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 being Element of L holds v0 "/\" v0 = v0 )
assume A2: 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 being Element of L holds v0 "/\" v0 = v0 )
assume A3: for v0, v2, v1 being Element of L holds v0 "/\" (v1 "\/" v2) = (v2 "/\" v0) "\/" (v1 "/\" v0) ; :: thesis: for v0 being Element of L holds v0 "/\" v0 = v0
A7: 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 A2;
hence v65 = (v66 "/\" v65) "\/" (v65 "/\" v65) by A3; :: thesis: verum
end;
A11: 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 A7;
hence (v0 "/\" v1) "/\" v1 = v0 "/\" v1 by A2; :: thesis: verum
end;
A18: 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 A11;
hence (v0 "/\" v64) "\/" (v65 "/\" v64) = v64 "/\" (v65 "\/" (v0 "/\" v64)) by A3; :: thesis: verum
end;
A35: 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 A3;
hence v65 "/\" (v66 "\/" v64) = v65 "/\" (v66 "\/" (v64 "/\" v65)) by A18; :: thesis: verum
end;
A40: 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 A7;
hence v64 "/\" v64 = v64 "/\" ((v0 "/\" v64) "\/" v64) by A35; :: thesis: verum
end;
A49: 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 A3;
hence (v64 "/\" v64) "\/" ((v65 "/\" v64) "/\" v64) = v64 "/\" v64 by A40; :: thesis: verum
end;
A51: 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 A11;
hence (v64 "/\" v64) "\/" (v65 "/\" v64) = v64 "/\" v64 by A49; :: thesis: verum
end;
let v65 be Element of L; :: thesis: v65 "/\" v65 = v65
(v65 "/\" v65) "\/" (v65 "/\" v65) = v65 by A7;
hence v65 "/\" v65 = v65 by A51; :: thesis: verum