text: let K be Field; :: thesis: for M1, M2 being Matrix of K st len M1 = len M2 & width M1 = width M2 holds
  - (M1 + M2) = (- M1) + (- M2)
  
  let M1, M2 be Matrix of K; :: thesis: ( len M1 = len M2 & width M1 = width M2 implies - (M1 + M2) = (- M1) + (- M2) )
  assume A1: ( len M1 = len M2 & width M1 = width M2 ) ; :: thesis: - (M1 + M2) = (- M1) + (- M2)
  A2: width (- M1) = width M1 by MATRIX_3:def 2;
  then A3: width ((- M1) + (- M2)) = width M1 by MATRIX_3:def 3;
  A4: ( len (M1 + M2) = len M1 & width (M1 + M2) = width M1 ) by MATRIX_3:def 3;
  A5: len (- M1) = len M1 by MATRIX_3:def 2;
  then A6: len ((- M1) + (- M2)) = len M1 by MATRIX_3:def 3;
  A7: ( len (- M2) = len M2 & width (- M2) = width M2 ) by MATRIX_3:def 2;
  per cases ( len M1 > 0 or len M1 = 0 ) by NAT_1:3;
  suppose A8: len M1 > 0 ; :: thesis: - (M1 + M2) = (- M1) + (- M2)
  then A9: M2 is Matrix of len M1, width M1,K by ;
  A10: M1 is Matrix of len M1, width M1,K by ;
  (M1 + M2) + ((- M1) + (- M2)) = (M1 + M2) + ((- M2) + (- M1)) by
  .= ((M1 + M2) + (- M2)) + (- M1) by
  .= (M1 + (M2 + (- M2))) + (- M1) by
  .= (M1 + (0. (K,(len M1),(width M1)))) + (- M1) by
  .= M1 + (- M1) by
  .= 0. (K,(len M1),(width M1)) by ;
  hence - (M1 + M2) = (- M1) + (- M2) by A4, A6, A3, Th8; :: thesis: verum
  end;
  suppose A11: len M1 = 0 ; :: thesis: - (M1 + M2) = (- M1) + (- M2)
  then len (- M1) = 0 by MATRIX_3:def 2;
  then A12: len ((- M1) + (- M2)) = 0 by MATRIX_3:def 3;
  len (M1 + M2) = 0 by ;
  then len (- (M1 + M2)) = 0 by MATRIX_3:def 2;
  hence - (M1 + M2) = (- M1) + (- M2) by ; :: thesis: verum
  end;
  end;
explanation: The provided text is a formal proof in a theorem-proving language, likely Mizar. It proves a property of matrices over a field \( K \). The theorem states that for two matrices \( M1 \) and \( M2 \) of the same dimensions, the negative of their sum is equal to the sum of their negatives, i.e., \( - (M1 + M2) = (- M1) + (- M2) \).
  
  The text does not contain a question but rather a detailed proof of the theorem.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: let K be Field; :: thesis: for M1, M2 being Matrix of K st len M1 = len M2 & width M1 = width M2 holds
  - (M1 + M2) = (- M1) + (- M2)
  
  let M1, M2 be Matrix of K; :: thesis: ( len M1 = len M2 & width M1 = width M2 implies - (M1 + M2) = (- M1) + (- M2) )
  assume A1: ( len M1 = len M2 & width M1 = width M2 ) ; :: thesis: - (M1 + M2) = (- M1) + (- M2)
  A2: width (- M1) = width M1 by MATRIX_3:def 2;
  then A3: width ((- M1) + (- M2)) = width M1 by MATRIX_3:def 3;
  A4: ( len (M1 + M2) = len M1 & width (M1 + M2) = width M1 ) by MATRIX_3:def 3;
  A5: len (- M1) = len M1 by MATRIX_3:def 2;
  then A6: len ((- M1) + (- M2)) = len M1 by MATRIX_3:def 3;
  A7: ( len (- M2) = len M2 & width (- M2) = width M2 ) by MATRIX_3:def 2;
  per cases ( len M1 > 0 or len M1 = 0 ) by NAT_1:3;
  suppose A8: len M1 > 0 ; :: thesis: - (M1 + M2) = (- M1) + (- M2)
  then A9: M2 is Matrix of len M1, width M1,K by ;
  A10: M1 is Matrix of len M1, width M1,K by ;
  (M1 + M2) + ((- M1) + (- M2)) = (M1 + M2) + ((- M2) + (- M1)) by
  .= ((M1 + M2) + (- M2)) + (- M1) by
  .= (M1 + (M2 + (- M2))) + (- M1) by
  .= (M1 + (0. (K,(len M1),(width M1)))) + (- M1) by
  .= M1 + (- M1) by
  .= 0. (K,(len M1),(width M1)) by ;
  hence - (M1 + M2) = (- M1) + (- M2) by A4, A6, A3, Th8; :: thesis: verum
  end;
  suppose A11: len M1 = 0 ; :: thesis: - (M1 + M2) = (- M1) + (- M2)
  then len (- M1) = 0 by MATRIX_3:def 2;
  then A12: len ((- M1) + (- M2)) = 0 by MATRIX_3:def 3;
  len (M1 + M2) = 0 by ;
  then len (- (M1 + M2)) = 0 by MATRIX_3:def 2;
  hence - (M1 + M2) = (- M1) + (- M2) by ; :: thesis: verum
  end;
  end;
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is a formal proof in a theorem-proving language, likely Mizar. It proves a property of matrices over a field \( K \). The theorem states that for two matrices \( M1 \) and \( M2 \) of the same dimensions, the negative of their sum is equal to the sum of their negatives, i.e., \( - (M1 + M2) = (- M1) + (- M2) \).
  
  The text does not contain a question but rather a detailed proof of the theorem.
  
  Label: no-question