reserve x,N for set,
        k for Nat;
reserve N for with_zero set;
reserve S for IC-Ins-separated non empty with_non-empty_values
     Mem-Struct over N;

theorem
  for p being data-only PartState of S, q being PartState of S
  holds p c= q iff p c= DataPart q
proof
  let p be data-only PartState of S, q be PartState of S;
  set X = (the carrier of S) \ ({IC S});
A1: q|X c= q by RELAT_1:59;
  hereby
A2:  X \/ ({IC S}) = (the carrier of S) \/ ({IC S})
     by XBOOLE_1:39;
    dom p c= the carrier of S by RELAT_1:def 18;
    then
A3: dom p c= X \/ ({IC S}) by A2,XBOOLE_1:10;
    assume
    p c= q;
    then
A4: p|X c= DataPart q by RELAT_1:76;
    dom p misses {IC S} by Def9;
    hence p c= DataPart q by A4,A3,RELAT_1:68,XBOOLE_1:73;
  end;
 thus thesis by A1,XBOOLE_1:1;
end;
