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 Th73:
 for p being PartState of S, k being Nat st IC S in dom p
  holds DecIC(IncIC(p,k),k) = p
proof let p be PartState of S, k be Nat such that
A1: IC S in dom p;
 thus DecIC(IncIC(p,k),k)
     = IncIC(p,k) +* Start-At(IC p + k -'k,S) by Th53
    .= IncIC(p,k) +* Start-At(IC p, S) by NAT_D:34
    .= p +* Start-At(IC p, S) by Th36
    .= p by A1,FUNCT_4:7,98;
end;
