reserve x,N for set,
        k for Nat;
reserve N for with_zero set;

theorem
 for S being IC-Ins-separated 1-element with_non-empty_values Mem-Struct over N
  for s1, s2 being State of S st IC s1 = IC s2 holds s1= s2
proof
 let T be IC-Ins-separated 1-element with_non-empty_values Mem-Struct over N;
  let s1,s2 be State of T such that
A1: IC s1 = IC s2;
A2: dom s1 = the carrier of T by PARTFUN1:def 2;
    then
A3: dom s1 =dom s2 by PARTFUN1:def 2;
 now
    let x be object;
    assume
A4: x in dom s1;
A5:  x = IC T  by A4,A2,STRUCT_0:def 10;
     hence s1.x = IC s1
        .= s2.x by A1,A5;
  end;
  hence thesis by A3;
end;
