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
  {a,b,c} c= V & a,b,c are_mutually_distinct &
  d in dom f & d in dom g & d in dom h implies
  NDentry(<*f,g,h*>,<*a,b,c*>,d) is NonatomicND of V,A
  proof
    assume that
A1: {a,b,c} c= V and
A2: a,b,c are_mutually_distinct and
A3: d in dom f & d in dom g & d in dom h;
    reconsider O = <*a,b,c*> as one-to-one FinSequence by A2,FINSEQ_3:95;
    set F = NDentry(<*f,g,h*>,O,d);
A4: F = {[a,f.d],[b,g.d],[c,h.d]} by Th23;
    then
A5: dom F = {a,b,c} by Th2;
A6: rng F = {f.d,g.d,h.d} by A4,Th3;
    f.d in ND(V,A) & g.d in ND(V,A) & h.d in ND(V,A) by A3,PARTFUN1:4;
    then rng F c= ND(V,A) by A6,Th1;
    hence thesis by A1,A5,Th6;
  end;
