reserve D for non empty set;
reserve m,n,N for Nat;
reserve size for non zero Nat;
reserve f1,f2,f3,f4,f5,f6 for BinominativeFunction of D;
reserve p1,p2,p3,p4,p5,p6,p7 for PartialPredicate of D;
reserve d,v for object;
reserve V,A for set;
reserve z for Element of V;
reserve val for Function;
reserve loc for V-valued Function;
reserve d1 for NonatomicND of V,A;
reserve T for TypeSCNominativeData of V,A;

theorem Th10:
  loc,val,size are_correct_wrt d1 implies
  for m,n being Nat st (n in dom val or 1 <= n <= size) & 1 <= m <= size holds
  val.n in dom(LocalOverlapSeq(A,loc,val,d1,size).m)
  proof
    set F = LocalOverlapSeq(A,loc,val,d1,size);
    assume
A1: loc,val,size are_correct_wrt d1;
    then
A2: val is_valid_wrt d1;
    let m,n be Nat;
    assume
A3: n in dom val or 1 <= n <= size;
    now
      assume
A4:   1 <= n <= size;
A5:   dom F c= dom val by A1;
      len F = size by Def4;
      hence n in dom val by A5,A4,FINSEQ_3:25;
    end;
    then
A6: val.n in rng val by A3,FUNCT_1:def 3;
    assume 1 <= m <= size;
    then dom(d1) c= dom(F.m) by A1,Th7;
    hence thesis by A2,A6;
  end;
