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 Th25:
  rng NDdataSeq(<*f*>,<*v*>,d) = v.-->f.d
  proof
    set g = <*f*>;
    set X = <*v*>;
    set N = NDdataSeq(g,X,d);
    set F = v.-->f.d;
A1: dom N = dom X by Def4;
A2: dom X = {1} by FINSEQ_1:2,38;
A5: F = {[v,f.d]} by ZFMISC_1:29;
    thus rng N c= F
    proof
      let y be object;
      assume y in rng N;
      then consider z being object such that
A6:   z in dom N and
A7:   N.z = y by FUNCT_1:def 3;
A8:   z = 1 by A1,A2,A6,TARSKI:def 1;
      N.z = [X.z,g.z.d] by A1,A6,Def4;
      hence thesis by A7,A8,A5,TARSKI:def 1;
    end;
    let m,n be object;
    assume [m,n] in F;
    then
A9: [m,n] = [v,f.d] by A5,TARSKI:def 1;
A10: 1 in dom N by A1,A2,TARSKI:def 1;
    then N.1 = [X.1,g.1.d] by A1,Def4;
    hence thesis by A9,A10,FUNCT_1:def 3;
  end;
