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,q being PartState of S, k being Nat st IC S in dom q
  holds DecIC(p+*IncIC(q,k),k) = p +* q
proof let p,q be PartState of S, k being Nat such that
A1: IC S in dom q;
  IC S in dom IncIC(q,k) by Th52;
 hence DecIC(p+*IncIC(q,k),k) = p +* DecIC(IncIC(q,k),k) by Th72
      .= p +* q by A1,Th73;
end;
