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 DecIC(DecIC(p,i),j) = DecIC(p,i+j)
 proof let p be PartState of S, i,j being Nat;
  thus DecIC(DecIC(p,i),j)
        = p +* Start-At(IC p-'i,S) +* Start-At(IC p -' i -' j,S) by Th65
       .= p +* Start-At(IC p-'i,S) +* Start-At(IC p -' (i + j),S) by NAT_2:30
       .= DecIC(p,i+j) by FUNCT_4:114;
 end;
