reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;

theorem Th81:
  len p = k + 1 & q = p | Seg k implies (p.(k + 1) in A iff p - A = q - A)
proof
  assume that
A1: len p = k + 1 and
A2: q = p | Seg k;
  thus p.(k + 1) in A implies p - A = q - A
  proof
    assume
A3: p.(k + 1) in A;
    thus p - A = (q ^ <* p.(k + 1) *>) - A by A1,A2,Th53
      .= (q - A) ^ (<* p.(k + 1) *> - A) by Lm11
      .= (q - A) ^ {} by A3,Lm7
      .= q - A by FINSEQ_1:34;
  end;
  assume that
A4: p - A = q - A and
A5: not p.(k + 1) in A;
  q - A = (q ^ <* p.(k + 1) *>) - A by A1,A2,A4,Th53
    .= (q - A) ^ (<* p.(k + 1) *> - A) by Lm11
    .= (q - A) ^ <* p.(k + 1) *> by A5,Lm6;
  hence thesis by FINSEQ_1:87;
end;
