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, k being Nat holds
  Start-At(IC p+k,S) c= IncIC (p,k)
proof
  let p be PartState of S, k be Nat;
A1: IC IncIC(p,k) = IC p + k by Th53;
A2: IC S in dom (IncIC(p,k)) by Th52;
A3: Start-At(IC p+k,S) = {[IC S,IC p + k]} & [IC S,IC p + k] in
  IncIC(p,k) by A2,A1,FUNCT_1:def 2,FUNCT_4:82;
  for x being object st x in Start-At(IC p+k,S)
    holds x in IncIC (p,k) by A3,TARSKI:def 1;
 hence thesis by TARSKI:def 3;
end;
