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 Th51:
  for p being PartState of S, k being Nat
  holds DataPart IncIC(p,k) = DataPart p
proof
  let p be PartState of S, k be Nat;
  thus DataPart IncIC(p,k)
  = DataPart p +* DataPart Start-At(IC p+k,S) by FUNCT_4:71
    .= DataPart p +* {} by Th20
    .= DataPart p;
end;
