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 Th53:
  for p being PartState of S, k being Nat
   holds IC IncIC (p,k) = IC p + k
proof
  let p be PartState of S, k be Nat;
   IC S in dom Start-At(IC p+k,S) by TARSKI:def 1;
  hence IC IncIC (p,k) = (Start-At((IC p)+k,S)).IC S by FUNCT_4:13
    .= IC p +k by FUNCOP_1:72;
end;
