reserve k, k1, n, n1, m for Nat;
reserve X, y for set;
reserve p for Real;
reserve r for Real;
reserve a, a1, a2, b, b1, b2, x, x0, z, z0 for Complex;
reserve s1, s3, seq, seq1 for Complex_Sequence;
reserve Y for Subset of COMPLEX;
reserve f, f1, f2 for PartFunc of COMPLEX,COMPLEX;
reserve Nseq for increasing sequence of NAT;
reserve h for 0-convergent non-zero Complex_Sequence;
reserve c for constant Complex_Sequence;
reserve R, R1, R2 for C_RestFunc;
reserve L, L1, L2 for C_LinearFunc;
reserve Z for open Subset of COMPLEX;

theorem Th19:
  (seq-seq1)^\k = (seq^\k)-(seq1^\k)
proof
  now
    let n be Element of NAT;
A1:  n+k in NAT by ORDINAL1:def 12;
    thus ((seq-seq1)^\k).n = (seq+-seq1).(n+k) by NAT_1:def 3
      .= seq.(n+k)+(-seq1).(n+k) by VALUED_1:1,A1
      .= seq.(n+k)+-seq1.(n+k) by VALUED_1:8
      .= (seq^\k).n-seq1.(n+k) by NAT_1:def 3
      .= (seq^\k).n+-(seq1^\k).n by NAT_1:def 3
      .= (seq^\k).n+(-(seq1^\k)).n by VALUED_1:8
      .= ((seq^\k)-(seq1^\k)).n by VALUED_1:1;
  end;
  hence thesis;
end;
