let v be object ; :: thesis: for V, A being set
for d1 being NonatomicND of V,A
for d2 being TypeSCNominativeData of V,A st v in V & not v in dom d1 & not d1 in A & not naming (V,A,v,d2) in A holds
dom (local_overlapping (V,A,d1,d2,v)) = {v} \/ (dom d1)
let V, A be set ; :: thesis: for d1 being NonatomicND of V,A
for d2 being TypeSCNominativeData of V,A st v in V & not v in dom d1 & not d1 in A & not naming (V,A,v,d2) in A holds
dom (local_overlapping (V,A,d1,d2,v)) = {v} \/ (dom d1)
let d1 be NonatomicND of V,A; :: thesis: for d2 being TypeSCNominativeData of V,A st v in V & not v in dom d1 & not d1 in A & not naming (V,A,v,d2) in A holds
dom (local_overlapping (V,A,d1,d2,v)) = {v} \/ (dom d1)
let d2 be TypeSCNominativeData of V,A; :: thesis: ( v in V & not v in dom d1 & not d1 in A & not naming (V,A,v,d2) in A implies dom (local_overlapping (V,A,d1,d2,v)) = {v} \/ (dom d1) )
assume that
A1: v in V and
A2: not v in dom d1 and
A3: ( not d1 in A & not naming (V,A,v,d2) in A ) ; :: thesis: dom (local_overlapping (V,A,d1,d2,v)) = {v} \/ (dom d1)
set n = naming (V,A,v,d2);
A4: naming (V,A,v,d2) = v .--> d2 by A1, NOMIN_1:def 13;
A5: dom (d1 | ((dom d1) \ (dom (naming (V,A,v,d2))))) = dom d1

proof

thus dom (d1 | ((dom d1) \ (dom (naming (V,A,v,d2))))) c= dom d1 by RELAT_1:60; :: according to XBOOLE_0:def 10 :: thesis: dom d1 c= dom (d1 | ((dom d1) \ (dom (naming (V,A,v,d2)))))
dom d1 c= (dom d1) \ (dom (naming (V,A,v,d2))) by A2, A4, ZFMISC_1:34;
hence dom d1 c= dom (d1 | ((dom d1) \ (dom (naming (V,A,v,d2))))) by RELAT_1:62; :: thesis: verum

end;

local_overlapping (V,A,d1,d2,v) = (naming (V,A,v,d2)) \/ (d1 | ((dom d1) \ (dom (naming (V,A,v,d2))))) by A3, NOMIN_1:64;
hence dom (local_overlapping (V,A,d1,d2,v)) = {v} \/ (dom d1) by A4, A5, XTUPLE_0:23; :: thesis: verum