reserve v for object;
reserve V,A for set;
reserve f for SCBinominativeFunction of V,A;
reserve d for TypeSCNominativeData of V,A;
reserve d1 for NonatomicND of V,A;
reserve a,b,c,z for Element of V;
reserve x,y for object;
reserve p,q,r,s for SCPartialNominativePredicate of V,A;

theorem Th11:
  (for d holds a is_a_value_on d) & (for d holds b is_a_value_on d) implies
  dom Equality(A,a,b) = dom denaming(V,A,a) /\ dom denaming(V,A,b)
  proof
    set Da = denaming(V,A,a);
    set Db = denaming(V,A,b);
    assume
A1: (for d holds a is_a_value_on d) & (for d holds b is_a_value_on d);
    dom Equality(A) = [:A,A:] by FUNCT_2:def 1;
    then rng <:Da,Db:> c= dom Equality(A) by A1,Th9;
    then dom Equality(A,a,b) = dom <:Da,Db:> by RELAT_1:27;
    hence thesis by FUNCT_3:def 7;
  end;
