reserve a,b,c,v,v1,x,y for object;
reserve V,A for set;
reserve d for TypeSCNominativeData of V,A;
reserve p,q,r for SCPartialNominativePredicate of V,A;
reserve n for Nat;
reserve X for Function;
reserve f,g,h for SCBinominativeFunction of V,A;

theorem Th34:
  d in dom(SC_Psuperpos(p,f,v)) implies
   SC_Psuperpos(p,f,v).d = p.local_overlapping(V,A,d,f.d,v) & d in dom f
  proof
    set F = SC_Psuperpos(p,f,v);
    assume
A1: d in dom(F);
    dom(F) = {d where d is TypeSCNominativeData of V,A:
     local_overlapping(V,A,d,f.d,v) in dom p & d in dom f} by Def11;
    then
A2: ex d1 being TypeSCNominativeData of V,A st d1 = d &
    local_overlapping(V,A,d1,f.d1,v) in dom p & d1 in dom f
    by A1;
    then F,d =~ p,local_overlapping(V,A,d,f.d,v) by Def11;
    hence thesis by A2;
  end;
