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;
reserve s for State of S;
reserve p for PartState of S;

theorem
  for s1,s2 be State of S holds
  DataPart s1 = DataPart s2 implies
    Initialize s1 =  Initialize s2
proof
  let s1,s2 be State of S;
  assume
A1: DataPart s1 = DataPart s2;
  set S1 = Initialize s1, S2 = Initialize s2;
A2: IC S1 =  0 & IC S2 =  0 by Th47;
  DataPart S1 = DataPart s1 by Th45
    .= DataPart S2 by A1,Th45;
  hence thesis by A2,Th78;
end;
