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

theorem JoinIdem:
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 A3: L is join-absorbing;
assume A4: for v0,v2,v1 holds (v0"/\"(v1"\/"v2))=((v2"/\"v0)"\/"(v1"/\"v0));
A8: for v65,v66 holds v65=((v66"/\"v65)"\/"(v65"/\"v65))
proof let v65,v66;
(v65"/\"(v65"\/"v66))=v65 by A3;
hence thesis by A4;
end;

A12: for v1,v0 holds ((v0"/\"v1)"/\"v1)=(v0"/\"v1)
proof let v1,v0;
((v0"/\"v1)"\/"(v1"/\"v1))=v1 by A8;
hence thesis by A3;
end;

A15: for v65,v64,v0 holds
 ((v0"/\"v64)"\/"(v65"/\"v64))=(v64"/\"(v65"\/"(v0"/\"v64)))
proof let v65,v64,v0;
((v0"/\"v64)"/\"v64)=(v0"/\"v64) by A12;
hence thesis by A4;
end;

A19: 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 A4;
hence thesis by A15;
end;

A24: for v64,v0 holds (v64"/\"v64)=(v64"/\"((v0"/\"v64)"\/"v64))
proof let v64,v0;
((v0"/\"v64)"\/"(v64"/\"v64))=v64 by A8;
hence thesis by A19;
end;

A35: for v64,v1 holds
 ((v64"/\"v64)"\/"(((v1"/\"v64)"\/"v64)"/\"((v1"/\"v64)"\/"v64)))=
   ((v1"/\"v64)"\/"v64)
proof let v64,v1;
(v64"/\"((v1"/\"v64)"\/"v64))=(v64"/\"v64) by A24;
hence thesis by A8;
end;

A42: for v0,v1 holds
 (v0"\/"(((v1"/\"v0)"\/"v0)"/\"((v1"/\"v0)"\/"v0)))=((v1"/\"v0)"\/"v0)
proof let v0,v1;
(v0"/\"v0)=v0 by A3,A4,Lemma1;
hence thesis by A35;
end;

A44: for v0,v1 holds (v0"\/"((v1"/\"v0)"\/"v0))=((v1"/\"v0)"\/"v0)
proof let v0,v1;
(((v1"/\"v0)"\/"v0)"/\"((v1"/\"v0)"\/"v0))=((v1"/\"v0)"\/"v0) by A3,A4,Lemma1;
hence thesis by A42;
end;

A46: for v1,v0 holds ((v0"/\"v1)"\/"v1)=v1
proof let v1,v0;
(v1"/\"v1)=v1 by A3,A4,Lemma1;
hence thesis by A8;
end;

A48: for v0,v1 holds (v0"\/"v0)=((v1"/\"v0)"\/"v0)
proof let v0,v1;
((v1"/\"v0)"\/"v0)=v0 by A46;
hence thesis by A44;
end;

now let v0,v1;
((v1"/\"v0)"\/"v0)=v0 by A46;
hence (v0"\/"v0)=v0 by A48;
end;
hence thesis;
end;
