let L be non empty LattStr ; :: thesis: ( L is join-absorbing & ( for v0, v2, v1 being Element of L holds v0 "/\" (v1 "\/" v2) = (v2 "/\" v0) "\/" (v1 "/\" v0) ) implies for v0, v1 being Element of L holds v0 "\/" v1 = v1 "\/" v0 )
assume A3: L is join-absorbing ; :: thesis: ( ex v0, v2, v1 being Element of L st not v0 "/\" (v1 "\/" v2) = (v2 "/\" v0) "\/" (v1 "/\" v0) or for v0, v1 being Element of L holds v0 "\/" v1 = v1 "\/" v0 )
assume A4: for v0, v2, v1 being Element of L holds v0 "/\" (v1 "\/" v2) = (v2 "/\" v0) "\/" (v1 "/\" v0) ; :: thesis: for v0, v1 being Element of L holds v0 "\/" v1 = v1 "\/" v0
A8: for v65, v66 being Element of L holds v65 = (v66 "/\" v65) "\/" (v65 "/\" v65)

proof

let v65, v66 be Element of L; :: thesis: v65 = (v66 "/\" v65) "\/" (v65 "/\" v65)
v65 "/\" (v65 "\/" v66) = v65 by A3;
hence v65 = (v66 "/\" v65) "\/" (v65 "/\" v65) by A4; :: thesis: verum

end;

A12: for v1, v0 being Element of L holds (v0 "/\" v1) "/\" v1 = v0 "/\" v1

proof

let v1, v0 be Element of L; :: thesis: (v0 "/\" v1) "/\" v1 = v0 "/\" v1
(v0 "/\" v1) "\/" (v1 "/\" v1) = v1 by A8;
hence (v0 "/\" v1) "/\" v1 = v0 "/\" v1 by A3; :: thesis: verum

end;

A15: for v65, v64, v0 being Element of L holds (v0 "/\" v64) "\/" (v65 "/\" v64) = v64 "/\" (v65 "\/" (v0 "/\" v64))

proof

let v65, v64, v0 be Element of L; :: thesis: (v0 "/\" v64) "\/" (v65 "/\" v64) = v64 "/\" (v65 "\/" (v0 "/\" v64))
(v0 "/\" v64) "/\" v64 = v0 "/\" v64 by A12;
hence (v0 "/\" v64) "\/" (v65 "/\" v64) = v64 "/\" (v65 "\/" (v0 "/\" v64)) by A4; :: thesis: verum

end;

A19: for v65, v64, v66 being Element of L holds v65 "/\" (v66 "\/" v64) = v65 "/\" (v66 "\/" (v64 "/\" v65))

proof

let v65, v64, v66 be Element of L; :: thesis: v65 "/\" (v66 "\/" v64) = v65 "/\" (v66 "\/" (v64 "/\" v65))
(v64 "/\" v65) "\/" (v66 "/\" v65) = v65 "/\" (v66 "\/" v64) by A4;
hence v65 "/\" (v66 "\/" v64) = v65 "/\" (v66 "\/" (v64 "/\" v65)) by A15; :: thesis: verum

end;

A24: for v64, v0 being Element of L holds v64 "/\" v64 = v64 "/\" ((v0 "/\" v64) "\/" v64)

proof

let v64, v0 be Element of L; :: thesis: v64 "/\" v64 = v64 "/\" ((v0 "/\" v64) "\/" v64)
(v0 "/\" v64) "\/" (v64 "/\" v64) = v64 by A8;
hence v64 "/\" v64 = v64 "/\" ((v0 "/\" v64) "\/" v64) by A19; :: thesis: verum

end;

A29: for v64, v65 being Element of L holds (v64 "/\" v64) "\/" ((v65 "/\" v64) "/\" v64) = v64 "/\" v64

proof

let v64, v65 be Element of L; :: thesis: (v64 "/\" v64) "\/" ((v65 "/\" v64) "/\" v64) = v64 "/\" v64
v64 "/\" ((v65 "/\" v64) "\/" v64) = (v64 "/\" v64) "\/" ((v65 "/\" v64) "/\" v64) by A4;
hence (v64 "/\" v64) "\/" ((v65 "/\" v64) "/\" v64) = v64 "/\" v64 by A24; :: thesis: verum

end;

A31: for v64, v65 being Element of L holds (v64 "/\" v64) "\/" (v65 "/\" v64) = v64 "/\" v64

proof

let v64, v65 be Element of L; :: thesis: (v64 "/\" v64) "\/" (v65 "/\" v64) = v64 "/\" v64
(v65 "/\" v64) "/\" v64 = v65 "/\" v64 by A12;
hence (v64 "/\" v64) "\/" (v65 "/\" v64) = v64 "/\" v64 by A29; :: thesis: verum

end;

A42: for v64, v65 being Element of L holds v64 "/\" (v65 "\/" v64) = v64

proof

let v64, v65 be Element of L; :: thesis: v64 "/\" (v65 "\/" v64) = v64
(v64 "/\" v64) "\/" (v65 "/\" v64) = v64 "/\" (v65 "\/" v64) by A4;
then v64 "/\" (v65 "\/" v64) = v64 "/\" v64 by A31
.= v64 by A4, A3, Lemma1 ;
hence v64 "/\" (v65 "\/" v64) = v64 ; :: thesis: verum

end;

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

proof

let v2, v1 be Element of L; :: thesis: (v2 "/\" (v1 "\/" v2)) "\/" (v1 "/\" (v1 "\/" v2)) = v1 "\/" v2
(v1 "\/" v2) "/\" (v1 "\/" v2) = (v2 "/\" (v1 "\/" v2)) "\/" (v1 "/\" (v1 "\/" v2)) by A4;
hence (v2 "/\" (v1 "\/" v2)) "\/" (v1 "/\" (v1 "\/" v2)) = v1 "\/" v2 by A4, A3, Lemma1; :: thesis: verum

end;

A48: for v2, v1 being Element of L holds v2 "\/" (v1 "/\" (v1 "\/" v2)) = v1 "\/" v2

proof

let v2, v1 be Element of L; :: thesis: v2 "\/" (v1 "/\" (v1 "\/" v2)) = v1 "\/" v2
v2 "/\" (v1 "\/" v2) = v2 by A42;
hence v2 "\/" (v1 "/\" (v1 "\/" v2)) = v1 "\/" v2 by A46; :: thesis: verum

end;

let v1, v2 be Element of L; :: thesis: v1 "\/" v2 = v2 "\/" v1
v1 "/\" (v1 "\/" v2) = v1 by A3;
hence v1 "\/" v2 = v2 "\/" v1 by A48; :: thesis: verum