let L be non empty LattStr ; :: thesis: ( ( for v1, v0 being Element of L holds v0 "/\" v1 = v1 "/\" v0 ) & ( for v0, v2, v1 being Element of L holds (v0 "/\" (v1 "\/" v2)) "\/" (v0 "/\" v1) = v0 "/\" (v1 "\/" v2) ) & ( for v0 being Element of L holds v0 "\/" v0 = v0 ) & ( for v2, v1, v0 being Element of L holds (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) ) & ( for v1, v0 being Element of L holds v0 "\/" v1 = v1 "\/" v0 ) & ( for v0, v2, v1 being Element of L holds (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) ) & ( for v1, v2, v0 being Element of L holds v0 "/\" (v1 "\/" (v0 "/\" v2)) = v0 "/\" (v1 "\/" v2) ) implies for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" v3) )
assume A1: for v1, v0 being Element of L holds v0 "/\" v1 = v1 "/\" v0 ; :: thesis: ( ex v0, v2, v1 being Element of L st not (v0 "/\" (v1 "\/" v2)) "\/" (v0 "/\" v1) = v0 "/\" (v1 "\/" v2) or ex v0 being Element of L st not v0 "\/" v0 = v0 or ex v2, v1, v0 being Element of L st not (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) or ex v1, v0 being Element of L st not v0 "\/" v1 = v1 "\/" v0 or ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v1, v2, v0 being Element of L st not v0 "/\" (v1 "\/" (v0 "/\" v2)) = v0 "/\" (v1 "\/" v2) or for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" v3) )
assume A2: for v0, v2, v1 being Element of L holds (v0 "/\" (v1 "\/" v2)) "\/" (v0 "/\" v1) = v0 "/\" (v1 "\/" v2) ; :: thesis: ( ex v0 being Element of L st not v0 "\/" v0 = v0 or ex v2, v1, v0 being Element of L st not (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) or ex v1, v0 being Element of L st not v0 "\/" v1 = v1 "\/" v0 or ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v1, v2, v0 being Element of L st not v0 "/\" (v1 "\/" (v0 "/\" v2)) = v0 "/\" (v1 "\/" v2) or for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" v3) )
assume A3: for v0 being Element of L holds v0 "\/" v0 = v0 ; :: thesis: ( ex v2, v1, v0 being Element of L st not (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) or ex v1, v0 being Element of L st not v0 "\/" v1 = v1 "\/" v0 or ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v1, v2, v0 being Element of L st not v0 "/\" (v1 "\/" (v0 "/\" v2)) = v0 "/\" (v1 "\/" v2) or for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" v3) )
assume A4: for v2, v1, v0 being Element of L holds (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) ; :: thesis: ( ex v1, v0 being Element of L st not v0 "\/" v1 = v1 "\/" v0 or ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v1, v2, v0 being Element of L st not v0 "/\" (v1 "\/" (v0 "/\" v2)) = v0 "/\" (v1 "\/" v2) or for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" v3) )
assume A5: for v1, v0 being Element of L holds v0 "\/" v1 = v1 "\/" v0 ; :: thesis: ( ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v1, v2, v0 being Element of L st not v0 "/\" (v1 "\/" (v0 "/\" v2)) = v0 "/\" (v1 "\/" v2) or for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" v3) )
assume A6: for v0, v2, v1 being Element of L holds (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) ; :: thesis: ( ex v1, v2, v0 being Element of L st not v0 "/\" (v1 "\/" (v0 "/\" v2)) = v0 "/\" (v1 "\/" v2) or for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" v3) )
A8: for v0, v2, v1 being Element of L holds (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" v2)) = v0 "\/" (v1 "/\" v2)

proof

let v0, v2, v1 be Element of L; :: thesis: (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" v2)) = v0 "\/" (v1 "/\" v2)
(v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" v2)) by A1;
hence (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" v2)) = v0 "\/" (v1 "/\" v2) by A6; :: thesis: verum

end;

assume A9: for v1, v2, v0 being Element of L holds v0 "/\" (v1 "\/" (v0 "/\" v2)) = v0 "/\" (v1 "\/" v2) ; :: thesis: for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" v3)
A12: for v0, v2, v1 being Element of L holds (v0 "/\" v1) "\/" (v0 "/\" (v1 "\/" v2)) = v0 "/\" (v1 "\/" v2)

proof

let v0, v2, v1 be Element of L; :: thesis: (v0 "/\" v1) "\/" (v0 "/\" (v1 "\/" v2)) = v0 "/\" (v1 "\/" v2)
(v0 "/\" (v1 "\/" v2)) "\/" (v0 "/\" v1) = (v0 "/\" v1) "\/" (v0 "/\" (v1 "\/" v2)) by A5;
hence (v0 "/\" v1) "\/" (v0 "/\" (v1 "\/" v2)) = v0 "/\" (v1 "\/" v2) by A2; :: thesis: verum

end;

A15: for v102, v100 being Element of L holds (v100 "\/" v102) "\/" v102 = v100 "\/" v102

proof

let v102, v100 be Element of L; :: thesis: (v100 "\/" v102) "\/" v102 = v100 "\/" v102
v102 "\/" v102 = v102 by A3;
hence (v100 "\/" v102) "\/" v102 = v100 "\/" v102 by A4; :: thesis: verum

end;

A18: for v1, v0 being Element of L holds v1 "\/" (v0 "\/" v1) = v0 "\/" v1

proof

let v1, v0 be Element of L; :: thesis: v1 "\/" (v0 "\/" v1) = v0 "\/" v1
(v0 "\/" v1) "\/" v1 = v1 "\/" (v0 "\/" v1) by A5;
hence v1 "\/" (v0 "\/" v1) = v0 "\/" v1 by A15; :: thesis: verum

end;

A21: for v2, v0, v1 being Element of L holds (v1 "\/" v0) "\/" v2 = v0 "\/" (v1 "\/" v2)

proof

let v2, v0, v1 be Element of L; :: thesis: (v1 "\/" v0) "\/" v2 = v0 "\/" (v1 "\/" v2)
v0 "\/" v1 = v1 "\/" v0 by A5;
hence (v1 "\/" v0) "\/" v2 = v0 "\/" (v1 "\/" v2) by A4; :: thesis: verum

end;

A24: for v0, v2, v1 being Element of L holds v0 "\/" (v1 "\/" v2) = v1 "\/" (v0 "\/" v2)

proof

let v0, v2, v1 be Element of L; :: thesis: v0 "\/" (v1 "\/" v2) = v1 "\/" (v0 "\/" v2)
(v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) by A4;
hence v0 "\/" (v1 "\/" v2) = v1 "\/" (v0 "\/" v2) by A21; :: thesis: verum

end;

A27: for v102, v101 being Element of L holds v101 "/\" (v101 "\/" (v101 "/\" v102)) = v101 "\/" (v101 "/\" v102)

proof

let v102, v101 be Element of L; :: thesis: v101 "/\" (v101 "\/" (v101 "/\" v102)) = v101 "\/" (v101 "/\" v102)
v101 "\/" v101 = v101 by A3;
hence v101 "/\" (v101 "\/" (v101 "/\" v102)) = v101 "\/" (v101 "/\" v102) by A8; :: thesis: verum

end;

A30: for v1, v0 being Element of L holds v0 "/\" (v0 "\/" v1) = v0 "\/" (v0 "/\" v1)

proof

let v1, v0 be Element of L; :: thesis: v0 "/\" (v0 "\/" v1) = v0 "\/" (v0 "/\" v1)
v0 "/\" (v0 "\/" (v0 "/\" v1)) = v0 "/\" (v0 "\/" v1) by A9;
hence v0 "/\" (v0 "\/" v1) = v0 "\/" (v0 "/\" v1) by A27; :: thesis: verum

end;

A33: for v1, v0, v2 being Element of L holds v0 "/\" (v1 "\/" (v2 "/\" v0)) = v0 "/\" (v1 "\/" v2)

proof

let v1, v0, v2 be Element of L; :: thesis: v0 "/\" (v1 "\/" (v2 "/\" v0)) = v0 "/\" (v1 "\/" v2)
v0 "/\" v2 = v2 "/\" v0 by A1;
hence v0 "/\" (v1 "\/" (v2 "/\" v0)) = v0 "/\" (v1 "\/" v2) by A9; :: thesis: verum

end;

A35: for v0, v1, v2 being Element of L holds (v0 "/\" v1) "\/" (v0 "/\" (v2 "\/" v1)) = v0 "/\" (v1 "\/" v2)

proof

let v0, v1, v2 be Element of L; :: thesis: (v0 "/\" v1) "\/" (v0 "/\" (v2 "\/" v1)) = v0 "/\" (v1 "\/" v2)
v1 "\/" v2 = v2 "\/" v1 by A5;
hence (v0 "/\" v1) "\/" (v0 "/\" (v2 "\/" v1)) = v0 "/\" (v1 "\/" v2) by A12; :: thesis: verum

end;

A37: for v0, v1 being Element of L holds v0 "\/" (v1 "/\" v0) = v0 "/\" (v0 "\/" v1)

proof

let v0, v1 be Element of L; :: thesis: v0 "\/" (v1 "/\" v0) = v0 "/\" (v0 "\/" v1)
v0 "/\" v1 = v1 "/\" v0 by A1;
hence v0 "\/" (v1 "/\" v0) = v0 "/\" (v0 "\/" v1) by A30; :: thesis: verum

end;

A40: for v1, v2, v101 being Element of L holds (v1 "\/" (v101 "/\" v2)) "\/" (v101 "/\" (v1 "\/" v2)) = (v1 "\/" (v101 "/\" v2)) "/\" ((v1 "\/" (v101 "/\" v2)) "\/" v101)

proof

let v1, v2, v101 be Element of L; :: thesis: (v1 "\/" (v101 "/\" v2)) "\/" (v101 "/\" (v1 "\/" v2)) = (v1 "\/" (v101 "/\" v2)) "/\" ((v1 "\/" (v101 "/\" v2)) "\/" v101)
v101 "/\" (v1 "\/" (v101 "/\" v2)) = v101 "/\" (v1 "\/" v2) by A9;
hence (v1 "\/" (v101 "/\" v2)) "\/" (v101 "/\" (v1 "\/" v2)) = (v1 "\/" (v101 "/\" v2)) "/\" ((v1 "\/" (v101 "/\" v2)) "\/" v101) by A37; :: thesis: verum

end;

A43: for v1, v2, v0 being Element of L holds (v1 "/\" (v0 "\/" v2)) "\/" (v0 "\/" (v1 "/\" v2)) = (v0 "\/" (v1 "/\" v2)) "/\" ((v0 "\/" (v1 "/\" v2)) "\/" v1)

proof

let v1, v2, v0 be Element of L; :: thesis: (v1 "/\" (v0 "\/" v2)) "\/" (v0 "\/" (v1 "/\" v2)) = (v0 "\/" (v1 "/\" v2)) "/\" ((v0 "\/" (v1 "/\" v2)) "\/" v1)
(v0 "\/" (v1 "/\" v2)) "\/" (v1 "/\" (v0 "\/" v2)) = (v1 "/\" (v0 "\/" v2)) "\/" (v0 "\/" (v1 "/\" v2)) by A5;
hence (v1 "/\" (v0 "\/" v2)) "\/" (v0 "\/" (v1 "/\" v2)) = (v0 "\/" (v1 "/\" v2)) "/\" ((v0 "\/" (v1 "/\" v2)) "\/" v1) by A40; :: thesis: verum

end;

A46: for v0, v2, v1 being Element of L holds v1 "\/" ((v0 "/\" (v1 "\/" v2)) "\/" (v0 "/\" v2)) = (v1 "\/" (v0 "/\" v2)) "/\" ((v1 "\/" (v0 "/\" v2)) "\/" v0)

proof

let v0, v2, v1 be Element of L; :: thesis: v1 "\/" ((v0 "/\" (v1 "\/" v2)) "\/" (v0 "/\" v2)) = (v1 "\/" (v0 "/\" v2)) "/\" ((v1 "\/" (v0 "/\" v2)) "\/" v0)
(v0 "/\" (v1 "\/" v2)) "\/" (v1 "\/" (v0 "/\" v2)) = v1 "\/" ((v0 "/\" (v1 "\/" v2)) "\/" (v0 "/\" v2)) by A24;
hence v1 "\/" ((v0 "/\" (v1 "\/" v2)) "\/" (v0 "/\" v2)) = (v1 "\/" (v0 "/\" v2)) "/\" ((v1 "\/" (v0 "/\" v2)) "\/" v0) by A43; :: thesis: verum

end;

A49: for v1, v2, v0 being Element of L holds v0 "\/" ((v1 "/\" v2) "\/" (v1 "/\" (v0 "\/" v2))) = (v0 "\/" (v1 "/\" v2)) "/\" ((v0 "\/" (v1 "/\" v2)) "\/" v1)

proof

let v1, v2, v0 be Element of L; :: thesis: v0 "\/" ((v1 "/\" v2) "\/" (v1 "/\" (v0 "\/" v2))) = (v0 "\/" (v1 "/\" v2)) "/\" ((v0 "\/" (v1 "/\" v2)) "\/" v1)
(v1 "/\" (v0 "\/" v2)) "\/" (v1 "/\" v2) = (v1 "/\" v2) "\/" (v1 "/\" (v0 "\/" v2)) by A5;
hence v0 "\/" ((v1 "/\" v2) "\/" (v1 "/\" (v0 "\/" v2))) = (v0 "\/" (v1 "/\" v2)) "/\" ((v0 "\/" (v1 "/\" v2)) "\/" v1) by A46; :: thesis: verum

end;

A51: for v1, v0, v2 being Element of L holds v0 "\/" (v1 "/\" (v2 "\/" v0)) = (v0 "\/" (v1 "/\" v2)) "/\" ((v0 "\/" (v1 "/\" v2)) "\/" v1)

proof

let v1, v0, v2 be Element of L; :: thesis: v0 "\/" (v1 "/\" (v2 "\/" v0)) = (v0 "\/" (v1 "/\" v2)) "/\" ((v0 "\/" (v1 "/\" v2)) "\/" v1)
(v1 "/\" v2) "\/" (v1 "/\" (v0 "\/" v2)) = v1 "/\" (v2 "\/" v0) by A35;
hence v0 "\/" (v1 "/\" (v2 "\/" v0)) = (v0 "\/" (v1 "/\" v2)) "/\" ((v0 "\/" (v1 "/\" v2)) "\/" v1) by A49; :: thesis: verum

end;

A53: for v1, v0, v2 being Element of L holds v0 "\/" (v1 "/\" (v2 "\/" v0)) = (v0 "\/" (v1 "/\" v2)) "/\" (v1 "\/" (v0 "\/" (v1 "/\" v2)))

proof

let v1, v0, v2 be Element of L; :: thesis: v0 "\/" (v1 "/\" (v2 "\/" v0)) = (v0 "\/" (v1 "/\" v2)) "/\" (v1 "\/" (v0 "\/" (v1 "/\" v2)))
(v0 "\/" (v1 "/\" v2)) "\/" v1 = v1 "\/" (v0 "\/" (v1 "/\" v2)) by A5;
hence v0 "\/" (v1 "/\" (v2 "\/" v0)) = (v0 "\/" (v1 "/\" v2)) "/\" (v1 "\/" (v0 "\/" (v1 "/\" v2))) by A51; :: thesis: verum

end;

A56: for v101, v0, v2, v1 being Element of L holds (v0 "\/" (v1 "/\" v2)) "/\" (v101 "\/" (v0 "\/" (v1 "/\" v2))) = (v0 "\/" (v1 "/\" v2)) "/\" (v101 "\/" (v0 "\/" v1))

proof

let v101, v0, v2, v1 be Element of L; :: thesis: (v0 "\/" (v1 "/\" v2)) "/\" (v101 "\/" (v0 "\/" (v1 "/\" v2))) = (v0 "\/" (v1 "/\" v2)) "/\" (v101 "\/" (v0 "\/" v1))
(v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" v2)) = v0 "\/" (v1 "/\" v2) by A8;
hence (v0 "\/" (v1 "/\" v2)) "/\" (v101 "\/" (v0 "\/" (v1 "/\" v2))) = (v0 "\/" (v1 "/\" v2)) "/\" (v101 "\/" (v0 "\/" v1)) by A33; :: thesis: verum

end;

A59: for v1, v0, v2 being Element of L holds v0 "\/" (v1 "/\" (v2 "\/" v0)) = (v0 "\/" (v1 "/\" v2)) "/\" (v1 "\/" (v0 "\/" v1))

proof

let v1, v0, v2 be Element of L; :: thesis: v0 "\/" (v1 "/\" (v2 "\/" v0)) = (v0 "\/" (v1 "/\" v2)) "/\" (v1 "\/" (v0 "\/" v1))
(v0 "\/" (v1 "/\" v2)) "/\" (v1 "\/" (v0 "\/" (v1 "/\" v2))) = (v0 "\/" (v1 "/\" v2)) "/\" (v1 "\/" (v0 "\/" v1)) by A56;
hence v0 "\/" (v1 "/\" (v2 "\/" v0)) = (v0 "\/" (v1 "/\" v2)) "/\" (v1 "\/" (v0 "\/" v1)) by A53; :: thesis: verum

end;

A61: for v1, v0, v2 being Element of L holds v0 "\/" (v1 "/\" (v2 "\/" v0)) = (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1)

proof

let v1, v0, v2 be Element of L; :: thesis: v0 "\/" (v1 "/\" (v2 "\/" v0)) = (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1)
v1 "\/" (v0 "\/" v1) = v0 "\/" v1 by A18;
hence v0 "\/" (v1 "/\" (v2 "\/" v0)) = (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) by A59; :: thesis: verum

end;

A63: for v1, v0, v2 being Element of L holds v0 "\/" (v1 "/\" (v2 "\/" v0)) = (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" v2))

proof

let v1, v0, v2 be Element of L; :: thesis: v0 "\/" (v1 "/\" (v2 "\/" v0)) = (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" v2))
(v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" v2)) by A1;
hence v0 "\/" (v1 "/\" (v2 "\/" v0)) = (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" v2)) by A61; :: thesis: verum

end;

A65: for v1, v0, v2 being Element of L holds v0 "\/" (v1 "/\" (v2 "\/" v0)) = v0 "\/" (v1 "/\" v2)

proof

let v1, v0, v2 be Element of L; :: thesis: v0 "\/" (v1 "/\" (v2 "\/" v0)) = v0 "\/" (v1 "/\" v2)
(v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" v2)) = v0 "\/" (v1 "/\" v2) by A8;
hence v0 "\/" (v1 "/\" (v2 "\/" v0)) = v0 "\/" (v1 "/\" v2) by A63; :: thesis: verum

end;

A68: for v101, v2, v102 being Element of L holds (v101 "/\" v2) "\/" (v101 "/\" (v102 "\/" v2)) = (v101 "/\" v2) "\/" (v101 "/\" v102)

proof

let v101, v2, v102 be Element of L; :: thesis: (v101 "/\" v2) "\/" (v101 "/\" (v102 "\/" v2)) = (v101 "/\" v2) "\/" (v101 "/\" v102)
v101 "/\" (v102 "\/" (v101 "/\" v2)) = v101 "/\" (v102 "\/" v2) by A9;
hence (v101 "/\" v2) "\/" (v101 "/\" (v102 "\/" v2)) = (v101 "/\" v2) "\/" (v101 "/\" v102) by A65; :: thesis: verum

end;

for v0, v2, v1 being Element of L holds v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2)

proof

let v0, v2, v1 be Element of L; :: thesis: v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2)
(v0 "/\" v1) "\/" (v0 "/\" (v2 "\/" v1)) = v0 "/\" (v1 "\/" v2) by A35;
hence v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2) by A68; :: thesis: verum

end;

hence for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" v3) ; :: thesis: verum