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 p being PartState of S st IC S in dom p
  holds IncIC( p,k) = DataPart p +* Start-At ((IC p) +k,S)
 proof let p be PartState of S;
A1: dom Start-At ((IC p) +k,S) = {IC S}
     .= dom Start-At (IC p,S);
  assume
A2: IC S in dom p;
  thus IncIC( p,k)
     = DataPart p +* Start-At (IC p,S) +* Start-At ((IC p) +k,S) by A2,Th26
    .= DataPart p +* Start-At ((IC p) +k,S) by A1,FUNCT_4:74;
 end;
