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 Th52:
 for p being PartState of S, k being Nat holds IC S in dom IncIC(p,k)
proof let p be PartState of S, k be Nat;
A1: dom IncIC(p,k) = dom p \/ dom Start-At(IC p+k,S) by FUNCT_4:def 1;
  IC S in dom Start-At(IC p+k,S) by Th15;
  hence thesis by A1,XBOOLE_0:def 3;
end;
