reserve x,y,P,Q for Integer;
reserve a,b,n for Nat;
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 for object;
reserve z for Element of V;
reserve T for TypeSCNominativeData of V,A;
reserve size for non zero Nat;
reserve x0, y0, p0, q0 for Integer;
reserve n0 for Nat;

theorem Th12:
  Seg 10 c= dom loc & loc is_valid_wrt d1 implies
  { loc/.1, loc/.2, loc/.3, loc/.4, loc/.5,
    loc/.6, loc/.7, loc/.8, loc/.9, loc/.10 } c= dom d1
  proof
    assume that
A1: Seg 10 c= dom loc and
A2: rng loc c= dom d1;
    set i = loc/.1, j = loc/.2, n = loc/.3, s = loc/.4, b = loc/.5, c = loc/.6;
    set p = loc/.7, q = loc/.8, ps = loc/.9, qc = loc/.10;
    let x be object;
    assume x in {i,j,n,s,b,c,p,q,ps,qc};
    then
A3: x = i or x = j or x = n or x = s or x = b or x = c or x = p or x = q
    or x = ps or x = qc by ENUMSET1:def 8;
A4: 1 in Seg 10 & ... & 10 in Seg 10;
    then
A5: loc.1 in rng loc & ... & loc.10 in rng loc by A1,FUNCT_1:def 3;
    loc.1 = loc/.1 & ... & loc.10 = loc/.10 by A1,A4,PARTFUN1:def 6;
    hence thesis by A2,A3,A5;
  end;
