reserve D for non empty set;
reserve f1,f2,f3,f4,f5,f6,f7,f8,f9,f10 for BinominativeFunction of D;
reserve p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11 for PartialPredicate of D;
reserve q1,q2,q3,q4,q5,q6,q7,q8,q9,q10 for total PartialPredicate of D;
reserve n,m,N for Nat;
reserve fD for PFuncs(D,D)-valued FinSequence;
reserve fB for PFuncs(D,BOOLEAN)-valued FinSequence;
reserve V,A for set;
reserve val for Function;
reserve loc for V-valued Function;
reserve d1 for NonatomicND of V,A;
reserve p for SCPartialNominativePredicate of V,A;
reserve d,v for object;
reserve size for non zero Nat;
reserve inp,pos for FinSequence;
reserve prg for non empty FPrg(ND(V,A))-valued FinSequence;

theorem Th20:
  1 <= n <= len prg & PrgLocalOverlapSeq(A,loc,d1,prg,pos).m in dom(prg.n)
  implies
  prg.n.(PrgLocalOverlapSeq(A,loc,d1,prg,pos).m)
    is TypeSCNominativeData of V,A
  proof
    set F = PrgLocalOverlapSeq(A,loc,d1,prg,pos);
    set P = prg.n;
    assume that
A1: 1 <= n and
A2: n <= len prg and
A3: F.m in dom(P);
    n in dom prg by A1,A2,FINSEQ_3:25;
    then P in rng prg by FUNCT_1:def 3;
    then
A4: rng(P) c= ND(V,A) by RELAT_1:def 19;
    P.(F.m) in rng P by A3,FUNCT_1:def 3;
    hence thesis by A4,NOMIN_1:39;
  end;
