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, j,k being Nat
 holds DecIC(p +* Start-At(j,S),k) = p +* Start-At(j-'k,S)
proof
 let p be PartState of S, j,k be Nat;
 thus DecIC(p +* Start-At(j,S),k)
   = p +* Start-At(IC(p +* Start-At(j,S))-'k,S) by FUNCT_4:114
  .= p +* Start-At(j-'k,S) by Th16;
end;
