reserve x, y for object, X for set,
  i, j, k, l, n, m for Nat,
  D for non empty set,
  K for commutative Ring,
  a,b for Element of K,
  perm, p, q for Element of Permutations(n),
  Perm,P for Permutation of Seg n,
  F for Function of Seg n,Seg n,
  perm2, p2, q2, pq2 for Element of Permutations(n+2),
  Perm2 for Permutation of Seg (n+2);
reserve s for Element of 2Set Seg (n+2);
reserve pD for FinSequence of D,
  M for Matrix of n,m,D,
  pK,qK for FinSequence of K,
  A for Matrix of n,K;

theorem Th49:
  for i,j st i in Seg n & j in Seg n & i < j for M be (Matrix of n
,K) st Line(M,i) = Line(M,j) for p,q,tr be Element of Permutations(n) st q = p*
  tr & tr is being_transposition & tr.i=j holds (Path_product M).q = - (
  Path_product M).p
proof
  let i,j such that
A1: i in Seg n and
A2: j in Seg n and
A3: i < j;
  {i,j} in 2Set Seg n by A1,A2,A3,Th1;
  then reconsider n2=n-2 as Nat by Th2,NAT_1:21,23;
  let M be (Matrix of n,K) such that
A4: Line(M,i) = Line(M,j);
  reconsider M9=M as Matrix of n2+2,K;
  let p,q,tr be Element of Permutations(n) such that
A5: q = p*tr and
A6: tr is being_transposition and
A7: tr.i=j;
  reconsider TR=tr as Permutation of Seg (n2+2) by MATRIX_1:def 12;
  set Mt=M9*TR;
A8: for k be Nat st 1 <=k & k <= len M9 holds M9.k=Mt.k
  proof
    let k be Nat such that
A9: 1 <=k and
A10: k <= len M9;
A11: k in Seg (len M9) by A9,A10;
A12: Line(M,j)=M.j by A2,MATRIX_0:52;
A13: dom TR=Seg n by FUNCT_2:52;
A14: Line(M,i)=M.i by A1,MATRIX_0:52;
A15: len M9=n by MATRIX_0:def 2;
    then
A16: Line(Mt,k)=M.(tr.k) by A11,Th38;
    per cases;
    suppose
      k=i;
      hence thesis by A1,A4,A7,A16,A14,A12,MATRIX_0:52;
    end;
    suppose
A17:  k=j;
      then
A18:  M.k=M.i by A2,A4,A14,MATRIX_0:52;
      Line(Mt,k)=M.i by A3,A6,A7,A16,A17,Th8;
      hence thesis by A2,A17,A18,MATRIX_0:52;
    end;
    suppose
      k<>i & k<>j;
      then Line(Mt,k)=M.k by A3,A6,A7,A11,A15,A13,A16,Th8;
      hence thesis by A11,A15,MATRIX_0:52;
    end;
  end;
  len Mt=len M9 by Def4;
  then
A19: Mt=M by A8;
  reconsider Tr=tr,p2=p as Element of Permutations(n2+2);
A20: sgn(Tr,K)=-1_K by A6,Th14;
  tr=tr" by A6,Th20;
  hence (Path_product M).q = (-1_K)*((Path_product M9).p2)
    by A5,A19,A20,Th43
    .= -1_K*((Path_product M9).p2) by VECTSP_1:9
    .= -(Path_product M).p;
end;
