attr L is QLT-selfdistributive means
   for v2,v1,v0 holds
    (((v0"/\"v1)"\/"v2)"/\"v1) "\/" (v2"/\"v0) =
      (((v0"\/"v1)"/\"v2)"\/"v1) "/\" (v2"\/"v0);
end;

theorem ThQLT3:
(for v0 holds v0"/\"v0 = v0) &
(for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2)) &
(for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
(for v0,v2,v1 holds (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2)) &
(for v0 holds v0"\/"v0 = v0) &
(for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
(for v1,v0 holds v0"\/"v1 = v1"\/"v0) &
(for v0,v2,v1 holds (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
(for v2,v1,v0 holds
 (((v0"/\"v1)"\/"v2)"/\"v1)"\/"(v2"/\"v0) =
    (((v0"\/"v1)"/\"v2)"\/"v1)"/\"(v2"\/"v0)) implies
  for v1,v2,v3 holds
    v1"\/"(v2"/\"v3) = (v1"\/"v2)"/\"(v1"\/"v3)
proof
assume A2: for v0 holds v0"/\"v0 = v0;
assume A3: for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2);
assume A4: for v1,v0 holds v0"/\"v1 = v1"/\"v0;
assume A5: for v0,v2,v1 holds
  (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2);
assume A6: for v0 holds v0"\/"v0 = v0;
assume A7: for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2);
assume A8: for v1,v0 holds v0"\/"v1 = v1"\/"v0;
assume A9: for v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2);

A11: for v0,v2,v1 holds
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) by A4;
  hence thesis by A9;
end;

assume A12: for v2,v1,v0 holds
 (((v0"/\"v1)"\/"v2)"/\"v1)"\/"(v2"/\"v0) =
    (((v0"\/"v1)"/\"v2)"\/"v1)"/\"(v2"\/"v0);

A14: for v2,v1,v0 holds
 (v1"/\"((v0"/\"v1)"\/"v2))"\/"(v2"/\"v0) =
   (((v0"\/"v1)"/\"v2)"\/"v1)"/\"(v2"\/"v0)
proof let v2,v1,v0;
  ((v0"/\"v1)"\/"v2)"/\"v1 = v1"/\"((v0"/\"v1)"\/"v2) by A4;
  hence thesis by A12;
end;

A17: for v2,v0,v1 holds
 (v2"/\"v1)"\/"(v0"/\"((v1"/\"v0)"\/"v2)) =
    (((v1"\/"v0)"/\"v2)"\/"v0)"/\"(v2"\/"v1)
proof let v2,v0,v1;
  (v0"/\"((v1"/\"v0)"\/"v2))"\/"(v2"/\"v1) =
    (v2"/\"v1)"\/"(v0"/\"((v1"/\"v0)"\/"v2)) by A8;
  hence thesis by A14;
end;

A20: for v0,v2,v1 holds
 (v0"/\"v1)"\/"(v2"/\"((v1"/\"v2)"\/"v0)) =
    (v2"\/"((v1"\/"v2)"/\"v0))"/\"(v0"\/"v1)
proof let v0,v2,v1;
  ((v1"\/"v2)"/\"v0)"\/"v2 = v2"\/"((v1"\/"v2)"/\"v0) by A8;
  hence thesis by A17;
end;

A23: for v0,v2,v1 holds
  (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) by A8;
  hence thesis by A5;
end;

A26: for v102,v101 holds v101"/\"v102 = v101"/\"(v101"/\"v102)
proof let v102,v101;
  v101"/\"v101 = v101 by A2;
  hence thesis by A3;
end;

A31: for v102,v100 holds
  (v100"/\"v102)"/\"v102 = v100"/\"v102
proof let v102,v100;
  v102"/\"v102 = v102 by A2;
  hence thesis by A3;
end;

A34: for v1,v0 holds v1"/\"(v0"/\"v1) = v0"/\"v1
proof let v1,v0;
  (v0"/\"v1)"/\"v1 = v1"/\"(v0"/\"v1) by A4;
  hence thesis by A31;
end;

A37: for v2,v0,v1 holds
  (v1"/\"v0)"/\"v2 = v0"/\"(v1"/\"v2)
proof let v2,v0,v1;
  v0"/\"v1 = v1"/\"v0 by A4;
  hence thesis by A3;
end;

A40: for v0,v2,v1 holds
  v0"/\"(v1"/\"v2) = v1"/\"(v0"/\"v2)
proof let v0,v2,v1;
  (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2) by A3;
  hence thesis by A37;
end;

A43: for v102,v101 holds
  v101"\/"v102 = v101"\/"(v101"\/"v102)
proof let v102,v101;
  v101"\/"v101 = v101 by A6;
  hence thesis by A7;
end;

A48: for v102,v100 holds
  (v100"\/"v102)"\/"v102 = v100"\/"v102
proof let v102,v100;
  v102"\/"v102 = v102 by A6;
  hence thesis by A7;
end;

A51: for v1,v0 holds v1"\/"(v0"\/"v1) = v0"\/"v1
proof let v1,v0;
  (v0"\/"v1)"\/"v1 = v1"\/"(v0"\/"v1) by A8;
  hence thesis by A48;
end;

A54: for v2,v0,v1 holds (v1"\/"v0)"\/"v2 = v0"\/"(v1"\/"v2)
proof let v2,v0,v1;
  v0"\/"v1 = v1"\/"v0 by A8;
  hence thesis by A7;
end;

A57: for v0,v2,v1 holds v0"\/"(v1"\/"v2) = v1"\/"(v0"\/"v2)
proof let v0,v2,v1;
  (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2) by A7;
  hence thesis by A54;
end;

A60: for v102,v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"v102 = (v0"\/"v1)"/\"((v0"\/"(v1"/\"v2))"/\"v102)
proof let v102,v0,v2,v1;
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2) by A11;
  hence thesis by A3;
end;

A65: for v100,v0,v102,v1 holds
 (v100"\/"(v0"/\"v1))"/\"(v100"\/"(v0"/\"(v1"/\"v102))) =
    v100"\/"((v0"/\"v1)"/\"v102)
proof let v100,v0,v102,v1;
  (v0"/\"v1)"/\"v102 = v0"/\"(v1"/\"v102) by A3;
  hence thesis by A11;
end;

A68: for v0,v1,v3,v2 holds
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"(v1"/\"(v2"/\"v3))) = v0"\/"(v1"/\"(v2"/\"v3))
proof let v0,v1,v3,v2;
  (v1"/\"v2)"/\"v3 = v1"/\"(v2"/\"v3) by A3;
  hence thesis by A65;
end;

A71: for v102,v101 holds
  v101"/\"(v101"\/"(v101"/\"v102)) = v101"\/"(v101"/\"v102)
proof let v102,v101;
  v101"\/"v101 = v101 by A6;
  hence thesis by A11;
end;

A75: for v102,v101 holds
  ((v101"/\"v102)"\/"v101)"/\"(v101"/\"v102) =
     (v101"/\"v102)"\/"(v101"/\"v102)
proof let v102,v101;
  (v101"/\"v102)"\/"(v101"/\"v102) = v101"/\"v102 by A6;
  hence thesis by A11;
end;

A78: for v1,v0 holds
  (v0"\/"(v0"/\"v1))"/\"(v0"/\"v1) = (v0"/\"v1)"\/"(v0"/\"v1)
proof let v1,v0;
  (v0"/\"v1)"\/"v0 = v0"\/"(v0"/\"v1) by A8;
  hence thesis by A75;
end;

A80: for v1,v0 holds
  (v0"/\"v1)"/\"(v0"\/"(v0"/\"v1)) = (v0"/\"v1)"\/"(v0"/\"v1)
proof let v1,v0;
  (v0"\/"(v0"/\"v1))"/\"(v0"/\"v1) =
    (v0"/\"v1)"/\"(v0"\/"(v0"/\"v1)) by A4;
  hence thesis by A78;
end;

A82: for v1,v0 holds
  v0"/\"(v1"/\"(v0"\/"(v0"/\"v1))) = (v0"/\"v1)"\/"(v0"/\"v1)
proof let v1,v0;
  (v0"/\"v1)"/\"(v0"\/"(v0"/\"v1)) =
    v0"/\"(v1"/\"(v0"\/"(v0"/\"v1))) by A3;
  hence thesis by A80;
end;

A84: for v1,v0 holds v0"/\"(v1"/\"(v0"\/"(v0"/\"v1))) = v0"/\"v1
proof let v1,v0;
  (v0"/\"v1)"\/"(v0"/\"v1) = v0"/\"v1 by A6;
  hence thesis by A82;
end;

A86: for v0,v2,v1 holds
  (v1"\/"v0)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  v0"\/"v1 = v1"\/"v0 by A8;
  hence thesis by A11;
end;

A90: for v102,v101 holds
 v101"\/"(v102"/\"((v101"/\"v102)"\/"v101)) =
  (v102"\/"((v101"\/"v102)"/\"v101))"/\"(v101"\/"v101)
proof let v102,v101;
  v101"/\"v101 = v101 by A2;
  hence thesis by A20;
end;

A93: for v1,v0 holds
  v0"\/"(v1"/\"(v0"\/"(v0"/\"v1))) = (v1"\/"((v0"\/"v1)"/\"v0))"/\"(v0"\/"v0)
proof let v1,v0;
  (v0"/\"v1)"\/"v0 = v0"\/"(v0"/\"v1) by A8;
  hence thesis by A90;
end;

A95: for v1,v0 holds
 v0"\/"(v1"/\"(v0"\/"(v0"/\"v1))) = (v1"\/"(v0"/\"(v0"\/"v1)))"/\"(v0"\/"v0)
proof let v1,v0;
  (v0"\/"v1)"/\"v0 = v0"/\"(v0"\/"v1) by A4;
  hence thesis by A93;
end;

A97: for v1,v0 holds
  v0"\/"(v1"/\"(v0"\/"(v0"/\"v1))) = (v1"\/"(v0"/\"(v0"\/"v1)))"/\"v0
proof let v1,v0;
  v0"\/"v0 = v0 by A6;
  hence thesis by A95;
end;

A99: for v1,v0 holds
  v0"\/"(v1"/\"(v0"\/"(v0"/\"v1))) = v0"/\"(v1"\/"(v0"/\"(v0"\/"v1)))
proof let v1,v0;
  (v1"\/"(v0"/\"(v0"\/"v1)))"/\"v0 = v0"/\"(v1"\/"(v0"/\"(v0"\/"v1))) by A4;
  hence thesis by A97;
end;

A101: for v0,v2,v1 holds
  (v1"/\"v0)"\/"(v2"/\"((v1"/\"v2)"\/"v0)) =
    (v2"\/"((v1"\/"v2)"/\"v0))"/\"(v0"\/"v1)
proof let v0,v2,v1;
  v0"/\"v1 = v1"/\"v0 by A4;
  hence thesis by A20;
end;

A104: for v0,v1,v2 holds
  (v0"/\"v1)"\/"(v2"/\"((v2"/\"v1)"\/"v0)) =
     (v2"\/"((v1"\/"v2)"/\"v0))"/\"(v0"\/"v1)
proof let v0,v1,v2;
  v1"/\"v2 = v2"/\"v1 by A4;
  hence thesis by A20;
end;

A107: for v101,v1,v0 holds
  ((v0"/\"v1)"\/"v101)"/\"((v101"\/"((v1"\/"v101)"/\"v0))"/\"(v0"\/"v1)) =
    (v0"/\"v1)"\/"(v101"/\"((v1"/\"v101)"\/"v0))
proof let v101,v1,v0;
  (v0"/\"v1)"\/"(v101"/\"((v1"/\"v101)"\/"v0)) =
    (v101"\/"((v1"\/"v101)"/\"v0))"/\"(v0"\/"v1) by A20;
  hence thesis by A11;
end;

A110: for v2,v1,v0 holds
 ((v0"/\"v1)"\/"v2)"/\"((v2"\/"((v1"\/"v2)"/\"v0))"/\"(v0"\/"v1)) =
    (v2"\/"((v1"\/"v2)"/\"v0))"/\"(v0"\/"v1)
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v2"/\"((v1"/\"v2)"\/"v0)) =
    (v2"\/"((v1"\/"v2)"/\"v0))"/\"(v0"\/"v1) by A20;
  hence thesis by A107;
end;

A113: for v102,v101 holds
  v101"\/"(v101"/\"(v101"\/"v102)) = v101"/\"(v101"\/"v102)
proof let v102,v101;
  v101"/\"v101 = v101 by A2;
  hence thesis by A23;
end;

A117: for v102,v101 holds
  ((v101"\/"v102)"/\"v101)"\/"(v101"\/"v102) = (v101"\/"v102)"/\"(v101"\/"v102)
proof let v102,v101;
  (v101"\/"v102)"/\"(v101"\/"v102) = v101"\/"v102 by A2;
  hence thesis by A23;
end;

A120: for v1,v0 holds
  (v0"/\"(v0"\/"v1))"\/"(v0"\/"v1) = (v0"\/"v1)"/\"(v0"\/"v1)
proof let v1,v0;
  (v0"\/"v1)"/\"v0 = v0"/\"(v0"\/"v1) by A4;
  hence thesis by A117;
end;

A122: for v1,v0 holds
  (v0"\/"v1)"\/"(v0"/\"(v0"\/"v1)) = (v0"\/"v1)"/\"(v0"\/"v1)
proof let v1,v0;
  (v0"/\"(v0"\/"v1))"\/"(v0"\/"v1) = (v0"\/"v1)"\/"(v0"/\"(v0"\/"v1)) by A8;
  hence thesis by A120;
end;

A124: for v1,v0 holds
  v0"\/"(v1"\/"(v0"/\"(v0"\/"v1))) = (v0"\/"v1)"/\"(v0"\/"v1)
proof let v1,v0;
  (v0"\/"v1)"\/"(v0"/\"(v0"\/"v1)) = v0"\/"(v1"\/"(v0"/\"(v0"\/"v1))) by A7;
  hence thesis by A122;
end;

A126: for v1,v0 holds v0"\/"(v1"\/"(v0"/\"(v0"\/"v1))) = v0"\/"v1
proof let v1,v0;
  (v0"\/"v1)"/\"(v0"\/"v1) = v0"\/"v1 by A2;
  hence thesis by A124;
end;

A128: for v0,v2,v1 holds
  (v1"/\"v0)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  v0"/\"v1 = v1"/\"v0 by A4;
  hence thesis by A23;
end;

A131: for v0,v2,v1 holds
  (v0"/\"v1)"\/"((v1"\/"v2)"/\"v0) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  v0"/\"(v1"\/"v2) = (v1"\/"v2)"/\"v0 by A4;
  hence thesis by A23;
end;

A133: for v0,v1,v2 holds
  (v0"/\"v1)"\/"(v0"/\"(v2"\/"v1)) = v0"/\"(v1"\/"v2)
proof let v0,v1,v2;
  v1"\/"v2 = v2"\/"v1 by A8;
  hence thesis by A23;
end;

A136: for v101,v2,v1 holds
 ((v101"/\"v1)"\/"v101)"/\"(v101"/\"(v1"\/"v2)) =
    (v101"/\"v1)"\/"(v101"/\"(v1"\/"v2))
proof let v101,v2,v1;
  (v101"/\"v1)"\/"(v101"/\"(v1"\/"v2)) = v101"/\"(v1"\/"v2) by A23;
  hence thesis by A11;
end;

A139: for v0,v2,v1 holds
 (v0"\/"(v0"/\"v1))"/\"(v0"/\"(v1"\/"v2)) = (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2))
proof let v0,v2,v1;
  (v0"/\"v1)"\/"v0 = v0"\/"(v0"/\"v1) by A8;
  hence thesis by A136;
end;

A141: for v2,v1,v0 holds
 v0"/\"((v0"\/"(v0"/\"v1))"/\"(v1"\/"v2)) = (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2))
proof let v2,v1,v0;
  (v0"\/"(v0"/\"v1))"/\"(v0"/\"(v1"\/"v2)) =
    v0"/\"((v0"\/"(v0"/\"v1))"/\"(v1"\/"v2)) by A40;
  hence thesis by A139;
end;

A143: for v2,v1,v0 holds
  v0"/\"((v0"\/"(v0"/\"v1))"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2) by A23;
  hence thesis by A141;
end;

A146: for v101,v2,v1 holds
 ((v101"\/"v1)"/\"v101)"\/"(v101"\/"(v1"/\"v2)) =
    (v101"\/"v1)"/\"(v101"\/"(v1"/\"v2))
proof let v101,v2,v1;
  (v101"\/"v1)"/\"(v101"\/"(v1"/\"v2)) = v101"\/"(v1"/\"v2) by A11;
  hence thesis by A23;
end;

A149: for v0,v2,v1 holds
  (v0"/\"(v0"\/"v1))"\/"(v0"\/"(v1"/\"v2)) = (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2))
proof let v0,v2,v1;
  (v0"\/"v1)"/\"v0 = v0"/\"(v0"\/"v1) by A4;
  hence thesis by A146;
end;

A151: for v2,v1,v0 holds
  v0"\/"((v0"/\"(v0"\/"v1))"\/"(v1"/\"v2)) = (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2))
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"\/"(v0"\/"(v1"/\"v2)) =
    v0"\/"((v0"/\"(v0"\/"v1))"\/"(v1"/\"v2)) by A57;
  hence thesis by A149;
end;

A153: for v2,v1,v0 holds
  v0"\/"((v0"/\"(v0"\/"v1))"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2) by A11;
  hence thesis by A151;
end;

A156: for v1,v101,v100 holds
  (v100"/\"v101)"/\"(v101"/\"v1) = v100"/\"(v101"/\"v1)
proof let v1,v101,v100;
  v101"/\"(v101"/\"v1) = v101"/\"v1 by A26;
  hence thesis by A3;
end;

A159: for v2,v1,v0 holds
  v1"/\"((v0"/\"v1)"/\"v2) = v0"/\"(v1"/\"v2)
proof let v2,v1,v0;
  (v0"/\"v1)"/\"(v1"/\"v2) = v1"/\"((v0"/\"v1)"/\"v2) by A40;
  hence thesis by A156;
end;

A162: for v1,v2,v0 holds
  v0"/\"(v1"/\"(v0"/\"v2)) = v1"/\"(v0"/\"v2)
proof let v1,v2,v0;
  (v1"/\"v0)"/\"v2 = v1"/\"(v0"/\"v2) by A3;
  hence thesis by A159;
end;

A165: for v102,v1,v100 holds
  (v100"/\"v1)"\/"(v102"/\"(((v100"/\"v1)"/\"v102)"\/"v100)) =
  (v102"\/"(((v100"/\"v1)"\/"v102)"/\"v100))"/\"(v100"\/"(v100"/\"v1))
proof let v102,v1,v100;
  v100"/\"(v100"/\"v1) = v100"/\"v1 by A26;
  hence thesis by A20;
end;

A168: for v0,v2,v1 holds
 (v0"/\"v1)"\/"(v2"/\"((v0"/\"(v1"/\"v2))"\/"v0)) =
   (v2"\/"(((v0"/\"v1)"\/"v2)"/\"v0))"/\"(v0"\/"(v0"/\"v1))
proof let v0,v2,v1;
  (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2) by A3;
  hence thesis by A165;
end;

A170: for v0,v2,v1 holds
 (v0"/\"v1)"\/"(v2"/\"(v0"\/"(v0"/\"(v1"/\"v2)))) =
    (v2"\/"(((v0"/\"v1)"\/"v2)"/\"v0))"/\"(v0"\/"(v0"/\"v1))
proof let v0,v2,v1;
  (v0"/\"(v1"/\"v2))"\/"v0 = v0"\/"(v0"/\"(v1"/\"v2)) by A8;
  hence thesis by A168;
end;

A172: for v0,v2,v1 holds
 (v0"/\"v1)"\/"(v2"/\"(v0"\/"(v0"/\"(v1"/\"v2)))) =
  (v2"\/"(v0"/\"((v0"/\"v1)"\/"v2)))"/\"(v0"\/"(v0"/\"v1))
proof let v0,v2,v1;
  ((v0"/\"v1)"\/"v2)"/\"v0 = v0"/\"((v0"/\"v1)"\/"v2) by A4;
  hence thesis by A170;
end;

A174: for v0,v2,v1 holds
 (v0"/\"v1)"\/"(v2"/\"(v0"\/"(v0"/\"(v1"/\"v2)))) =
   (v0"\/"(v0"/\"v1))"/\"(v2"\/"(v0"/\"((v0"/\"v1)"\/"v2)))
proof let v0,v2,v1;
  (v2"\/"(v0"/\"((v0"/\"v1)"\/"v2)))"/\"(v0"\/"(v0"/\"v1)) =
    (v0"\/"(v0"/\"v1))"/\"(v2"\/"(v0"/\"((v0"/\"v1)"\/"v2))) by A4;
  hence thesis by A172;
end;

A177: for v0,v100,v1 holds
  v100"/\"(v0"/\"(v1"/\"v100)) = (v0"/\"v1)"/\"v100
proof let v0,v100,v1;
  (v0"/\"v1)"/\"v100 = v0"/\"(v1"/\"v100) by A3;
  hence thesis by A34;
end;

A180: for v1,v0,v2 holds
  v0"/\"(v1"/\"(v2"/\"v0)) = v1"/\"(v2"/\"v0)
proof let v1,v0,v2;
  (v1"/\"v2)"/\"v0 = v1"/\"(v2"/\"v0) by A3;
  hence thesis by A177;
end;

A183: for v100,v101,v1 holds
 (v100"\/"v101)"/\"(v100"\/"(v1"/\"v101)) = v100"\/"(v101"/\"(v1"/\"v101))
proof let v100,v101,v1;
  v101"/\"(v1"/\"v101) = v1"/\"v101 by A34;
  hence thesis by A11;
end;

A186: for v0,v1,v2 holds
  (v0"\/"v1)"/\"(v0"\/"(v2"/\"v1)) = v0"\/"(v2"/\"v1)
proof let v0,v1,v2;
  v1"/\"(v2"/\"v1) = v2"/\"v1 by A34;
  hence thesis by A183;
end;

A189: for v1,v101,v100 holds
  (v100"\/"v101)"\/"(v101"\/"v1) = v100"\/"(v101"\/"v1)
proof let v1,v101,v100;
  v101"\/"(v101"\/"v1) = v101"\/"v1 by A43;
  hence thesis by A7;
end;

A192: for v2,v1,v0 holds
  v1"\/"((v0"\/"v1)"\/"v2) = v0"\/"(v1"\/"v2)
proof let v2,v1,v0;
  (v0"\/"v1)"\/"(v1"\/"v2) = v1"\/"((v0"\/"v1)"\/"v2) by A57;
  hence thesis by A189;
end;

A195: for v1,v2,v0 holds
  v0"\/"(v1"\/"(v0"\/"v2)) = v1"\/"(v0"\/"v2)
proof let v1,v2,v0;
  (v1"\/"v0)"\/"v2 = v1"\/"(v0"\/"v2) by A7;
  hence thesis by A192;
end;

A198: for v102,v1,v100 holds
  (v100"\/"v1)"/\"(v100"\/"((v100"\/"v1)"/\"v102)) =
     v100"\/"((v100"\/"v1)"/\"v102)
proof let v102,v1,v100;
  v100"\/"(v100"\/"v1) = v100"\/"v1 by A43;
  hence thesis by A11;
end;

A202: for v0,v100,v1 holds
  v100"\/"(v0"\/"(v1"\/"v100)) = (v0"\/"v1)"\/"v100
proof let v0,v100,v1;
  (v0"\/"v1)"\/"v100 = v0"\/"(v1"\/"v100) by A7;
  hence thesis by A51;
end;

A205: for v1,v0,v2 holds
  v0"\/"(v1"\/"(v2"\/"v0)) = v1"\/"(v2"\/"v0)
proof let v1,v0,v2;
  (v1"\/"v2)"\/"v0 = v1"\/"(v2"\/"v0) by A7;
  hence thesis by A202;
end;

A208: for v102,v100,v1 holds
 (v1"\/"v100)"/\"(v100"\/"((v1"\/"v100)"/\"v102)) =
    v100"\/"((v1"\/"v100)"/\"v102)
proof let v102,v100,v1;
  v100"\/"(v1"\/"v100) = v1"\/"v100 by A51;
  hence thesis by A11;
end;

A212: for v102,v1,v100 holds
  (v100"\/"(v100"/\"v1))"/\"v102 = v100"/\"((v100"\/"(v100"/\"v1))"/\"v102)
proof let v102,v1,v100;
  v100"/\"(v100"\/"(v100"/\"v1)) = v100"\/"(v100"/\"v1) by A71;
  hence thesis by A3;
end;

A217: for v1,v0 holds
 (v0"/\"v1)"/\"((v0"/\"v1)"\/"((v0"/\"v1)"/\"((v1"/\"(v0"/\"v1))"\/"v0))) =
    ((v0"/\"v1)"\/"((v1"\/"(v0"/\"v1))"/\"v0))"/\"(v0"\/"v1)
proof let v1,v0;
  (v0"/\"v1)"\/"((v0"/\"v1)"/\"((v1"/\"(v0"/\"v1))"\/"v0)) =
    ((v0"/\"v1)"\/"((v1"\/"(v0"/\"v1))"/\"v0))"/\"(v0"\/"v1) by A20;
  hence thesis by A71;
end;

A219: for v1,v0 holds
 (v0"/\"v1)"/\"((v0"/\"v1)"\/"((v0"/\"v1)"/\"((v0"/\"v1)"\/"v0))) =
   ((v0"/\"v1)"\/"((v1"\/"(v0"/\"v1))"/\"v0))"/\"(v0"\/"v1)
proof let v1,v0;
  v1"/\"(v0"/\"v1) = v0"/\"v1 by A34;
  hence thesis by A217;
end;

A221: for v1,v0 holds
 (v0"/\"v1)"/\"((v0"/\"v1)"\/"((v0"/\"v1)"/\"(v0"\/"(v0"/\"v1)))) =
   ((v0"/\"v1)"\/"((v1"\/"(v0"/\"v1))"/\"v0))"/\"(v0"\/"v1)
proof let v1,v0;
  (v0"/\"v1)"\/"v0 = v0"\/"(v0"/\"v1) by A8;
  hence thesis by A219;
end;

A223: for v1,v0 holds
 (v0"/\"v1)"/\"((v0"/\"v1)"\/"(v0"/\"(v1"/\"(v0"\/"(v0"/\"v1))))) =
    (((v0"/\"v1)"\/"((v1"\/"(v0"/\"v1))"/\"v0))"/\"(v0"\/"v1))
proof let v1,v0;
  (v0"/\"v1)"/\"(v0"\/"(v0"/\"v1)) = v0"/\"(v1"/\"(v0"\/"(v0"/\"v1))) by A3;
  hence thesis by A221;
end;

A225: for v1,v0 holds
 (v0"/\"v1)"/\"((v0"/\"v1)"\/"(v0"/\"v1)) =
   ((v0"/\"v1)"\/"((v1"\/"(v0"/\"v1))"/\"v0))"/\"(v0"\/"v1)
proof let v1,v0;
  v0"/\"(v1"/\"(v0"\/"(v0"/\"v1))) = v0"/\"v1 by A84;
  hence thesis by A223;
end;

A227: for v1,v0 holds
 (v0"/\"v1)"/\"(v0"/\"v1) =
    ((v0"/\"v1)"\/"((v1"\/"(v0"/\"v1))"/\"v0))"/\"(v0"\/"v1)
proof let v1,v0;
  (v0"/\"v1)"\/"(v0"/\"v1) = v0"/\"v1 by A6;
  hence thesis by A225;
end;

A229: for v1,v0 holds
  v0"/\"v1 = ((v0"/\"v1)"\/"((v1"\/"(v0"/\"v1))"/\"v0))"/\"(v0"\/"v1)
proof let v1,v0;
  (v0"/\"v1)"/\"(v0"/\"v1) = v0"/\"v1 by A2;
  hence thesis by A227;
end;

A231: for v1,v0 holds
  v0"/\"v1 = ((v0"/\"v1)"\/"(v0"/\"(v1"\/"(v0"/\"v1))))"/\"(v0"\/"v1)
proof let v1,v0;
  (v1"\/"(v0"/\"v1))"/\"v0 = v0"/\"(v1"\/"(v0"/\"v1)) by A4;
  hence thesis by A229;
end;

A233: for v1,v0 holds
  v0"/\"v1 = (v0"/\"(v1"\/"(v0"/\"v1)))"/\"(v0"\/"v1)
proof let v1,v0;
  (v0"/\"v1)"\/"(v0"/\"(v1"\/"(v0"/\"v1))) = v0"/\"(v1"\/"(v0"/\"v1)) by A23;
  hence thesis by A231;
end;

A235: for v1,v0 holds
  v0"/\"v1 = v0"/\"((v1"\/"(v0"/\"v1))"/\"(v0"\/"v1))
proof let v1,v0;
  (v0"/\"(v1"\/"(v0"/\"v1)))"/\"(v0"\/"v1) =
    v0"/\"((v1"\/"(v0"/\"v1))"/\"(v0"\/"v1)) by A3;
  hence thesis by A233;
end;

A239: for v100,v1,v101 holds
 v100"/\"(v101"\/"(v101"/\"v1)) = v101"/\"(v100"/\"(v101"\/"(v101"/\"v1)))
proof let v100,v1,v101;
  v101"/\"(v101"\/"(v101"/\"v1)) = v101"\/"(v101"/\"v1)
    by A71;
  hence thesis by A40;
end;

A244: for v2,v1,v0 holds
  (v0"\/"(v0"/\"v1))"/\"(v1"\/"v2) = v0"/\"(v1"\/"v2)
proof let v2,v1,v0;
  v0"/\"((v0"\/"(v0"/\"v1))"/\"(v1"\/"v2)) =
    (v0"\/"(v0"/\"v1))"/\"(v1"\/"v2) by A212;
  hence thesis by A143;
end;

A246: for v1,v0 holds
  v1"/\"(v0"\/"(v0"/\"v1)) = v0"/\"v1
proof let v1,v0;
  v0"/\"(v1"/\"(v0"\/"(v0"/\"v1))) = v1"/\"(v0"\/"(v0"/\"v1)) by A239;
  hence thesis by A84;
end;

A249: for v1,v0 holds
  v0"\/"(v0"/\"v1) = v0"/\"(v1"\/"(v0"/\"(v0"\/"v1)))
proof let v1,v0;
  v1"/\"(v0"\/"(v0"/\"v1)) = v0"/\"v1 by A246;
  hence thesis by A99;
end;

A251: for v1,v0 holds v0"/\"(v1"\/"(v0"/\"v1)) = v1"/\"v0
proof let v1,v0;
  v1"/\"v0 = v0"/\"v1 by A4;
  hence thesis by A246;
end;

A254: for v1,v0 holds
 ((v1"/\"(v0"/\"v1))"\/"v0)"/\"(((v0"/\"v1)"\/"
    ((v1"\/"(v0"/\"v1))"/\"v0))"/\"(v0"\/"v1)) =
      (v0"/\"v1)"/\"((v1"/\"(v0"/\"v1))"\/"v0)
proof let v1,v0;
  (v0"/\"v1)"\/"((v0"/\"v1)"/\"((v1"/\"(v0"/\"v1))"\/"v0)) =
    ((v0"/\"v1)"\/"((v1"\/"(v0"/\"v1))"/\"v0))"/\"(v0"\/"v1) by A20;
  hence thesis by A246;
end;

A257: for v0,v1 holds
  ((v1"/\"v0)"\/"v1)"/\"(((v1"/\"v0)"\/"
    ((v0"\/"(v1"/\"v0))"/\"v1))"/\"(v1"\/"v0)) =
      (v1"/\"v0)"/\"((v0"/\"(v1"/\"v0))"\/"v1)
proof let v0,v1;
  v0"/\"(v1"/\"v0) = v1"/\"v0 by A34;
  hence thesis by A254;
end;

A260: for v1,v0 holds
  (v0"\/"(v0"/\"v1))"/\"(((v0"/\"v1)"\/"
    ((v1"\/"(v0"/\"v1))"/\"v0))"/\"(v0"\/"v1)) =
       (v0"/\"v1)"/\"((v1"/\"(v0"/\"v1))"\/"v0)
proof let v1,v0;
  (v0"/\"v1)"\/"v0 = v0"\/"(v0"/\"v1) by A8;
  hence thesis by A257;
end;

A262: for v1,v0 holds
  (v0"\/"(v0"/\"v1))"/\"(((v0"/\"v1)"\/"
    (v0"/\"(v1"\/"(v0"/\"v1))))"/\"(v0"\/"v1)) =
     (v0"/\"v1)"/\"((v1"/\"(v0"/\"v1))"\/"v0)
proof let v1,v0;
  (v1"\/"(v0"/\"v1))"/\"v0 = v0"/\"(v1"\/"(v0"/\"v1)) by A4;
  hence thesis by A260;
end;

A264: for v1,v0 holds
  (v0"\/"(v0"/\"v1))"/\"(((v0"/\"v1)"\/"(v1"/\"v0))"/\"(v0"\/"v1)) =
    (v0"/\"v1)"/\"((v1"/\"(v0"/\"v1))"\/"v0)
proof let v1,v0;
  v0"/\"(v1"\/"(v0"/\"v1)) = v1"/\"v0 by A251;
  hence thesis by A262;
end;

A266: for v1,v0 holds
  (v0"\/"(v0"/\"v1))"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"(v1"/\"v0))) =
    (v0"/\"v1)"/\"((v1"/\"(v0"/\"v1))"\/"v0)
proof let v1,v0;
  ((v0"/\"v1)"\/"(v1"/\"v0))"/\"(v0"\/"v1) =
     (v0"\/"v1)"/\"((v0"/\"v1)"\/"(v1"/\"v0)) by A4;
  hence thesis by A264;
end;

A268: for v1,v0 holds
  (v0"\/"v1)"/\"((v0"\/"(v0"/\"v1))"/\"((v0"/\"v1)"\/"(v1"/\"v0))) =
    (v0"/\"v1)"/\"((v1"/\"(v0"/\"v1))"\/"v0)
proof let v1,v0;
  (v0"\/"(v0"/\"v1))"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"(v1"/\"v0))) =
     (v0"\/"v1)"/\"((v0"\/"(v0"/\"v1))"/\"((v0"/\"v1)"\/"(v1"/\"v0))) by A40;
  hence thesis by A266;
end;

A270: for v1,v0 holds
  (v0"\/"v1)"/\"((v0"\/"(v0"/\"v1))"/\"((v0"/\"v1)"\/"(v1"/\"v0))) =
    (v0"/\"v1)"/\"((v0"/\"v1)"\/"v0)
proof let v1,v0;
  v1"/\"(v0"/\"v1) = v0"/\"v1 by A34;
  hence thesis by A268;
end;

A272: for v1,v0 holds
 (v0"\/"v1)"/\"((v0"\/"(v0"/\"v1))"/\"((v0"/\"v1)"\/"(v1"/\"v0))) =
   (v0"/\"v1)"/\"(v0"\/"(v0"/\"v1))
proof let v1,v0;
  (v0"/\"v1)"\/"v0 = v0"\/"(v0"/\"v1) by A8;
  hence thesis by A270;
end;

A274: for v1,v0 holds
  (v0"\/"v1)"/\"((v0"\/"(v0"/\"v1))"/\"((v0"/\"v1)"\/"(v1"/\"v0))) =
     v0"/\"(v1"/\"(v0"\/"(v0"/\"v1)))
proof let v1,v0;
  (v0"/\"v1)"/\"(v0"\/"(v0"/\"v1)) = v0"/\"(v1"/\"(v0"\/"(v0"/\"v1))) by A3;
  hence thesis by A272;
end;

A276: for v1,v0 holds
  (v0"\/"v1)"/\"((v0"\/"(v0"/\"v1))"/\"((v0"/\"v1)"\/"(v1"/\"v0))) =
     v0"/\"(v0"/\"v1)
proof let v1,v0;
  v1"/\"(v0"\/"(v0"/\"v1)) = v0"/\"v1 by A246;
  hence thesis by A274;
end;

A278: for v1,v0 holds
 (v0"\/"v1)"/\"((v0"\/"(v0"/\"v1))"/\"((v0"/\"v1)"\/"(v1"/\"v0))) = v0"/\"v1
proof let v1,v0;
  v0"/\"(v0"/\"v1) = v0"/\"v1 by A26;
  hence thesis by A276;
end;

A281: for v101,v100 holds
 (v100"/\"v101)"\/"(v101"/\"v100) = v100"/\"(v101"\/"(v101"/\"v100))
proof let v101,v100;
  v100"/\"(v101"\/"(v101"/\"v100)) = v101"/\"v100 by A246;
  hence thesis by A23;
end;

A284: for v1,v0 holds (v0"/\"v1)"\/"(v1"/\"v0) = v1"/\"v0
proof let v1,v0;
  v0"/\"(v1"\/"(v1"/\"v0)) = v1"/\"v0 by A246;
  hence thesis by A281;
end;

A286: for v1,v0 holds
  (v0"\/"v1)"/\"((v0"\/"(v0"/\"v1))"/\"(v1"/\"v0)) = v0"/\"v1
proof let v1,v0;
  (v0"/\"v1)"\/"(v1"/\"v0) = v1"/\"v0 by A284;
  hence thesis by A278;
end;

A288: for v1,v0 holds
  (v0"\/"v1)"/\"((v1"/\"v0)"/\"(v0"\/"(v0"/\"v1))) = v0"/\"v1
proof let v1,v0;
  (v0"\/"(v0"/\"v1))"/\"(v1"/\"v0) = (v1"/\"v0)"/\"(v0"\/"(v0"/\"v1)) by A4;
  hence thesis by A286;
end;

A290: for v1,v0 holds
  (v0"\/"v1)"/\"(v1"/\"(v0"/\"(v0"\/"(v0"/\"v1)))) = v0"/\"v1
proof let v1,v0;
  (v1"/\"v0)"/\"(v0"\/"(v0"/\"v1)) = v1"/\"(v0"/\"(v0"\/"(v0"/\"v1))) by A3;
  hence thesis by A288;
end;

A292: for v1,v0 holds
  (v0"\/"v1)"/\"(v1"/\"(v0"\/"(v0"/\"v1))) = v0"/\"v1
proof let v1,v0;
  v0"/\"(v0"\/"(v0"/\"v1)) = v0"\/"(v0"/\"v1) by A71;
  hence thesis by A290;
end;

A294: for v1,v0 holds
  (v0"\/"v1)"/\"(v0"/\"v1) = v0"/\"v1
proof let v1,v0;
  v1"/\"(v0"\/"(v0"/\"v1)) = v0"/\"v1 by A246;
  hence thesis by A292;
end;

A296: for v1,v0 holds
  (v0"/\"v1)"/\"(v0"\/"v1) = v0"/\"v1
proof let v1,v0;
  (v0"\/"v1)"/\"(v0"/\"v1) = (v0"/\"v1)"/\"(v0"\/"v1) by A4;
  hence thesis by A294;
end;

A298: for v1,v0 holds
  v0"/\"(v1"/\"(v0"\/"v1)) = v0"/\"v1
proof let v1,v0;
  (v0"/\"v1)"/\"(v0"\/"v1) = v0"/\"(v1"/\"(v0"\/"v1)) by A3;
  hence thesis by A296;
end;

A301: for v102,v100,v1 holds
  (v1"/\"v100)"/\"v102 = v100"/\"((v1"\/"(v100"/\"v1))"/\"v102)
proof let v102,v100,v1;
  v100"/\"(v1"\/"(v100"/\"v1)) = v1"/\"v100 by A251;
  hence thesis by A3;
end;

A304: for v0,v2,v1 holds
  v0"/\"(v1"/\"v2) = v1"/\"((v0"\/"(v1"/\"v0))"/\"v2)
proof let v0,v2,v1;
  (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2) by A3;
  hence thesis by A301;
end;

A308: for v1,v0 holds
  v1"/\"(v0"/\"(v0"\/"v1)) = v0"/\"v1
proof let v1,v0;
  v0"/\"((v1"\/"(v0"/\"v1))"/\"(v0"\/"v1)) =
    v1"/\"(v0"/\"(v0"\/"v1)) by A304;
  hence thesis by A235;
end;

A312: for v103,v100,v101,v1 holds
  (v100"\/"v101)"/\"((v100"\/"(v1"/\"v101))"/\"v103) =
    (v100"\/"(v101"/\"(v1"/\"v101)))"/\"v103
proof let v103,v100,v101,v1;
  v101"/\"(v1"/\"v101) = v1"/\"v101 by A34;
  hence thesis by A60;
end;

A315: for v3,v0,v1,v2 holds
 (v0"\/"v1)"/\"((v0"\/"(v2"/\"v1))"/\"v3) = (v0"\/"(v2"/\"v1))"/\"v3
proof let v3,v0,v1,v2;
  v1"/\"(v2"/\"v1) = v2"/\"v1 by A34;
  hence thesis by A312;
end;

A318: for v102,v100,v1 holds
 (v1"/\"v100)"/\"v102 = v100"/\"((v1"/\"(v1"\/"v100))"/\"v102)
proof let v102,v100,v1;
  v100"/\"(v1"/\"(v1"\/"v100)) = v1"/\"v100 by A308;
  hence thesis by A3;
end;

A321: for v0,v2,v1 holds
  v0"/\"(v1"/\"v2) = v1"/\"((v0"/\"(v0"\/"v1))"/\"v2)
proof let v0,v2,v1;
  (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2) by A3;
  hence thesis by A318;
end;

A323: for v0,v2,v1 holds
  v0"/\"(v1"/\"v2) = v1"/\"(v0"/\"((v0"\/"v1)"/\"v2))
proof let v0,v2,v1;
  (v0"/\"(v0"\/"v1))"/\"v2 = v0"/\"((v0"\/"v1)"/\"v2) by A3;
  hence thesis by A321;
end;

A328: for v0,v100,v1 holds
 v100"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"v100))) = (v0"\/"v1)"/\"v100
proof let v0,v100,v1;
  (v0"\/"v1)"\/"v100 = v0"\/"(v1"\/"v100) by A7;
  hence thesis by A308;
end;

A332: for v102,v1,v100 holds
 (v100"/\"(v100"\/"v1))"\/"v102 = v100"\/"((v100"/\"(v100"\/"v1))"\/"v102)
proof let v102,v1,v100;
  v100"\/"(v100"/\"(v100"\/"v1)) = v100"/\"(v100"\/"v1) by A113;
  hence thesis by A7;
end;

A337: for v100,v1,v101 holds
  v100"\/"(v101"/\"(v101"\/"v1)) = v101"\/"(v100"\/"(v101"/\"(v101"\/"v1)))
proof let v100,v1,v101;
  v101"\/"(v101"/\"(v101"\/"v1)) = v101"/\"(v101"\/"v1) by A113;
  hence thesis by A57;
end;

A342: for v2,v1,v0 holds
  (v0"/\"(v0"\/"v1))"\/"(v1"/\"v2) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  v0"\/"((v0"/\"(v0"\/"v1))"\/"(v1"/\"v2)) =
    (v0"/\"(v0"\/"v1))"\/"(v1"/\"v2) by A332;
  hence thesis by A153;
end;

A344: for v1,v0 holds
  v1"\/"(v0"/\"(v0"\/"v1)) = v0"\/"v1
proof let v1,v0;
  v0"\/"(v1"\/"(v0"/\"(v0"\/"v1))) = v1"\/"(v0"/\"(v0"\/"v1)) by A337;
  hence thesis by A126;
end;

A347: for v1,v0 holds
  v0"\/"(v0"/\"v1) = v0"/\"(v0"\/"v1)
proof let v1,v0;
  v1"\/"(v0"/\"(v0"\/"v1)) = v0"\/"v1 by A344;
  hence thesis by A249;
end;

A349: for v2,v1,v0 holds
  (v0"/\"(v0"\/"v1))"/\"(v1"\/"v2) = v0"/\"(v1"\/"v2)
proof let v2,v1,v0;
  v0"\/"(v0"/\"v1) = v0"/\"(v0"\/"v1) by A347;
  hence thesis by A244;
end;

A351: for v2,v1,v0 holds
  v0"/\"((v0"\/"v1)"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"/\"(v1"\/"v2) = v0"/\"((v0"\/"v1)"/\"(v1"\/"v2)) by A3;
  hence thesis by A349;
end;

A353: for v0,v2,v1 holds
 (v0"/\"v1)"\/"(v2"/\"(v0"/\"(v0"\/"(v1"/\"v2)))) =
  (v0"\/"(v0"/\"v1))"/\"(v2"\/"(v0"/\"((v0"/\"v1)"\/"v2)))
proof let v0,v2,v1;
  v0"\/"(v0"/\"(v1"/\"v2)) = v0"/\"(v0"\/"(v1"/\"v2)) by A347;
  hence thesis by A174;
end;

A355: for v0,v2,v1 holds
 (v0"/\"v1)"\/"(v2"/\"(v0"/\"(v0"\/"(v1"/\"v2)))) =
   (v0"/\"(v0"\/"v1))"/\"(v2"\/"(v0"/\"((v0"/\"v1)"\/"v2)))
proof let v0,v2,v1;
  v0"\/"(v0"/\"v1) = v0"/\"(v0"\/"v1) by A347;
  hence thesis by A353;
end;

A357: for v0,v2,v1 holds
 (v0"/\"v1)"\/"(v2"/\"(v0"/\"(v0"\/"(v1"/\"v2)))) =
    v0"/\"((v0"\/"v1)"/\"(v2"\/"(v0"/\"((v0"/\"v1)"\/"v2))))
proof let v0,v2,v1;
  (v0"/\"(v0"\/"v1))"/\"(v2"\/"(v0"/\"((v0"/\"v1)"\/"v2))) =
    v0"/\"((v0"\/"v1)"/\"(v2"\/"(v0"/\"((v0"/\"v1)"\/"v2)))) by A3;
  hence thesis by A355;
end;

A360: for v101,v100 holds
 (v100"\/"v101)"/\"(v101"\/"v100) = v100"\/"(v101"/\"(v101"\/"v100))
proof let v101,v100;
  v100"\/"(v101"/\"(v101"\/"v100)) = v101"\/"v100 by A344;
  hence thesis by A11;
end;

A363: for v1,v0 holds
  (v0"\/"v1)"/\"(v1"\/"v0) = v1"\/"v0
proof let v1,v0;
  v0"\/"(v1"/\"(v1"\/"v0)) = v1"\/"v0 by A344;
  hence thesis by A360;
end;

A366: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"((v0"/\"v1)"/\"(v0"/\"(v1"\/"v2))) =
   (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2))
proof let v0,v2,v1;
  (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2) by A23;
  hence thesis by A344;
end;

A368: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v0"/\"((v0"/\"v1)"/\"(v1"\/"v2))) =
   (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2))
proof let v0,v2,v1;
  (v0"/\"v1)"/\"(v0"/\"(v1"\/"v2)) = v0"/\"((v0"/\"v1)"/\"(v1"\/"v2)) by A40;
  hence thesis by A366;
end;

A370: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v0"/\"(v0"/\"(v1"/\"(v1"\/"v2)))) =
   (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2))
proof let v0,v2,v1;
  (v0"/\"v1)"/\"(v1"\/"v2) = v0"/\"(v1"/\"(v1"\/"v2)) by A3;
  hence thesis by A368;
end;

A372: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v0"/\"(v1"/\"(v1"\/"v2))) =
   (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2))
proof let v0,v2,v1;
  v0"/\"(v0"/\"(v1"/\"(v1"\/"v2))) = v0"/\"(v1"/\"(v1"\/"v2)) by A26;
  hence thesis by A370;
end;

A374: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v0"/\"(v1"/\"(v1"\/"v2))) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2) by A23;
  hence thesis by A372;
end;

A377: for v101,v1 holds
 (v1"\/"v101)"\/"(v101"/\"(v1"\/"v101)) = v101"\/"(v1"\/"v101)
proof let v101,v1;
  v101"\/"(v1"\/"v101) = v1"\/"v101 by A51;
  hence thesis by A344;
end;

A380: for v1,v0 holds
  v0"\/"(v1"\/"(v1"/\"(v0"\/"v1))) = v1"\/"(v0"\/"v1)
proof let v1,v0;
  (v0"\/"v1)"\/"(v1"/\"(v0"\/"v1)) = v0"\/"(v1"\/"(v1"/\"(v0"\/"v1))) by A7;
  hence thesis by A377;
end;

A382: for v1,v0 holds
  v0"\/"(v1"/\"(v1"\/"(v0"\/"v1))) = v1"\/"(v0"\/"v1)
proof let v1,v0;
  v1"\/"(v1"/\"(v0"\/"v1)) = v1"/\"(v1"\/"(v0"\/"v1)) by A347;
  hence thesis by A380;
end;

A384: for v1,v0 holds
  v0"\/"(v1"/\"(v0"\/"v1)) = v1"\/"(v0"\/"v1)
proof let v1,v0;
  v1"\/"(v0"\/"v1) = v0"\/"v1 by A51;
  hence thesis by A382;
end;

A386: for v1,v0 holds v0"\/"(v1"/\"(v0"\/"v1)) = v0"\/"v1
proof let v1,v0;
  v1"\/"(v0"\/"v1) = v0"\/"v1 by A51;
  hence thesis by A384;
end;

A389: for v103,v100,v101 holds
 (v100"\/"v101)"/\"((v101"\/"v100)"/\"v103) =
    (v100"\/"(v101"/\"(v101"\/"v100)))"/\"v103
proof let v103,v100,v101;
  v100"\/"(v101"/\"(v101"\/"v100)) = v101"\/"v100 by A344;
  hence thesis by A60;
end;

A392: for v2,v0,v1 holds
  (v0"\/"v1)"/\"((v1"\/"v0)"/\"v2) = (v1"\/"v0)"/\"v2
proof let v2,v0,v1;
  v0"\/"(v1"/\"(v1"\/"v0)) = v1"\/"v0 by A344;
  hence thesis by A389;
end;

A395: for v0,v101,v1 holds
  (v0"/\"v1)"\/"(v0"/\"(v1"/\"v101)) =
    (v0"/\"v1)"/\"((v0"/\"v1)"\/"v101)
proof let v0,v101,v1;
  (v0"/\"v1)"/\"v101 = v0"/\"(v1"/\"v101) by A3;
  hence thesis by A347;
end;

A398: for v0,v2,v1 holds
  (v0"/\"v1)"\/"(v0"/\"(v1"/\"v2)) = v0"/\"(v1"/\"((v0"/\"v1)"\/"v2))
proof let v0,v2,v1;
  (v0"/\"v1)"/\"((v0"/\"v1)"\/"v2) =
     v0"/\"(v1"/\"((v0"/\"v1)"\/"v2)) by A3;
  hence thesis by A395;
end;

A400: for v0,v1 holds
  v0"\/"(v1"/\"v0) = v0"/\"(v0"\/"v1)
proof let v0,v1;
  v0"/\"v1 = v1"/\"v0 by A4;
  hence thesis by A347;
end;

A403: for v102,v1,v100 holds
 (v100"/\"(v100"\/"v1))"\/"v102 = v100"\/"((v100"/\"v1)"\/"v102)
proof let v102,v1,v100;
  v100"\/"(v100"/\"v1) = v100"/\"(v100"\/"v1) by A347;
  hence thesis by A7;
end;

A407: for v100,v1,v101 holds
 (v100"/\"v101)"\/"(v100"/\"(v101"/\"(v101"\/"v1))) =
    v100"/\"(v101"\/"(v101"/\"v1))
proof let v100,v1,v101;
  v101"\/"(v101"/\"v1) = v101"/\"(v101"\/"v1) by A347;
  hence thesis by A23;
end;

A410: for v2,v1,v0 holds
  v0"/\"(v1"/\"((v0"/\"v1)"\/"(v1"\/"v2))) = v0"/\"(v1"\/"(v1"/\"v2))
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v0"/\"(v1"/\"(v1"\/"v2))) =
    v0"/\"(v1"/\"((v0"/\"v1)"\/"(v1"\/"v2))) by A398;
  hence thesis by A407;
end;

A412: for v2,v1,v0 holds
  v0"/\"(v1"/\"(v1"\/"((v0"/\"v1)"\/"v2))) = v0"/\"(v1"\/"(v1"/\"v2))
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v1"\/"v2) = v1"\/"((v0"/\"v1)"\/"v2) by A57;
  hence thesis by A410;
end;

A414: for v2,v1,v0 holds
 v0"/\"(v1"/\"(v1"\/"((v0"/\"v1)"\/"v2))) = v0"/\"(v1"/\"(v1"\/"v2))
proof let v2,v1,v0;
  v1"\/"(v1"/\"v2) = v1"/\"(v1"\/"v2) by A347;
  hence thesis by A412;
end;

A417: for v1,v2,v100 holds
  v100"\/"(v1"/\"(v100"/\"v2)) = v100"/\"(v100"\/"(v1"/\"v2))
proof let v1,v2,v100;
  v100"/\"(v1"/\"v2) = v1"/\"(v100"/\"v2) by A40;
  hence thesis by A347;
end;

A421: for v100,v1,v101 holds
  v100"\/"(v101"/\"(v101"\/"v1)) = v101"\/"(v100"\/"(v101"/\"v1))
proof let v100,v1,v101;
  v101"\/"(v101"/\"v1) = v101"/\"(v101"\/"v1) by A347;
  hence thesis by A57;
end;

A426: for v2,v1,v0 holds
  v0"\/"((v0"/\"v1)"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"\/"(v1"/\"v2) =
    v0"\/"((v0"/\"v1)"\/"(v1"/\"v2)) by A403;
  hence thesis by A342;
end;

A429: for v100,v102,v101 holds
 (v100"\/"v101)"/\"((v101"/\"v102)"\/"v100) =
    (v100"\/"(v101"/\"v102))"/\"((v101"/\"v102)"\/"v100)
proof let v100,v102,v101;
  (v100"\/"(v101"/\"v102))"/\"((v101"/\"v102)"\/"v100)
  = (v101"/\"v102)"\/"v100 by A363;
  hence thesis by A60;
end;

A432: for v0,v2,v1 holds
  (v0"\/"v1)"/\"((v1"/\"v2)"\/"v0) = (v1"/\"v2)"\/"v0
proof let v0,v2,v1;
  (v0"\/"(v1"/\"v2))"/\"((v1"/\"v2)"\/"v0) = (v1"/\"v2)"\/"v0 by A363;
  hence thesis by A429;
end;

A435: for v100,v101,v1,v0 holds
 (v100"\/"(v101"/\"(v0"\/"v1)))"/\"(v100"\/"(v101"/\"(v1"\/"v0)))
   = v100"\/"(v101"/\"((v0"\/"v1)"/\"(v1"\/"v0)))
proof let v100,v101,v1,v0;
  (v0"\/"v1)"/\"(v1"\/"v0) = v1"\/"v0 by A363;
  hence thesis by A68;
end;

A438: for v0,v1,v3,v2 holds
 (v0"\/"(v1"/\"(v2"\/"v3)))"/\"(v0"\/"(v1"/\"(v3"\/"v2))) =
    v0"\/"(v1"/\"(v3"\/"v2))
proof let v0,v1,v3,v2;
  (v2"\/"v3)"/\"(v3"\/"v2) = v3"\/"v2 by A363;
  hence thesis by A435;
end;

A441: for v0,v100,v1 holds
  v100"\/"(v0"/\"(v1"/\"v100)) = v100"/\"(v100"\/"(v0"/\"v1))
proof let v0,v100,v1;
  (v0"/\"v1)"/\"v100 = v0"/\"(v1"/\"v100) by A3;
  hence thesis by A400;
end;

A445: for v1,v2,v101 holds
 (v1"/\"v2)"\/"(v1"/\"(v101"/\"v2)) =
    (v1"/\"v2)"/\"((v1"/\"v2)"\/"v101)
proof let v1,v2,v101;
  v101"/\"(v1"/\"v2) = v1"/\"(v101"/\"v2) by A40;
  hence thesis by A400;
end;

A448: for v0,v1,v2 holds
 (v0"/\"v1)"\/"(v0"/\"(v2"/\"v1)) = v0"/\"(v1"/\"((v0"/\"v1)"\/"v2))
proof let v0,v1,v2;
  (v0"/\"v1)"/\"((v0"/\"v1)"\/"v2) =
    v0"/\"(v1"/\"((v0"/\"v1)"\/"v2)) by A3;
  hence thesis by A445;
end;

A451: for v100,v1,v101 holds
  v100"\/"(v101"/\"(v101"\/"v1)) = v101"\/"(v100"\/"(v1"/\"v101))
proof let v100,v1,v101;
  v101"\/"(v1"/\"v101) = v101"/\"(v101"\/"v1) by A400;
  hence thesis by A57;
end;

A456: for v0,v2,v1 holds
 v0"/\"((v1"\/"v2)"/\"((v0"/\"(v1"\/"v2))"\/"v1)) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"(v0"/\"(v1"/\"(v1"\/"v2))) =
  v0"/\"((v1"\/"v2)"/\"((v0"/\"(v1"\/"v2))"\/"v1)) by A448;
  hence thesis by A374;
end;

A458: for v0,v2,v1 holds
 v0"/\"((v1"\/"v2)"/\"(v1"\/"(v0"/\"(v1"\/"v2)))) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"v1 = v1"\/"(v0"/\"(v1"\/"v2)) by A8;
  hence thesis by A456;
end;

A461: for v0,v1,v102,v100 holds
 v100"/\"(v0"/\"(v1"/\"(v100"/\"v102))) = (v0"/\"v1)"/\"(v100"/\"v102)
proof let v0,v1,v102,v100;
  (v0"/\"v1)"/\"(v100"/\"v102) = v0"/\"(v1"/\"(v100"/\"v102)) by A3;
  hence thesis by A162;
end;

A464: for v1,v2,v3,v0 holds
 v0"/\"(v1"/\"(v2"/\"(v0"/\"v3))) = v1"/\"(v2"/\"(v0"/\"v3))
proof let v1,v2,v3,v0;
  (v1"/\"v2)"/\"(v0"/\"v3) = v1"/\"(v2"/\"(v0"/\"v3)) by A3;
  hence thesis by A461;
end;

A467: for v101,v0,v100,v1 holds
 v100"/\"(v101"/\"(v0"/\"(v1"/\"v100))) = v101"/\"((v0"/\"v1)"/\"v100)
proof let v101,v0,v100,v1;
  (v0"/\"v1)"/\"v100 = v0"/\"(v1"/\"v100) by A3;
  hence thesis by A180;
end;

A470: for v1,v2,v0,v3 holds
 v0"/\"(v1"/\"(v2"/\"(v3"/\"v0))) = v1"/\"(v2"/\"(v3"/\"v0))
proof let v1,v2,v0,v3;
  (v2"/\"v3)"/\"v0 = v2"/\"(v3"/\"v0) by A3;
  hence thesis by A467;
end;

A473: for v101,v1,v102 holds
 (v1"/\"(v102"\/"v1))"/\"(v101"/\"(v102"/\"v1)) =
    v101"/\"(v102"/\"(v1"/\"(v102"\/"v1)))
proof let v101,v1,v102;
  v102"/\"(v1"/\"(v102"\/"v1)) = v102"/\"v1 by A298;
  hence thesis by A180;
end;

A476: for v2,v0,v1 holds
 v0"/\"((v1"\/"v0)"/\"(v2"/\"(v1"/\"v0))) =
   v2"/\"(v1"/\"(v0"/\"(v1"\/"v0)))
proof let v2,v0,v1;
  (v0"/\"(v1"\/"v0))"/\"(v2"/\"(v1"/\"v0)) =
  v0"/\"((v1"\/"v0)"/\"(v2"/\"(v1"/\"v0))) by A3;
  hence thesis by A473;
end;

A478: for v2,v0,v1 holds
 (v1"\/"v0)"/\"(v2"/\"(v1"/\"v0)) = v2"/\"(v1"/\"(v0"/\"(v1"\/"v0)))
proof let v2,v0,v1;
  v0"/\"((v1"\/"v0)"/\"(v2"/\"(v1"/\"v0))) =
    (v1"\/"v0)"/\"(v2"/\"(v1"/\"v0)) by A470;
  hence thesis by A476;
end;

A481: for v2,v1,v0 holds
  (v0"\/"v1)"/\"(v2"/\"(v0"/\"v1)) = v2"/\"(v0"/\"v1)
proof let v2,v1,v0;
  v0"/\"(v1"/\"(v0"\/"v1)) = v0"/\"v1 by A298;
  hence thesis by A478;
end;

A484: for v0,v1,v102,v100 holds
 v100"\/"(v0"\/"(v1"\/"(v100"\/"v102))) = (v0"\/"v1)"\/"(v100"\/"v102)
proof let v0,v1,v102,v100;
  (v0"\/"v1)"\/"(v100"\/"v102) = v0"\/"(v1"\/"(v100"\/"v102)) by A7;
  hence thesis by A195;
end;

A487: for v1,v2,v3,v0 holds
 v0"\/"(v1"\/"(v2"\/"(v0"\/"v3))) = v1"\/"(v2"\/"(v0"\/"v3))
proof let v1,v2,v3,v0;
  (v1"\/"v2)"\/"(v0"\/"v3) = v1"\/"(v2"\/"(v0"\/"v3)) by A7;
  hence thesis by A484;
end;

A490: for v101,v102,v1 holds
 (v1"/\"(v1"\/"v102))"\/"(v101"\/"(v1"\/"v102)) =
    v101"\/"(v102"\/"(v1"/\"(v1"\/"v102)))
proof let v101,v102,v1;
  v102"\/"(v1"/\"(v1"\/"v102)) = v1"\/"v102 by A344;
  hence thesis by A205;
end;

A493: for v2,v1,v0 holds
  v0"\/"((v0"/\"v1)"\/"(v2"\/"(v0"\/"v1))) =
    v2"\/"(v1"\/"(v0"/\"(v0"\/"v1)))
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"\/"(v2"\/"(v0"\/"v1)) =
    v0"\/"((v0"/\"v1)"\/"(v2"\/"(v0"\/"v1))) by A403;
  hence thesis by A490;
end;

A495: for v2,v1,v0 holds
 (v0"/\"v1)"\/"(v2"\/"(v0"\/"v1)) = v2"\/"(v1"\/"(v0"/\"(v0"\/"v1)))
proof let v2,v1,v0;
  v0"\/"((v0"/\"v1)"\/"(v2"\/"(v0"\/"v1))) =
    (v0"/\"v1)"\/"(v2"\/"(v0"\/"v1)) by A487;
  hence thesis by A493;
end;

A497: for v2,v1,v0 holds
  (v0"/\"v1)"\/"(v2"\/"(v0"\/"v1)) = v2"\/"(v0"\/"v1)
proof let v2,v1,v0;
  v1"\/"(v0"/\"(v0"\/"v1)) = v0"\/"v1 by A344;
  hence thesis by A495;
end;

A500: for v1,v2,v0 holds
  v1"\/"(v0"/\"v2) = (v1"\/"(v0"/\"v2))"/\"(v0"\/"v1)
proof let v1,v2,v0;
  (v0"\/"v1)"/\"(v1"\/"(v0"/\"v2)) = v1"\/"(v0"/\"v2) by A86;
  hence thesis by A4;
end;

A505: for v102,v1,v100 holds
 (v100"\/"v1)"/\"((v100"\/"v1)"\/"(v100"/\"v102)) =
    (v100"\/"v1)"\/"(v100"/\"v102)
proof let v102,v1,v100;
  v100"\/"(v100"\/"v1) = v100"\/"v1 by A43;
  hence thesis by A86;
end;

A508: for v1,v2,v0 holds
 (v0"\/"v1)"/\"(v0"\/"(v1"\/"(v0"/\"v2))) = (v0"\/"v1)"\/"(v0"/\"v2)
proof let v1,v2,v0;
  (v0"\/"v1)"\/"(v0"/\"v2) = v0"\/"(v1"\/"(v0"/\"v2)) by A7;
  hence thesis by A505;
end;

A510: for v1,v2,v0 holds
 (v0"\/"v1)"/\"(v0"\/"(v1"\/"(v0"/\"v2))) = v0"\/"(v1"\/"(v0"/\"v2))
proof let v1,v2,v0;
  (v0"\/"v1)"\/"(v0"/\"v2) = v0"\/"(v1"\/"(v0"/\"v2)) by A7;
  hence thesis by A508;
end;

A513: for v0,v2,v1 holds
  v0"/\"(v1"\/"v2) = ((v1"\/"v2)"/\"v0)"\/"(v0"/\"v1)
proof let v0,v2,v1;
  (v0"/\"v1)"\/"((v1"\/"v2)"/\"v0) = v0"/\"(v1"\/"v2) by A131;
  hence thesis by A8;
end;

A517: for v0,v1,v2 holds
  (v0"/\"v1)"\/"((v2"\/"v1)"/\"v0) = v0"/\"(v1"\/"v2)
proof let v0,v1,v2;
  v1"\/"v2=v2"\/"v1 by A8;
  hence thesis by A131;
end;

A520: for v101,v2,v1 holds
 ((v101"\/"v1)"/\"v101)"\/"((v101"\/"v1)"/\"((v1"/\"v2)"\/"v101)) =
    v101"\/"(v1"/\"v2)
proof let v101,v2,v1;
  (v101"\/"v1)"/\"(v101"\/"(v1"/\"v2)) = v101"\/"(v1"/\"v2) by A11;
  hence thesis by A133;
end;

A523: for v0,v2,v1 holds
 (v0"/\"(v0"\/"v1))"\/"((v0"\/"v1)"/\"((v1"/\"v2)"\/"v0)) =
   v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v0"\/"v1)"/\"v0 = v0"/\"(v0"\/"v1) by A4;
  hence thesis by A520;
end;

A525: for v0,v2,v1 holds
 (v0"/\"(v0"\/"v1))"\/"((v1"/\"v2)"\/"v0) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v0"\/"v1)"/\"((v1"/\"v2)"\/"v0) = (v1"/\"v2)"\/"v0 by A432;
  hence thesis by A523;
end;

A527: for v0,v2,v1 holds
  v0"\/"((v0"/\"v1)"\/"((v1"/\"v2)"\/"v0)) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v0"/\"(v0"\/"v1))"\/"((v1"/\"v2)"\/"v0) =
    v0"\/"((v0"/\"v1)"\/"((v1"/\"v2)"\/"v0)) by A403;
  hence thesis by A525;
end;

A529: for v0,v2,v1 holds
  (v0"/\"v1)"\/"((v1"/\"v2)"\/"v0) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  v0"\/"((v0"/\"v1)"\/"((v1"/\"v2)"\/"v0)) =
    (v0"/\"v1)"\/"((v1"/\"v2)"\/"v0) by A205;
  hence thesis by A527;
end;

A532: for v100,v1,v102 holds
  (v100"/\"(v1"/\"(v102"\/"v1)))"\/"(v100"/\"(v102"\/"v1)) =
     v100"/\"((v1"/\"(v102"\/"v1))"\/"v102)
proof let v100,v1,v102;
  v102"\/"(v1"/\"(v102"\/"v1)) = v102"\/"v1 by A386;
  hence thesis by A133;
end;

A535: for v0,v1,v2 holds
 (v0"/\"(v2"\/"v1))"\/"(v0"/\"(v1"/\"(v2"\/"v1))) =
    v0"/\"((v1"/\"(v2"\/"v1))"\/"v2)
proof let v0,v1,v2;
  (v0"/\"(v1"/\"(v2"\/"v1)))"\/"(v0"/\"(v2"\/"v1)) =
    (v0"/\"(v2"\/"v1))"\/"(v0"/\"(v1"/\"(v2"\/"v1))) by A8;
  hence thesis by A532;
end;

A538: for v0,v2,v1 holds
 v0"/\"((v1"\/"v2)"/\"((v0"/\"(v1"\/"v2))"\/"v2)) =
    v0"/\"((v2"/\"(v1"\/"v2))"\/"v1)
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"(v0"/\"(v2"/\"(v1"\/"v2))) =
    v0"/\"((v1"\/"v2)"/\"((v0"/\"(v1"\/"v2))"\/"v2)) by A448;
  hence thesis by A535;
end;

A540: for v0,v2,v1 holds
  v0"/\"((v1"\/"v2)"/\"(v2"\/"(v0"/\"(v1"\/"v2)))) =
    v0"/\"((v2"/\"(v1"\/"v2))"\/"v1)
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"v2 = v2"\/"(v0"/\"(v1"\/"v2)) by A8;
  hence thesis by A538;
end;

A542: for v0,v2,v1 holds
  v0"/\"((v1"\/"v2)"/\"(v2"\/"(v0"/\"(v1"\/"v2)))) =
    v0"/\"(v1"\/"(v2"/\"(v1"\/"v2)))
proof let v0,v2,v1;
  (v2"/\"(v1"\/"v2))"\/"v1 = v1"\/"(v2"/\"(v1"\/"v2)) by A8;
  hence thesis by A540;
end;

A544: for v0,v2,v1 holds
  v0"/\"((v1"\/"v2)"/\"(v2"\/"(v0"/\"(v1"\/"v2)))) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  v1"\/"(v2"/\"(v1"\/"v2)) = v1"\/"v2 by A386;
  hence thesis by A542;
end;

A547: for v101,v1,v2 holds
 ((v101"\/"v1)"/\"v101)"\/"(v101"\/"(v2"/\"v1)) =
    (v101"\/"v1)"/\"(v101"\/"(v2"/\"v1))
proof let v101,v1,v2;
  (v101"\/"v1)"/\"(v101"\/"(v2"/\"v1)) = v101"\/"(v2"/\"v1) by A186;
  hence thesis by A23;
end;

A550: for v0,v1,v2 holds
  (v0"/\"(v0"\/"v1))"\/"(v0"\/"(v2"/\"v1)) = (v0"\/"v1)"/\"(v0"\/"(v2"/\"v1))
proof let v0,v1,v2;
  (v0"\/"v1)"/\"v0 = v0"/\"(v0"\/"v1) by A4;
  hence thesis by A547;
end;

A552: for v2,v1,v0 holds
  v0"\/"((v0"/\"(v0"\/"v1))"\/"(v2"/\"v1)) = (v0"\/"v1)"/\"(v0"\/"(v2"/\"v1))
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"\/"(v0"\/"(v2"/\"v1)) =
     v0"\/"((v0"/\"(v0"\/"v1))"\/"(v2"/\"v1)) by A57;
  hence thesis by A550;
end;

A554: for v2,v1,v0 holds
 v0"\/"(v0"\/"((v0"/\"v1)"\/"(v2"/\"v1))) = (v0"\/"v1)"/\"(v0"\/"(v2"/\"v1))
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"\/"(v2"/\"v1) =
    v0"\/"((v0"/\"v1)"\/"(v2"/\"v1)) by A403;
  hence thesis by A552;
end;

A556: for v2,v1,v0 holds
  v0"\/"((v0"/\"v1)"\/"(v2"/\"v1)) = (v0"\/"v1)"/\"(v0"\/"(v2"/\"v1))
proof let v2,v1,v0;
  v0"\/"(v0"\/"((v0"/\"v1)"\/"(v2"/\"v1))) =
    v0"\/"((v0"/\"v1)"\/"(v2"/\"v1)) by A43;
  hence thesis by A554;
end;

A558: for v2,v1,v0 holds
  v0"\/"((v0"/\"v1)"\/"(v2"/\"v1)) = v0"\/"(v2"/\"v1)
proof let v2,v1,v0;
  (v0"\/"v1)"/\"(v0"\/"(v2"/\"v1)) = v0"\/"(v2"/\"v1) by A186;
  hence thesis by A556;
end;

A561: for v102,v1,v100 holds
 (v100"\/"v1)"/\"(v100"\/"(v102"/\"(v100"\/"v1))) =
    v100"\/"(v102"/\"(v100"\/"v1))
proof let v102,v1,v100;
  v100"\/"(v100"\/"v1) = v100"\/"v1 by A43;
  hence thesis by A186;
end;

A565: for v102,v100,v1 holds
 (v1"\/"v100)"/\"(v100"\/"(v102"/\"(v1"\/"v100))) =
    v100"\/"(v102"/\"(v1"\/"v100))
proof let v102,v100,v1;
  v100"\/"(v1"\/"v100) = v1"\/"v100 by A51;
  hence thesis by A186;
end;

A568: for v0,v2,v1 holds
  v0"/\"(v1"\/"(v0"/\"(v1"\/"v2))) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  (v1"\/"v2)"/\"(v1"\/"(v0"/\"(v1"\/"v2))) = v1"\/"(v0"/\"(v1"\/"v2)) by A561;
  hence thesis by A458;
end;

A570: for v0,v2,v1 holds
  v0"/\"(v2"\/"(v0"/\"(v1"\/"v2))) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  (v1"\/"v2)"/\"(v2"\/"(v0"/\"(v1"\/"v2))) = v2"\/"(v0"/\"(v1"\/"v2)) by A565;
  hence thesis by A544;
end;

A574: for v101,v1,v2,v102 holds
 (v1"/\"((v1"\/"v102)"/\"v2))"/\"(v101"/\"(v1"/\"(v102"/\"v2))) =
   v101"/\"(v102"/\"(v1"/\"((v1"\/"v102)"/\"v2)))
proof let v101,v1,v2,v102;
  v102"/\"(v1"/\"((v1"\/"v102)"/\"v2)) = v1"/\"(v102"/\"v2) by A323;
  hence thesis by A180;
end;

A577: for v3,v0,v2,v1 holds
 v0"/\"(((v0"\/"v1)"/\"v2)"/\"(v3"/\"(v0"/\"(v1"/\"v2)))) =
    v3"/\"(v1"/\"(v0"/\"((v0"\/"v1)"/\"v2)))
proof let v3,v0,v2,v1;
  (v0"/\"((v0"\/"v1)"/\"v2))"/\"(v3"/\"(v0"/\"(v1"/\"v2))) =
     v0"/\"(((v0"\/"v1)"/\"v2)"/\"(v3"/\"(v0"/\"(v1"/\"v2)))) by A3;
  hence thesis by A574;
end;

A579: for v3,v0,v2,v1 holds
 v0"/\"((v0"\/"v1)"/\"(v2"/\"(v3"/\"(v0"/\"(v1"/\"v2))))) =
   v3"/\"(v1"/\"(v0"/\"((v0"\/"v1)"/\"v2)))
proof let v3,v0,v2,v1;
  ((v0"\/"v1)"/\"v2)"/\"(v3"/\"(v0"/\"(v1"/\"v2))) =
    (v0"\/"v1)"/\"(v2"/\"(v3"/\"(v0"/\"(v1"/\"v2)))) by A3;
  hence thesis by A577;
end;

A581: for v3,v0,v2,v1 holds
 v0"/\"((v0"\/"v1)"/\"(v3"/\"(v0"/\"(v1"/\"v2)))) =
    v3"/\"(v1"/\"(v0"/\"((v0"\/"v1)"/\"v2)))
proof let v3,v0,v2,v1;
  v2"/\"(v3"/\"(v0"/\"(v1"/\"v2))) =  v3"/\"(v0"/\"(v1"/\"v2)) by A470;
  hence thesis by A579;
end;

A584: for v2,v0,v3,v1 holds
 (v0"\/"v1)"/\"(v2"/\"(v0"/\"(v1"/\"v3))) =
    v2"/\"(v1"/\"(v0"/\"((v0"\/"v1)"/\"v3)))
proof let v2,v0,v3,v1;
  v0"/\"((v0"\/"v1)"/\"(v2"/\"(v0"/\"(v1"/\"v3)))) =
    (v0"\/"v1)"/\"(v2"/\"(v0"/\"(v1"/\"v3))) by A464;
  hence thesis by A581;
end;

A586: for v2,v0,v3,v1 holds
 (v0"\/"v1)"/\"(v2"/\"(v0"/\"(v1"/\"v3))) = v2"/\"(v0"/\"(v1"/\"v3))
proof let v2,v0,v3,v1;
  v1"/\"(v0"/\"((v0"\/"v1)"/\"v3)) = v0"/\"(v1"/\"v3) by A323;
  hence thesis by A584;
end;

A589: for v102,v1,v100 holds
 v100"/\"((v100"/\"(v100"\/"v1))"/\"((v100"/\"v1)"\/"v102)) =
   (v100"/\"((v100"/\"v1)"\/"v102))
proof let v102,v1,v100;
  v100"\/"(v100"/\"v1) = v100"/\"(v100"\/"v1) by A347;
  hence thesis by A351;
end;

A592: for v2,v1,v0 holds
 v0"/\"(v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2))) = v0"/\"((v0"/\"v1)"\/"v2)
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"/\"((v0"/\"v1)"\/"v2) =
    v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2)) by A3;
  hence thesis by A589;
end;

A594: for v2,v1,v0 holds
 v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2)) = v0"/\"((v0"/\"v1)"\/"v2)
proof let v2,v1,v0;
  v0"/\"(v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2))) =
    v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2)) by A26;
  hence thesis by A592;
end;

A597: for v100,v1,v2,v101 holds
  v100"/\"((v100"\/"v101)"/\"(v1"\/"(v101"\/"v2))) =
   v100"/\"(v101"\/"(v1"\/"(v101"\/"v2)))
proof let v100,v1,v2,v101;
  v101"\/"(v1"\/"(v101"\/"v2)) = v1"\/"(v101"\/"v2) by A195;
  hence thesis by A351;
end;

A600: for v0,v2,v3,v1 holds
  v0"/\"((v0"\/"v1)"/\"(v2"\/"(v1"\/"v3))) = v0"/\"(v2"\/"(v1"\/"v3))
proof let v0,v2,v3,v1;
  v1"\/"(v2"\/"(v1"\/"v3)) = v2"\/"(v1"\/"v3) by A195;
  hence thesis by A597;
end;

A603: for v101,v1,v0 holds
  v0"\/"((v0"/\"v1)"\/"v101) = v101"\/"(v0"/\"(v0"\/"v1))
proof let v101,v1,v0;
  (v0"/\"(v0"\/"v1))"\/"v101 = v0"\/"((v0"/\"v1)"\/"v101) by A403;
  hence thesis by A8;
end;

A607: for v101,v1,v0 holds
  v0"\/"((v0"/\"v1)"\/"((v0"/\"(v0"\/"v1))"/\"v101)) =
    (v0"/\"(v0"\/"v1))"/\"((v0"/\"(v0"\/"v1))"\/"v101)
proof let v101,v1,v0;
  (v0"/\"(v0"\/"v1))"\/"((v0"/\"(v0"\/"v1))"/\"v101) =
     v0"\/"((v0"/\"v1)"\/"((v0"/\"(v0"\/"v1))"/\"v101)) by A403;
  hence thesis by A347;
end;

A610: for v2,v1,v0 holds
  v0"\/"((v0"/\"v1)"\/"(v0"/\"((v0"\/"v1)"/\"v2))) =
    (v0"/\"(v0"\/"v1))"/\"((v0"/\"(v0"\/"v1))"\/"v2)
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"/\"v2 = v0"/\"((v0"\/"v1)"/\"v2) by A3;
  hence thesis by A607;
end;

A612: for v2,v1,v0 holds
  v0"\/"((v0"/\"v1)"\/"(v0"/\"((v0"\/"v1)"/\"v2))) =
    (v0"/\"(v0"\/"v1))"/\"(v0"\/"((v0"/\"v1)"\/"v2))
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"\/"v2 = v0"\/"((v0"/\"v1)"\/"v2) by A403;
  hence thesis by A610;
end;

A614: for v2,v1,v0 holds
  v0"\/"((v0"/\"v1)"\/"(v0"/\"((v0"\/"v1)"/\"v2))) =
    v0"/\"((v0"\/"v1)"/\"(v0"\/"((v0"/\"v1)"\/"v2)))
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"/\"(v0"\/"((v0"/\"v1)"\/"v2)) =
     v0"/\"((v0"\/"v1)"/\"(v0"\/"((v0"/\"v1)"\/"v2))) by A3;
  hence thesis by A612;
end;

A617: for v101,v1,v0 holds
 v0"\/"((v0"/\"v1)"\/"(v101"/\"(v0"/\"(v0"\/"v1)))) =
   (v0"/\"(v0"\/"v1))"/\"((v0"/\"(v0"\/"v1))"\/"v101)
proof let v101,v1,v0;
  (v0"/\"(v0"\/"v1))"\/"(v101"/\"(v0"/\"(v0"\/"v1))) =
     v0"\/"((v0"/\"v1)"\/"(v101"/\"(v0"/\"(v0"\/"v1)))) by A403;
  hence thesis by A400;
end;

A620: for v2,v1,v0 holds
 v0"\/"((v0"/\"v1)"\/"(v2"/\"(v0"/\"(v0"\/"v1)))) =
   (v0"/\"(v0"\/"v1))"/\"(v0"\/"((v0"/\"v1)"\/"v2))
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"\/"v2 = v0"\/"((v0"/\"v1)"\/"v2) by A403;
  hence thesis by A617;
end;

A622: for v2,v1,v0 holds
 v0"\/"((v0"/\"v1)"\/"(v2"/\"(v0"/\"(v0"\/"v1)))) =
   v0"/\"((v0"\/"v1)"/\"(v0"\/"((v0"/\"v1)"\/"v2)))
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"/\"(v0"\/"((v0"/\"v1)"\/"v2)) =
     v0"/\"((v0"\/"v1)"/\"(v0"\/"((v0"/\"v1)"\/"v2))) by A3;
  hence thesis by A620;
end;

A625: for v101,v0,v102,v1 holds
 (v0"/\"v1)"\/"(v101"/\"(v0"/\"(v1"/\"v102))) =
    (v0"/\"v1)"/\"((v0"/\"v1)"\/"(v101"/\"v102))
proof let v101,v0,v102,v1;
  (v0"/\"v1)"/\"v102 = v0"/\"(v1"/\"v102) by A3;
  hence thesis by A417;
end;

A628: for v2,v0,v3,v1 holds
 (v0"/\"v1)"\/"(v2"/\"(v0"/\"(v1"/\"v3))) =
    v0"/\"(v1"/\"((v0"/\"v1)"\/"(v2"/\"v3)))
proof let v2,v0,v3,v1;
  (v0"/\"v1)"/\"((v0"/\"v1)"\/"(v2"/\"v3)) =
    v0"/\"(v1"/\"((v0"/\"v1)"\/"(v2"/\"v3))) by A3;
  hence thesis by A625;
end;

A631: for v0,v1,v102,v100 holds
 v100"\/"(v0"/\"(v1"/\"(v100"/\"v102))) =
   v100"/\"(v100"\/"((v0"/\"v1)"/\"v102))
proof let v0,v1,v102,v100;
  (v0"/\"v1)"/\"(v100"/\"v102) = v0"/\"(v1"/\"(v100"/\"v102)) by A3;
  hence thesis by A417;
end;

A634: for v1,v2,v3,v0 holds
 v0"\/"(v1"/\"(v2"/\"(v0"/\"v3))) = v0"/\"(v0"\/"(v1"/\"(v2"/\"v3)))
proof let v1,v2,v3,v0;
  (v1"/\"v2)"/\"v3 = v1"/\"(v2"/\"v3) by A3;
  hence thesis by A631;
end;

A637: for v1,v2,v102,v101 holds
 ((v1"/\"((v101"/\"v102)"/\"v2))"/\"v101)"\/"(v102"/\"
   ((v101"/\"v102)"/\"((v101"/\"v102)"\/"(v1"/\"v2)))) =
      (v102"\/"((v101"\/"v102)"/\"(v1"/\"((v101"/\"v102)"/\"
         v2))))"/\"((v1"/\"((v101"/\"v102)"/\"v2))"\/"v101)
proof let v1,v2,v102,v101;
  (v101"/\"v102)"\/"(v1"/\"((v101"/\"v102)"/\"v2)) =
    (v101"/\"v102)"/\"((v101"/\"v102)"\/"(v1"/\"v2)) by A417;
  hence thesis by A20;
end;

A640: for v0,v1,v3,v2 holds
 ((v0"/\"(v1"/\"(v2"/\"v3)))"/\"v1)"\/"(v2"/\"((v1"/\"v2)"/\"
    ((v1"/\"v2)"\/"(v0"/\"v3)))) =
  (v2"\/"((v1"\/"v2)"/\"(v0"/\"((v1"/\"v2)"/\"v3))))"/\"
    ((v0"/\"((v1"/\"v2)"/\"v3))"\/"v1)
proof let v0,v1,v3,v2;
  (v1"/\"v2)"/\"v3 = v1"/\"(v2"/\"v3) by A3;
  hence thesis by A637;
end;

A642: for v0,v1,v3,v2 holds
 (v1"/\"(v0"/\"(v1"/\"(v2"/\"v3))))"\/"(v2"/\"((v1"/\"v2)"/\"
    ((v1"/\"v2)"\/"(v0"/\"v3)))) =
  (v2"\/"((v1"\/"v2)"/\"(v0"/\"((v1"/\"v2)"/\"v3))))"/\"
    ((v0"/\"((v1"/\"v2)"/\"v3))"\/"v1)
proof let v0,v1,v3,v2;
  (v0"/\"(v1"/\"(v2"/\"v3)))"/\"v1 = v1"/\"(v0"/\"(v1"/\"(v2"/\"v3))) by A4;
  hence thesis by A640;
end;

A645: for v1,v0,v3,v2 holds
 (v1"/\"(v0"/\"(v2"/\"v3)))"\/"(v2"/\"((v0"/\"v2)"/\"
  ((v0"/\"v2)"\/"(v1"/\"v3)))) =
   (v2"\/"((v0"\/"v2)"/\"(v1"/\"((v0"/\"v2)"/\"v3))))"/\"
     ((v1"/\"((v0"/\"v2)"/\"v3))"\/"v0)
proof let v1,v0,v3,v2;
  v0"/\"(v1"/\"(v0"/\"(v2"/\"v3))) = v1"/\"(v0"/\"(v2"/\"v3)) by A162;
  hence thesis by A642;
end;

A648: for v0,v1,v3,v2 holds
 (v0"/\"(v1"/\"(v2"/\"v3)))"\/"(v2"/\"(v1"/\"(v2"/\"((v1"/\"v2)"\/"
  (v0"/\"v3))))) =
 (v2"\/"((v1"\/"v2)"/\"(v0"/\"((v1"/\"v2)"/\"v3))))"/\"
    ((v0"/\"((v1"/\"v2)"/\"v3))"\/"v1)
proof let v0,v1,v3,v2;
  (v1"/\"v2)"/\"((v1"/\"v2)"\/"(v0"/\"v3)) =
    v1"/\"(v2"/\"((v1"/\"v2)"\/"(v0"/\"v3))) by A3;
  hence thesis by A645;
end;

A650: for v0,v1,v3,v2 holds
 (v0"/\"(v1"/\"(v2"/\"v3)))"\/"(v1"/\"(v2"/\"((v1"/\"v2)"\/"(v0"/\"v3)))) =
   (v2"\/"((v1"\/"v2)"/\"(v0"/\"((v1"/\"v2)"/\"v3))))"/\"
      ((v0"/\"((v1"/\"v2)"/\"v3))"\/"v1)
proof let v0,v1,v3,v2;
  v2"/\"(v1"/\"(v2"/\"((v1"/\"v2)"\/"(v0"/\"v3)))) =
    v1"/\"(v2"/\"((v1"/\"v2)"\/"(v0"/\"v3))) by A162;
  hence thesis by A648;
end;

A652: for v0,v1,v3,v2 holds
 (v0"/\"(v1"/\"(v2"/\"v3)))"\/"(v1"/\"(v2"/\"((v1"/\"v2)"\/"(v0"/\"v3)))) =
   (v2"\/"((v1"\/"v2)"/\"(v0"/\"(v1"/\"(v2"/\"v3)))))"/\"
      ((v0"/\"((v1"/\"v2)"/\"v3))"\/"v1)
proof let v0,v1,v3,v2;
  (v1"/\"v2)"/\"v3 = v1"/\"(v2"/\"v3) by A3;
  hence thesis by A650;
end;

A654: for v0,v1,v3,v2 holds
 (v0"/\"(v1"/\"(v2"/\"v3)))"\/"(v1"/\"(v2"/\"((v1"/\"v2)"\/"(v0"/\"v3)))) =
   (v2"\/"(v0"/\"(v1"/\"(v2"/\"v3))))"/\"((v0"/\"((v1"/\"v2)"/\"v3))"\/"v1)
proof let v0,v1,v3,v2;
  (v1"\/"v2)"/\"(v0"/\"(v1"/\"(v2"/\"v3))) = v0"/\"(v1"/\"(v2"/\"v3)) by A586;
  hence thesis by A652;
end;

A656: for v0,v1,v3,v2 holds
 (v0"/\"(v1"/\"(v2"/\"v3)))"\/"(v1"/\"(v2"/\"((v1"/\"v2)"\/"(v0"/\"v3)))) =
   (v2"/\"(v2"\/"(v0"/\"(v1"/\"v3))))"/\"((v0"/\"((v1"/\"v2)"/\"v3))"\/"v1)
proof let v0,v1,v3,v2;
  v2"\/"(v0"/\"(v1"/\"(v2"/\"v3))) = v2"/\"(v2"\/"(v0"/\"(v1"/\"v3))) by A634;
  hence thesis by A654;
end;

A658: for v0,v1,v3,v2 holds
 (v0"/\"(v1"/\"(v2"/\"v3)))"\/"(v1"/\"(v2"/\"((v1"/\"v2)"\/"(v0"/\"v3)))) =
   (v2"/\"(v2"\/"(v0"/\"(v1"/\"v3))))"/\"((v0"/\"(v1"/\"(v2"/\"v3)))"\/"v1)
proof let v0,v1,v3,v2;
  (v1"/\"v2)"/\"v3 = v1"/\"(v2"/\"v3) by A3;
  hence thesis by A656;
end;

A660: for v0,v1,v3,v2 holds
 (v0"/\"(v1"/\"(v2"/\"v3)))"\/"(v1"/\"(v2"/\"((v1"/\"v2)"\/"(v0"/\"v3)))) =
    (v2"/\"(v2"\/"(v0"/\"(v1"/\"v3))))"/\"(v1"\/"(v0"/\"(v1"/\"(v2"/\"v3))))
proof let v0,v1,v3,v2;
  (v0"/\"(v1"/\"(v2"/\"v3)))"\/"v1 = v1"\/"(v0"/\"(v1"/\"(v2"/\"v3))) by A8;
  hence thesis by A658;
end;

A662: for v0,v1,v3,v2 holds
 (v0"/\"(v1"/\"(v2"/\"v3)))"\/"(v1"/\"(v2"/\"
    ((v1"/\"v2)"\/"(v0"/\"v3)))) = (v2"/\"(v2"\/"
      (v0"/\"(v1"/\"v3))))"/\"(v1"/\"(v1"\/"(v0"/\"(v2"/\"v3))))
proof let v0,v1,v3,v2;
  v1"\/"(v0"/\"(v1"/\"(v2"/\"v3))) = v1"/\"(v1"\/"(v0"/\"(v2"/\"v3))) by A417;
  hence thesis by A660;
end;

A664: for v0,v1,v3,v2 holds
 (v0"/\"(v1"/\"(v2"/\"v3)))"\/"(v1"/\"(v2"/\"((v1"/\"v2)"\/"(v0"/\"v3)))) =
    v1"/\"((v2"/\"(v2"\/"(v0"/\"(v1"/\"v3))))"/\"(v1"\/"(v0"/\"(v2"/\"v3))))
proof let v0,v1,v3,v2;
  (v2"/\"(v2"\/"(v0"/\"(v1"/\"v3))))"/\"(v1"/\"(v1"\/"(v0"/\"(v2"/\"v3))))
    = v1"/\"((v2"/\"(v2"\/"(v0"/\"(v1"/\"v3))))"/\"(v1"\/"(v0"/\"(v2"/\"v3))))
      by A40;
  hence thesis by A662;
end;

A666: for v0,v1,v3,v2 holds
 (v0"/\"(v1"/\"(v2"/\"v3)))"\/"(v1"/\"(v2"/\"((v1"/\"v2)"\/"(v0"/\"v3)))) =
    v1"/\"(v2"/\"((v2"\/"(v0"/\"(v1"/\"v3)))"/\"(v1"\/"(v0"/\"(v2"/\"v3)))))
proof let v0,v1,v3,v2;
  (v2"/\"(v2"\/"(v0"/\"(v1"/\"v3))))"/\"(v1"\/"
     (v0"/\"(v2"/\"v3))) = v2"/\"((v2"\/"(v0"/\"
      (v1"/\"v3)))"/\"(v1"\/"(v0"/\"(v2"/\"v3)))) by A3;
  hence thesis by A664;
end;

A669: for v0,v100,v1 holds
  v100"\/"((v100"\/"((v1"\/"v100)"/\"v0))"/\"(v0"\/"v1)) =
    (v0"/\"v1)"\/"(v100"/\"(v100"\/"((v1"/\"v100)"\/"v0)))
proof let v0,v100,v1;
  (v0"/\"v1)"\/"(v100"/\"((v1"/\"v100)"\/"v0)) =
    (v100"\/"((v1"\/"v100)"/\"v0))"/\"(v0"\/"v1) by A20;
  hence thesis by A421;
end;

A675: for v101,v1,v2,v100 holds
 (v100"/\"v101)"\/"(v101"/\"(v1"\/"(v100"/\"(v100"\/"v2))))
   =  v101"/\"(v100"\/"(v1"\/"(v100"/\"v2)))
proof let v101,v1,v2,v100;
  v100"\/"(v1"\/"(v100"/\"v2)) = v1"\/"(v100"/\"(v100"\/"v2)) by A421;
  hence thesis by A128;
end;

A679: for v101,v2,v100 holds
  (v2"\/"((v100"\/"v2)"/\"(v100"\/"v101)))"/\"((v100"\/"v101)"\/"v100) =
    v100"\/"((v100"/\"v101)"\/"(v2"/\"((v100"/\"v2)"\/"(v100"\/"v101))))
proof let v101,v2,v100;
  (v100"/\"(v100"\/"v101))"\/"(v2"/\"((v100"/\"v2)"\/"(v100"\/"v101))) =
    (v2"\/"((v100"\/"v2)"/\"(v100"\/"v101)))"/\"((v100"\/"v101)"\/"v100)
       by A101;
  hence thesis by A403;
end;

A682: for v2,v0,v1 holds
 (v0"\/"((v1"\/"v0)"/\"(v1"\/"v2)))"/\"(v1"\/"(v1"\/"v2)) =
   v1"\/"((v1"/\"v2)"\/"(v0"/\"((v1"/\"v0)"\/"(v1"\/"v2))))
proof let v2,v0,v1;
  (v1"\/"v2)"\/"v1 = v1"\/"(v1"\/"v2) by A8;
  hence thesis by A679;
end;

A684: for v2,v0,v1 holds
  (v0"\/"((v1"\/"v0)"/\"(v1"\/"v2)))"/\"(v1"\/"v2) =
    v1"\/"((v1"/\"v2)"\/"(v0"/\"((v1"/\"v0)"\/"(v1"\/"v2))))
proof let v2,v0,v1;
  v1"\/"(v1"\/"v2) = v1"\/"v2 by A43;
  hence thesis by A682;
end;

A686: for v2,v0,v1 holds
 (v1"\/"v2)"/\"(v0"\/"((v1"\/"v0)"/\"(v1"\/"v2))) =
   v1"\/"((v1"/\"v2)"\/"(v0"/\"((v1"/\"v0)"\/"(v1"\/"v2))))
proof let v2,v0,v1;
  (v0"\/"((v1"\/"v0)"/\"(v1"\/"v2)))"/\"(v1"\/"v2) =
    (v1"\/"v2)"/\"(v0"\/"((v1"\/"v0)"/\"(v1"\/"v2))) by A4;
  hence thesis by A684;
end;

A689: for v1,v2,v0 holds
 (v0"\/"v1)"/\"(v2"\/"((v0"\/"v2)"/\"(v0"\/"v1))) =
   v0"\/"((v0"/\"v1)"\/"(v2"/\"(v0"\/"((v0"/\"v2)"\/"v1))))
proof let v1,v2,v0;
  (v0"/\"v2)"\/"(v0"\/"v1) = v0"\/"((v0"/\"v2)"\/"v1) by A57;
  hence thesis by A686;
end;

A693: for v102,v1,v100 holds
  v100"\/"(v100"\/"((v100"/\"v1)"\/"((v100"\/"v1)"/\"v102))) =
    v100"\/"((v100"\/"v1)"/\"v102)
proof let v102,v1,v100;
  (v100"/\"(v100"\/"v1))"\/"((v100"\/"v1)"/\"v102) =
    v100"\/"((v100"/\"v1)"\/"((v100"\/"v1)"/\"v102)) by A403;
  hence thesis by A426;
end;

A696: for v2,v1,v0 holds
 v0"\/"((v0"/\"v1)"\/"((v0"\/"v1)"/\"v2)) = v0"\/"((v0"\/"v1)"/\"v2)
proof let v2,v1,v0;
  v0"\/"(v0"\/"((v0"/\"v1)"\/"((v0"\/"v1)"/\"v2))) =
    v0"\/"((v0"/\"v1)"\/"((v0"\/"v1)"/\"v2)) by A43;
  hence thesis by A693;
end;

A699: for v100,v1,v102 holds
 (v100"/\"(v102"\/"v1))"\/"(v102"/\"(v102"\/"((v102"/\"v1)"\/"v100))) =
   ((v102"\/"(((v102"\/"v1)"\/"v102)"/\"v100))"/\"(v100"\/"(v102"\/"v1)))
proof let v100,v1,v102;
  (v102"/\"(v102"\/"v1))"\/"v100 = v102"\/"((v102"/\"v1)"\/"v100) by A403;
  hence thesis by A104;
end;

A702: for v0,v2,v1 holds
  (v0"/\"(v1"\/"v2))"\/"(v1"/\"(v1"\/"((v1"/\"v2)"\/"v0))) =
    (v1"\/"((v1"\/"(v1"\/"v2))"/\"v0))"/\"(v0"\/"(v1"\/"v2))
proof let v0,v2,v1;
  (v1"\/"v2)"\/"v1 = v1"\/"(v1"\/"v2) by A8;
  hence thesis by A699;
end;

A704: for v0,v2,v1 holds
  (v0"/\"(v1"\/"v2))"\/"(v1"/\"(v1"\/"((v1"/\"v2)"\/"v0))) =
    (v1"\/"((v1"\/"v2)"/\"v0))"/\"(v0"\/"(v1"\/"v2))
proof let v0,v2,v1;
  v1"\/"(v1"\/"v2) = v1"\/"v2 by A43;
  hence thesis by A702;
end;

A707: for v102,v2,v101 holds
  (v101"/\"v2)"\/"(v102"/\"(v101"/\"v2)) =
    (v101"/\"v2)"/\"((v101"/\"v2)"\/"(v101"/\"v102))
proof let v102,v2,v101;
  v101"/\"(v102"/\"(v101"/\"v2)) = v102"/\"(v101"/\"v2) by A162;
  hence thesis by A441;
end;

A710: for v2,v1,v0 holds
  (v0"/\"v1)"/\"((v0"/\"v1)"\/"v2) =
    (v0"/\"v1)"/\"((v0"/\"v1)"\/"(v0"/\"v2))
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v2"/\"(v0"/\"v1)) = (v0"/\"v1)"/\"((v0"/\"v1)"\/"v2) by A400;
  hence thesis by A707;
end;

A712: for v2,v1,v0 holds
  v0"/\"(v1"/\"((v0"/\"v1)"\/"v2)) = (v0"/\"v1)"/\"((v0"/\"v1)"\/"(v0"/\"v2))
proof let v2,v1,v0;
  (v0"/\"v1)"/\"((v0"/\"v1)"\/"v2) =
     v0"/\"(v1"/\"((v0"/\"v1)"\/"v2)) by A3;
  hence thesis by A710;
end;

A714: for v2,v1,v0 holds
 v0"/\"(v1"/\"((v0"/\"v1)"\/"v2)) =
   v0"/\"(v1"/\"((v0"/\"v1)"\/"(v0"/\"v2)))
proof let v2,v1,v0;
  (v0"/\"v1)"/\"((v0"/\"v1)"\/"(v0"/\"v2)) =
    v0"/\"(v1"/\"((v0"/\"v1)"\/"(v0"/\"v2))) by A3;
  hence thesis by A712;
end;

A717: for v2,v1,v0 holds
  (v1"\/"v0)"/\"(v2"/\"(v0"/\"v1)) = v2"/\"(v0"/\"v1)
proof let v2,v1,v0;
  v0"\/"v1 = v1"\/"v0 by A8;
  hence thesis by A481;
end;

A721: for v102,v1,v0 holds
 (v0"/\"v1)"\/"(v102"/\"(v0"/\"v1)) =
   (v0"/\"v1)"/\"((v0"/\"v1)"\/"((v0"\/"v1)"/\"v102))
proof let v102,v1,v0;
  (v0"\/"v1)"/\"(v102"/\"(v0"/\"v1)) = v102"/\"(v0"/\"v1) by A481;
  hence thesis by A441;
end;

A724: for v2,v1,v0 holds
 (v0"/\"v1)"/\"((v0"/\"v1)"\/"v2) =
    (v0"/\"v1)"/\"((v0"/\"v1)"\/"((v0"\/"v1)"/\"v2))
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v2"/\"(v0"/\"v1)) = (v0"/\"v1)"/\"((v0"/\"v1)"\/"v2) by A400;
  hence thesis by A721;
end;

A726: for v2,v1,v0 holds
  v0"/\"(v1"/\"((v0"/\"v1)"\/"v2)) =
    (v0"/\"v1)"/\"((v0"/\"v1)"\/"((v0"\/"v1)"/\"v2))
proof let v2,v1,v0;
  (v0"/\"v1)"/\"((v0"/\"v1)"\/"v2) =
    v0"/\"(v1"/\"((v0"/\"v1)"\/"v2)) by A3;
  hence thesis by A724;
end;

A728: for v2,v1,v0 holds
 v0"/\"(v1"/\"((v0"/\"v1)"\/"v2)) =
   v0"/\"(v1"/\"((v0"/\"v1)"\/"((v0"\/"v1)"/\"v2)))
proof let v2,v1,v0;
  (v0"/\"v1)"/\"((v0"/\"v1)"\/"((v0"\/"v1)"/\"v2)) =
     v0"/\"(v1"/\"((v0"/\"v1)"\/"((v0"\/"v1)"/\"v2))) by A3;
  hence thesis by A726;
end;

A732: for v2,v100,v1,v0 holds
 v100"/\"((v100"\/"(v0"/\"v1))"/\"(v2"\/"(v0"\/"v1))) =
   v100"/\"((v0"/\"v1)"\/"(v2"\/"(v0"\/"v1)))
proof let v2,v100,v1,v0;
  (v0"/\"v1)"\/"(v2"\/"(v0"\/"v1)) = v2"\/"(v0"\/"v1) by A497;
  hence thesis by A351;
end;

A735: for v3,v0,v2,v1 holds
 v0"/\"((v0"\/"(v1"/\"v2))"/\"(v3"\/"(v1"\/"v2))) = v0"/\"(v3"\/"(v1"\/"v2))
proof let v3,v0,v2,v1;
  (v1"/\"v2)"\/"(v3"\/"(v1"\/"v2)) = v3"\/"(v1"\/"v2) by A497;
  hence thesis by A732;
end;

A738: for v102,v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"v102 = (v0"\/"(v1"/\"v2))"/\"((v1"\/"v0)"/\"v102)
proof let v102,v0,v2,v1;
  (v0"\/"(v1"/\"v2))"/\"(v1"\/"v0) = v0"\/"(v1"/\"v2) by A500;
  hence thesis by A3;
end;

A743: for v102,v1,v2,v100 holds
 ((v1"\/"(v100"/\"(v100"\/"v2)))"/\"v102)"\/"(v102"/\"v100) =
   v102"/\"(v100"\/"(v1"\/"(v100"/\"v2)))
proof let v102,v1,v2,v100;
  v100"\/"(v1"\/"(v100"/\"v2)) = v1"\/"(v100"/\"(v100"\/"v2)) by A421;
  hence thesis by A513;
end;

A747: for v101,v1,v102 holds
 (v102"\/"v1)"\/"(v101"/\"(v102"\/"v1)) = (v102"\/"v1)"\/"(v101"/\"v102)
proof let v101,v1,v102;
  ((v102"\/"v1)"/\"v101)"\/"(v101"/\"v102) = v101"/\"(v102"\/"v1) by A513;
  hence thesis by A426;
end;

A750: for v2,v1,v0 holds
 (v0"\/"v1)"/\"((v0"\/"v1)"\/"v2) = (v0"\/"v1)"\/"(v2"/\"v0)
proof let v2,v1,v0;
  (v0"\/"v1)"\/"(v2"/\"(v0"\/"v1)) = (v0"\/"v1)"/\"((v0"\/"v1)"\/"v2) by A400;
  hence thesis by A747;
end;

A752: for v0,v2,v1 holds
 (v0"\/"v1)"/\"(v0"\/"(v1"\/"v2)) = (v0"\/"v1)"\/"(v2"/\"v0)
proof let v0,v2,v1;
  (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2) by A7;
  hence thesis by A750;
end;

A754: for v0,v2,v1 holds
 (v0"\/"v1)"/\"(v0"\/"(v1"\/"v2)) = v0"\/"(v1"\/"(v2"/\"v0))
proof let v0,v2,v1;
  (v0"\/"v1)"\/"(v2"/\"v0) = v0"\/"(v1"\/"(v2"/\"v0)) by A7;
  hence thesis by A752;
end;

A758: for v102,v1,v100 holds
 v100"\/"(v102"/\"(v100"\/"v1)) =
   ((v100"\/"v1)"/\"v102)"\/"(v100"/\"(v100"\/"v102))
proof let v102,v1,v100;
  ((v100"\/"v1)"/\"v102)"\/"(v102"/\"v100) = v102"/\"(v100"\/"v1) by A513;
  hence thesis by A451;
end;

A764: for v102,v1,v2,v100 holds
 ((v1"\/"(v100"/\"(v100"\/"v2)))"/\"v102)"\/"(v102"/\"v100) =
   v102"/\"(v100"\/"(v1"\/"(v2"/\"v100)))
proof let v102,v1,v2,v100;
  v100"\/"(v1"\/"(v2"/\"v100)) = v1"\/"(v100"/\"(v100"\/"v2)) by A451;
  hence thesis by A513;
end;

A767: for v3,v0,v2,v1 holds
  v3"/\"(v1"\/"(v0"\/"(v1"/\"v2))) = v3"/\"(v1"\/"(v0"\/"(v2"/\"v1)))
proof let v3,v0,v2,v1;
  ((v0"\/"(v1"/\"(v1"\/"v2)))"/\"v3)"\/"(v3"/\"v1) =
    v3"/\"(v1"\/"(v0"\/"(v1"/\"v2))) by A743;
  hence thesis by A764;
end;

A770: for v0,v2,v3,v1 holds
 v0"/\"(v1"\/"(v2"\/"(v1"/\"v3))) = v0"/\"((v1"\/"v2)"/\"(v1"\/"(v2"\/"v3)))
proof let v0,v2,v3,v1;
  v1"\/"(v2"\/"(v3"/\"v1)) = (v1"\/"v2)"/\"(v1"\/"(v2"\/"v3)) by A754;
  hence thesis by A767;
end;

A772: for v1,v2,v3,v0 holds
 (v0"/\"v1)"\/"(v1"/\"(v2"\/"(v0"/\"(v0"\/"v3)))) =
    v1"/\"((v0"\/"v2)"/\"(v0"\/"(v2"\/"v3)))
proof let v1,v2,v3,v0;
  v1"/\"(v0"\/"(v2"\/"(v0"/\"v3))) = v1"/\"((v0"\/"v2)"/\"(v0"\/"(v2"\/"v3)))
    by A770;
  hence thesis by A675;
end;

A774: for v0,v2,v1 holds
 (v0"\/"v1)"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"v2))) = v0"\/"(v1"\/"(v0"/\"v2))
proof let v0,v2,v1;
  (v0"\/"v1)"/\"(v0"\/"(v1"\/"(v0"/\"v2))) =
    (v0"\/"v1)"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"v2))) by A770;
  hence thesis by A510;
end;

A776: for v0,v2,v1 holds
 (v0"\/"v1)"/\"(v0"\/"(v1"\/"v2)) = v0"\/"(v1"\/"(v0"/\"v2))
proof let v0,v2,v1;
  (v0"\/"v1)"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"v2))) =
    (v0"\/"v1)"/\"(v0"\/"(v1"\/"v2)) by A26;
  hence thesis by A774;
end;

A779: for v2,v1,v0 holds
 (v0"\/"(v0"/\"v1))"/\"(v0"\/"((v0"/\"v1)"\/"((v0"\/"v1)"/\"v2))) =
    v0"/\"((v0"\/"v1)"/\"(v0"\/"((v0"/\"v1)"\/"v2)))
proof let v2,v1,v0;
  v0"\/"((v0"/\"v1)"\/"(v0"/\"((v0"\/"v1)"/\"v2))) =
    (v0"\/"(v0"/\"v1))"/\"(v0"\/"((v0"/\"v1)"\/"((v0"\/"v1)"/\"v2))) by A776;
  hence thesis by A614;
end;

A781: for v2,v1,v0 holds
 (v0"/\"(v0"\/"v1))"/\"(v0"\/"((v0"/\"v1)"\/"((v0"\/"v1)"/\"v2))) =
    v0"/\"((v0"\/"v1)"/\"(v0"\/"((v0"/\"v1)"\/"v2)))
proof let v2,v1,v0;
  v0"\/"(v0"/\"v1) = v0"/\"(v0"\/"v1) by A347;
  hence thesis by A779;
end;

A783: for v2,v1,v0 holds
  (v0"/\"(v0"\/"v1))"/\"(v0"\/"((v0"\/"v1)"/\"v2)) =
    v0"/\"((v0"\/"v1)"/\"(v0"\/"((v0"/\"v1)"\/"v2)))
proof let v2,v1,v0;
  v0"\/"((v0"/\"v1)"\/"((v0"\/"v1)"/\"v2)) = v0"\/"((v0"\/"v1)"/\"v2) by A696;
  hence thesis by A781;
end;

A785: for v2,v1,v0 holds
  v0"/\"((v0"\/"v1)"/\"(v0"\/"((v0"\/"v1)"/\"v2))) =
    v0"/\"((v0"\/"v1)"/\"(v0"\/"((v0"/\"v1)"\/"v2)))
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"/\"(v0"\/"((v0"\/"v1)"/\"v2)) =
    v0"/\"((v0"\/"v1)"/\"(v0"\/"((v0"\/"v1)"/\"v2))) by A3;
  hence thesis by A783;
end;

A787: for v2,v1,v0 holds
  v0"/\"(v0"\/"((v0"\/"v1)"/\"v2)) =
    v0"/\"((v0"\/"v1)"/\"(v0"\/"((v0"/\"v1)"\/"v2)))
proof let v2,v1,v0;
  (v0"\/"v1)"/\"(v0"\/"((v0"\/"v1)"/\"v2)) = v0"\/"((v0"\/"v1)"/\"v2) by A198;
  hence thesis by A785;
end;

A790: for v2,v1,v0 holds
 v0"\/"((v0"/\"v1)"\/"(v2"/\"(v0"/\"(v0"\/"v1)))) =
    v0"/\"(v0"\/"((v0"\/"v1)"/\"v2))
proof let v2,v1,v0;
  v0"/\"((v0"\/"v1)"/\"(v0"\/"((v0"/\"v1)"\/"v2))) =
    v0"/\"(v0"\/"((v0"\/"v1)"/\"v2)) by A787;
  hence thesis by A622;
end;

A793: for v102,v100,v1 holds
 ((v1"/\"v100)"\/"((v102"\/"(v1"/\"(v1"\/"v100)))"/\"v100))=(v100"/\"((v1"/\"
(v1"\/"v100))"\/"v102))
proof let v102,v100,v1;
  v100"/\"(v1"/\"(v1"\/"v100)) = v1"/\"v100 by A308;
  hence thesis by A517;
end;

A796: for v2,v1,v0 holds
  (v0"/\"v1)"\/"(v1"/\"(v2"\/"(v0"/\"(v0"\/"v1)))) =
     v1"/\"((v0"/\"(v0"\/"v1))"\/"v2)
proof let v2,v1,v0;
  (v2"\/"(v0"/\"(v0"\/"v1)))"/\"v1 = v1"/\"(v2"\/"(v0"/\"(v0"\/"v1))) by A4;
  hence thesis by A793;
end;

A798: for v0,v1,v2 holds
  v1"/\"((v0"\/"v2)"/\"(v0"\/"(v2"\/"v1))) = v1"/\"((v0"/\"(v0"\/"v1))"\/"v2)
proof let v0,v1,v2;
  (v0"/\"v1)"\/"(v1"/\"(v2"\/"(v0"/\"(v0"\/"v1)))) =
     v1"/\"((v0"\/"v2)"/\"(v0"\/"(v2"\/"v1))) by A772;
  hence thesis by A796;
end;

A801: for v0,v2,v1 holds
  (v1"\/"v2)"/\"v0 = v0"/\"((v1"/\"(v1"\/"v0))"\/"v2)
proof let v0,v2,v1;
  v0"/\"((v1"\/"v2)"/\"(v1"\/"(v2"\/"v0))) = (v1"\/"v2)"/\"v0 by A328;
  hence thesis by A798;
end;

A804: for v2,v1,v0 holds (v0"\/"v1)"/\"v2 = v2"/\"(v0"\/"((v0"/\"v2)"\/"v1))
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v2))"\/"v1 = v0"\/"((v0"/\"v2)"\/"v1) by A403;
  hence thesis by A801;
end;

A808: for v2,v1,v0 holds
  v0"\/"((v0"/\"v1)"\/"((v0"\/"v1)"/\"v2)) =
    (v0"\/"v1)"/\"(v2"\/"((v0"\/"v2)"/\"(v0"\/"v1)))
proof let v2,v1,v0;
  v2"/\"(v0"\/"((v0"/\"v2)"\/"v1)) = (v0"\/"v1)"/\"v2 by A804;
  hence thesis by A689;
end;

A810: for v2,v1,v0 holds
  v0"\/"((v0"\/"v1)"/\"v2) = (v0"\/"v1)"/\"(v2"\/"((v0"\/"v2)"/\"(v0"\/"v1)))
proof let v2,v1,v0;
  v0"\/"((v0"/\"v1)"\/"((v0"\/"v1)"/\"v2)) =
    v0"\/"((v0"\/"v1)"/\"v2) by A696;
  hence thesis by A808;
end;

A813: for v1,v2,v102,v101 holds
 ((v1"/\"((v101"/\"v102)"/\"v2))"/\"v101)"\/"
    ((v101"/\"v102)"/\"((v101"/\"v102)"\/"(v1"/\"v2))) =
  (v1"/\"((v101"/\"v102)"/\"v2))"\/"(v101"/\"v102)
proof let v1,v2,v102,v101;
  (v101"/\"v102)"\/"(v1"/\"((v101"/\"v102)"/\"v2)) =
    (v101"/\"v102)"/\"((v101"/\"v102)"\/"(v1"/\"v2)) by A417;
  hence thesis by A529;
end;

A816: for v0,v1,v3,v2 holds
 ((v0"/\"(v1"/\"(v2"/\"v3)))"/\"v1)"\/"((v1"/\"v2)"/\"
    ((v1"/\"v2)"\/"(v0"/\"v3))) =
  (v0"/\"((v1"/\"v2)"/\"v3))"\/"(v1"/\"v2)
proof let v0,v1,v3,v2;
  (v1"/\"v2)"/\"v3 = v1"/\"(v2"/\"v3) by A3;
  hence thesis by A813;
end;

A818: for v0,v1,v3,v2 holds
 (v1"/\"(v0"/\"(v1"/\"(v2"/\"v3))))"\/"
   ((v1"/\"v2)"/\"((v1"/\"v2)"\/"(v0"/\"v3))) =
    (v0"/\"((v1"/\"v2)"/\"v3))"\/"(v1"/\"v2)
proof let v0,v1,v3,v2;
  (v0"/\"(v1"/\"(v2"/\"v3)))"/\"v1 = v1"/\"(v0"/\"(v1"/\"(v2"/\"v3))) by A4;
  hence thesis by A816;
end;

A821: for v1,v0,v3,v2 holds
 (v1"/\"(v0"/\"(v2"/\"v3)))"\/"((v0"/\"v2)"/\"((v0"/\"v2)"\/"(v1"/\"v3))) =
    (v1"/\"((v0"/\"v2)"/\"v3))"\/"(v0"/\"v2)
proof let v1,v0,v3,v2;
  v0"/\"(v1"/\"(v0"/\"(v2"/\"v3))) = v1"/\"(v0"/\"(v2"/\"v3)) by A162;
  hence thesis by A818;
end;

A824: for v0,v1,v3,v2 holds
 (v0"/\"(v1"/\"(v2"/\"v3)))"\/"(v1"/\"(v2"/\"((v1"/\"v2)"\/"(v0"/\"v3))))
    = (v0"/\"((v1"/\"v2)"/\"v3))"\/"(v1"/\"v2)
proof let v0,v1,v3,v2;
  (v1"/\"v2)"/\"((v1"/\"v2)"\/"(v0"/\"v3)) =
    v1"/\"(v2"/\"((v1"/\"v2)"\/"(v0"/\"v3))) by A3;
  hence thesis by A821;
end;

A826: for v2,v0,v3,v1 holds
 v1"/\"(v2"/\"((v2"\/"(v0"/\"(v1"/\"v3)))"/\"(v1"\/"(v0"/\"(v2"/\"v3)))))
   = (v0"/\"((v1"/\"v2)"/\"v3))"\/"(v1"/\"v2)
proof let v2,v0,v3,v1;
  (v0"/\"(v1"/\"(v2"/\"v3)))"\/"(v1"/\"(v2"/\"((v1"/\"v2)"\/"(v0"/\"v3))))
     = v1"/\"(v2"/\"((v2"\/"(v0"/\"(v1"/\"v3)))"/\"(v1"\/"(v0"/\"(v2"/\"v3)))))
       by A666;
  hence thesis by A824;
end;

A829: for v1,v2,v3,v0 holds
 v0"/\"(v1"/\"((v1"\/"(v2"/\"(v0"/\"v3)))"/\"(v0"\/"(v2"/\"(v1"/\"v3))))) =
   (v2"/\"(v0"/\"(v1"/\"v3)))"\/"(v0"/\"v1)
proof let v1,v2,v3,v0;
  (v0"/\"v1)"/\"v3 = v0"/\"(v1"/\"v3) by A3;
  hence thesis by A826;
end;

A831: for v1,v2,v3,v0 holds
 v0"/\"(v1"/\"((v1"\/"(v2"/\"(v0"/\"v3)))"/\"(v0"\/"(v2"/\"(v1"/\"v3))))) =
  (v0"/\"v1)"\/"(v2"/\"(v0"/\"(v1"/\"v3)))
proof let v1,v2,v3,v0;
  (v2"/\"(v0"/\"(v1"/\"v3)))"\/"(v0"/\"v1) =
    (v0"/\"v1)"\/"(v2"/\"(v0"/\"(v1"/\"v3))) by A8;
  hence thesis by A829;
end;

A833: for v1,v2,v3,v0 holds
 v0"/\"(v1"/\"((v1"\/"(v2"/\"(v0"/\"v3)))"/\"(v0"\/"(v2"/\"(v1"/\"v3)))))
   = v0"/\"(v1"/\"((v0"/\"v1)"\/"(v2"/\"v3)))
proof let v1,v2,v3,v0;
  (v0"/\"v1)"\/"(v2"/\"(v0"/\"(v1"/\"v3))) =
    v0"/\"(v1"/\"((v0"/\"v1)"\/"(v2"/\"v3))) by A628;
  hence thesis by A831;
end;

A836: for v2,v1,v0 holds
 v0"/\"(v1"/\"((v1"\/"((v0"\/"v1)"/\"(v0"/\"v2)))"/\"
    (v0"\/"((v0"\/"v1)"/\"(v1"/\"v2))))) =
 v0"/\"(v1"/\"((v0"/\"v1)"\/"v2))
proof let v2,v1,v0;
  v0"/\"(v1"/\"((v0"/\"v1)"\/"((v0"\/"v1)"/\"v2))) =
     v0"/\"(v1"/\"((v1"\/"((v0"\/"v1)"/\"(v0"/\"v2)))"/\"
    (v0"\/"((v0"\/"v1)"/\"(v1"/\"v2))))) by A833;
  hence thesis by A728;
end;

A838: for v2,v1,v0 holds
 v0"/\"(v1"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"(v0"\/"
    ((v0"\/"v1)"/\"(v1"/\"v2))))) =
   v0"/\"(v1"/\"((v0"/\"v1)"\/"v2))
proof let v2,v1,v0;
  (v0"\/"v1)"/\"(v0"/\"v2) = v0"/\"((v0"\/"v1)"/\"v2) by A40;
  hence thesis by A836;
end;

A840: for v2,v1,v0 holds
 v0"/\"(v1"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
   (v0"\/"(v1"/\"((v0"\/"v1)"/\"v2))))) =
      v0"/\"(v1"/\"((v0"/\"v1)"\/"v2))
proof let v2,v1,v0;
  (v0"\/"v1)"/\"(v1"/\"v2) = v1"/\"((v0"\/"v1)"/\"v2) by A40;
  hence thesis by A838;
end;

A843: for v2,v1,v0 holds
 v0"/\"(v1"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"(v0"/\"v2))))"/\"
   (v0"\/"(v1"/\"((v0"\/"v1)"/\"(v0"/\"v2)))))) =
       v0"/\"(v1"/\"((v0"/\"v1)"\/"v2))
proof let v2,v1,v0;
  v0"/\"(v1"/\"((v0"/\"v1)"\/"(v0"/\"v2))) =
     v0"/\"(v1"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"
  (v0"/\"v2))))"/\"(v0"\/"(v1"/\"((v0"\/"v1)"/\"(v0"/\"v2)))))) by A840;
  hence thesis by A714;
end;

A845: for v2,v1,v0 holds
 v0"/\"(v1"/\"((v1"\/"(v0"/\"(v0"/\"((v0"\/"v1)"/\"v2))))"/\"
   (v0"\/"(v1"/\"((v0"\/"v1)"/\"(v0"/\"v2)))))) =
    v0"/\"(v1"/\"((v0"/\"v1)"\/"v2))
proof let v2,v1,v0;
  (v0"\/"v1)"/\"(v0"/\"v2) = v0"/\"((v0"\/"v1)"/\"v2) by A40;
  hence thesis by A843;
end;

A847: for v2,v1,v0 holds
  v0"/\"(v1"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
    (v0"\/"(v1"/\"((v0"\/"v1)"/\"(v0"/\"v2)))))) =
      v0"/\"(v1"/\"((v0"/\"v1)"\/"v2))
proof let v2,v1,v0;
  v0"/\"(v0"/\"((v0"\/"v1)"/\"v2)) = v0"/\"((v0"\/"v1)"/\"v2) by A26;
  hence thesis by A845;
end;

A849: for v2,v1,v0 holds
   v0"/\"(v1"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
     (v0"\/"(v1"/\"(v0"/\"((v0"\/"v1)"/\"v2)))))) =
        v0"/\"(v1"/\"((v0"/\"v1)"\/"v2))
proof let v2,v1,v0;
  (v0"\/"v1)"/\"(v0"/\"v2) = v0"/\"((v0"\/"v1)"/\"v2) by A40;
  hence thesis by A847;
end;

A851: for v2,v1,v0 holds
 v0"/\"(v1"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
    (v0"\/"(v0"/\"(v1"/\"v2))))) = v0"/\"(v1"/\"((v0"/\"v1)"\/"v2))
proof let v2,v1,v0;
  v1"/\"(v0"/\"((v0"\/"v1)"/\"v2)) = v0"/\"(v1"/\"v2) by A323;
  hence thesis by A849;
end;

A853: for v2,v1,v0 holds
 v0"/\"(v1"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
    (v0"/\"(v0"\/"(v1"/\"v2))))) = v0"/\"(v1"/\"((v0"/\"v1)"\/"v2))
proof let v2,v1,v0;
  v0"\/"(v0"/\"(v1"/\"v2)) = v0"/\"(v0"\/"(v1"/\"v2)) by A347;
  hence thesis by A851;
end;

A855: for v2,v1,v0 holds
  v0"/\"(v1"/\"(v0"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
   (v0"\/"(v1"/\"v2))))) =
     v0"/\"(v1"/\"((v0"/\"v1)"\/"v2))
proof let v2,v1,v0;
  (v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"(v0"/\"(v0"\/"(v1"/\"v2))) =
    v0"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"(v0"\/"(v1"/\"v2))) by A40;
  hence thesis by A853;
end;

A857: for v2,v1,v0 holds
 v1"/\"(v0"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
   (v0"\/"(v1"/\"v2)))) = v0"/\"(v1"/\"((v0"/\"v1)"\/"v2))
proof let v2,v1,v0;
  v0"/\"(v1"/\"(v0"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
    (v0"\/"(v1"/\"v2))))) =
       v1"/\"(v0"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"(v0"\/"(v1"/\"v2))))
   by A162;
  hence thesis by A855;
end;

A860: for v2,v0,v1 holds
 v0"/\"(v1"/\"((v0"\/"(v1"/\"((v1"\/"v0)"/\"v2)))"/\"(v1"\/"(v0"/\"v2)))) =
    v1"/\"(v0"/\"((v0"\/"(v1"/\"((v1"\/"v0)"/\"v2)))"/\"
     (v1"\/"(v0"/\"((v1"\/"v0)"/\"v2)))))
proof let v2,v0,v1;
  v1"/\"(v0"/\"((v1"/\"v0)"\/"v2)) =
    v1"/\"(v0"/\"((v0"\/"(v1"/\"((v1"\/"v0)"/\"v2)))"/\"
      (v1"\/"(v0"/\"((v1"\/"v0)"/\"v2))))) by A840;
  hence thesis by A857;
end;

A864: for v2,v1,v0 holds
  v0"/\"(v1"/\"((v0"/\"v1)"\/"v2)) =
    v1"/\"(v0"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"(v0"\/"(v1"/\"v2))))
proof let v2,v1,v0;
  v0"/\"(v1"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
     (v0"\/"(v1"/\"((v0"\/"v1)"/\"v2))))) =
  v1"/\"(v0"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
    (v0"\/"(v1"/\"v2)))) by A860;
  hence thesis by A840;
end;

A867: for v102,v1,v100 holds
 v100"\/"((v100"/\"v1)"\/"(v102"/\"(v100"/\"v1))) =
    v100"\/"(v102"/\"(v100"/\"v1))
proof let v102,v1,v100;
  v100"/\"(v100"/\"v1) = v100"/\"v1 by A26;
  hence thesis by A558;
end;

A870: for v2,v1,v0 holds
 v0"\/"((v0"/\"v1)"/\"((v0"/\"v1)"\/"v2)) = v0"\/"(v2"/\"(v0"/\"v1))
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v2"/\"(v0"/\"v1)) = (v0"/\"v1)"/\"((v0"/\"v1)"\/"v2) by A400;
  hence thesis by A867;
end;

A872: for v2,v1,v0 holds
 v0"\/"(v0"/\"(v1"/\"((v0"/\"v1)"\/"v2))) = v0"\/"(v2"/\"(v0"/\"v1))
proof let v2,v1,v0;
  (v0"/\"v1)"/\"((v0"/\"v1)"\/"v2) = v0"/\"(v1"/\"((v0"/\"v1)"\/"v2)) by A3;
  hence thesis by A870;
end;

A874: for v2,v1,v0 holds
 v0"\/"(v1"/\"(v0"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
    (v0"\/"(v1"/\"v2))))) = v0"\/"(v2"/\"(v0"/\"v1))
proof let v2,v1,v0;
  v0"/\"(v1"/\"((v0"/\"v1)"\/"v2)) =
    v1"/\"(v0"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"(v0"\/"(v1"/\"v2))))
      by A864;
  hence thesis by A872;
end;

A876: for v2,v1,v0 holds
 v0"/\"(v0"\/"(v1"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
   (v0"\/"(v1"/\"v2))))) = v0"\/"(v2"/\"(v0"/\"v1))
proof let v2,v1,v0;
  v0"\/"(v1"/\"(v0"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
  (v0"\/"(v1"/\"v2))))) =
    v0"/\"(v0"\/"(v1"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
      (v0"\/"(v1"/\"v2))))) by A417;
  hence thesis by A874;
end;

A878: for v2,v1,v0 holds
  v0"/\"(v0"\/"(v1"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
  (v0"\/"(v1"/\"v2))))) = v0"/\"(v0"\/"(v2"/\"v1))
proof let v2,v1,v0;
  v0"\/"(v2"/\"(v0"/\"v1)) = v0"/\"(v0"\/"(v2"/\"v1)) by A417;
  hence thesis by A876;
end;

A881: for v102,v1,v100 holds
 v100"\/"(v100"\/"((v100"/\"v1)"\/"(v102"/\"(v100"\/"v1)))) =
   v100"\/"(v102"/\"(v100"\/"v1))
proof let v102,v1,v100;
  (v100"/\"(v100"\/"v1))"\/"(v102"/\"(v100"\/"v1)) =
     v100"\/"((v100"/\"v1)"\/"(v102"/\"(v100"\/"v1))) by A403;
  hence thesis by A558;
end;

A884: for v2,v1,v0 holds
 v0"\/"((v0"/\"v1)"\/"(v2"/\"(v0"\/"v1))) = v0"\/"(v2"/\"(v0"\/"v1))
proof let v2,v1,v0;
  v0"\/"(v0"\/"((v0"/\"v1)"\/"(v2"/\"(v0"\/"v1)))) =
    v0"\/"((v0"/\"v1)"\/"(v2"/\"(v0"\/"v1))) by A43;
  hence thesis by A881;
end;

A887: for v101,v2,v1 holds
 (v1"\/"(v101"/\"(v1"\/"v2)))"\/"(v101"/\"(v1"\/"v2)) =
   (v1"\/"(v101"/\"(v1"\/"v2)))"/\"((v1"\/"(v101"/\"(v1"\/"v2)))"\/"v101)
proof let v101,v2,v1;
  v101"/\"(v1"\/"(v101"/\"(v1"\/"v2))) = v101"/\"(v1"\/"v2) by A568;
  hence thesis by A400;
end;

A890: for v1,v2,v0 holds
 (v1"/\"(v0"\/"v2))"\/"(v0"\/"(v1"/\"(v0"\/"v2))) =
    (v0"\/"(v1"/\"(v0"\/"v2)))"/\"((v0"\/"(v1"/\"(v0"\/"v2)))"\/"v1)
proof let v1,v2,v0;
  (v0"\/"(v1"/\"(v0"\/"v2)))"\/"(v1"/\"(v0"\/"v2)) =
    (v1"/\"(v0"\/"v2))"\/"(v0"\/"(v1"/\"(v0"\/"v2))) by A8;
  hence thesis by A887;
end;

A893: for v0,v2,v1 holds
 v1"\/"(v0"/\"(v1"\/"v2)) =
    (v1"\/"(v0"/\"(v1"\/"v2)))"/\"((v1"\/"(v0"/\"(v1"\/"v2)))"\/"v0)
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"(v1"\/"(v0"/\"(v1"\/"v2))) =
    v1"\/"(v0"/\"(v1"\/"v2)) by A51;
  hence thesis by A890;
end;

A896: for v1,v2,v0 holds
 v0"\/"(v1"/\"(v0"\/"v2)) =
  (v0"\/"(v1"/\"(v0"\/"v2)))"/\"(v1"\/"(v0"\/"(v1"/\"(v0"\/"v2))))
proof let v1,v2,v0;
  (v0"\/"(v1"/\"(v0"\/"v2)))"\/"v1 = v1"\/"(v0"\/"(v1"/\"(v0"\/"v2))) by A8;
  hence thesis by A893;
end;

A898: for v1,v2,v0 holds
 v0"\/"(v1"/\"(v0"\/"v2)) =
    (v0"\/"(v1"/\"(v0"\/"v2)))"/\"((v1"\/"v0)"/\"(v1"\/"(v0"\/"(v0"\/"v2))))
proof let v1,v2,v0;
  v1"\/"(v0"\/"(v1"/\"(v0"\/"v2))) =
    (v1"\/"v0)"/\"(v1"\/"(v0"\/"(v0"\/"v2))) by A776;
  hence thesis by A896;
end;

A900: for v1,v2,v0 holds
 v0"\/"(v1"/\"(v0"\/"v2)) =
   (v0"\/"(v1"/\"(v0"\/"v2)))"/\"((v1"\/"v0)"/\"(v1"\/"(v0"\/"v2)))
proof let v1,v2,v0;
  v0"\/"(v0"\/"v2) = v0"\/"v2 by A43;
  hence thesis by A898;
end;

A902: for v1,v2,v0 holds
 v0"\/"(v1"/\"(v0"\/"v2)) =
   (v0"\/"(v1"/\"(v0"\/"v2)))"/\"(v1"\/"(v0"\/"v2))
proof let v1,v2,v0;
  (v0"\/"(v1"/\"(v0"\/"v2)))"/\"((v1"\/"v0)"/\"(v1"\/"(v0"\/"v2))) =
    (v0"\/"(v1"/\"(v0"\/"v2)))"/\"(v1"\/"(v0"\/"v2)) by A738;
  hence thesis by A900;
end;

A905: for v0,v1,v2 holds
  v0"/\"(v1"\/"((v2"\/"v1)"/\"v0)) = v0"/\"(v2"\/"v1)
proof let v0,v1,v2;
  v0"/\"(v2"\/"v1) = (v2"\/"v1)"/\"v0 by A4;
  hence thesis by A570;
end;

A908: for v100,v101,v1,v2 holds
 v100"/\"(v101"/\"(v2"\/"v1)) =
   v101"/\"(v100"/\"(v1"\/"(v101"/\"(v2"\/"v1))))
proof let v100,v101,v1,v2;
  v101"/\"(v1"\/"(v101"/\"(v2"\/"v1))) = v101"/\"(v2"\/"v1) by A570;
  hence thesis by A40;
end;

A913: for v2,v1,v0 holds
  v0"\/"((v0"\/"v1)"/\"v2) = (v0"\/"v1)"/\"(v0"\/"v2)
proof let v2,v1,v0;
  (v0"\/"v1)"/\"(v2"\/"((v0"\/"v2)"/\"(v0"\/"v1))) =
     (v0"\/"v1)"/\"(v0"\/"v2) by A905;
  hence thesis by A810;
end;

A915: for v0,v2,v1 holds
 (v0"/\"v1)"\/"(v2"/\"(v0"/\"(v0"\/"(v1"/\"v2)))) =
   (v0"\/"v1)"/\"(v0"/\"((v0"/\"v1)"\/"v2))
proof let v0,v2,v1;
  v0"/\"((v0"\/"v1)"/\"(v2"\/"(v0"/\"((v0"/\"v1)"\/"v2)))) =
    (v0"\/"v1)"/\"(v0"/\"((v0"/\"v1)"\/"v2)) by A908;
  hence thesis by A357;
end;

A917: for v0,v2,v1 holds
 (v0"/\"v1)"\/"(v2"/\"(v0"/\"(v0"\/"(v1"/\"v2)))) =
    v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2))
proof let v0,v2,v1;
  (v0"\/"v1)"/\"(v0"/\"((v0"/\"v1)"\/"v2)) =
    v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2)) by A40;
  hence thesis by A915;
end;

A919: for v0,v2,v1 holds
 (v0"/\"v1)"\/"(v2"/\"(v0"/\"(v0"\/"(v1"/\"v2)))) =
    v0"/\"((v0"/\"v1)"\/"v2)
proof let v0,v2,v1;
  v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2)) =
    v0"/\"((v0"/\"v1)"\/"v2) by A594;
  hence thesis by A917;
end;

A921: for v2,v1,v0 holds
 v0"\/"((v0"/\"v1)"\/"(v2"/\"(v0"/\"(v0"\/"v1)))) =
   v0"/\"((v0"\/"v1)"/\"(v0"\/"v2))
proof let v2,v1,v0;
  v0"\/"((v0"\/"v1)"/\"v2) = (v0"\/"v1)"/\"(v0"\/"v2) by A913;
  hence thesis by A790;
end;

A923: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v1"/\"(v1"\/"((v1"/\"v2)"\/"v0))) =
    ((v1"\/"v2)"/\"(v1"\/"v0))"/\"(v0"\/"(v1"\/"v2))
proof let v0,v2,v1;
  v1"\/"((v1"\/"v2)"/\"v0) = (v1"\/"v2)"/\"(v1"\/"v0) by A913;
  hence thesis by A704;
end;

A925: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v1"/\"(v1"\/"((v1"/\"v2)"\/"v0))) =
    (v1"\/"v2)"/\"((v1"\/"v0)"/\"(v0"\/"(v1"\/"v2)))
proof let v0,v2,v1;
  ((v1"\/"v2)"/\"(v1"\/"v0))"/\"(v0"\/"(v1"\/"v2)) =
    (v1"\/"v2)"/\"((v1"\/"v0)"/\"(v0"\/"(v1"\/"v2))) by A3;
  hence thesis by A923;
end;

A927: for v0,v2,v1 holds
 (v0"/\"v1)"\/"(v2"/\"(v2"\/"((v1"/\"v2)"\/"v0))) =
    (v2"\/"((v1"\/"v2)"/\"v0))"/\"(v2"\/"(v0"\/"v1))
proof let v0,v2,v1;
  v2"\/"((v2"\/"((v1"\/"v2)"/\"v0))"/\"(v0"\/"v1)) =
    (v2"\/"((v1"\/"v2)"/\"v0))"/\"(v2"\/"(v0"\/"v1)) by A913;
  hence thesis by A669;
end;

A929: for v2,v1,v0 holds
 ((v0"/\"v1)"\/"v2)"/\"((v2"\/"((v1"\/"v2)"/\"v0))"/\"(v0"\/"v1)) =
    (v2"\/"((v2"\/"v1)"/\"v0))"/\"(v0"\/"v1)
proof let v2,v1,v0;
  v1"\/"v2 = v2"\/"v1 by A8;
  hence thesis by A110;
end;

A931: for v0,v2,v1 holds
  (v2"\/"((v1"\/"v2)"/\"v0))"/\"(v0"\/"v1) =
     (v2"\/"((v2"\/"v1)"/\"v0))"/\"(v0"\/"v1)
proof let v0,v2,v1;
  ((v0"/\"v1)"\/"v2)"/\"((v2"\/"((v1"\/"v2)"/\"v0))"/\"(v0"\/"v1)) =
    (v2"\/"((v1"\/"v2)"/\"v0))"/\"(v0"\/"v1) by A110;
  hence thesis by A929;
end;

A934: for v2,v0,v1 holds
 (v0"\/"((v1"\/"v0)"/\"v2))"/\"(v2"\/"v1) =
    ((v0"\/"v1)"/\"(v0"\/"v2))"/\"(v2"\/"v1)
proof let v2,v0,v1;
  v0"\/"((v0"\/"v1)"/\"v2) = (v0"\/"v1)"/\"(v0"\/"v2) by A913;
  hence thesis by A931;
end;

A936: for v2,v0,v1 holds
 (v0"\/"((v1"\/"v0)"/\"v2))"/\"(v2"\/"v1) =
   (v0"\/"v1)"/\"((v0"\/"v2)"/\"(v2"\/"v1))
proof let v2,v0,v1;
  ((v0"\/"v1)"/\"(v0"\/"v2))"/\"(v2"\/"v1) =
      (v0"\/"v1)"/\"((v0"\/"v2)"/\"(v2"\/"v1)) by A3;
  hence thesis by A934;
end;

A938: for v2,v1,v0 holds
 ((v0"/\"v1)"\/"v2)"/\"((v2"\/"((v1"\/"v2)"/\"v0))"/\"(v0"\/"v1)) =
    (((v1"\/"v2)"/\"v0)"\/"v2)"/\"(v0"\/"v1)
proof let v2,v1,v0;
  v2"\/"((v1"\/"v2)"/\"v0) = ((v1"\/"v2)"/\"v0)"\/"v2 by A8;
  hence thesis by A110;
end;

A940: for v2,v1,v0 holds
 ((v0"/\"v1)"\/"v2)"/\"((v2"\/"v1)"/\"((v2"\/"v0)"/\"(v0"\/"v1))) =
    (((v1"\/"v2)"/\"v0)"\/"v2)"/\"(v0"\/"v1)
proof let v2,v1,v0;
  (v2"\/"((v1"\/"v2)"/\"v0))"/\"(v0"\/"v1) =
    (v2"\/"v1)"/\"((v2"\/"v0)"/\"(v0"\/"v1)) by A936;
  hence thesis by A938;
end;

A942: for v2,v1,v0 holds
 ((v0"/\"v1)"\/"v2)"/\"((v2"\/"v1)"/\"((v2"\/"v0)"/\"(v0"\/"v1))) =
    (v2"\/"((v1"\/"v2)"/\"v0))"/\"(v0"\/"v1)
proof let v2,v1,v0;
  ((v1"\/"v2)"/\"v0)"\/"v2 = v2"\/"((v1"\/"v2)"/\"v0) by A8;
  hence thesis by A940;
end;

A944: for v2,v1,v0 holds
 ((v0"/\"v1)"\/"v2)"/\"((v2"\/"v1)"/\"((v2"\/"v0)"/\"(v0"\/"v1))) =
    (v2"\/"v1)"/\"((v2"\/"v0)"/\"(v0"\/"v1))
proof let v2,v1,v0;
  (v2"\/"((v1"\/"v2)"/\"v0))"/\"(v0"\/"v1) =
    (v2"\/"v1)"/\"((v2"\/"v0)"/\"(v0"\/"v1)) by A936;
  hence thesis by A942;
end;

A947: for v100,v1,v0 holds
 (v100"\/"(v0"/\"v1))"/\"((v100"\/"((v1"\/"v100)"/\"v0))"/\"(v0"\/"v1)) =
   ((v0"/\"v1)"\/"v100)"/\"((v100"\/"((v1"\/"v100)"/\"v0))"/\"(v0"\/"v1))
proof let v100,v1,v0;
  ((v0"/\"v1)"\/"v100)"/\"((v100"\/"((v1"\/"v100)"/\"v0))"/\"(v0"\/"v1)) =
     (v100"\/"((v1"\/"v100)"/\"v0))"/\"(v0"\/"v1) by A110;
  hence thesis by A392;
end;

A950: for v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"((v0"\/"v2)"/\"((v0"\/"v1)"/\"(v1"\/"v2))) =
    ((v1"/\"v2)"\/"v0)"/\"((v0"\/"((v2"\/"v0)"/\"v1))"/\"(v1"\/"v2))
proof let v0,v2,v1;
  (v0"\/"((v2"\/"v0)"/\"v1))"/\"(v1"\/"v2) =
    (v0"\/"v2)"/\"((v0"\/"v1)"/\"(v1"\/"v2)) by A936;
  hence thesis by A947;
end;

A952: for v0,v2,v1 holds
 (v0"\/"v2)"/\"((v0"\/"(v1"/\"v2))"/\"((v0"\/"v1)"/\"(v1"\/"v2))) =
    ((v1"/\"v2)"\/"v0)"/\"((v0"\/"((v2"\/"v0)"/\"v1))"/\"(v1"\/"v2))
proof let v0,v2,v1;
  (v0"\/"(v1"/\"v2))"/\"((v0"\/"v2)"/\"((v0"\/"v1)"/\"(v1"\/"v2))) =
     (v0"\/"v2)"/\"((v0"\/"(v1"/\"v2))"/\"((v0"\/"v1)"/\"(v1"\/"v2))) by A40;
  hence thesis by A950;
end;

A955: for v0,v1,v2 holds
 (v0"\/"v1)"/\"((v0"\/"v2)"/\"((v0"\/"(v2"/\"v1))"/\"(v2"\/"v1))) =
    ((v2"/\"v1)"\/"v0)"/\"((v0"\/"((v1"\/"v0)"/\"v2))"/\"(v2"\/"v1))
proof let v0,v1,v2;
  (v0"\/"(v2"/\"v1))"/\"((v0"\/"v2)"/\"(v2"\/"v1)) =
    (v0"\/"v2)"/\"((v0"\/"(v2"/\"v1))"/\"(v2"\/"v1)) by A40;
  hence thesis by A952;
end;

A957: for v0,v1,v2 holds
 (v0"\/"v1)"/\"((v0"\/"(v2"/\"v1))"/\"(v2"\/"v1)) =
    ((v2"/\"v1)"\/"v0)"/\"((v0"\/"((v1"\/"v0)"/\"v2))"/\"(v2"\/"v1))
proof let v0,v1,v2;
  (v0"\/"v2)"/\"((v0"\/"(v2"/\"v1))"/\"(v2"\/"v1)) =
    (v0"\/"(v2"/\"v1))"/\"(v2"\/"v1) by A60;
  hence thesis by A955;
end;

A959: for v0,v1,v2 holds
 (v0"\/"(v2"/\"v1))"/\"(v2"\/"v1) =
    ((v2"/\"v1)"\/"v0)"/\"((v0"\/"((v1"\/"v0)"/\"v2))"/\"(v2"\/"v1))
proof let v0,v1,v2;
  (v0"\/"v1)"/\"((v0"\/"(v2"/\"v1))"/\"(v2"\/"v1)) =
    (v0"\/"(v2"/\"v1))"/\"(v2"\/"v1) by A315;
  hence thesis by A957;
end;

A962: for v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"(v1"\/"v2) =
   ((v1"/\"v2)"\/"v0)"/\"((v0"\/"v2)"/\"((v0"\/"v1)"/\"(v1"\/"v2)))
proof let v0,v2,v1;
  (v0"\/"((v2"\/"v0)"/\"v1))"/\"(v1"\/"v2) =
    (v0"\/"v2)"/\"((v0"\/"v1)"/\"(v1"\/"v2)) by A936;
  hence thesis by A959;
end;

A964: for v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"(v1"\/"v2) =
   (v0"\/"v2)"/\"((v0"\/"v1)"/\"(v1"\/"v2))
proof let v0,v2,v1;
  ((v1"/\"v2)"\/"v0)"/\"((v0"\/"v2)"/\"((v0"\/"v1)"/\"(v1"\/"v2))) =
     (v0"\/"v2)"/\"((v0"\/"v1)"/\"(v1"\/"v2)) by A944;
  hence thesis by A962;
end;

A966: for v1,v2,v0 holds
 (v0"\/"(v0"\/"v2))"/\"((v0"\/"v1)"/\"(v1"\/"(v0"\/"v2))) =
    v0"\/"(v1"/\"(v0"\/"v2))
proof let v1,v2,v0;
  (v0"\/"(v1"/\"(v0"\/"v2)))"/\"(v1"\/"(v0"\/"v2)) =
    (v0"\/"(v0"\/"v2))"/\"((v0"\/"v1)"/\"(v1"\/"(v0"\/"v2))) by A964;
  hence thesis by A902;
end;

A969: for v2,v1,v0 holds
 (v0"\/"v1)"/\"((v0"\/"v2)"/\"(v2"\/"(v0"\/"v1))) =
   v0"\/"(v2"/\"(v0"\/"v1))
proof let v2,v1,v0;
  v0"\/"(v0"\/"v1) = v0"\/"v1 by A43;
  hence thesis by A966;
end;

A974: for v1,v101,v100 holds
  v100"\/"((v100"/\"v101)"/\"((v100"/\"v101)"\/"v1)) =
    ((v100"/\"v101)"/\"v1)"\/"(v100"/\"(v100"\/"v101))
proof let v1,v101,v100;
  (v100"/\"v101)"\/"((v100"/\"v101)"/\"v1) =
    (v100"/\"v101)"/\"((v100"/\"v101)"\/"v1) by A347;
  hence thesis by A603;
end;

A977: for v2,v1,v0 holds
  v0"\/"(v0"/\"(v1"/\"((v0"/\"v1)"\/"v2))) =
    ((v0"/\"v1)"/\"v2)"\/"(v0"/\"(v0"\/"v1))
proof let v2,v1,v0;
  (v0"/\"v1)"/\"((v0"/\"v1)"\/"v2) =
    v0"/\"(v1"/\"((v0"/\"v1)"\/"v2)) by A3;
  hence thesis by A974;
end;

A979: for v2,v1,v0 holds
 v0"\/"(v1"/\"(v0"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
   (v0"\/"(v1"/\"v2))))) = ((v0"/\"v1)"/\"v2)"\/"(v0"/\"(v0"\/"v1))
proof let v2,v1,v0;
  v0"/\"(v1"/\"((v0"/\"v1)"\/"v2)) =
    v1"/\"(v0"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"(v0"\/"(v1"/\"v2))))
      by A864;
  hence thesis by A977;
end;

A981: for v2,v1,v0 holds
 v0"/\"(v0"\/"(v1"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
    (v0"\/"(v1"/\"v2))))) = ((v0"/\"v1)"/\"v2)"\/"(v0"/\"(v0"\/"v1))
proof let v2,v1,v0;
  v0"\/"(v1"/\"(v0"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
    (v0"\/"(v1"/\"v2))))) =
  v0"/\"(v0"\/"(v1"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))
     "/\"(v0"\/"(v1"/\"v2))))) by A417;
  hence thesis by A979;
end;

A983: for v0,v1,v2 holds
 v0"/\"(v0"\/"(v2"/\"v1)) = ((v0"/\"v1)"/\"v2)"\/"(v0"/\"(v0"\/"v1))
proof let v0,v1,v2;
  v0"/\"(v0"\/"(v1"/\"((v1"\/"(v0"/\"((v0"\/"v1)"/\"v2)))"/\"
     (v0"\/"(v1"/\"v2))))) = v0"/\"(v0"\/"(v2"/\"v1)) by A878;
  hence thesis by A981;
end;

A986: for v0,v2,v1 holds
  v0"/\"(v0"\/"(v1"/\"v2)) = (v0"/\"(v2"/\"v1))"\/"(v0"/\"(v0"\/"v2))
proof let v0,v2,v1;
  (v0"/\"v2)"/\"v1 = v0"/\"(v2"/\"v1) by A3;
  hence thesis by A983;
end;

A991: for v100,v1,v102 holds
 v100"\/"(v102"\/"((v102"/\"v1)"\/"(v100"/\"(v102"\/"v1))))
   = v100"\/"(v102"/\"(v102"\/"v1))
proof let v100,v1,v102;
  v102"\/"((v102"/\"v1)"\/"(v100"/\"(v102"\/"v1))) =
    (v100"/\"(v102"\/"v1))"\/"(v102"/\"(v102"\/"v1)) by A603;
  hence thesis by A558;
end;

A994: for v0,v2,v1 holds
 v0"\/"(v1"\/"(v0"/\"(v1"\/"v2))) = v0"\/"(v1"/\"(v1"\/"v2))
proof let v0,v2,v1;
  v1"\/"((v1"/\"v2)"\/"(v0"/\"(v1"\/"v2))) =
    v1"\/"(v0"/\"(v1"\/"v2)) by A884;
  hence thesis by A991;
end;

A996: for v0,v2,v1 holds
 (v0"\/"v1)"/\"(v0"\/"(v1"\/"(v1"\/"v2))) = v0"\/"(v1"/\"(v1"\/"v2))
proof let v0,v2,v1;
  v0"\/"(v1"\/"(v0"/\"(v1"\/"v2))) =
    (v0"\/"v1)"/\"(v0"\/"(v1"\/"(v1"\/"v2))) by A776;
  hence thesis by A994;
end;

A998: for v0,v2,v1 holds
 (v0"\/"v1)"/\"(v0"\/"(v1"\/"v2)) = v0"\/"(v1"/\"(v1"\/"v2))
proof let v0,v2,v1;
  v1"\/"(v1"\/"v2) = v1"\/"v2 by A43;
  hence thesis by A996;
end;

A1001: for v0,v2,v1 holds
 ((v0"/\"(v1"/\"v2))"\/"v0)"/\"((v0"/\"(v1"/\"v2))"\/"(v0"\/"v1)) =
    v0"/\"(v0"\/"(v2"/\"v1))
proof let v0,v2,v1;
  (v0"/\"(v1"/\"v2))"\/"(v0"/\"(v0"\/"v1)) =
    ((v0"/\"(v1"/\"v2))"\/"v0)"/\"((v0"/\"(v1"/\"v2))"\/"(v0"\/"v1)) by A998;
  hence thesis by A986;
end;

A1003: for v0,v2,v1 holds
 (v0"\/"(v0"/\"(v1"/\"v2)))"/\"((v0"/\"(v1"/\"v2))"\/"(v0"\/"v1)) =
   v0"/\"(v0"\/"(v2"/\"v1))
proof let v0,v2,v1;
  (v0"/\"(v1"/\"v2))"\/"v0 = v0"\/"(v0"/\"(v1"/\"v2)) by A8;
  hence thesis by A1001;
end;

A1005: for v0,v2,v1 holds
 (v0"/\"(v0"\/"(v1"/\"v2)))"/\"((v0"/\"(v1"/\"v2))"\/"(v0"\/"v1)) =
    v0"/\"(v0"\/"(v2"/\"v1))
proof let v0,v2,v1;
  v0"\/"(v0"/\"(v1"/\"v2)) = v0"/\"(v0"\/"(v1"/\"v2)) by A347;
  hence thesis by A1003;
end;

A1007: for v0,v2,v1 holds
 (v0"/\"(v0"\/"(v1"/\"v2)))"/\"(v0"\/"((v0"/\"(v1"/\"v2))"\/"v1)) =
    v0"/\"(v0"\/"(v2"/\"v1))
proof let v0,v2,v1;
  (v0"/\"(v1"/\"v2))"\/"(v0"\/"v1) = v0"\/"((v0"/\"(v1"/\"v2))"\/"v1) by A57;
  hence thesis by A1005;
end;

A1009: for v0,v2,v1 holds
 (v0"/\"(v0"\/"(v1"/\"v2)))"/\"(v0"\/"(v1"\/"(v0"/\"(v1"/\"v2)))) =
    v0"/\"(v0"\/"(v2"/\"v1))
proof let v0,v2,v1;
  (v0"/\"(v1"/\"v2))"\/"v1 = v1"\/"(v0"/\"(v1"/\"v2)) by A8;
  hence thesis by A1007;
end;

A1011: for v0,v2,v1 holds
 (v0"/\"(v0"\/"(v1"/\"v2)))"/\"(v0"\/"(v1"/\"(v1"\/"(v0"/\"v2)))) =
    v0"/\"(v0"\/"(v2"/\"v1))
proof let v0,v2,v1;
  v1"\/"(v0"/\"(v1"/\"v2)) = v1"/\"(v1"\/"(v0"/\"v2)) by A417;
  hence thesis by A1009;
end;

A1013: for v0,v2,v1 holds
 (v0"/\"(v0"\/"(v1"/\"v2)))"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"(v0"/\"v2))))
  = v0"/\"(v0"\/"(v2"/\"v1))
proof let v0,v2,v1;
  v0"\/"(v1"/\"(v1"\/"(v0"/\"v2))) =
    (v0"\/"v1)"/\"(v0"\/"(v1"\/"(v0"/\"v2))) by A998;
  hence thesis by A1011;
end;

A1015: for v0,v2,v1 holds
 (v0"/\"(v0"\/"(v1"/\"v2)))"/\"((v0"\/"v1)"/\"
    ((v0"\/"v1)"/\"(v0"\/"(v1"\/"v2)))) = v0"/\"(v0"\/"(v2"/\"v1))
proof let v0,v2,v1;
  v0"\/"(v1"\/"(v0"/\"v2)) = (v0"\/"v1)"/\"(v0"\/"(v1"\/"v2)) by A776;
  hence thesis by A1013;
end;

A1017: for v0,v2,v1 holds
 (v0"/\"(v0"\/"(v1"/\"v2)))"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"v2))) =
    v0"/\"(v0"\/"(v2"/\"v1))
proof let v0,v2,v1;
  (v0"\/"v1)"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"v2))) =
    (v0"\/"v1)"/\"(v0"\/"(v1"\/"v2)) by A26;
  hence thesis by A1015;
end;

A1019: for v0,v2,v1 holds
 (v0"\/"v1)"/\"((v0"/\"(v0"\/"(v1"/\"v2)))"/\"(v0"\/"(v1"\/"v2))) =
   v0"/\"(v0"\/"(v2"/\"v1))
proof let v0,v2,v1;
  (v0"/\"(v0"\/"(v1"/\"v2)))"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"v2))) =
    (v0"\/"v1)"/\"((v0"/\"(v0"\/"(v1"/\"v2)))"/\"(v0"\/"(v1"\/"v2))) by A40;
  hence thesis by A1017;
end;

A1021: for v0,v2,v1 holds
 (v0"\/"v1)"/\"(v0"/\"((v0"\/"(v1"/\"v2))"/\"(v0"\/"(v1"\/"v2)))) =
    v0"/\"(v0"\/"(v2"/\"v1))
proof let v0,v2,v1;
  (v0"/\"(v0"\/"(v1"/\"v2)))"/\"(v0"\/"(v1"\/"v2)) =
    v0"/\"((v0"\/"(v1"/\"v2))"/\"(v0"\/"(v1"\/"v2))) by A3;
  hence thesis by A1019;
end;

A1023: for v0,v2,v1 holds
 (v0"\/"v1)"/\"(v0"/\"(v0"\/"(v1"\/"v2))) = v0"/\"(v0"\/"(v2"/\"v1))
proof let v0,v2,v1;
  v0"/\"((v0"\/"(v1"/\"v2))"/\"(v0"\/"(v1"\/"v2))) =
    v0"/\"(v0"\/"(v1"\/"v2)) by A735;
  hence thesis by A1021;
end;

A1025: for v0,v2,v1 holds
 v0"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"v2))) = v0"/\"(v0"\/"(v2"/\"v1))
proof let v0,v2,v1;
  (v0"\/"v1)"/\"(v0"/\"(v0"\/"(v1"\/"v2))) =
    v0"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"v2))) by A40;
  hence thesis by A1023;
end;

A1027: for v0,v2,v1 holds
  v0"/\"(v0"\/"(v1"\/"v2)) = v0"/\"(v0"\/"(v2"/\"v1))
proof let v0,v2,v1;
  v0"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"v2))) = v0"/\"(v0"\/"(v1"\/"v2)) by A600;
  hence thesis by A1025;
end;

A1029: for v0,v2,v1 holds
 (v0"/\"v1)"\/"(v2"/\"(v2"\/"(v0"/\"(v1"/\"v2)))) =
   (v2"\/"((v1"\/"v2)"/\"v0))"/\"(v2"\/"(v0"\/"v1))
proof let v0,v2,v1;
  v2"/\"(v2"\/"((v1"/\"v2)"\/"v0)) = v2"/\"(v2"\/"(v0"/\"(v1"/\"v2))) by A1027;
  hence thesis by A927;
end;

A1031: for v2,v1,v0 holds
 (v0"/\"v1)"\/"(v2"/\"(v2"/\"(v2"\/"(v0"/\"v1)))) =
    (v2"\/"((v1"\/"v2)"/\"v0))"/\"(v2"\/"(v0"\/"v1))
proof let v2,v1,v0;
  v2"\/"(v0"/\"(v1"/\"v2)) = v2"/\"(v2"\/"(v0"/\"v1)) by A441;
  hence thesis by A1029;
end;

A1033: for v2,v1,v0 holds
  ((v0"/\"v1)"\/"(v2"/\"(v2"\/"(v0"/\"v1))) =
    (v2"\/"((v1"\/"v2)"/\"v0))"/\"(v2"\/"(v0"\/"v1)))
proof let v2,v1,v0;
  v2"/\"(v2"/\"(v2"\/"(v0"/\"v1))) = v2"/\"(v2"\/"(v0"/\"v1)) by A26;
  hence thesis by A1031;
end;

A1035: for v2,v1,v0 holds
 v2"\/"(v0"/\"v1) = (v2"\/"((v1"\/"v2)"/\"v0))"/\"(v2"\/"(v0"\/"v1))
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v2"/\"(v2"\/"(v0"/\"v1))) = v2"\/"(v0"/\"v1) by A344;
  hence thesis by A1033;
end;

A1040: for v0,v2,v1 holds
  (v0"/\"(v1"\/"v2))"\/"(v1"/\"(v1"\/"(v0"/\"(v1"/\"v2)))) =
    (v1"\/"v2)"/\"((v1"\/"v0)"/\"(v0"\/"(v1"\/"v2)))
proof let v0,v2,v1;
  v1"/\"(v1"\/"((v1"/\"v2)"\/"v0)) = v1"/\"(v1"\/"(v0"/\"(v1"/\"v2))) by A1027;
  hence thesis by A925;
end;

A1042: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v1"/\"(v1"/\"(v1"\/"(v0"/\"v2)))) =
    (v1"\/"v2)"/\"((v1"\/"v0)"/\"(v0"\/"(v1"\/"v2)))
proof let v0,v2,v1;
  v1"\/"(v0"/\"(v1"/\"v2)) = v1"/\"(v1"\/"(v0"/\"v2)) by A417;
  hence thesis by A1040;
end;

A1044: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v1"/\"(v1"\/"(v0"/\"v2))) =
   (v1"\/"v2)"/\"((v1"\/"v0)"/\"(v0"\/"(v1"\/"v2)))
proof let v0,v2,v1;
  v1"/\"(v1"/\"(v1"\/"(v0"/\"v2))) = v1"/\"(v1"\/"(v0"/\"v2)) by A26;
  hence thesis by A1042;
end;

A1046: for v0,v2,v1 holds
 ((v0"/\"(v1"\/"v2))"\/"v1)"/\"((v0"/\"(v1"\/"v2))"\/"(v1"\/"(v0"/\"v2)))
  = (v1"\/"v2)"/\"((v1"\/"v0)"/\"(v0"\/"(v1"\/"v2)))
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"(v1"/\"(v1"\/"(v0"/\"v2))) =
   ((v0"/\"(v1"\/"v2))"\/"v1)"/\"((v0"/\"(v1"\/"v2))"\/"(v1"\/"(v0"/\"v2)))
     by A998;
  hence thesis by A1044;
end;

A1048: for v0,v2,v1 holds
 (v1"\/"(v0"/\"(v1"\/"v2)))"/\"((v0"/\"(v1"\/"v2))"\/"(v1"\/"(v0"/\"v2)))
    = (v1"\/"v2)"/\"((v1"\/"v0)"/\"(v0"\/"(v1"\/"v2)))
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"v1 = v1"\/"(v0"/\"(v1"\/"v2)) by A8;
  hence thesis by A1046;
end;

A1051: for v1,v2,v0 holds
 (v0"\/"(v1"/\"(v0"\/"v2)))"/\"(v0"\/"((v1"/\"(v0"\/"v2))"\/"(v1"/\"v2)))
    = (v0"\/"v2)"/\"((v0"\/"v1)"/\"(v1"\/"(v0"\/"v2)))
proof let v1,v2,v0;
  (v1"/\"(v0"\/"v2))"\/"(v0"\/"(v1"/\"v2)) =
    v0"\/"((v1"/\"(v0"\/"v2))"\/"(v1"/\"v2)) by A57;
  hence thesis by A1048;
end;

A1053: for v1,v2,v0 holds
 (v0"\/"(v1"/\"(v0"\/"v2)))"/\"(v0"\/"((v1"/\"v2)"\/"(v1"/\"(v0"\/"v2))))
   = (v0"\/"v2)"/\"((v0"\/"v1)"/\"(v1"\/"(v0"\/"v2)))
proof let v1,v2,v0;
  (v1"/\"(v0"\/"v2))"\/"(v1"/\"v2) = (v1"/\"v2)"\/"(v1"/\"(v0"\/"v2)) by A8;
  hence thesis by A1051;
end;

A1055: for v1,v2,v0 holds
 (v0"\/"(v1"/\"(v0"\/"v2)))"/\"(v0"\/"(v1"/\"(v2"\/"v0))) =
    (v0"\/"v2)"/\"((v0"\/"v1)"/\"(v1"\/"(v0"\/"v2)))
proof let v1,v2,v0;
  (v1"/\"v2)"\/"(v1"/\"(v0"\/"v2)) = v1"/\"(v2"\/"v0) by A133;
  hence thesis by A1053;
end;

A1057: for v1,v0,v2 holds
  v0"\/"(v1"/\"(v2"\/"v0)) = (v0"\/"v2)"/\"((v0"\/"v1)"/\"(v1"\/"(v0"\/"v2)))
proof let v1,v0,v2;
  (v0"\/"(v1"/\"(v0"\/"v2)))"/\"(v0"\/"(v1"/\"(v2"\/"v0))) =
    v0"\/"(v1"/\"(v2"\/"v0)) by A438;
  hence thesis by A1055;
end;

A1059: for v2,v1,v0 holds
 (((v0"\/"v1)"/\"v2)"\/"v0)"/\"(((v0"\/"v1)"/\"v2)"\/"(v0"\/"v2))
    = v0"\/"(v2"/\"(v0"\/"v1))
proof let v2,v1,v0;
  ((v0"\/"v1)"/\"v2)"\/"(v0"/\"(v0"\/"v2)) =
    (((v0"\/"v1)"/\"v2)"\/"v0)"/\"(((v0"\/"v1)"/\"v2)"\/"(v0"\/"v2)) by A998;
  hence thesis by A758;
end;

A1061: for v2,v1,v0 holds
 (v0"\/"((v0"\/"v1)"/\"v2))"/\"(((v0"\/"v1)"/\"v2)"\/"(v0"\/"v2)) =
   v0"\/"(v2"/\"(v0"\/"v1))
proof let v2,v1,v0;
  ((v0"\/"v1)"/\"v2)"\/"v0 = v0"\/"((v0"\/"v1)"/\"v2) by A8;
  hence thesis by A1059;
end;

A1063: for v2,v1,v0 holds
 ((v0"\/"v1)"/\"(v0"\/"v2))"/\"(((v0"\/"v1)"/\"v2)"\/"(v0"\/"v2)) =
   v0"\/"(v2"/\"(v0"\/"v1))
proof let v2,v1,v0;
  v0"\/"((v0"\/"v1)"/\"v2) = (v0"\/"v1)"/\"(v0"\/"v2) by A913;
  hence thesis by A1061;
end;

A1065: for v2,v1,v0 holds
 ((v0"\/"v1)"/\"(v0"\/"v2))"/\"(v0"\/"(((v0"\/"v1)"/\"v2)"\/"v2)) =
    v0"\/"(v2"/\"(v0"\/"v1))
proof let v2,v1,v0;
  ((v0"\/"v1)"/\"v2)"\/"(v0"\/"v2) =
     v0"\/"(((v0"\/"v1)"/\"v2)"\/"v2) by A57;
  hence thesis by A1063;
end;

A1067: for v2,v1,v0 holds
 ((v0"\/"v1)"/\"(v0"\/"v2))"/\"(v0"\/"(v2"\/"((v0"\/"v1)"/\"v2))) =
    v0"\/"(v2"/\"(v0"\/"v1))
proof let v2,v1,v0;
  ((v0"\/"v1)"/\"v2)"\/"v2 = v2"\/"((v0"\/"v1)"/\"v2) by A8;
  hence thesis by A1065;
end;

A1069: for v2,v1,v0 holds
 ((v0"\/"v1)"/\"(v0"\/"v2))"/\"(v0"\/"(v2"/\"(v2"\/"(v0"\/"v1)))) =
    v0"\/"(v2"/\"(v0"\/"v1))
proof let v2,v1,v0;
  v2"\/"((v0"\/"v1)"/\"v2) = v2"/\"(v2"\/"(v0"\/"v1)) by A400;
  hence thesis by A1067;
end;

A1071: for v2,v1,v0 holds
 ((v0"\/"v1)"/\"(v0"\/"v2))"/\"(v0"\/"(v2"/\"(v2"\/"(v1"/\"v0)))) =
    v0"\/"(v2"/\"(v0"\/"v1))
proof let v2,v1,v0;
  v2"/\"(v2"\/"(v0"\/"v1)) = v2"/\"(v2"\/"(v1"/\"v0)) by A1027;
  hence thesis by A1069;
end;

A1073: for v2,v1,v0 holds
 ((v0"\/"v1)"/\"(v0"\/"v2))"/\"((v0"\/"v2)"/\"(v0"\/"(v2"\/"(v1"/\"v0)))) =
    v0"\/"(v2"/\"(v0"\/"v1))
proof let v2,v1,v0;
  v0"\/"(v2"/\"(v2"\/"(v1"/\"v0))) =
    (v0"\/"v2)"/\"(v0"\/"(v2"\/"(v1"/\"v0))) by A998;
  hence thesis by A1071;
end;

A1075: for v2,v1,v0 holds
 ((v0"\/"v1)"/\"(v0"\/"v2))"/\"((v0"\/"v2)"/\"
    ((v0"\/"v2)"/\"(v0"\/"(v2"\/"v1)))) =
   v0"\/"(v2"/\"(v0"\/"v1))
proof let v2,v1,v0;
  v0"\/"(v2"\/"(v1"/\"v0)) = (v0"\/"v2)"/\"(v0"\/"(v2"\/"v1)) by A754;
  hence thesis by A1073;
end;

A1077: for v2,v1,v0 holds
 ((v0"\/"v1)"/\"(v0"\/"v2))"/\"((v0"\/"v2)"/\"(v0"\/"(v2"\/"v1))) =
    v0"\/"(v2"/\"(v0"\/"v1))
proof let v2,v1,v0;
  (v0"\/"v2)"/\"((v0"\/"v2)"/\"(v0"\/"(v2"\/"v1))) =
     (v0"\/"v2)"/\"(v0"\/"(v2"\/"v1)) by A26;
  hence thesis by A1075;
end;

A1079: for v2,v1,v0 holds
 (v0"\/"v2)"/\"(((v0"\/"v1)"/\"(v0"\/"v2))"/\"(v0"\/"(v2"\/"v1))) =
    v0"\/"(v2"/\"(v0"\/"v1))
proof let v2,v1,v0;
  ((v0"\/"v1)"/\"(v0"\/"v2))"/\"((v0"\/"v2)"/\"(v0"\/"(v2"\/"v1))) =
    (v0"\/"v2)"/\"(((v0"\/"v1)"/\"(v0"\/"v2))"/\"(v0"\/"(v2"\/"v1))) by A40;
  hence thesis by A1077;
end;

A1082: for v0,v2,v1 holds
 (v0"\/"v1)"/\"((v0"\/"v2)"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"v2)))) =
    v0"\/"(v1"/\"(v0"\/"v2))
proof let v0,v2,v1;
  ((v0"\/"v2)"/\"(v0"\/"v1))"/\"(v0"\/"(v1"\/"v2)) =
    (v0"\/"v2)"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"v2))) by A3;
  hence thesis by A1079;
end;

A1084: for v0,v2,v1 holds
 (v0"\/"v2)"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"v2))) = v0"\/"(v1"/\"(v0"\/"v2))
proof let v0,v2,v1;
  (v0"\/"v1)"/\"((v0"\/"v2)"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"v2)))) =
    (v0"\/"v2)"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"v2))) by A162;
  hence thesis by A1082;
end;

A1089: for v2,v1,v0 holds
  v0"/\"(v1"/\"(v1"\/"(v2"/\"(v0"/\"v1)))) = v0"/\"(v1"/\"(v1"\/"v2))
proof let v2,v1,v0;
  v1"/\"(v1"\/"((v0"/\"v1)"\/"v2)) = v1"/\"(v1"\/"(v2"/\"(v0"/\"v1))) by A1027;
  hence thesis by A414;
end;

A1091: for v1,v0,v2 holds
 v0"/\"(v1"/\"(v1"/\"(v1"\/"(v2"/\"v0)))) = v0"/\"(v1"/\"(v1"\/"v2))
proof let v1,v0,v2;
  v1"\/"(v2"/\"(v0"/\"v1)) = v1"/\"(v1"\/"(v2"/\"v0)) by A441;
  hence thesis by A1089;
end;

A1093: for v1,v0,v2 holds
 v0"/\"(v1"/\"(v1"\/"(v2"/\"v0))) = v0"/\"(v1"/\"(v1"\/"v2))
proof let v1,v0,v2;
  v1"/\"(v1"/\"(v1"\/"(v2"/\"v0))) = v1"/\"(v1"\/"(v2"/\"v0)) by A26;
  hence thesis by A1091;
end;

A1095: for v2,v1,v0 holds
 (v0"/\"v1)"\/"(v2"/\"(v0"/\"(v0"\/"v1))) = v0"/\"((v0"/\"v1)"\/"v2)
proof let v2,v1,v0;
  v2"/\"(v0"/\"(v0"\/"(v1"/\"v2))) = v2"/\"(v0"/\"(v0"\/"v1)) by A1093;
  hence thesis by A919;
end;

A1097: for v2,v1,v0 holds
  v0"\/"(v0"/\"((v0"/\"v1)"\/"v2)) = v0"/\"((v0"\/"v1)"/\"(v0"\/"v2))
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v2"/\"(v0"/\"(v0"\/"v1))) = v0"/\"((v0"/\"v1)"\/"v2)
     by A1095;
  hence thesis by A921;
end;

A1099: for v2,v1,v0 holds
 v0"/\"(v0"\/"((v0"/\"v1)"\/"v2)) = v0"/\"((v0"\/"v1)"/\"(v0"\/"v2))
proof let v2,v1,v0;
  v0"\/"(v0"/\"((v0"/\"v1)"\/"v2)) =
    v0"/\"(v0"\/"((v0"/\"v1)"\/"v2)) by A347;
  hence thesis by A1097;
end;

A1101: for v2,v1,v0 holds
  v0"/\"(v0"\/"(v2"/\"(v0"/\"v1))) = v0"/\"((v0"\/"v1)"/\"(v0"\/"v2))
proof let v2,v1,v0;
  v0"/\"(v0"\/"((v0"/\"v1)"\/"v2)) =
     v0"/\"(v0"\/"(v2"/\"(v0"/\"v1))) by A1027;
  hence thesis by A1099;
end;

A1104: for v0,v2,v1 holds
 v0"/\"(v0"/\"(v0"\/"(v1"/\"v2))) = v0"/\"((v0"\/"v2)"/\"(v0"\/"v1))
proof let v0,v2,v1;
  v0"\/"(v1"/\"(v0"/\"v2)) = v0"/\"(v0"\/"(v1"/\"v2)) by A417;
  hence thesis by A1101;
end;

A1106: for v0,v2,v1 holds
  v0"/\"(v0"\/"(v1"/\"v2)) = v0"/\"((v0"\/"v2)"/\"(v0"\/"v1))
proof let v0,v2,v1;
  v0"/\"(v0"/\"(v0"\/"(v1"/\"v2))) = v0"/\"(v0"\/"(v1"/\"v2)) by A26;
  hence thesis by A1104;
end;

A1108: for v0,v2,v1 holds
 v0"/\"(v0"\/"(v1"\/"v2)) = v0"/\"((v0"\/"v1)"/\"(v0"\/"v2))
proof let v0,v2,v1;
  v0"/\"(v0"\/"(v2"/\"v1)) = v0"/\"((v0"\/"v1)"/\"(v0"\/"v2)) by A1106;
  hence thesis by A1027;
end;

A1111: for v100,v0,v1 holds
 (v100"\/"(v1"\/"v0))"/\"((v1"\/"v0)"/\"v100) =
    (v0"\/"v1)"/\"((v1"\/"v0)"/\"v100)
proof let v100,v0,v1;
  (v0"\/"v1)"/\"((v1"\/"v0)"/\"v100) = (v1"\/"v0)"/\"v100 by A392;
  hence thesis by A717;
end;

A1114: for v0,v2,v1 holds
 ((v1"\/"v2)"/\"v0)"/\"(v0"\/"(v1"\/"v2)) =
    (v2"\/"v1)"/\"((v1"\/"v2)"/\"v0)
proof let v0,v2,v1;
  (v0"\/"(v1"\/"v2))"/\"((v1"\/"v2)"/\"v0) =
    ((v1"\/"v2)"/\"v0)"/\"(v0"\/"(v1"\/"v2)) by A4;
  hence thesis by A1111;
end;

A1117: for v2,v1,v0 holds
 (v0"\/"v1)"/\"(v2"/\"(v2"\/"(v0"\/"v1))) =
   (v1"\/"v0)"/\"((v0"\/"v1)"/\"v2)
proof let v2,v1,v0;
  ((v0"\/"v1)"/\"v2)"/\"(v2"\/"(v0"\/"v1)) =
    (v0"\/"v1)"/\"(v2"/\"(v2"\/"(v0"\/"v1))) by A3;
  hence thesis by A1114;
end;

A1119: for v1,v0,v2 holds
 (v0"\/"v1)"/\"(v2"/\"((v2"\/"v0)"/\"(v2"\/"v1))) =
    (v1"\/"v0)"/\"((v0"\/"v1)"/\"v2)
proof let v1,v0,v2;
  v2"/\"(v2"\/"(v0"\/"v1)) = v2"/\"((v2"\/"v0)"/\"(v2"\/"v1)) by A1108;
  hence thesis by A1117;
end;

A1121: for v1,v0,v2 holds
 (v0"\/"v1)"/\"(v2"/\"((v2"\/"v0)"/\"(v2"\/"v1))) = (v0"\/"v1)"/\"v2
proof let v1,v0,v2;
  (v1"\/"v0)"/\"((v0"\/"v1)"/\"v2) = (v0"\/"v1)"/\"v2 by A392;
  hence thesis by A1119;
end;

A1123: for v2,v0,v1 holds
 (v0"\/"v1)"/\"(v1"\/"((v1"\/"v0)"/\"v2)) = v1"\/"((v0"\/"v1)"/\"v2)
proof let v2,v0,v1;
  v0"\/"v1 = v1"\/"v0 by A8;
  hence thesis by A208;
end;

A1125: for v2,v0,v1 holds
 (v0"\/"v1)"/\"((v1"\/"v0)"/\"(v1"\/"v2)) = v1"\/"((v0"\/"v1)"/\"v2)
proof let v2,v0,v1;
  v1"\/"((v1"\/"v0)"/\"v2) = (v1"\/"v0)"/\"(v1"\/"v2) by A913;
  hence thesis by A1123;
end;

A1127: for v2,v0,v1 holds
   (v1"\/"v0)"/\"(v1"\/"v2) = v1"\/"((v0"\/"v1)"/\"v2)
proof let v2,v0,v1;
  (v0"\/"v1)"/\"((v1"\/"v0)"/\"(v1"\/"v2)) =
    (v1"\/"v0)"/\"(v1"\/"v2) by A392;
  hence thesis by A1125;
end;

A1132: for v1,v101,v100 holds
 (v100"\/"v101)"/\"(v101"\/"(v1"/\"(v100"\/"v101))) =
    v101"\/"((v100"\/"v101)"/\"(v1"/\"(v1"\/"(v100"\/"v101))))
proof let v1,v101,v100;
  (v100"\/"v101)"/\"(v1"/\"(v1"\/"(v100"\/"v101))) =
    v1"/\"(v100"\/"v101) by A308;
  hence thesis by A208;
end;

A1135: for v2,v1,v0 holds
  v1"\/"(v2"/\"(v0"\/"v1)) =
    v1"\/"((v0"\/"v1)"/\"(v2"/\"(v2"\/"(v0"\/"v1))))
proof let v2,v1,v0;
  (v0"\/"v1)"/\"(v1"\/"(v2"/\"(v0"\/"v1))) =
    v1"\/"(v2"/\"(v0"\/"v1)) by A565;
  hence thesis by A1132;
end;

A1138: for v1,v0,v2 holds
 v0"\/"(v1"/\"(v2"\/"v0)) =
   v0"\/"((v2"\/"v0)"/\"(v1"/\"((v1"\/"v2)"/\"(v1"\/"v0))))
proof let v1,v0,v2;
  v1"/\"(v1"\/"(v2"\/"v0)) = v1"/\"((v1"\/"v2)"/\"(v1"\/"v0)) by A1108;
  hence thesis by A1135;
end;

A1140: for v1,v0,v2 holds
  v0"\/"(v1"/\"(v2"\/"v0)) = v0"\/"((v2"\/"v0)"/\"v1)
proof let v1,v0,v2;
  (v2"\/"v0)"/\"(v1"/\"((v1"\/"v2)"/\"(v1"\/"v0))) =
    (v2"\/"v0)"/\"v1 by A1121;
  hence thesis by A1138;
end;

A1142: for v1,v0,v2 holds
  v0"\/"(v1"/\"(v2"\/"v0)) = (v0"\/"v2)"/\"(v0"\/"v1)
proof let v1,v0,v2;
  v0"\/"((v2"\/"v0)"/\"v1) = (v0"\/"v2)"/\"(v0"\/"v1) by A1127;
  hence thesis by A1140;
end;

A1144: for v2,v1,v0 holds
 ((v0"\/"v1)"/\"(v0"\/"v2))"/\"(v0"\/"(v2"\/"v1)) = v0"\/"(v2"/\"v1)
proof let v2,v1,v0;
  v0"\/"((v1"\/"v0)"/\"v2) = (v0"\/"v1)"/\"(v0"\/"v2) by A1127;
  hence thesis by A1035;
end;

A1146: for v0,v1,v2 holds
 (v0"\/"v1)"/\"((v0"\/"v2)"/\"(v0"\/"(v2"\/"v1))) =
   v0"\/"(v2"/\"v1)
proof let v0,v1,v2;
  ((v0"\/"v1)"/\"(v0"\/"v2))"/\"(v0"\/"(v2"\/"v1)) =
     (v0"\/"v1)"/\"((v0"\/"v2)"/\"(v0"\/"(v2"\/"v1))) by A3;
  hence thesis by A1144;
end;

A1148: for v1,v2,v0 holds
 (v0"\/"v2)"/\"(v0"\/"v1) =
   (v0"\/"v2)"/\"((v0"\/"v1)"/\"(v1"\/"(v0"\/"v2)))
proof let v1,v2,v0;
  v0"\/"(v1"/\"(v2"\/"v0)) = (v0"\/"v2)"/\"(v0"\/"v1) by A1142;
  hence thesis by A1057;
end;

A1152: for v1,v2,v0 holds
   v0"\/"(v1"/\"(v0"\/"v2)) = v0"\/"(v1"/\"v2)
proof let v1,v2,v0;
  (v0"\/"v2)"/\"((v0"\/"v1)"/\"(v0"\/"(v1"\/"v2))) =
     v0"\/"(v1"/\"v2) by A1146;
  hence thesis by A1084;
end;

A1154: for v0,v2,v1 holds
 v0"\/"(v1"/\"v2) = (v0"\/"v2)"/\"((v0"\/"v1)"/\"(v1"\/"(v0"\/"v2)))
proof let v0,v2,v1;
  v0"\/"(v1"/\"(v0"\/"v2)) = v0"\/"(v1"/\"v2) by A1152;
  hence thesis by A969;
end;

Z:
for v0,v2,v1 holds v0"\/"(v1"/\"v2) = (v0"\/"v2)"/\"(v0"\/"v1)
proof let v0,v2,v1;
  (v0"\/"v2)"/\"((v0"\/"v1)"/\"(v1"\/"(v0"\/"v2))) =
    (v0"\/"v2)"/\"(v0"\/"v1) by A1148;
  hence thesis by A1154;
end;
  let v1,v2,v3;
  v1"\/"(v2"/\"v3) = (v1"\/"v3)"/\"(v1"\/"v2) by Z;
  hence thesis by A4;
end;

Cluster3:
  L is meet-idempotent meet-associative meet-commutative
       satisfying_QLT1 join-idempotent join-associative join-commutative
        satisfying_QLT2 QLT-selfdistributive implies
    L is distributive'
  proof
    assume
    L is meet-idempotent meet-associative meet-commutative satisfying_QLT1
       join-idempotent join-associative join-commutative
        satisfying_QLT2 QLT-selfdistributive; then
    (for v0 holds v0"/\"v0 = v0) &
    (for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2)) &
    (for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
    (for v0,v2,v1 holds (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2)) &
    (for v0 holds v0"\/"v0 = v0) &
    (for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
    (for v1,v0 holds v0"\/"v1 = v1"\/"v0) &
    (for v0,v2,v1 holds (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
    (for v2,v1,v0 holds
       (((v0"/\"v1)"\/"v2)"/\"v1)"\/"(v2"/\"v0) =
    (((v0"\/"v1)"/\"v2)"\/"v1)"/\"(v2"\/"v0)) by LATTICES:def 4, def 5, def 6,
       def 7, SHEFFER1:def 9, ROBBINS1:def 7; then
    for v1,v2,v3 holds
      v1"\/"(v2"/\"v3) = (v1"\/"v2)"/\"(v1"\/"v3) by ThQLT3;
    hence thesis by SHEFFER1:def 5;
  end;

registration
  cluster meet-idempotent meet-associative meet-commutative
       satisfying_QLT1 join-idempotent join-associative join-commutative
        satisfying_QLT2 QLT-selfdistributive ->
     distributive' for non empty LattStr;
  coherence by Cluster3;
end;

begin :: QLT-4: Bowden Inequality

definition let L;
  attr L is satisfying_Bowden_inequality means
    for x,y,z being Element of L holds
      (x "\/" y) "/\" z [= x "\/" (y "/\" z);
end;

definition let L be join-commutative non empty LattStr;
  redefine attr L is satisfying_Bowden_inequality means :BowdenRedef:  :: redefines satisfying_Bowden_inequality (LATQUASI, definition)
    for v0,v2,v1 being Element of L holds
      (v0"\/"(v1"/\"v2))"\/"((v0"\/"v1)"/\"v2) = v0"\/"(v1"/\"v2);
  compatibility
  proof
    thus L is satisfying_Bowden_inequality implies
    for v0,v2,v1 being Element of L holds
      (v0"\/"(v1"/\"v2))"\/"((v0"\/"v1)"/\"v2) = v0"\/"(v1"/\"v2)
        by LATTICES:def 3;
    assume
B1: for v0,v2,v1 being Element of L holds
      (v0"\/"(v1"/\"v2))"\/"((v0"\/"v1)"/\"v2) = v0"\/"(v1"/\"v2);
    let x,y,z be Element of L;
    (x"\/"(y"/\"z))"\/"((x"\/"y)"/\"z) = x"\/"(y"/\"z) by B1;
    hence thesis by LATTICES:def 3;
  end;
end;

theorem ThQLT4:
(for v0 holds v0"/\"v0 = v0) &
(for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
(for v0,v2,v1 holds
  (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2)) &
(for v0 holds v0"\/"v0 = v0) &
(for v2,v1,v0 holds
  (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
(for v1,v0 holds v0"\/"v1 = v1"\/"v0) &
(for v0,v2,v1 holds
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
(for v0,v2,v1 holds
   (v0"\/"(v1"/\"v2))"\/"((v0"\/"v1)"/\"v2) = v0"\/"(v1"/\"v2)) implies
for v1,v2,v3 holds v1"\/"(v2"/\"v3) = (v1"\/"v2)"/\"(v1"\/"v3)
proof
assume A1: for v0 holds v0"/\"v0 = v0;
assume A2: for v1,v0 holds v0"/\"v1 = v1"/\"v0;
assume A3: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2);
assume A4: for v0 holds v0"\/"v0 = v0;
assume A5: for v2,v1,v0 holds
  (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2);
assume A6: for v1,v0 holds v0"\/"v1 = v1"\/"v0;
assume A7: for v0,v2,v1 holds
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2);

A9: for v0,v2,v1 holds
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) =
    (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) by A2;
  hence thesis by A7;
end;

assume A10: for v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"\/"((v0"\/"v1)"/\"v2) = v0"\/"(v1"/\"v2);

A12: for v2,v1,v0 holds
  v0"\/"((v1"/\"v2)"\/"((v0"\/"v1)"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  (v0"\/"(v1"/\"v2))"\/"((v0"\/"v1)"/\"v2) =
    v0"\/"((v1"/\"v2)"\/"((v0"\/"v1)"/\"v2)) by A5;
hence thesis by A10;
end;

A16: for v0,v2,v1 holds (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) by A6;
  hence thesis by A3;
end;

A19: for v102,v100 holds (v100"\/"v102)"\/"v102 = v100"\/"v102
proof let v102,v100;
  v102"\/"v102 = v102 by A4;
  hence thesis by A5;
end;

A22: for v1,v0 holds v1"\/"(v0"\/"v1) = v0"\/"v1
proof let v1,v0;
  (v0"\/"v1)"\/"v1 = v1"\/"(v0"\/"v1) by A6;
  hence thesis by A19;
end;

A25: for v2,v0,v1 holds (v1"\/"v0)"\/"v2 = v0"\/"(v1"\/"v2)
proof let v2,v0,v1;
  v0"\/"v1 = v1"\/"v0 by A6;
  hence thesis by A5;
end;

A28: for v0,v2,v1 holds
  v0"\/"(v1"\/"v2) = v1"\/"(v0"\/"v2)
proof let v0,v2,v1;
  (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2) by A5;
  hence thesis by A25;
end;

A31: for v102,v101 holds
  v101"/\"(v101"\/"(v101"/\"v102)) = v101"\/"(v101"/\"v102)
proof let v102,v101;
  v101"\/"v101 = v101 by A4;
  hence thesis by A9;
end;

A34: for v2,v1,v0 holds
  v0"\/"((v1"/\"v2)"\/"(v2"/\"(v0"\/"v1))) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  (v0"\/"v1)"/\"v2 = v2"/\"(v0"\/"v1) by A2;
  hence thesis by A12;
end;

A37: for v102,v101 holds
  v101"\/"(v101"/\"(v101"\/"v102)) = v101"/\"(v101"\/"v102)
proof let v102,v101;
  v101"/\"v101 = v101 by A1;
  hence thesis by A16;
end;

A40: for v0,v2,v1 holds
  (v1"/\"v0)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  v0"/\"v1 = v1"/\"v0 by A2;
  hence thesis by A16;
end;

A44: for v101,v2,v1 holds
 ((v101"\/"v1)"/\"v101)"\/"(v101"\/"(v1"/\"v2)) =
    (v101"\/"v1)"/\"(v101"\/"(v1"/\"v2))
proof let v101,v2,v1;
  (v101"\/"v1)"/\"(v101"\/"(v1"/\"v2)) = v101"\/"(v1"/\"v2) by A9;
  hence thesis by A16;
end;

A47: for v0,v2,v1 holds
  (v0"/\"(v0"\/"v1))"\/"(v0"\/"(v1"/\"v2)) =
    (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2))
proof let v0,v2,v1;
  (v0"\/"v1)"/\"v0 = v0"/\"(v0"\/"v1) by A2;
  hence thesis by A44;
end;

A49: for v2,v1,v0 holds
 v0"\/"((v0"/\"(v0"\/"v1))"\/"(v1"/\"v2)) = (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2))
proof let v2,v1,v0;
 (v0"/\"(v0"\/"v1))"\/"(v0"\/"(v1"/\"v2)) =
    v0"\/"((v0"/\"(v0"\/"v1))"\/"(v1"/\"v2)) by A28;
hence thesis by A47;
end;

A51: for v2,v1,v0 holds
  v0"\/"((v0"/\"(v0"\/"v1))"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2) by A9;
  hence thesis by A49;
end;

A53: for v0,v1 holds
  v0"/\"(v0"\/"(v1"/\"v0)) = v0"\/"(v0"/\"v1)
proof let v0,v1;
  v0"/\"v1 = v1"/\"v0 by A2;
  hence thesis by A31;
end;

A56: for v102,v1,v100 holds
 (v100"/\"(v100"\/"v1))"\/"v102 = v100"\/"((v100"/\"(v100"\/"v1))"\/"v102)
proof let v102,v1,v100;
  v100"\/"(v100"/\"(v100"\/"v1)) = v100"/\"(v100"\/"v1) by A37;
  hence thesis by A5;
end;

A60: for v2,v1,v0 holds (v0"/\"(v0"\/"v1))"\/"(v1"/\"v2) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  v0"\/"((v0"/\"(v0"\/"v1))"\/"(v1"/\"v2)) = (v0"/\"(v0"\/"v1))"\/"(v1"/\"v2)
    by A56;
  hence thesis by A51;
end;

A63: for v101,v100,v1 holds
  (v100"/\"v101)"\/"(v101"/\"(v1"\/"v100)) = v101"/\"(v100"\/"(v1"\/"v100))
proof let v101,v100,v1;
  v100"\/"(v1"\/"v100) = v1"\/"v100 by A22;
  hence thesis by A40;
end;

A66: for v1,v0,v2 holds
  (v0"/\"v1)"\/"(v1"/\"(v2"\/"v0)) = v1"/\"(v2"\/"v0)
proof let v1,v0,v2;
  v0"\/"(v2"\/"v0) = v2"\/"v0 by A22;
  hence thesis by A63;
end;

A68: for v2,v1,v0 holds
  v0"\/"(v2"/\"(v0"\/"v1)) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  (v1"/\"v2)"\/"(v2"/\"(v0"\/"v1)) = v2"/\"(v0"\/"v1) by A66;
  hence thesis by A34;
end;

A71: for v1,v2,v0 holds v0"\/"((v0"\/"v2)"/\"v1) = v0"\/"(v2"/\"v1)
proof let v1,v2,v0;
  v1"/\"(v0"\/"v2) = (v0"\/"v2)"/\"v1 by A2;
  hence thesis by A68;
end;

A74: for v1,v2,v0 holds
  v0"\/"(v1"/\"(v0"\/"v2)) = v0"\/"(v1"/\"v2)
proof let v1,v2,v0;
  v2"/\"v1 = v1"/\"v2 by A2;
  hence thesis by A68;
end;

A77: for v100,v2 holds
 v100"/\"(v100"\/"(v2"/\"v100)) = v100"\/"(v100"/\"(v100"\/"v2))
proof let v100,v2;
  v100"\/"(v100"/\"(v100"\/"v2)) = v100"\/"(v2"/\"v100) by A68;
  hence thesis by A31;
end;

A80: for v1,v0 holds v0"\/"(v0"/\"v1) = v0"\/"(v0"/\"(v0"\/"v1))
proof let v1,v0;
  v0"/\"(v0"\/"(v1"/\"v0)) = v0"\/"(v0"/\"v1) by A53;
  hence thesis by A77;
end;

A82: for v1,v0 holds v0"\/"(v0"/\"v1) = v0"/\"(v0"\/"v1)
proof let v1,v0;
  v0"\/"(v0"/\"(v0"\/"v1)) = v0"/\"(v0"\/"v1) by A37;
  hence thesis by A80;
end;

A85: for v2,v1,v0 holds
  (v0"\/"(v0"/\"v1))"\/"(v1"/\"v2) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  v0"/\"(v0"\/"v1) = v0"\/"(v0"/\"v1) by A82;
  hence thesis by A60;
end;

A87: for v2,v1,v0 holds
  v0"\/"((v0"/\"v1)"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  (v0"\/"(v0"/\"v1))"\/"(v1"/\"v2) = v0"\/"((v0"/\"v1)"\/"(v1"/\"v2)) by A5;
  hence thesis by A85;
end;

A90: for v102,v1,v100 holds
 (v100"\/"(v100"/\"v1))"\/"((v100"\/"v1)"/\"(v100"\/"v102)) =
   (v100"\/"v1)"/\"(v100"\/"v102)
proof let v102,v1,v100;
  v100"/\"(v100"\/"v1) = v100"\/"(v100"/\"v1) by A82;
  hence thesis by A40;
end;

A93: for v2,v1,v0 holds
 v0"\/"((v0"/\"v1)"\/"((v0"\/"v1)"/\"(v0"\/"v2))) =
   (v0"\/"v1)"/\"(v0"\/"v2)
proof let v2,v1,v0;
  (v0"\/"(v0"/\"v1))"\/"((v0"\/"v1)"/\"(v0"\/"v2)) =
     v0"\/"((v0"/\"v1)"\/"((v0"\/"v1)"/\"(v0"\/"v2))) by A5;
  hence thesis by A90;
end;

A96: for v100,v101,v2,v1 holds
 v100"\/"(v101"\/"(v1"/\"v2)) = v101"\/"(v100"\/"((v101"\/"v1)"/\"v2))
proof let v100,v101,v2,v1;
  v101"\/"((v101"\/"v1)"/\"v2) = v101"\/"(v1"/\"v2) by A71;
  hence thesis by A28;
end;

A101: for v1,v2,v0 holds
 (v0"/\"v1)"\/"(v0"\/"(v1"/\"(v0"\/"v2))) = (v0"\/"v1)"/\"(v0"\/"v2)
proof let v1,v2,v0;
  v0"\/"((v0"/\"v1)"\/"((v0"\/"v1)"/\"(v0"\/"v2))) =
    (v0"/\"v1)"\/"(v0"\/"(v1"/\"(v0"\/"v2))) by A96;
  hence thesis by A93;
end;

A103: for v0,v2,v1 holds
 (v0"/\"v1)"\/"(v0"\/"(v1"/\"v2)) = (v0"\/"v1)"/\"(v0"\/"v2)
proof let v0,v2,v1;
  v0"\/"(v1"/\"(v0"\/"v2)) = v0"\/"(v1"/\"v2) by A74;
  hence thesis by A101;
end;

A105: for v2,v1,v0 holds
 v0"\/"((v0"/\"v1)"\/"(v1"/\"v2)) = (v0"\/"v1)"/\"(v0"\/"v2)
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v0"\/"(v1"/\"v2)) = v0"\/"((v0"/\"v1)"\/"(v1"/\"v2)) by A28;
  hence thesis by A103;
end;

for v0,v2,v1 holds v0"\/"(v1"/\"v2) = (v0"\/"v1)"/\"(v0"\/"v2)
proof let v0,v2,v1;
  v0"\/"((v0"/\"v1)"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2) by A87;
  hence thesis by A105;
end;
hence thesis;
end;

Cluster4:
  L is meet-idempotent meet-associative meet-commutative
       satisfying_QLT1 join-idempotent join-associative join-commutative
        satisfying_QLT2 satisfying_Bowden_inequality implies
    L is distributive'
  proof
    assume
A0: L is meet-idempotent meet-associative meet-commutative satisfying_QLT1
      join-idempotent join-associative join-commutative
        satisfying_QLT2 satisfying_Bowden_inequality; then
A2: (for v0 holds v0"/\"v0 = v0) &
    (for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
    (for v0,v2,v1 holds
      (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2)) &
    (for v0 holds v0"\/"v0 = v0) &
    (for v2,v1,v0 holds
      (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
    (for v1,v0 holds v0"\/"v1 = v1"\/"v0) &
    (for v0,v2,v1 holds
      (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2))
        by LATTICES:def 4, def 5, def 6, SHEFFER1:def 9, ROBBINS1:def 7;
    (for v0,v2,v1 holds
      (v0"\/"(v1"/\"v2))"\/"((v0"\/"v1)"/\"v2) = v0"\/"(v1"/\"v2))
        by BowdenRedef,A0; then
    for v1,v2,v3 holds v1"\/"(v2"/\"v3) = (v1"\/"v2)"/\"(v1"\/"v3)
      by ThQLT4,A2;
    hence thesis by SHEFFER1:def 5;
  end;

registration
  cluster meet-idempotent meet-associative meet-commutative satisfying_QLT1
      join-idempotent join-associative join-commutative
        satisfying_QLT2 satisfying_Bowden_inequality ->
    distributive' for non empty LattStr;
  coherence by Cluster4;
end;

begin :: Preliminaries to QLT-5: Modularity for Quasilattices

definition let L;
  attr L is QLT-selfmodular means
   for v2,v1,v0 holds
     (v0"/\"v1)"\/"(v2"/\"(v0"\/"v1)) = (v0"\/"v1)"/\"(v2"\/"(v0"/\"v1));
end;

definition let L be join-idempotent non empty LattStr;
  let a,b be Element of L;
  redefine pred a [= b;  :: redefines [= (LATTICES, definition)
  reflexivity
  proof
    let a be Element of L;
    a "\/" a = a by ROBBINS1:def 7;
    hence thesis by LATTICES:def 3;
  end;
end;

:: Modularity for Quasilattices

theorem QLTMod: :: Modularity for QuasiLattices
(for v0 holds v0"/\"v0 = v0) &
(for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2)) &
(for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
(for v0,v2,v1 holds (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2)) &
(for v0 holds v0"\/"v0 = v0) &
(for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
(for v1,v0 holds v0"\/"v1 = v1"\/"v0) &
(for v0,v2,v1 holds (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
(for v0,v1,v2 st v0 "\/" v1 = v1 holds
 v0"\/"(v2"/\"v1) = (v0"\/"v2)"/\"v1) implies
for v1,v2,v3 holds (v1"/\"v2)"\/"(v1"/\"v3) = v1"/\"(v2"\/"(v1"/\"v3))
proof
assume A2: for v0 holds v0"/\"v0 = v0;
assume A3: for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2);
assume A4: for v1,v0 holds v0"/\"v1 = v1"/\"v0;
assume A5: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2);
assume A6: for v0 holds v0"\/"v0 = v0;
assume A7: for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2);
assume A8: for v1,v0 holds v0"\/"v1 = v1"\/"v0;
assume A9: for v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2);

A11: for v0,v2,v1 holds (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) by A4;
  hence thesis by A9;
end;

assume A12: for v0,v1,v2 st v0 "\/" v1 = v1 holds
 v0"\/"(v2"/\"v1) = (v0"\/"v2)"/\"v1;

assume not thesis; then
consider c1,c2,c3 being Element of L such that
A14:  (c1"/\"c2)"\/"(c1"/\"c3) <> c1"/\"(c2"\/"(c1"/\"c3));

A17: for v0,v2,v1 holds (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) by A8;
  hence thesis by A5;
end;

A20: for v102,v101 holds v101"/\"v102 = v101"/\"(v101"/\"v102)
proof let v102,v101;
  v101"/\"v101 = v101 by A2;
  hence thesis by A3;
end;

A24: for v2,v0,v1 holds (v1"/\"v0)"/\"v2 = v0"/\"(v1"/\"v2)
proof let v2,v0,v1;
  v0"/\"v1 = v1"/\"v0 by A4;
  hence thesis by A3;
end;

A27: for v0,v2,v1 holds v0"/\"(v1"/\"v2) = v1"/\"(v0"/\"v2)
proof let v0,v2,v1;
  (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2) by A3;
  hence thesis by A24;
end;

A30: for v102,v101 holds v101"\/"v102 = v101"\/"(v101"\/"v102)
proof let v102,v101;
  v101"\/"v101 = v101 by A6;
  hence thesis by A7;
end;

A35: for v102,v100 holds (v100"\/"v102)"\/"v102 = v100"\/"v102
proof let v102,v100;
  v102"\/"v102 = v102 by A6;
  hence thesis by A7;
end;

A38: for v1,v0 holds v1"\/"(v0"\/"v1) = v0"\/"v1
proof let v1,v0;
  (v0"\/"v1)"\/"v1 = v1"\/"(v0"\/"v1) by A8;
  hence thesis by A35;
end;

A41: for v0,v2,v1 holds (v1"\/"v0)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  v0"\/"v1 = v1"\/"v0 by A8;
  hence thesis by A11;
end;

A45: for v2,v100 holds (v100"\/"v2)"/\"v100 = v100"\/"(v2"/\"v100)
proof let v2,v100;
  v100"\/"v100 = v100 implies (v100"\/"v2)"/\"v100 = v100"\/"(v2"/\"v100)
    by A12;
  hence thesis by A6;
end;

A48: for v1,v0 holds v0"/\"(v0"\/"v1) = v0"\/"(v1"/\"v0)
proof let v1,v0;
  (v0"\/"v1)"/\"v0 = v0"/\"(v0"\/"v1) by A4;
  hence thesis by A45;
end;

A52: for v102,v1,v100 holds
 (v100"/\"v1)"\/"(v100"/\"((v100"/\"v1)"\/"v102)) =
    v100"/\"((v100"/\"v1)"\/"v102)
proof let v102,v1,v100;
  v100"/\"(v100"/\"v1) = v100"/\"v1 by A20;
  hence thesis by A17;
end;

A56: for v101,v2,v100 holds
 (v100"\/"v2)"/\"(v100"\/"v101) = v100"\/"(v2"/\"(v100"\/"v101))
proof let v101,v2,v100;
  v100"\/"(v100"\/"v101) = v100"\/"v101 implies
    (v100"\/"v2)"/\"(v100"\/"v101) = v100"\/"(v2"/\"(v100"\/"v101)) by A12;
  hence thesis by A30;
end;

A60: for v101,v2,v100 holds
 (v100"\/"v2)"/\"(v101"\/"v100) = v100"\/"(v2"/\"(v101"\/"v100))
proof let v101,v2,v100;
  v100"\/"(v101"\/"v100) = v101"\/"v100 implies
    (v100"\/"v2)"/\"(v101"\/"v100) = v100"\/"(v2"/\"(v101"\/"v100)) by A12;
  hence thesis by A38;
end;

A63: for v1,v0 holds v0"\/"(v0"/\"v1) = v0"/\"(v0"\/"v1)
proof let v1,v0;
  v1"/\"v0 = v0"/\"v1 by A4;
  hence thesis by A48;
end;

A67: for v100,v1,v101 holds
 v100"/\"(v101"\/"(v101"/\"v1)) = v101"/\"(v100"/\"(v101"\/"v1))
proof let v100,v1,v101;
  v101"/\"(v101"\/"v1) = v101"\/"(v101"/\"v1) by A63;
  hence thesis by A27;
end;

A71: for v1,v2,v0 holds v1"\/"(v0"/\"v2) = (v1"\/"(v0"/\"v2))"/\"(v0"\/"v1)
proof let v1,v2,v0;
  (v0"\/"v1)"/\"(v1"\/"(v0"/\"v2)) = v1"\/"(v0"/\"v2) by A41;
  hence thesis by A4;
end;

A74: for v0,v2,v1 holds v0"\/"(v1"/\"v2) = v0"\/"((v1"/\"v2)"/\"(v1"\/"v0))
proof let v0,v2,v1;
  (v0"\/"(v1"/\"v2))"/\"(v1"\/"v0) = v0"\/"((v1"/\"v2)"/\"(v1"\/"v0)) by A60;
  hence thesis by A71;
end;

A76: for v0,v2,v1 holds v0"\/"(v1"/\"v2) = v0"\/"(v1"/\"(v2"/\"(v1"\/"v0)))
proof let v0,v2,v1;
  (v1"/\"v2)"/\"(v1"\/"v0) = v1"/\"(v2"/\"(v1"\/"v0)) by A3;
  hence thesis by A74;
end;

A79: for v2,v0,v1 holds (v1"\/"v0)"/\"(v0"\/"v2) = v0"\/"(v1"/\"(v0"\/"v2))
proof let v2,v0,v1;
  v0"\/"v1 = v1"\/"v0 by A8;
  hence thesis by A56;
end;

A83: for v101,v2,v102 holds
 ((v102"/\"v2)"\/"v101)"/\"(v102"\/"(v102"/\"v2)) =
   (v102"/\"v2)"\/"(v102"/\"(v101"/\"(v102"\/"v2)))
proof let v101,v2,v102;
  v101"/\"(v102"\/"(v102"/\"v2)) = v102"/\"(v101"/\"(v102"\/"v2)) by A67;
  hence thesis by A60;
end;

A86: for v2,v1,v0 holds
 (v0"\/"(v0"/\"v1))"/\"((v0"/\"v1)"\/"v2) =
   (v0"/\"v1)"\/"(v0"/\"(v2"/\"(v0"\/"v1)))
proof let v2,v1,v0;
  ((v0"/\"v1)"\/"v2)"/\"(v0"\/"(v0"/\"v1)) =
    (v0"\/"(v0"/\"v1))"/\"((v0"/\"v1)"\/"v2) by A4;
  hence thesis by A83;
end;

A88: for v2,v1,v0 holds
 (v0"/\"v1)"\/"(v0"/\"((v0"/\"v1)"\/"v2)) =
   (v0"/\"v1)"\/"(v0"/\"(v2"/\"(v0"\/"v1)))
proof let v2,v1,v0;
  (v0"\/"(v0"/\"v1))"/\"((v0"/\"v1)"\/"v2) =
    (v0"/\"v1)"\/"(v0"/\"((v0"/\"v1)"\/"v2)) by A79;
  hence thesis by A86;
end;

A90: for v2,v1,v0 holds
  v0"/\"((v0"/\"v1)"\/"v2) = (v0"/\"v1)"\/"(v0"/\"(v2"/\"(v0"\/"v1)))
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v0"/\"((v0"/\"v1)"\/"v2)) = v0"/\"((v0"/\"v1)"\/"v2) by A52;
  hence thesis by A88;
end;

A94: for v102,v2,v101 holds
 (v101"/\"v2)"\/"(v101"/\"(v101"/\"(v102"/\"(v101"\/"v2)))) =
   (v101"/\"v2)"\/"(v101"/\"v102)
proof let v102,v2,v101;
  v102"/\"(v101"\/"(v101"/\"v2)) = v101"/\"(v102"/\"(v101"\/"v2)) by A67;
  hence thesis by A76;
end;

A97: for v2,v1,v0 holds
 (v0"/\"v1)"\/"(v0"/\"(v2"/\"(v0"\/"v1))) = (v0"/\"v1)"\/"(v0"/\"v2)
proof let v2,v1,v0;
  v0"/\"(v0"/\"(v2"/\"(v0"\/"v1))) = v0"/\"(v2"/\"(v0"\/"v1)) by A20;
  hence thesis by A94;
end;

A99: for v2,v1,v0 holds (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"((v0"/\"v1)"\/"v2)
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v0"/\"(v2"/\"(v0"\/"v1))) = (v0"/\"v1)"\/"(v0"/\"v2) by A97;
  hence thesis by A90;
end;

A102: for v2,v1,v0 holds v0"/\"(v2"\/"(v0"/\"v1)) = (v0"/\"v1)"\/"(v0"/\"v2)
proof let v2,v1,v0;
  (v0"/\"v1)"\/"v2 = v2"\/"(v0"/\"v1) by A8;
  hence thesis by A99;
end;

(c1"/\"c3)"\/"(c1"/\"c2) <> (c1"/\"c2)"\/"(c1"/\"c3) by A102,A14;
hence thesis by A8;
end;

ClusterA:
  L is meet-idempotent join-idempotent meet-commutative join-commutative
       meet-associative join-associative satisfying_QLT1 satisfying_QLT2
       modular implies
    for v1,v2,v3 holds
      (v1"/\"v2)"\/"(v1"/\"v3) = v1"/\"(v2"\/"(v1"/\"v3))
  proof
    assume
A0: L is meet-idempotent join-idempotent meet-commutative join-commutative
       meet-associative join-associative satisfying_QLT1
       satisfying_QLT2 modular; then
A1: (for v0 holds v0"/\"v0 = v0) &
    (for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2)) &
    (for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
    (for v0,v2,v1 holds (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2)) &
    (for v0 holds v0"\/"v0 = v0) &
    (for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
    (for v1,v0 holds v0"\/"v1 = v1"\/"v0) &
    (for v0,v2,v1 holds (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2))
      by LATTICES:def 4,def 5,def 6,def 7,SHEFFER1:def 9,ROBBINS1:def 7;
    for v0,v1,v2 st v0 "\/" v1 = v1 holds
       v0"\/"(v2"/\"v1) = (v0"\/"v2)"/\"v1 by A0,LATTICES:def 3,def 12;
    hence thesis by QLTMod,A1;
  end;

theorem QLTMod2:
(for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
(for v0,v2,v1 holds (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2)) &
(for v0 holds v0"\/"v0 = v0) &
(for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
(for v1,v0 holds v0"\/"v1 = v1"\/"v0) &
(for v0,v2,v1 holds (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
(for v2,v1,v0 holds (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v1"\/"(v0"/\"v2)))
implies
for v1,v2,v3 st v1"\/"v2 = v2 holds v1"\/"(v3"/\"v2) = (v1"\/"v3)"/\"v2
proof
assume A1: for v1,v0 holds v0"/\"v1 = v1"/\"v0;
assume A2: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2);
assume A3: for v0 holds v0"\/"v0 = v0;
assume A4: for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2);
assume A5: for v1,v0 holds v0"\/"v1 = v1"\/"v0;
assume A6: for v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2);

A8: for v0,v2,v1 holds (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) by A1;
  hence thesis by A6;
end;

assume A9: for v2,v1,v0 holds
 (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v1"\/"(v0"/\"v2));

assume not thesis; then
consider c1, c2, c3 being Element of L such that
A10: c1"\/"c2 = c2 and
A11:  c1"\/"(c3"/\"c2) <> (c1"\/"c3)"/\"c2;

c1"\/"(c2"/\"c3) <> (c1"\/"c3)"/\"c2 by A1,A11; then
A15:  c1"\/"(c2"/\"c3) <> c2"/\"(c1"\/"c3) by A1;

A17: for v0,v2,v1 holds (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) by A5;
  hence thesis by A2;
end;

A19: for v0,v2,v1 holds v0"/\"(v1"\/"(v0"/\"(v1"\/"v2))) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"(v0"/\"(v1"\/"v2))) by A9;
  hence thesis by A17;
end;

A22: for v102,v101 holds v101"\/"v102 = v101"\/"(v101"\/"v102)
proof let v102,v101;
  v101"\/"v101 = v101 by A3;
  hence thesis by A4;
end;

A27: for v1,v2,v0 holds v0"/\"(v1"\/"(v0"/\"v2)) = (v0"/\"v2)"\/"(v0"/\"v1)
proof let v1,v2,v0;
  (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v1"\/"(v0"/\"v2)) by A9;
  hence thesis by A5;
end;

A29: for v1,v2,v0 holds v0"/\"(v1"\/"(v0"/\"v2)) = v0"/\"(v2"\/"(v0"/\"v1))
proof let v1,v2,v0;
  (v0"/\"v2)"\/"(v0"/\"v1) = v0"/\"(v2"\/"(v0"/\"v1)) by A9;
  hence thesis by A27;
end;

A35: for v102 holds c1"\/"(c2"/\"(c1"\/"v102)) = c2"/\"(c1"\/"v102)
proof let v102;
  c2"/\"(c1"\/"(c2"/\"(c1"\/"v102))) = c1"\/"(c2"/\"(c1"\/"v102)) by A10,A8;
  hence thesis by A19;
end;

A38: for v0 holds c1"\/"(c2"/\"(v0"\/"c1)) = c2"/\"(c1"\/"v0)
proof let v0;
  c1"\/"v0 = v0"\/"c1 by A5;
  hence thesis by A35;
end;

A40: for v0,v2,v1 holds v0"/\"((v1"\/"v2)"\/"(v0"/\"v1)) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  v0"/\"(v1"\/"(v0"/\"(v1"\/"v2))) = v0"/\"((v1"\/"v2)"\/"(v0"/\"v1)) by A29;
  hence thesis by A19;
end;

A42: for v2,v1,v0 holds v0"/\"(v1"\/"(v2"\/"(v0"/\"v1))) = v0"/\"(v1"\/"v2)
proof let v2,v1,v0;
  (v1"\/"v2)"\/"(v0"/\"v1) = v1"\/"(v2"\/"(v0"/\"v1)) by A4;
  hence thesis by A40;
end;

A44: for v0 holds c2"/\"(v0"\/"(c2"/\"c1)) = c1"\/"(c2"/\"v0)
proof let v0;
  c2"/\"(c1"\/"(c2"/\"v0)) = c2"/\"(v0"\/"(c2"/\"c1)) by A29;
  hence thesis by A10,A8;
end;

A45: c2"/\"c1 = c1"/\"c2 by A1;

A50: for v101 holds
 c2"/\"(v101"\/"(c1"\/"(c2"/\"v101))) = c2"/\"(v101"\/"(c1"/\"c2))
proof let v101;
  c1"\/"(c2"/\"v101) = c2"/\"(v101"\/"(c1"/\"c2)) by A45,A44;
  hence thesis by A19;
end;

A53: for v0 holds c2"/\"(v0"\/"c1) = c2"/\"(v0"\/"(c1"/\"c2))
proof let v0;
  c2"/\"(v0"\/"(c1"\/"(c2"/\"v0))) = c2"/\"(v0"\/"c1) by A42;
  hence thesis by A50;
end;

A57: for v0 holds c1"\/"(c2"/\"(v0"\/"(c1"/\"c2))) = c1"\/"(c2"/\"v0)
proof let v0;
  c1"\/"(c2"/\"v0) = c2"/\"(v0"\/"(c1"/\"c2)) by A45,A44;
  hence thesis by A22;
end;

A59: for v0 holds c1"\/"(c2"/\"(v0"\/"c1)) = c1"\/"(c2"/\"v0)
proof let v0;
  c2"/\"(v0"\/"(c1"/\"c2)) = c2"/\"(v0"\/"c1) by A53;
  hence thesis by A57;
end;

for v0 holds c2"/\"(c1"\/"v0) = c1"\/"(c2"/\"v0)
proof let v0;
  c1"\/"(c2"/\"(v0"\/"c1)) = c2"/\"(c1"\/"v0) by A38;
  hence thesis by A59;
end;
hence thesis by A15;
end;

ClusterB:
  L is meet-idempotent join-idempotent meet-commutative join-commutative
       meet-associative join-associative satisfying_QLT1 satisfying_QLT2 &
   (for v2,v1,v0 being Element of L holds
     (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v1"\/"(v0"/\"v2)))
  implies L is modular
  proof
    assume
A0: L is meet-idempotent join-idempotent meet-commutative join-commutative
       meet-associative join-associative satisfying_QLT1 satisfying_QLT2;
A1: (for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
    (for v0,v2,v1 holds (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2)) &
    (for v0 holds v0"\/"v0 = v0) &
    (for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
    (for v1,v0 holds v0"\/"v1 = v1"\/"v0) &
    (for v0,v2,v1 holds (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
    (for v2,v1,v0 holds (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v1"\/"(v0"/\"v2)))
      implies
    for v1,v2,v3 st v1"\/"v2 = v2 holds v1"\/"(v3"/\"v2) = (v1"\/"v3)"/\"v2
      by QLTMod2;
    assume
AA: for v2,v1,v0 being Element of L holds
     (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v1"\/"(v0"/\"v2));
    for a,b,c being Element of L st a [= c holds a"\/"(b"/\"c) = (a"\/"b)"/\"c
    proof
      let a,b,c be Element of L;
      assume a [= c; then
      a "\/" c = c by LATTICES:def 3;
      hence thesis by A1,AA,A0,LATTICES:def 4,def 5,def 6,ROBBINS1:def 7;
    end;
    hence thesis by LATTICES:def 12;
  end;

definition let L be meet-idempotent join-idempotent meet-commutative
       join-commutative meet-associative join-associative satisfying_QLT1
    satisfying_QLT2 non empty LattStr;
  redefine attr L is modular means :ModRedef:  :: redefines modular (LATTICES, definition)
  for v1,v2,v3 being Element of L holds
      (v1"/\"v2)"\/"(v1"/\"v3) = v1"/\"(v2"\/"(v1"/\"v3));
  compatibility
  proof
    thus L is modular implies
    for v1,v2,v3 being Element of L holds
      (v1"/\"v2)"\/"(v1"/\"v3) = v1"/\"(v2"\/"(v1"/\"v3)) by ClusterA;
SS: L is meet-idempotent join-idempotent meet-commutative
       join-commutative meet-associative join-associative
       satisfying_QLT1 satisfying_QLT2 &
    (for v2,v1,v0 being Element of L holds
      (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v1"\/"(v0"/\"v2)))
        implies L is modular by ClusterB;
    assume for v1,v2,v3 being Element of L holds
      (v1"/\"v2)"\/"(v1"/\"v3) = v1"/\"(v2"\/"(v1"/\"v3));
    hence thesis by SS;
  end;
end;

begin :: QLT-5

theorem ThQLT5: :: QLT-5
(for v0 holds v0"/\"v0 = v0) &
(for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2)) &
(for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
(for v0,v2,v1 holds (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2)) &
(for v0 holds v0"\/"v0 = v0) &
(for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
(for v1,v0 holds v0"\/"v1 = v1"\/"v0) &
(for v0,v2,v1 holds (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
(for v2,v1,v0 holds
 (v0"/\"v1)"\/"(v2"/\"(v0"\/"v1)) = (v0"\/"v1)"/\"(v2"\/"(v0"/\"v1))) implies
  for v1,v2,v3 holds
    (v1"/\"v2)"\/"(v1"/\"v3) = v1"/\"(v2"\/"(v1"/\"v3))
proof
assume A1: for v0 holds v0"/\"v0 = v0;
assume A2: for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2);
assume A3: for v1,v0 holds v0"/\"v1 = v1"/\"v0;
assume A4: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2);
assume A5: for v0 holds v0"\/"v0 = v0;
assume A6: for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2);
assume A7: for v1,v0 holds v0"\/"v1 = v1"\/"v0;
assume A8: for v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2);

A10: for v0,v2,v1 holds (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) by A3;
  hence thesis by A8;
end;

assume A11: for v2,v1,v0 holds
 (v0"/\"v1)"\/"(v2"/\"(v0"\/"v1)) = (v0"\/"v1)"/\"(v2"\/"(v0"/\"v1));

A14: for v0,v2,v1 holds (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) by A7;
  hence thesis by A4;
end;

A17: for v102,v101 holds v101"/\"v102 = v101"/\"(v101"/\"v102)
proof let v102,v101;
  v101"/\"v101 = v101 by A1;
  hence thesis by A2;
end;

A22: for v102,v100 holds (v100"/\"v102)"/\"v102 = v100"/\"v102
proof let v102,v100;
  v102"/\"v102 = v102 by A1;
  hence thesis by A2;
end;

A25: for v1,v0 holds v1"/\"(v0"/\"v1) = v0"/\"v1
proof let v1,v0;
  (v0"/\"v1)"/\"v1 = v1"/\"(v0"/\"v1) by A3;
  hence thesis by A22;
end;

A28: for v2,v0,v1 holds (v1"/\"v0)"/\"v2 = v0"/\"(v1"/\"v2)
proof let v2,v0,v1;
  v0"/\"v1 = v1"/\"v0 by A3;
  hence thesis by A2;
end;

A31: for v0,v2,v1 holds v0"/\"(v1"/\"v2) = v1"/\"(v0"/\"v2)
proof let v0,v2,v1;
  (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2) by A2;
  hence thesis by A28;
end;

A34: for v102,v101 holds v101"\/"v102 = v101"\/"(v101"\/"v102)
proof let v102,v101;
  v101"\/"v101 = v101 by A5;
  hence thesis by A6;
end;

A39: for v102,v100 holds (v100"\/"v102)"\/"v102 = v100"\/"v102
proof let v102,v100;
  v102"\/"v102 = v102 by A5;
  hence thesis by A6;
end;

A42: for v1,v0 holds v1"\/"(v0"\/"v1) = v0"\/"v1
proof let v1,v0;
  (v0"\/"v1)"\/"v1 = v1"\/"(v0"\/"v1) by A7;
  hence thesis by A39;
end;

A45: for v2,v0,v1 holds (v1"\/"v0)"\/"v2 = v0"\/"(v1"\/"v2)
proof let v2,v0,v1;
  v0"\/"v1 = v1"\/"v0 by A7;
  hence thesis by A6;
end;

A48: for v0,v2,v1 holds v0"\/"(v1"\/"v2) = v1"\/"(v0"\/"v2)
proof let v0,v2,v1;
  (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2) by A6;
  hence thesis by A45;
end;

A51: for v102,v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"v102 = (v0"\/"v1)"/\"((v0"\/"(v1"/\"v2))"/\"v102)
proof let v102,v0,v2,v1;
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2) by A10;
  hence thesis by A2;
end;

A55: for v0,v2,v1 holds (v1"\/"v0)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  v0"\/"v1 = v1"\/"v0 by A7;
  hence thesis by A10;
end;

A59: for v101,v102 holds
 v101"\/"(v102"/\"(v101"\/"v101)) =
  (v101"\/"v101)"/\"(v102"\/"(v101"/\"v101))
proof let v101,v102;
  v101"/\"v101 = v101 by A1;
  hence thesis by A11;
end;

A62: for v0,v1 holds v0"\/"(v1"/\"v0) = (v0"\/"v0)"/\"(v1"\/"(v0"/\"v0))
proof let v0,v1;
  v0"\/"v0 = v0 by A5;
  hence thesis by A59;
end;

A64: for v0,v1 holds v0"\/"(v1"/\"v0) = v0"/\"(v1"\/"(v0"/\"v0))
proof let v0,v1;
  v0"\/"v0 = v0 by A5;
  hence thesis by A62;
end;

A66: for v0,v1 holds v0"\/"(v1"/\"v0) = v0"/\"(v1"\/"v0)
proof let v0,v1;
  v0"/\"v0 = v0 by A1;
  hence thesis by A64;
end;

A69: for v0,v1,v101,v100 holds
 (v100"/\"v101)"\/"(v0"/\"(v1"/\"(v100"\/"v101))) =
   (v100"\/"v101)"/\"((v0"/\"v1)"\/"(v100"/\"v101))
proof let v0,v1,v101,v100;
  (v0"/\"v1)"/\"(v100"\/"v101) = v0"/\"(v1"/\"(v100"\/"v101)) by A2;
  hence thesis by A11;
end;

A72: for v2,v1,v0 holds
 (v0"/\"v1)"\/"(v2"/\"(v0"\/"v1)) = (v0"\/"v1)"/\"((v0"/\"v1)"\/"v2)
proof let v2,v1,v0;
  v2"\/"(v0"/\"v1) = (v0"/\"v1)"\/"v2 by A7;
  hence thesis by A11;
end;

A74: for v0,v2,v1 holds (v1"/\"v0)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  v0"/\"v1 = v1"/\"v0 by A3;
  hence thesis by A14;
end;

A77: for v0,v1,v2 holds (v0"/\"v1)"\/"(v0"/\"(v2"\/"v1)) = v0"/\"(v1"\/"v2)
proof let v0,v1,v2;
  v1"\/"v2 = v2"\/"v1 by A7;
  hence thesis by A14;
end;

A80: for v101,v2,v1 holds
 ((v101"/\"v1)"\/"v101)"/\"(v101"/\"(v1"\/"v2)) =
  (v101"/\"v1)"\/"(v101"/\"(v1"\/"v2))
proof let v101,v2,v1;
  (v101"/\"v1)"\/"(v101"/\"(v1"\/"v2)) = v101"/\"(v1"\/"v2) by A14;
  hence thesis by A10;
end;

A83: for v0,v2,v1 holds
 (v0"\/"(v0"/\"v1))"/\"(v0"/\"(v1"\/"v2)) = (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2))
proof let v0,v2,v1;
  (v0"/\"v1)"\/"v0 = v0"\/"(v0"/\"v1) by A7;
  hence thesis by A80;
end;

A85: for v2,v1,v0 holds
 v0"/\"((v0"\/"(v0"/\"v1))"/\"(v1"\/"v2)) = (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2))
proof let v2,v1,v0;
  (v0"\/"(v0"/\"v1))"/\"(v0"/\"(v1"\/"v2)) =
    v0"/\"((v0"\/"(v0"/\"v1))"/\"(v1"\/"v2)) by A31;
  hence thesis by A83;
end;

A87: for v2,v1,v0 holds
  v0"/\"((v0"\/"(v0"/\"v1))"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2) by A14;
  hence thesis by A85;
end;

A90: for v102,v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"\/"((v0"\/"v1)"/\"((v0"\/"(v1"/\"v2))"\/"v102)) =
  (v0"\/"v1)"/\"((v0"\/"(v1"/\"v2))"\/"v102)
proof let v102,v0,v2,v1;
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2) by A10;
  hence thesis by A14;
end;

A93: for v0,v3,v2,v1 holds
 (v0"\/"(v1"/\"v2))"\/"((v0"\/"v1)"/\"(v0"\/"((v1"/\"v2)"\/"v3))) =
  (v0"\/"v1)"/\"((v0"\/"(v1"/\"v2))"\/"v3)
proof let v0,v3,v2,v1;
  (v0"\/"(v1"/\"v2))"\/"v3 = v0"\/"((v1"/\"v2)"\/"v3) by A6;
  hence thesis by A90;
end;

A95: for v0,v3,v2,v1 holds
 v0"\/"((v1"/\"v2)"\/"((v0"\/"v1)"/\"(v0"\/"((v1"/\"v2)"\/"v3)))) =
   (v0"\/"v1)"/\"((v0"\/"(v1"/\"v2))"\/"v3)
proof let v0,v3,v2,v1;
  (v0"\/"(v1"/\"v2))"\/"((v0"\/"v1)"/\"(v0"\/"((v1"/\"v2)"\/"v3))) =
    v0"\/"((v1"/\"v2)"\/"((v0"\/"v1)"/\"(v0"\/"((v1"/\"v2)"\/"v3)))) by A6;
  hence thesis by A93;
end;

A97: for v0,v3,v2,v1 holds
 v0"\/"((v1"/\"v2)"\/"((v0"\/"v1)"/\"(v0"\/"((v1"/\"v2)"\/"v3)))) =
   (v0"\/"v1)"/\"(v0"\/"((v1"/\"v2)"\/"v3))
proof let v0,v3,v2,v1;
  (v0"\/"(v1"/\"v2))"\/"v3 = v0"\/"((v1"/\"v2)"\/"v3) by A6;
  hence thesis by A95;
end;

A100: for v102,v1,v100 holds
 (v100"/\"v1)"\/"(v100"/\"((v100"/\"v1)"\/"v102)) =
   v100"/\"((v100"/\"v1)"\/"v102)
proof let v102,v1,v100;
  v100"/\"(v100"/\"v1) = v100"/\"v1 by A17;
  hence thesis by A14;
end;

A104: for v0,v100,v1 holds v100"\/"(v0"\/"(v1"\/"v100)) = (v0"\/"v1)"\/"v100
proof let v0,v100,v1;
  (v0"\/"v1)"\/"v100 = v0"\/"(v1"\/"v100) by A6;
  hence thesis by A42;
end;

A107: for v1,v0,v2 holds v0"\/"(v1"\/"(v2"\/"v0)) = v1"\/"(v2"\/"v0)
proof let v1,v0,v2;
  (v1"\/"v2)"\/"v0 = v1"\/"(v2"\/"v0) by A6;
  hence thesis by A104;
end;

A109: for v1,v0 holds v0"\/"(v0"/\"v1) = v0"/\"(v1"\/"v0)
proof let v1,v0;
  v1"/\"v0 = v0"/\"v1 by A3;
  hence thesis by A66;
end;

A112: for v102,v100,v1 holds
 (v100"/\"(v1"\/"v100))"\/"v102 = v100"\/"((v1"/\"v100)"\/"v102)
proof let v102,v100,v1;
  v100"\/"(v1"/\"v100) = v100"/\"(v1"\/"v100) by A66;
  hence thesis by A6;
end;

A115: for v0,v1 holds v0"\/"(v1"/\"v0) = v0"/\"(v0"\/"v1)
proof let v0,v1;
  v1"\/"v0 = v0"\/"v1 by A7;
  hence thesis by A66;
end;

A117: for v1,v0 holds v0"\/"(v0"/\"v1) = v0"/\"(v0"\/"v1)
proof let v1,v0;
  v1"\/"v0 = v0"\/"v1 by A7;
  hence thesis by A109;
end;

A119: for v2,v1,v0 holds
 v0"/\"((v0"/\"(v0"\/"v1))"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v2,v1,v0;
  v0"\/"(v0"/\"v1) = v0"/\"(v0"\/"v1) by A117;
  hence thesis by A87;
end;

A121: for v2,v1,v0 holds
 v0"/\"(v0"/\"((v0"\/"v1)"/\"(v1"\/"v2))) = v0"/\"(v1"\/"v2)
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"/\"(v1"\/"v2) = v0"/\"((v0"\/"v1)"/\"(v1"\/"v2)) by A2;
  hence thesis by A119;
end;

A123: for v2,v1,v0 holds v0"/\"((v0"\/"v1)"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v2,v1,v0;
  v0"/\"(v0"/\"((v0"\/"v1)"/\"(v1"\/"v2))) =
    v0"/\"((v0"\/"v1)"/\"(v1"\/"v2)) by A17;
  hence thesis by A121;
end;

A126: for v102,v1,v100 holds
 (v100"/\"(v100"\/"v1))"\/"v102 = v100"\/"((v1"/\"v100)"\/"v102)
proof let v102,v1,v100;
  v100"\/"(v1"/\"v100) = v100"/\"(v100"\/"v1) by A115;
  hence thesis by A6;
end;

A130: for v101,v102 holds
 (v102"\/"v101)"/\"(v101"\/"v102) = v101"\/"(v102"/\"v102)
proof let v101,v102;
  v102"/\"v102 = v102 by A1;
  hence thesis by A55;
end;

A133: for v1,v0 holds (v0"\/"v1)"/\"(v1"\/"v0) = v1"\/"v0
proof let v1,v0;
  v0"/\"v0 = v0 by A1;
  hence thesis by A130;
end;

A136: for v101,v100,v1 holds
 (v100"\/"v101)"/\"(v101"\/"(v1"/\"v100)) = v101"\/"(v100"/\"(v1"/\"v100))
proof let v101,v100,v1;
  v100"/\"(v1"/\"v100) = v1"/\"v100 by A25;
  hence thesis by A55;
end;

A139: for v1,v0,v2 holds (v0"\/"v1)"/\"(v1"\/"(v2"/\"v0)) = v1"\/"(v2"/\"v0)
proof let v1,v0,v2;
  v0"/\"(v2"/\"v0) = v2"/\"v0 by A25;
  hence thesis by A136;
end;

A142: for v102,v1,v100 holds
 (v100"/\"(v100"\/"v1))"/\"((v100"/\"v1)"\/"(v100"/\"v102)) =
   (v100"/\"v1)"\/"(v100"/\"v102)
proof let v102,v1,v100;
  v100"\/"(v100"/\"v1) = v100"/\"(v100"\/"v1) by A117;
  hence thesis by A55;
end;

A145: for v2,v1,v0 holds
 v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"(v0"/\"v2))) = (v0"/\"v1)"\/"(v0"/\"v2)
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"/\"((v0"/\"v1)"\/"(v0"/\"v2)) =
    v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"(v0"/\"v2))) by A2;
  hence thesis by A142;
end;

A148: for v100,v102,v101 holds
 (v100"\/"v101)"/\"((v101"/\"v102)"\/"v100) =
   (v100"\/"(v101"/\"v102))"/\"((v101"/\"v102)"\/"v100)
proof let v100,v102,v101;
  (v100"\/"(v101"/\"v102))"/\"((v101"/\"v102)"\/"v100) =
    (v101"/\"v102)"\/"v100 by A133;
  hence thesis by A51;
end;

A151: for v0,v2,v1 holds (v0"\/"v1)"/\"((v1"/\"v2)"\/"v0) = (v1"/\"v2)"\/"v0
proof let v0,v2,v1;
  (v0"\/"(v1"/\"v2))"/\"((v1"/\"v2)"\/"v0) = (v1"/\"v2)"\/"v0 by A133;
  hence thesis by A148;
end;

A154: for v101,v102 holds
 (v102"/\"v101)"\/"(v101"/\"v102) = v101"/\"(v102"\/"v102)
proof let v101,v102;
  v102"\/"v102 = v102 by A5;
  hence thesis by A74;
end;

A157: for v1,v0 holds (v0"/\"v1)"\/"(v1"/\"v0) = v1"/\"v0
proof let v1,v0;
  v0"\/"v0 = v0 by A5;
  hence thesis by A154;
end;

A159: for v1,v0,v2 holds (v0"/\"v1)"\/"(v1"/\"(v2"\/"v0)) = v1"/\"(v0"\/"v2)
proof let v1,v0,v2;
  v0"\/"v2 = v2"\/"v0 by A7;
  hence thesis by A74;
end;

A162: for v102,v0,v1 holds
 (v1"/\"v0)"\/"v102 = (v0"/\"v1)"\/"((v1"/\"v0)"\/"v102)
proof let v102,v0,v1;
  (v0"/\"v1)"\/"(v1"/\"v0) = v1"/\"v0 by A157;
  hence thesis by A6;
end;

A168: for v101,v2,v1 holds
 ((v101"\/"v1)"/\"v101)"\/"((v101"\/"v1)"/\"((v1"/\"v2)"\/"v101)) =
  v101"\/"(v1"/\"v2)
proof let v101,v2,v1;
  (v101"\/"v1)"/\"(v101"\/"(v1"/\"v2)) = v101"\/"(v1"/\"v2) by A10;
  hence thesis by A77;
end;

A171: for v0,v2,v1 holds
 (v0"/\"(v0"\/"v1))"\/"((v0"\/"v1)"/\"((v1"/\"v2)"\/"v0)) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v0"\/"v1)"/\"v0 = v0"/\"(v0"\/"v1) by A3;
  hence thesis by A168;
end;

A173: for v0,v2,v1 holds
 (v0"/\"(v0"\/"v1))"\/"((v1"/\"v2)"\/"v0) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v0"\/"v1)"/\"((v1"/\"v2)"\/"v0) = (v1"/\"v2)"\/"v0 by A151;
  hence thesis by A171;
end;

A175: for v0,v2,v1 holds
  v0"\/"((v1"/\"v0)"\/"((v1"/\"v2)"\/"v0)) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v0"/\"(v0"\/"v1))"\/"((v1"/\"v2)"\/"v0) =
    v0"\/"((v1"/\"v0)"\/"((v1"/\"v2)"\/"v0)) by A126;
  hence thesis by A173;
end;

A177: for v0,v2,v1 holds (v1"/\"v0)"\/"((v1"/\"v2)"\/"v0) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  v0"\/"((v1"/\"v0)"\/"((v1"/\"v2)"\/"v0)) =
    (v1"/\"v0)"\/"((v1"/\"v2)"\/"v0) by A107;
  hence thesis by A175;
end;

A180: for v2,v3,v1,v0 holds
 (v0"/\"v1)"\/"(v2"/\"(v3"/\"(v0"\/"v1))) =
  (v0"\/"v1)"/\"((v0"/\"v1)"\/"(v2"/\"v3))
proof let v2,v3,v1,v0;
  (v2"/\"v3)"\/"(v0"/\"v1) = (v0"/\"v1)"\/"(v2"/\"v3) by A7;
  hence thesis by A69;
end;

A183: for v102,v1,v100 holds
  v100"/\"((v100"/\"(v100"\/"v1))"/\"((v100"/\"v1)"\/"v102)) =
    v100"/\"((v100"/\"v1)"\/"v102)
proof let v102,v1,v100;
  v100"\/"(v100"/\"v1) = v100"/\"(v100"\/"v1) by A117;
  hence thesis by A123;
end;

A186: for v2,v1,v0 holds
 v0"/\"(v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2))) = v0"/\"((v0"/\"v1)"\/"v2)
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"/\"((v0"/\"v1)"\/"v2) =
    v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2)) by A2;
  hence thesis by A183;
end;

A188: for v2,v1,v0 holds
 v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2)) = v0"/\"((v0"/\"v1)"\/"v2)
proof let v2,v1,v0;
  v0"/\"(v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2))) =
    v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2)) by A17;
  hence thesis by A186;
end;

A190: for v2,v1,v0 holds
 v0"/\"((v0"/\"v1)"\/"(v0"/\"v2)) = (v0"/\"v1)"\/"(v0"/\"v2)
proof let v2,v1,v0;
  v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"(v0"/\"v2))) =
    v0"/\"((v0"/\"v1)"\/"(v0"/\"v2)) by A188;
  hence thesis by A145;
end;

A193: for v102,v1,v100 holds
 (v100"/\"v1)"\/"(v102"/\"(v100"\/"(v100"/\"v1))) =
   (v100"\/"(v100"/\"v1))"/\"((v100"/\"(v100"/\"v1))"\/"v102)
proof let v102,v1,v100;
  v100"/\"(v100"/\"v1) = v100"/\"v1 by A17;
  hence thesis by A72;
end;

A196: for v2,v1,v0 holds
 (v0"/\"v1)"\/"(v2"/\"(v0"/\"(v0"\/"v1))) =
   (v0"\/"(v0"/\"v1))"/\"((v0"/\"(v0"/\"v1))"\/"v2)
proof let v2,v1,v0;
  v0"\/"(v0"/\"v1) = v0"/\"(v0"\/"v1) by A117;
  hence thesis by A193;
end;

A198: for v2,v1,v0 holds
 (v0"\/"v1)"/\"((v0"/\"v1)"\/"(v2"/\"v0)) =
   (v0"\/"(v0"/\"v1))"/\"((v0"/\"(v0"/\"v1))"\/"v2)
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v2"/\"(v0"/\"(v0"\/"v1))) =
    (v0"\/"v1)"/\"((v0"/\"v1)"\/"(v2"/\"v0)) by A180;
  hence thesis by A196;
end;

A200: for v2,v1,v0 holds
 (v0"\/"v1)"/\"((v0"/\"v1)"\/"(v2"/\"v0)) =
   (v0"/\"(v0"\/"v1))"/\"((v0"/\"(v0"/\"v1))"\/"v2)
proof let v2,v1,v0;
  v0"\/"(v0"/\"v1) = v0"/\"(v0"\/"v1) by A117;
  hence thesis by A198;
end;

A202: for v2,v1,v0 holds
 (v0"\/"v1)"/\"((v0"/\"v1)"\/"(v2"/\"v0)) =
    (v0"/\"(v0"\/"v1))"/\"((v0"/\"v1)"\/"v2)
proof let v2,v1,v0;
  v0"/\"(v0"/\"v1) = v0"/\"v1 by A17;
  hence thesis by A200;
end;

A204: for v2,v1,v0 holds
 (v0"\/"v1)"/\"((v0"/\"v1)"\/"(v2"/\"v0)) =
    v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2))
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"/\"((v0"/\"v1)"\/"v2) =
   v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2)) by A2;
  hence thesis by A202;
end;

A206: for v2,v1,v0 holds
 (v0"\/"v1)"/\"((v0"/\"v1)"\/"(v2"/\"v0)) = v0"/\"((v0"/\"v1)"\/"v2)
proof let v2,v1,v0;
  v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2)) = v0"/\"((v0"/\"v1)"\/"v2) by A188;
  hence thesis by A204;
end;

A209: for v102,v1,v100 holds
 (v100"/\"(v100"\/"v1))"/\"((v100"/\"v1)"\/"(v102"/\"v100)) =
   (v100"/\"v1)"\/"(v102"/\"v100)
proof let v102,v1,v100;
  v100"\/"(v100"/\"v1) = v100"/\"(v100"\/"v1) by A117;
  hence thesis by A139;
end;

A212: for v2,v1,v0 holds
 v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"(v2"/\"v0))) = (v0"/\"v1)"\/"(v2"/\"v0)
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"/\"((v0"/\"v1)"\/"(v2"/\"v0)) =
    v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"(v2"/\"v0))) by A2;
  hence thesis by A209;
end;

A214: for v2,v1,v0 holds
 v0"/\"(v0"/\"((v0"/\"v1)"\/"v2)) = (v0"/\"v1)"\/"(v2"/\"v0)
proof let v2,v1,v0;
  (v0"\/"v1)"/\"((v0"/\"v1)"\/"(v2"/\"v0)) = v0"/\"((v0"/\"v1)"\/"v2) by A206;
  hence thesis by A212;
end;

A216: for v2,v1,v0 holds v0"/\"((v0"/\"v1)"\/"v2) = (v0"/\"v1)"\/"(v2"/\"v0)
proof let v2,v1,v0;
  v0"/\"(v0"/\"((v0"/\"v1)"\/"v2)) = v0"/\"((v0"/\"v1)"\/"v2) by A17;
  hence thesis by A214;
end;

A220: for v100,v0,v2 holds
 (v100"/\"(v0"\/"v100))"\/"(v100"\/"(v2"/\"v0)) =
   (v0"\/"v100)"/\"(v100"\/"(v2"/\"v0))
proof let v100,v0,v2;
  (v0"\/"v100)"/\"(v100"\/"(v2"/\"v0)) = v100"\/"(v2"/\"v0) by A139;
  hence thesis by A74;
end;

A223: for v2,v0,v1 holds
 v0"\/"((v0"/\"(v1"\/"v0))"\/"(v2"/\"v1)) = (v1"\/"v0)"/\"(v0"\/"(v2"/\"v1))
proof let v2,v0,v1;
  (v0"/\"(v1"\/"v0))"\/"(v0"\/"(v2"/\"v1)) =
    v0"\/"((v0"/\"(v1"\/"v0))"\/"(v2"/\"v1)) by A48;
  hence thesis by A220;
end;

A225: for v2,v0,v1 holds
  v0"\/"(v0"\/"((v1"/\"v0)"\/"(v2"/\"v1))) = (v1"\/"v0)"/\"(v0"\/"(v2"/\"v1))
proof let v2,v0,v1;
  (v0"/\"(v1"\/"v0))"\/"(v2"/\"v1) = v0"\/"((v1"/\"v0)"\/"(v2"/\"v1)) by A112;
  hence thesis by A223;
end;

A227: for v2,v0,v1 holds
 v0"\/"(v0"\/"(v1"/\"((v1"/\"v0)"\/"v2))) = (v1"\/"v0)"/\"(v0"\/"(v2"/\"v1))
proof let v2,v0,v1;
  (v1"/\"v0)"\/"(v2"/\"v1) = v1"/\"((v1"/\"v0)"\/"v2) by A216;
  hence thesis by A225;
end;

A229: for v2,v0,v1 holds
 v0"\/"(v1"/\"((v1"/\"v0)"\/"v2)) = (v1"\/"v0)"/\"(v0"\/"(v2"/\"v1))
proof let v2,v0,v1;
  v0"\/"(v0"\/"(v1"/\"((v1"/\"v0)"\/"v2))) =
    v0"\/"(v1"/\"((v1"/\"v0)"\/"v2)) by A34;
  hence thesis by A227;
end;

A231: for v2,v0,v1 holds v0"\/"(v1"/\"((v1"/\"v0)"\/"v2)) = v0"\/"(v2"/\"v1)
proof let v2,v0,v1;
  (v1"\/"v0)"/\"(v0"\/"(v2"/\"v1)) = v0"\/"(v2"/\"v1) by A139;
  hence thesis by A229;
end;

A234: for v1,v102,v100 holds
 (v1"/\"v100)"\/"((v100"/\"v102)"\/"(v1"/\"v100)) =
    (v1"/\"v100)"\/"(v100"/\"v102)
proof let v1,v102,v100;
  v100"/\"(v1"/\"v100) = v1"/\"v100 by A25;
  hence thesis by A177;
end;

A237: for v0,v2,v1 holds
 (v0"/\"v1)"\/"(v1"/\"((v1"/\"v2)"\/"v0)) = (v0"/\"v1)"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v1"/\"v2)"\/"(v0"/\"v1) = v1"/\"((v1"/\"v2)"\/"v0) by A216;
  hence thesis by A234;
end;

A239: for v0,v2,v1 holds v1"/\"(v0"\/"(v1"/\"v2)) = (v0"/\"v1)"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v0"/\"v1)"\/"(v1"/\"((v1"/\"v2)"\/"v0)) = v1"/\"(v0"\/"(v1"/\"v2)) by A159;
  hence thesis by A237;
end;

A245: for v103,v102,v101 holds
 (v102"/\"v101)"\/"((v101"/\"v102)"\/"(((v102"/\"v101)"\/"v101)"/\"
  ((v101"/\"v102)"\/"v103))) =
    ((v102"/\"v101)"\/"v101)"/\"((v102"/\"v101)"\/"
      ((v101"/\"v102)"\/"v103))
proof let v103,v102,v101;
  (v102"/\"v101)"\/"((v101"/\"v102)"\/"v103) = (v101"/\"v102)"\/"v103 by A162;
  hence thesis by A97;
end;

A248: for v2,v0,v1 holds
 (v0"/\"v1)"\/"((v1"/\"v0)"\/"((v1"\/"(v0"/\"v1))"/\"((v1"/\"v0)"\/"v2))) =
   ((v0"/\"v1)"\/"v1)"/\"((v0"/\"v1)"\/"((v1"/\"v0)"\/"v2))
proof let v2,v0,v1;
  (v0"/\"v1)"\/"v1 = v1"\/"(v0"/\"v1) by A7;
  hence thesis by A245;
end;

A250: for v2,v0,v1 holds
 (v0"/\"v1)"\/"((v1"/\"v0)"\/"((v1"/\"(v1"\/"v0))"/\"((v1"/\"v0)"\/"v2))) =
   ((v0"/\"v1)"\/"v1)"/\"((v0"/\"v1)"\/"((v1"/\"v0)"\/"v2))
proof let v2,v0,v1;
  v1"\/"(v0"/\"v1) = v1"/\"(v1"\/"v0) by A115;
  hence thesis by A248;
end;

A252: for v2,v0,v1 holds
 (v0"/\"v1)"\/"((v1"/\"v0)"\/"(v1"/\"((v1"\/"v0)"/\"((v1"/\"v0)"\/"v2)))) =
  ((v0"/\"v1)"\/"v1)"/\"((v0"/\"v1)"\/"((v1"/\"v0)"\/"v2))
proof let v2,v0,v1;
  (v1"/\"(v1"\/"v0))"/\"((v1"/\"v0)"\/"v2) =
    v1"/\"((v1"\/"v0)"/\"((v1"/\"v0)"\/"v2)) by A2;
  hence thesis by A250;
end;

A254: for v2,v0,v1 holds
  (v0"/\"v1)"\/"((v1"/\"v0)"\/"(v1"/\"((v1"/\"v0)"\/"v2))) =
    ((v0"/\"v1)"\/"v1)"/\"((v0"/\"v1)"\/"((v1"/\"v0)"\/"v2))
proof let v2,v0,v1;
  v1"/\"((v1"\/"v0)"/\"((v1"/\"v0)"\/"v2)) = v1"/\"((v1"/\"v0)"\/"v2) by A188;
  hence thesis by A252;
end;

A256: for v2,v0,v1 holds
 (v0"/\"v1)"\/"(v1"/\"((v1"/\"v0)"\/"v2)) =
    ((v0"/\"v1)"\/"v1)"/\"((v0"/\"v1)"\/"((v1"/\"v0)"\/"v2))
proof let v2,v0,v1;
  (v1"/\"v0)"\/"(v1"/\"((v1"/\"v0)"\/"v2)) = v1"/\"((v1"/\"v0)"\/"v2) by A100;
  hence thesis by A254;
end;

A258: for v2,v0,v1 holds
 v1"/\"(v0"\/"(v1"/\"((v1"/\"v0)"\/"v2))) =
   ((v0"/\"v1)"\/"v1)"/\"((v0"/\"v1)"\/"((v1"/\"v0)"\/"v2))
proof let v2,v0,v1;
  (v0"/\"v1)"\/"(v1"/\"((v1"/\"v0)"\/"v2)) =
    v1"/\"(v0"\/"(v1"/\"((v1"/\"v0)"\/"v2))) by A239;
  hence thesis by A256;
end;

A261: for v1,v0,v2 holds
 v0"/\"(v1"\/"(v2"/\"v0)) =
   ((v1"/\"v0)"\/"v0)"/\"((v1"/\"v0)"\/"((v0"/\"v1)"\/"v2))
proof let v1,v0,v2;
  v1"\/"(v0"/\"((v0"/\"v1)"\/"v2)) = v1"\/"(v2"/\"v0) by A231;
  hence thesis by A258;
end;

A263: for v1,v0,v2 holds
 v0"/\"(v1"\/"(v2"/\"v0)) =
  (v0"\/"(v1"/\"v0))"/\"((v1"/\"v0)"\/"((v0"/\"v1)"\/"v2))
proof let v1,v0,v2;
  (v1"/\"v0)"\/"v0 = v0"\/"(v1"/\"v0) by A7;
  hence thesis by A261;
end;

A265: for v1,v0,v2 holds
 v0"/\"(v1"\/"(v2"/\"v0)) =
  (v0"/\"(v0"\/"v1))"/\"((v1"/\"v0)"\/"((v0"/\"v1)"\/"v2))
proof let v1,v0,v2;
  v0"\/"(v1"/\"v0) = v0"/\"(v0"\/"v1) by A115;
  hence thesis by A263;
end;

A267: for v1,v0,v2 holds
 v0"/\"(v1"\/"(v2"/\"v0)) = (v0"/\"(v0"\/"v1))"/\"((v0"/\"v1)"\/"v2)
proof let v1,v0,v2;
  (v1"/\"v0)"\/"((v0"/\"v1)"\/"v2) = (v0"/\"v1)"\/"v2 by A162;
  hence thesis by A265;
end;

A269: for v1,v0,v2 holds
  v0"/\"(v1"\/"(v2"/\"v0)) = v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2))
proof let v1,v0,v2;
  (v0"/\"(v0"\/"v1))"/\"((v0"/\"v1)"\/"v2) =
    v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2)) by A2;
  hence thesis by A267;
end;

A271: for v1,v0,v2 holds v0"/\"(v1"\/"(v2"/\"v0)) = v0"/\"((v0"/\"v1)"\/"v2)
proof let v1,v0,v2;
  v0"/\"((v0"\/"v1)"/\"((v0"/\"v1)"\/"v2)) = v0"/\"((v0"/\"v1)"\/"v2) by A188;
  hence thesis by A269;
end;

A274: for v1,v2,v0 holds
 v0"/\"(v1"\/"((v0"/\"v2)"/\"v0)) = (v0"/\"v1)"\/"(v0"/\"v2)
proof let v1,v2,v0;
  v0"/\"((v0"/\"v1)"\/"(v0"/\"v2)) = v0"/\"(v1"\/"((v0"/\"v2)"/\"v0)) by A271;
  hence thesis by A190;
end;

A276: for v1,v2,v0 holds
 v0"/\"(v1"\/"(v0"/\"(v0"/\"v2))) = (v0"/\"v1)"\/"(v0"/\"v2)
proof let v1,v2,v0;
  (v0"/\"v2)"/\"v0 = v0"/\"(v0"/\"v2) by A3;
  hence thesis by A274;
end;

for v1,v2,v0 holds v0"/\"(v1"\/"(v0"/\"v2)) = (v0"/\"v1)"\/"(v0"/\"v2)
proof
  let v1,v2,v0;
  v0"/\"(v0"/\"v2) = v0"/\"v2 by A17;
  hence thesis by A276;
end;
hence thesis;
end;

Cluster5:
  L is meet-idempotent meet-associative meet-commutative
       satisfying_QLT1 join-idempotent join-associative join-commutative
        satisfying_QLT2 QLT-selfmodular implies
    L is modular
  proof
    assume
A1: L is meet-idempotent meet-associative meet-commutative
       satisfying_QLT1 join-idempotent join-associative join-commutative
        satisfying_QLT2 QLT-selfmodular; then
    (for v0 holds v0"/\"v0 = v0) &
    (for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2)) &
    (for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
    (for v0,v2,v1 holds
      (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2)) &
    (for v0 holds v0"\/"v0 = v0) &
    (for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
    (for v1,v0 holds v0"\/"v1 = v1"\/"v0) &
    (for v0,v2,v1 holds
      (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
    for v2,v1,v0 holds
      (v0"/\"v1)"\/"(v2"/\"(v0"\/"v1)) = (v0"\/"v1)"/\"(v2"\/"(v0"/\"v1))
         by LATTICES:def 4, def 5, def 6, def 7,
           SHEFFER1:def 9, ROBBINS1:def 7; then
    for v1,v2,v3 holds
      (v1"/\"v2)"\/"(v1"/\"v3) = v1"/\"(v2"\/"(v1"/\"v3)) by ThQLT5;
    hence thesis by ModRedef,A1;
  end;

registration
  cluster meet-idempotent meet-associative meet-commutative
       satisfying_QLT1 join-idempotent join-associative join-commutative
       satisfying_QLT2 QLT-selfmodular -> modular for non empty LattStr;
  coherence by Cluster5;
end;

begin :: QLT-6

theorem ThQLT6:
(for v0 holds v0"/\"v0 = v0) &
(for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2)) &
(for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
(for v0,v2,v1 holds
  (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2)) &
(for v0 holds v0"\/"v0 = v0) &
(for v2,v1,v0 holds
  (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
(for v1,v0 holds v0"\/"v1 = v1"\/"v0) &
(for v0,v2,v1 holds
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
(for v2,v1,v0 holds
  ((v0"\/"v1)"/\"v2)"\/"v1 = ((v2"\/"v1)"/\"v0)"\/"v1)
implies
for v1,v2,v3 holds
  (v1"/\"v2)"\/"(v1"/\"v3) = v1"/\"(v2"\/"(v1"/\"v3))
proof
assume A2: for v0 holds v0"/\"v0 = v0;
assume A3: for v2,v1,v0 holds
  (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2);
assume A4: for v1,v0 holds v0"/\"v1 = v1"/\"v0;
assume A5: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2);
assume A6: for v0 holds v0"\/"v0 = v0;
assume A7: for v2,v1,v0 holds
  (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2);
assume A8: for v1,v0 holds v0"\/"v1 = v1"\/"v0;
assume A9: for v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2);

A11: for v0,v2,v1 holds
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) =
    (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) by A4;
  hence thesis by A9;
end;

assume A12: for v2,v1,v0 holds
 ((v0"\/"v1)"/\"v2)"\/"v1 = ((v2"\/"v1)"/\"v0)"\/"v1;

A14: for v2,v1,v0 holds
  v1"\/"((v0"\/"v1)"/\"v2) = ((v2"\/"v1)"/\"v0)"\/"v1
proof let v2,v1,v0;
  ((v0"\/"v1)"/\"v2)"\/"v1 = v1"\/"((v0"\/"v1)"/\"v2) by A8;
  hence thesis by A12;
end;

A17: for v2,v0,v1 holds
  v0"\/"((v1"\/"v0)"/\"v2) = v0"\/"((v2"\/"v0)"/\"v1)
proof let v2,v0,v1;
  ((v2"\/"v0)"/\"v1)"\/"v0 = v0"\/"((v2"\/"v0)"/\"v1) by A8;
  hence thesis by A14;
end;

A21: for v0,v2,v1 holds
  (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) =
    (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) by A8;
  hence thesis by A5;
end;

A24: for v102,v101 holds v101"/\"v102 = v101"/\"(v101"/\"v102)
proof let v102,v101;
  v101"/\"v101 = v101 by A2;
  hence thesis by A3;
end;

A28: for v2,v0,v1 holds (v1"/\"v0)"/\"v2 = v0"/\"(v1"/\"v2)
proof let v2,v0,v1;
  v0"/\"v1 = v1"/\"v0 by A4;
  hence thesis by A3;
end;

A31: for v0,v2,v1 holds v0"/\"(v1"/\"v2) = v1"/\"(v0"/\"v2)
proof let v0,v2,v1;
  (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2) by A3;
  hence thesis by A28;
end;

A34: for v102,v100 holds (v100"\/"v102)"\/"v102 = v100"\/"v102
proof let v102,v100;
  v102"\/"v102 = v102 by A6;
  hence thesis by A7;
end;

A37: for v1,v0 holds
  v1"\/"(v0"\/"v1) = v0"\/"v1
proof let v1,v0;
  (v0"\/"v1)"\/"v1 = v1"\/"(v0"\/"v1) by A8;
  hence thesis by A34;
end;

A40: for v0,v1,v2 holds
  (v0"\/"v1)"/\"(v0"\/"(v2"/\"v1)) = v0"\/"(v1"/\"v2)
proof let v0,v1,v2;
  v1"/\"v2 = v2"/\"v1 by A4;
  hence thesis by A11;
end;

A43: for v100,v101 holds
 v100"\/"(v101"\/"v100) =
    v100"\/"(((v101"\/"v100)"\/"v100)"/\"v101)
proof let v100,v101;
  (v101"\/"v100)"/\"(v101"\/"v100) = v101"\/"v100 by A2;
  hence thesis by A17;
end;

A46: for v0,v1 holds v1"\/"v0 = v0"\/"(((v1"\/"v0)"\/"v0)"/\"v1)
proof let v0,v1;
  v0"\/"(v1"\/"v0) = v1"\/"v0 by A37;
  hence thesis by A43;
end;

A49: for v1,v0 holds
  v0"\/"v1 = v1"\/"((v1"\/"(v0"\/"v1))"/\"v0)
proof let v1,v0;
  (v0"\/"v1)"\/"v1 = v1"\/"(v0"\/"v1) by A8;
  hence thesis by A46;
end;

A51: for v1,v0 holds
  v0"\/"v1 = v1"\/"((v0"\/"v1)"/\"v0)
proof let v1,v0;
  v1"\/"(v0"\/"v1) = v0"\/"v1 by A37;
  hence thesis by A49;
end;

A53: for v1,v0 holds
  v0"\/"v1 = v1"\/"(v0"/\"(v0"\/"v1))
proof let v1,v0;
  (v0"\/"v1)"/\"v0 = v0"/\"(v0"\/"v1) by A4;
  hence thesis by A51;
end;

A57: for v2,v0,v1 holds
  v0"\/"(v2"/\"(v1"\/"v0)) = v0"\/"((v2"\/"v0)"/\"v1)
proof let v2,v0,v1;
  (v1"\/"v0)"/\"v2 = v2"/\"(v1"\/"v0) by A4;
  hence thesis by A17;
end;

A62: for v102,v100 holds
 v100"\/"(v100"/\"v102) = v100"\/"((v102"\/"v100)"/\"v100)
proof let v102,v100;
  v100"\/"v100 = v100 by A6;
  hence thesis by A17;
end;

A65: for v1,v0 holds v0"\/"(v0"/\"v1) = v0"\/"(v0"/\"(v1"\/"v0))
proof let v1,v0;
  (v1"\/"v0)"/\"v0 = v0"/\"(v1"\/"v0) by A4;
  hence thesis by A62;
end;

A69: for v102,v2,v100,v1 holds
 (v100"\/"((v2"\/"v100)"/\"v1))"/\"
   (v100"\/"(((v1"\/"v100)"/\"v2)"/\"v102)) =
    v100"\/"(((v1"\/"v100)"/\"v2)"/\"v102)
proof let v102,v2,v100,v1;
  v100"\/"((v1"\/"v100)"/\"v2) =
    v100"\/"((v2"\/"v100)"/\"v1) by A17;
  hence thesis by A11;
end;

A72: for v3,v1,v0,v2 holds
 (v0"\/"(v1"/\"(v2"\/"v0)))"/\"
   (v0"\/"(((v2"\/"v0)"/\"v1)"/\"v3)) =
     v0"\/"(((v2"\/"v0)"/\"v1)"/\"v3)
proof let v3,v1,v0,v2;
  v0"\/"((v1"\/"v0)"/\"v2) = v0"\/"(v1"/\"(v2"\/"v0)) by A57;
  hence thesis by A69;
end;

A74: for v1,v3,v0,v2 holds
 (v0"\/"(v1"/\"(v2"\/"v0)))"/\"(v0"\/"((v2"\/"v0)"/\"(v1"/\"v3))) =
    v0"\/"(((v2"\/"v0)"/\"v1)"/\"v3)
proof let v1,v3,v0,v2;
  ((v2"\/"v0)"/\"v1)"/\"v3 = (v2"\/"v0)"/\"(v1"/\"v3) by A3;
  hence thesis by A72;
end;

A76: for v2,v0,v3,v1 holds
 (v0"\/"(v1"/\"(v2"\/"v0)))"/\"(v0"\/"(v2"/\"((v1"/\"v3)"\/"v0))) =
   v0"\/"(((v2"\/"v0)"/\"v1)"/\"v3)
proof let v2,v0,v3,v1;
  v0"\/"((v2"\/"v0)"/\"(v1"/\"v3)) =
    v0"\/"(v2"/\"((v1"/\"v3)"\/"v0)) by A57;
  hence thesis by A74;
end;

A78: for v2,v0,v3,v1 holds
  (v0"\/"(v1"/\"(v2"\/"v0)))"/\"(v0"\/"(v2"/\"((v1"/\"v3)"\/"v0))) =
     v0"\/"((v2"\/"v0)"/\"(v1"/\"v3))
proof let v2,v0,v3,v1;
  ((v2"\/"v0)"/\"v1)"/\"v3 = (v2"\/"v0)"/\"(v1"/\"v3) by A3;
  hence thesis by A76;
end;

A80: for v2,v0,v3,v1 holds
 (v0"\/"(v1"/\"(v2"\/"v0)))"/\"(v0"\/"(v2"/\"((v1"/\"v3)"\/"v0))) =
    v0"\/"(v2"/\"((v1"/\"v3)"\/"v0))
proof let v2,v0,v3,v1;
  v0"\/"((v2"\/"v0)"/\"(v1"/\"v3)) = v0"\/"(v2"/\"((v1"/\"v3)"\/"v0)) by A57;
  hence thesis by A78;
end;

A83: for v102,v101 holds
  v101"\/"(v101"/\"(v101"\/"v102)) = v101"/\"(v101"\/"v102)
proof let v102,v101;
  v101"/\"v101 = v101 by A2;
  hence thesis by A21;
end;

A87: for v100,v101,v1 holds
 (v100"/\"v101)"\/"(v100"/\"(v1"\/"v101)) =
    v100"/\"(v101"\/"(v1"\/"v101))
proof let v100,v101,v1;
  v101"\/"(v1"\/"v101) = v1"\/"v101 by A37;
  hence thesis by A21;
end;

A90: for v0,v1,v2 holds
  (v0"/\"v1)"\/"(v0"/\"(v2"\/"v1)) = v0"/\"(v2"\/"v1)
proof let v0,v1,v2;
  v1"\/"(v2"\/"v1) = v2"\/"v1 by A37;
  hence thesis by A87;
end;

A93: for v101,v1 holds
 (v1"\/"v101)"\/"(v101"/\"(v1"\/"v101)) = v101"\/"(v1"\/"v101)
proof let v101,v1;
  v101"\/"(v1"\/"v101) = v1"\/"v101 by A37;
  hence thesis by A53;
end;

A96: for v1,v0 holds
  v0"\/"(v1"\/"(v1"/\"(v0"\/"v1))) = v1"\/"(v0"\/"v1)
proof let v1,v0;
  (v0"\/"v1)"\/"(v1"/\"(v0"\/"v1)) =
    v0"\/"(v1"\/"(v1"/\"(v0"\/"v1))) by A7;
  hence thesis by A93;
end;

A98: for v0,v1 holds
  v0"\/"(v1"\/"(v1"/\"v0)) = v1"\/"(v0"\/"v1)
proof let v0,v1;
  v1"\/"(v1"/\"(v0"\/"v1)) = v1"\/"(v1"/\"v0) by A65;
  hence thesis by A96;
end;

A100: for v0,v1 holds v0"\/"(v1"\/"(v1"/\"v0)) = v0"\/"v1
proof let v0,v1;
  v1"\/"(v0"\/"v1) = v0"\/"v1 by A37;
  hence thesis by A98;
end;

A103: for v0,v100,v1 holds
 v100"\/"((v0"/\"v1)"\/"(v0"/\"(v1"/\"v100))) =
   v100"\/"(v0"/\"v1)
proof let v0,v100,v1;
  (v0"/\"v1)"/\"v100 = v0"/\"(v1"/\"v100) by A3;
  hence thesis by A100;
end;

A106: for v1,v0 holds
  v0"\/"(v0"/\"(v0"\/"v1)) = v0"\/"(v0"/\"v1)
proof let v1,v0;
  v1"\/"v0 = v0"\/"v1 by A8;
  hence thesis by A65;
end;

A108: for v1,v0 holds
  v0"/\"(v0"\/"v1) = v0"\/"(v0"/\"v1)
proof let v1,v0;
  v0"\/"(v0"/\"(v0"\/"v1)) = v0"/\"(v0"\/"v1) by A83;
  hence thesis by A106;
end;

A111: for v100,v1,v101 holds
  v100"/\"(v101"\/"(v101"/\"v1)) = v101"/\"(v100"/\"(v101"\/"v1))
proof let v100,v1,v101;
  v101"/\"(v101"\/"v1) = v101"\/"(v101"/\"v1) by A108;
  hence thesis by A31;
end;

A115: for v1,v100,v101 holds
 (v100"\/"v101)"/\"(v100"\/"(v1"/\"(v101"\/"v100))) =
    v100"\/"(v101"/\"(v1"\/"v100))
proof let v1,v100,v101;
  v100"\/"((v1"\/"v100)"/\"v101) =
    v100"\/"(v1"/\"(v101"\/"v100)) by A57;
  hence thesis by A40;
end;

A119: for v102,v1,v100 holds
 (v100"/\"v1)"\/"(v100"/\"(v102"\/"(v100"/\"v1))) =
    v100"/\"(v102"\/"(v100"/\"v1))
proof let v102,v1,v100;
  v100"/\"(v100"/\"v1) = v100"/\"v1 by A24;
  hence thesis by A90;
end;

A123: for v100,v103,v101 holds
 (v100"\/"(v101"/\"((v101"/\"v103)"\/"v100)))"/\"
   (v100"\/"((v101"/\"v103)"\/"
    ((v101"/\"v103)"/\"v100))) =
      v100"\/"((v101"/\"v103)"/\"((v101"/\"v103)"\/"v100))
proof let v100,v103,v101;
  (v101"/\"v103)"/\"((v101"/\"v103)"\/"v100) =
    (v101"/\"v103)"\/"((v101"/\"v103)"/\"v100) by A108;
  hence thesis by A80;
end;

A126: for v0,v2,v1 holds
  (v0"\/"(v1"/\"((v1"/\"v2)"\/"v0)))"/\"
    (v0"\/"((v1"/\"v2)"\/"(v1"/\"(v2"/\"v0)))) =
      v0"\/"((v1"/\"v2)"/\"((v1"/\"v2)"\/"v0))
proof let v0,v2,v1;
  (v1"/\"v2)"/\"v0 = v1"/\"(v2"/\"v0) by A3;
  hence thesis by A123;
end;

A128: for v0,v2,v1 holds
 (v0"\/"(v1"/\"((v1"/\"v2)"\/"v0)))"/\"(v0"\/"(v1"/\"v2)) =
    v0"\/"((v1"/\"v2)"/\"((v1"/\"v2)"\/"v0))
proof let v0,v2,v1;
  v0"\/"((v1"/\"v2)"\/"(v1"/\"(v2"/\"v0))) = v0"\/"(v1"/\"v2) by A103;
  hence thesis by A126;
end;

A130: for v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"(v0"\/"(v1"/\"((v1"/\"v2)"\/"v0))) =
    v0"\/"((v1"/\"v2)"/\"((v1"/\"v2)"\/"v0))
proof let v0,v2,v1;
  (v0"\/"(v1"/\"((v1"/\"v2)"\/"v0)))"/\"(v0"\/"(v1"/\"v2)) =
    (v0"\/"(v1"/\"v2))"/\"(v0"\/"(v1"/\"((v1"/\"v2)"\/"v0))) by A4;
  hence thesis by A128;
end;

A132: for v0,v2,v1 holds
  v0"\/"((v1"/\"v2)"/\"(v1"\/"v0)) =
    v0"\/"((v1"/\"v2)"/\"((v1"/\"v2)"\/"v0))
proof let v0,v2,v1;
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"(v1"/\"((v1"/\"v2)"\/"v0))) =
    v0"\/"((v1"/\"v2)"/\"(v1"\/"v0)) by A115;
  hence thesis by A130;
end;

A134: for v2,v0,v1 holds
  v0"\/"(v1"/\"(v2"/\"(v1"\/"v0))) =
    v0"\/"((v1"/\"v2)"/\"((v1"/\"v2)"\/"v0))
proof let v2,v0,v1;
  (v1"/\"v2)"/\"(v1"\/"v0) = v1"/\"(v2"/\"(v1"\/"v0)) by A3;
  hence thesis by A132;
end;

A136: for v2,v0,v1 holds
  v0"\/"(v1"/\"(v2"/\"(v1"\/"v0))) =
    v0"\/"((v1"/\"v2)"\/"((v1"/\"v2)"/\"v0))
proof let v2,v0,v1;
  (v1"/\"v2)"/\"((v1"/\"v2)"\/"v0) =
    (v1"/\"v2)"\/"((v1"/\"v2)"/\"v0) by A108;
  hence thesis by A134;
end;

A138: for v2,v0,v1 holds
  v0"\/"(v1"/\"(v2"/\"(v1"\/"v0))) =
    v0"\/"((v1"/\"v2)"\/"(v1"/\"(v2"/\"v0)))
proof let v2,v0,v1;
  (v1"/\"v2)"/\"v0 = v1"/\"(v2"/\"v0) by A3;
  hence thesis by A136;
end;

A140: for v2,v0,v1 holds
  v0"\/"(v1"/\"(v2"/\"(v1"\/"v0))) = v0"\/"(v1"/\"v2)
proof let v2,v0,v1;
  v0"\/"((v1"/\"v2)"\/"(v1"/\"(v2"/\"v0))) = v0"\/"(v1"/\"v2) by A103;
  hence thesis by A138;
end;

A143: for v101,v2,v102 holds
 (v102"/\"v2)"\/"((v101"\/"(v102"/\"v2))"/\"v102) =
   (v102"/\"v2)"\/"(v102"/\"(v101"/\"(v102"\/"v2)))
proof let v101,v2,v102;
  v101"/\"(v102"\/"(v102"/\"v2)) =
    v102"/\"(v101"/\"(v102"\/"v2)) by A111;
  hence thesis by A57;
end;

A146: for v2,v1,v0 holds
  (v0"/\"v1)"\/"(v0"/\"(v2"\/"(v0"/\"v1))) =
     (v0"/\"v1)"\/"(v0"/\"(v2"/\"(v0"\/"v1)))
proof let v2,v1,v0;
  (v2"\/"(v0"/\"v1))"/\"v0 = v0"/\"(v2"\/"(v0"/\"v1)) by A4;
  hence thesis by A143;
end;

A148: for v2,v1,v0 holds
  v0"/\"(v2"\/"(v0"/\"v1)) =
    (v0"/\"v1)"\/"(v0"/\"(v2"/\"(v0"\/"v1)))
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v0"/\"(v2"\/"(v0"/\"v1))) =
    v0"/\"(v2"\/"(v0"/\"v1)) by A119;
  hence thesis by A146;
end;

A154: for v102,v2,v101 holds
 (v101"/\"v2)"\/"(v101"/\"(v101"/\"(v102"/\"(v101"\/"v2)))) =
    (v101"/\"v2)"\/"(v101"/\"v102)
proof let v102,v2,v101;
  v102"/\"(v101"\/"(v101"/\"v2)) = v101"/\"(v102"/\"(v101"\/"v2)) by A111;
  hence thesis by A140;
end;

A157: for v2,v1,v0 holds
 (v0"/\"v1)"\/"(v0"/\"(v2"/\"(v0"\/"v1))) = (v0"/\"v1)"\/"(v0"/\"v2)
proof let v2,v1,v0;
  v0"/\"(v0"/\"(v2"/\"(v0"\/"v1))) = v0"/\"(v2"/\"(v0"\/"v1)) by A24;
  hence thesis by A154;
end;

A159: for v2,v1,v0 holds
   (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v2"\/"(v0"/\"v1))
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v0"/\"(v2"/\"(v0"\/"v1))) =
    (v0"/\"v1)"\/"(v0"/\"v2) by A157;
  hence thesis by A148;
end;

  let v1,v2,v3;
  v1"/\"(v2"\/"(v1"/\"v3)) = (v1"/\"v3)"\/"(v1"/\"v2) by A159;
  hence thesis by A8;
end;

definition let L;
  attr L is QLT-selfmodular' means
    for v2,v1,v0 holds
      ((v0"\/"v1)"/\"v2)"\/"v1 = ((v2"\/"v1)"/\"v0)"\/"v1;
end;

Cluster6:
  L is meet-idempotent meet-associative meet-commutative
       satisfying_QLT1 join-idempotent join-associative join-commutative
        satisfying_QLT2 QLT-selfmodular' implies
    L is modular
  proof
    assume
A1: L is meet-idempotent meet-associative meet-commutative
       satisfying_QLT1 join-idempotent join-associative join-commutative
        satisfying_QLT2 QLT-selfmodular'; then
    (for v0 holds v0"/\"v0 = v0) &
    (for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2)) &
    (for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
    (for v0,v2,v1 holds
      (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2)) &
    (for v0 holds v0"\/"v0 = v0) &
    (for v2,v1,v0 holds
      (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
    (for v1,v0 holds v0"\/"v1 = v1"\/"v0) &
    (for v0,v2,v1 holds
      (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
    (for v2,v1,v0 holds
      ((v0"\/"v1)"/\"v2)"\/"v1 = ((v2"\/"v1)"/\"v0)"\/"v1)
         by LATTICES:def 4, def 5, def 6, def 7,
           SHEFFER1:def 9, ROBBINS1:def 7; then
    for v1,v2,v3 holds
     (v1"/\"v2)"\/"(v1"/\"v3) = v1"/\"(v2"\/"(v1"/\"v3)) by ThQLT6;
    hence thesis by ModRedef,A1;
  end;

registration
  cluster meet-idempotent meet-associative meet-commutative
       satisfying_QLT1 join-idempotent join-associative join-commutative
        satisfying_QLT2 QLT-selfmodular' -> modular for non empty LattStr;
  coherence by Cluster6;
end;

begin :: The Counterexample Needed to Prove QLT-7

theorem :: QLT-7
  ex L1,L2 being QuasiLattice st
    the carrier of L1 = the carrier of L2 &
    the L_join of L1 = the L_join of L2 &
    the L_meet of L1 <> the L_meet of L2
  proof
    take L1 = QLTLattice1;
    take L2 = QLTLattice2;
    thus the carrier of L1 = the carrier of L2;
    thus the L_join of L1 = the L_join of L2;
    thus thesis by WazneQLT7;
  end;