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= DecIC (p,k)
proof
  let p be PartState of S, k be Nat;
A1: IC DecIC(p,k) = IC p -' k by Th65;
A2: IC S in dom (DecIC(p,k)) by Th64;
A3: Start-At(IC p-'k,S) = {[IC S,IC p -' k]} & [IC S,IC p -' k] in
  DecIC(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 DecIC (p,k) by A3,TARSKI:def 1;
 hence thesis by TARSKI:def 3;
end;
