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

theorem JoinCom:
L is join-absorbing &
(for v0,v2,v1 holds (v0"/\"(v1"\/"v2))=((v2"/\"v0)"\/"(v1"/\"v0)))
implies for v0,v1 holds v0"\/"v1=v1"\/"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;

A29: 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 A4;
hence thesis by A24;
end;

A31: for v64,v65 holds ((v64"/\"v64)"\/"(v65"/\"v64))=(v64"/\"v64)
proof let v64,v65;
((v65"/\"v64)"/\"v64)=(v65"/\"v64) by A12;
hence thesis by A29;
end;

A42: for v64,v65 holds (v64"/\"(v65"\/"v64))=v64
proof let v64,v65;
((v64"/\"v64)"\/"(v65"/\"v64))=(v64"/\"(v65"\/"v64)) by A4; then
(v64"/\"(v65"\/"v64))=(v64"/\"v64) by A31 .= v64 by A4,A3,Lemma1;
hence thesis;
end;

A46: for v2,v1 holds ((v2"/\"(v1"\/"v2))"\/"(v1"/\"(v1"\/"v2)))=(v1"\/"v2)
proof let v2,v1;
((v1"\/"v2)"/\"(v1"\/"v2))=((v2"/\"(v1"\/"v2))"\/"(v1"/\"(v1"\/"v2))) by A4;
hence thesis by A4,A3,Lemma1;
end;

A48: for v2,v1 holds (v2"\/"(v1"/\"(v1"\/"v2)))=(v1"\/"v2)
proof let v2,v1;
(v2"/\"(v1"\/"v2))=v2 by A42;
hence thesis by A46;
end;
let v1,v2;
(v1"/\"(v1"\/"v2))=v1 by A3;
hence thesis by A48;
end;
