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, i,j being Nat
 holds IncIC(IncIC(p,i),j) = IncIC(p,i+j)
 proof let p be PartState of S, i,j being Nat;
  thus IncIC(IncIC(p,i),j)
        = p +* Start-At(IC p+i,S) +* Start-At(IC p + i +j,S) by Th53
       .= IncIC(p,i+j) by FUNCT_4:114;
 end;
