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;

theorem
  for f being Function, a,d being object holds
  NDentry(<*f*>,<*a*>,d) = {[a,f.d]}
  proof
    let f be Function;
    let a,d be object;
    set X = <*a*>;
    set G = <*f*>;
    set A = {[a,f.d]};
    set N = NDdataSeq(G,X,d);
    set F = NDentry(G,X,d);
A1: dom N = dom X by Def4;
A2: dom X = {1} by FINSEQ_1:2,38;
A3: 1 in {1} by TARSKI:def 1;
    then
A4: N.1 = [X.1,G.1.d] by A2,Def4;
    thus F c= A
    proof
      let y be object;
      assume y in F;
      then consider z being object such that
A6:   z in dom N and
A7:   N.z = y by FUNCT_1:def 3;
      z = 1 by A1,A2,A6,TARSKI:def 1;
      hence thesis by A4,A7,TARSKI:def 1;
    end;
    let y,z be object;
    assume [y,z] in A;
    then [y,z] = [a,f.d] by TARSKI:def 1;
    hence thesis by A1,A2,A3,A4,FUNCT_1:def 3;
  end;
