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 Th30:
  for g being (V,A)-FPrg-yielding FinSequence
  for X being one-to-one FinSequence
   st dom g = dom X & d in_doms g
  holds rng NDentry(g,X,d) c= ND(V,A)
  proof
    let g be (V,A)-FPrg-yielding FinSequence;
    let X be one-to-one FinSequence;
    assume that
A1: dom g = dom X and
A2: d in_doms g;
    set f = NDdataSeq(g,X,d);
    set D = NDentry(g,X,d);
A3: dom f = dom X by Def4;
    let y be object;
    assume y in rng D;
    then consider a such that
A4: a in dom D and
A5: D.a = y by FUNCT_1:def 3;
    [a,y] in D by A4,A5,FUNCT_1:1;
    then consider v such that
A6: v in dom f and
A7: f.v = [a,y] by FUNCT_1:def 3;
    reconsider v as Element of NAT by A6;
    f.v = [X.v,g.v.d] by A3,A6,Def4;
    then
A8: y = g.v.d by A7,XTUPLE_0:1;
    1 <= v <= len g by A1,A3,A6,FINSEQ_3:25;
    then g.v is SCBinominativeFunction of V,A by Def6;
    then
A9: rng(g.v) c= ND(V,A) by RELAT_1:def 19;
    d in dom(g.v) by A1,A3,A6,A2;
    hence thesis by A8,A9,FUNCT_1:3;
  end;
