 reserve L for non empty LattStr;
 reserve v64,v65,v66,v67,v103,v3,v102,v101,v100,v2,v1,v0 for Element of L;

theorem Lemma1:
L is join-absorbing &
(for v0,v2,v1 holds (v0"/\"(v1"\/"v2))=((v2"/\"v0)"\/"(v1"/\"v0)))
implies for v0 holds (v0"/\"v0)=v0
proof
assume A2: L is join-absorbing;
assume A3: for v0,v2,v1 holds (v0"/\"(v1"\/"v2))=((v2"/\"v0)"\/"(v1"/\"v0));

A7: for v65,v66 holds v65=((v66"/\"v65)"\/"(v65"/\"v65))
proof let v65,v66;
(v65"/\"(v65"\/"v66))=v65 by A2;
hence thesis by A3;
end;

A11: for v1,v0 holds ((v0"/\"v1)"/\"v1)=(v0"/\"v1)
proof let v1,v0;
((v0"/\"v1)"\/"(v1"/\"v1))=v1 by A7;
hence thesis by A2;
end;

A18: for v65,v64,v0 holds
 ((v0"/\"v64)"\/"(v65"/\"v64))=(v64"/\"(v65"\/"(v0"/\"v64)))
proof let v65,v64,v0;
((v0"/\"v64)"/\"v64)=(v0"/\"v64) by A11;
hence thesis by A3;
end;

A35: for v65,v64,v66 holds (v65"/\"(v66"\/"v64))=(v65"/\"(v66"\/"(v64"/\"v65)))
proof let v65,v64,v66;
((v64"/\"v65)"\/"(v66"/\"v65))=(v65"/\"(v66"\/"v64)) by A3;
hence thesis by A18;
end;

A40: for v64,v0 holds (v64"/\"v64)=(v64"/\"((v0"/\"v64)"\/"v64))
proof let v64,v0;
((v0"/\"v64)"\/"(v64"/\"v64))=v64 by A7;
hence thesis by A35;
end;

A49: for v64,v65 holds ((v64"/\"v64)"\/"((v65"/\"v64)"/\"v64))=(v64"/\"v64)
proof let v64,v65;
(v64"/\"((v65"/\"v64)"\/"v64))=((v64"/\"v64)"\/"((v65"/\"v64)"/\"v64)) by A3;
hence thesis by A40;
end;

A51: for v64,v65 holds ((v64"/\"v64)"\/"(v65"/\"v64))=(v64"/\"v64)
proof let v64,v65;
((v65"/\"v64)"/\"v64)=(v65"/\"v64) by A11;
hence thesis by A49;
end;
let v65;
((v65"/\"v65)"\/"(v65"/\"v65))=v65 by A7;
hence thesis by A51;
end;
