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 Th33:
  for g being (V,A)-FPrg-yielding FinSequence st product g <> {}
   for x being Element of product g st d in dom(SC_Psuperpos(p,x,X))
    holds d in_doms g &
    SC_Psuperpos(p,x,X).d = p.global_overlapping(V,A,d,NDentry(g,X,d))
  proof
    let g be (V,A)-FPrg-yielding FinSequence such that
A1: product g <> {};
    let x be Element of product g such that
A2: d in dom(SC_Psuperpos(p,x,X));
    SC_Psuperpos(p,x,X) = SCPsuperpos(g,X).(p,x) by A1,Def10;
    then dom(SC_Psuperpos(p,x,X)) = {d where d is TypeSCNominativeData of V,A:
    global_overlapping(V,A,d,NDentry(g,X,d)) in dom p & d in_doms g}
    by A1,Def9;
    then
A3: ex d1 being TypeSCNominativeData of V,A st d1 = d &
    global_overlapping(V,A,d1,NDentry(g,X,d1)) in dom p & d1 in_doms g by A2;
    then SCPsuperpos(g,X).(p,x),d =~ p,global_overlapping(V,A,d,NDentry(g,X,d))
    by A1,Def9;
    hence thesis by A1,A3,Def10;
  end;
