reserve X for ComplexUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;
reserve Rseq for Real_Sequence;
reserve Cseq,Cseq1,Cseq2 for Complex_Sequence;
reserve z,z1,z2 for Complex;
reserve r for Real;
reserve k,n,m for Nat;

theorem Th19:
  seq.(n+1) = Sum(seq,n+1) - Sum(seq,n)
proof
  thus Sum(seq,n+1)-Sum(seq,n) = (seq.(n+1)+Sum(seq,n)) - Sum(seq,n) by
BHSP_4:def 1
    .= seq.(n+1) + (Sum(seq,n)-Sum(seq,n)) by RLVECT_1:def 3
    .= seq.(n+1) + 0.X by RLVECT_1:15
    .= seq.(n+1) by RLVECT_1:4;
end;
