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 Th33:
  for l, pK, qK st l in Seg n & len pK = n & len qK = n for M be
  Matrix of n,K holds Det(RLine(M,l,a*pK+b*qK)) = a*Det(RLine(M,l,pK)) + b*Det(
  RLine(M,l,qK))
proof
  let l, pK, qK such that
A1: l in Seg n and
A2: len pK = n and
A3: len qK = n;
  set P=Permutations(n);
  set KK=the carrier of K;
  set aa=the addF of K;
  let M be Matrix of n,K;
  set Rpq=RLine(M,l,a*pK+b*qK);
  set Rp=RLine(M,l,pK);
  set Rq=RLine(M,l,qK);
  set Pathpq=Path_product(Rpq);
  set Pathp=Path_product(Rp);
  set Pathq=Path_product(Rq);
  set F=In (P,Fin P);
  P in Fin P by FINSUB_1:def 5; then
A4: F=P by SUBSET_1:def 8;
  then consider Gpq be Function of Fin P,KK such that
A5: Det Rpq = Gpq.F and
A6: for e being Element of KK st e is_a_unity_wrt aa holds Gpq.{} = e and
A7: for x being Element of P holds Gpq.{x} = Pathpq.x and
A8: for B9 being Element of Fin P st B9 c= F & B9 <> {} for x being
  Element of P st x in F \ B9 holds Gpq.(B9 \/ {x}) = aa.(Gpq.B9,Pathpq.x) by
SETWISEO:def 3;
  consider Gq be Function of Fin P,KK such that
A9: Det Rq = Gq.F and
A10: for e being Element of KK st e is_a_unity_wrt aa holds Gq.{} = e and
A11: for x being Element of P holds Gq.{x} = Pathq.x and
A12: for B9 being Element of Fin P st B9 c= F & B9 <> {} for x being
  Element of P st x in F \ B9 holds Gq.(B9 \/ {x}) = aa.(Gq.B9,Pathq.x) by A4,
SETWISEO:def 3;
  consider Gp be Function of Fin P,KK such that
A13: Det Rp = Gp.F and
A14: for e being Element of KK st e is_a_unity_wrt aa holds Gp.{} = e and
A15: for x being Element of P holds Gp.{x} = Pathp.x and
A16: for B9 being Element of Fin P st B9 c= F & B9 <> {} for x being
  Element of P st x in F \ B9 holds Gp.(B9 \/ {x}) = aa.(Gp.B9,Pathp.x) by A4,
SETWISEO:def 3;
  defpred P[Nat] means for B be Element of Fin P st card B=$1 holds Gpq.B=a*Gp
  .B+b*Gq.B;
A17: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A18: P[k];
    let B be Element of Fin P such that
A19: card B=k+1;
    now
      per cases;
      case
        k=0;
        then consider x being object such that
A20:    B={x} by A19,CARD_2:42;
A21:    x in B by A20,TARSKI:def 1;
        B c= P by FINSUB_1:def 5;
        then reconsider x as Element of P by A21;
A22:    Gp.B=Pathp.x by A15,A20;
A23:    Gq.B=Pathq.x by A11,A20;
        Gpq.B=Pathpq.x by A7,A20;
        hence thesis by A1,A2,A3,A22,A23,Th32;
      end;
      case
A24:    k>0;
        consider x being object such that
A25:    x in B by A19,CARD_1:27,XBOOLE_0:def 1;
        B c= P by FINSUB_1:def 5;
        then reconsider x as Element of P by A25;
        B c= P by FINSUB_1:def 5;
        then B\{x} c= P;
        then reconsider B9=B\{x} as Element of Fin P by FINSUB_1:def 5;
A26:    not x in B9 by ZFMISC_1:56;
        then
A27:    x in F\B9 by A4,XBOOLE_0:def 5;
A28:    {x} \/ B9=B by A25,ZFMISC_1:116;
        then
A29:    k+1=card B9+1 by A19,A26,CARD_2:41;
        then
A30:    Gpq.B9=a*Gp.B9+b*Gq.B9 by A18;
A31:    B9 c= F by A4,FINSUB_1:def 5;
        then Gpq.B =aa.(Gpq.B9,Pathpq.x) by A8,A24,A28,A29,A27,CARD_1:27;
        then
A32:    Gpq.B =(a*Gp.B9+b*Gq.B9)+(a*Pathp.x+b*Pathq.x) by A1,A2,A3,A30,Th32
          .=a*Gp.B9+(b*Gq.B9+(a*Pathp.x+b*Pathq.x)) by RLVECT_1:def 3
          .=a*Gp.B9+(a*Pathp.x+(b*Gq.B9+b*Pathq.x)) by RLVECT_1:def 3
          .=(a*Gp.B9+a*Pathp.x)+(b*Gq.B9+b*Pathq.x) by RLVECT_1:def 3
          .=(a*(Gp.B9+Pathp.x))+(b*Gq.B9+b*Pathq.x) by VECTSP_1:def 7
          .=(a*(aa.(Gp.B9,Pathp.x)))+(b*(Gq.B9+Pathq.x)) by VECTSP_1:def 7
          .=(a*(aa.(Gp.B9,Pathp.x)))+(b*(aa.(Gq.B9,Pathq.x)));
        Gp.B =aa.(Gp.B9,Pathp.x) by A16,A24,A28,A29,A27,A31,CARD_1:27;
        hence thesis by A12,A24,A28,A29,A27,A31,A32,CARD_1:27;
      end;
    end;
    hence thesis;
  end;
A33: P[0]
  proof
    let B be Element of Fin P;
    assume card B=0;
    then
A34: B={};
    then
A35: Gp.B=0.K by A14,FVSUM_1:6;
A36: Gq.B=0.K by A10,A34,FVSUM_1:6;
    Gpq.B=0.K by A6,A34,FVSUM_1:6;
    hence thesis by A35,A36,RLVECT_1:4;
  end;
  for k be Nat holds P[k] from NAT_1:sch 2(A33,A17);
  then P[card F];
  hence thesis by A5,A13,A9;
end;
