reserve

  k,n,m,i,j for Element of NAT,
  K for Field;
reserve L for non empty addLoopStr;
reserve G for non empty multLoopStr;

theorem Th37:
  for a being Element of K,P,Q being Matrix of n,K st n>0 & a<>0.K
& P*(1,1)= a" & (for i st 1<i & i<=n holds P.i=Base_FinSeq(K,n,i)) & Q*(1,1)=a
  & (for j st 1<j & j<=n holds Q*(1,j)= -a*(P*(1,j))) & (for i st 1<i & i<=n
  holds Q.i=Base_FinSeq(K,n,i)) holds P is invertible & Q=P~
proof
  let a be Element of K,P,Q be Matrix of n,K;
  assume that
A1: n>0 and
A2: a<>0.K and
A3: P*(1,1)= a" and
A4: for i st 1<i & i<=n holds P.i=Base_FinSeq(K,n,i) and
A5: Q*(1,1)=a and
A6: for j st 1<j & j<=n holds Q*(1,j)= -a*(P*(1,j)) and
A7: for i st 1<i & i<=n holds Q.i=Base_FinSeq(K,n,i);
A8: 0+1<=n by A1,NAT_1:13;
A9: len (Base_FinSeq(K,n,1))=n by Th23;
A10: len P=n by MATRIX_0:24;
A11: len ((a")*(Base_FinSeq(K,n,1)))=len (Base_FinSeq(K,n,1)) by MATRIXR1:16
    .= n by Th23;
A12: for k be Nat st 1<=k & k<=n holds (Col(P,1)).k=((a")*(Base_FinSeq(K,n,
  1))).k
  proof
    let k be Nat;
A13: k in NAT by ORDINAL1:def 12;
    assume that
A14: 1<=k and
A15: k<=n;
A16: k in Seg n by A14,A15,FINSEQ_1:1;
    then
A17: k in dom P by A10,FINSEQ_1:def 3;
A18: now
      [k,1] in Indices P by A8,A14,A15,MATRIX_0:31;
      then
A19:  ex p being FinSequence of K st p = P.k & P*(k,1) = p.1 by MATRIX_0:def 5;
      assume
A20:  k<>1;
      then 1<k by A14,XXREAL_0:1;
      then P*(k,1)=(Base_FinSeq(K,n,k)).1 by A4,A13,A15,A19
        .= 0.K by A8,A20,Th25;
      hence (Col(P,1)).k= 0.K by A17,MATRIX_0:def 8;
    end;
A21: ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A9,A14,A15,
FINSEQ_4:15;
A22: now
      assume
A23:  k=1;
      k in dom ((a")*(Base_FinSeq(K,n,1))) by A11,A16,FINSEQ_1:def 3;
      hence ((a")*(Base_FinSeq(K,n,1))).k =(a")*((Base_FinSeq(K,n,1))/.k) by
A21,FVSUM_1:50
        .=(a")*(1.K) by A15,A21,A23,Th24
        .= a";
    end;
A24: k in dom ((a")*(Base_FinSeq(K,n,1))) by A11,A14,A15,FINSEQ_3:25;
A25: now
      assume
A26:  k<>1;
      thus ((a")*(Base_FinSeq(K,n,1))).k = (a")*((Base_FinSeq(K,n,1))/.k) by
A24,A21,FVSUM_1:50
        .= (a")*(0.K) by A14,A15,A21,A26,Th25
        .= 0.K;
    end;
    1<=n by A14,A15,XXREAL_0:2;
    then 1 in dom P by A10,FINSEQ_3:25;
    hence thesis by A3,A25,A22,A18,MATRIX_0:def 8;
  end;
A27: 0+1<=n by A1,NAT_1:13;
A28: len Q=n by MATRIX_0:24;
  then
A29: 1 in Seg len Q by A27,FINSEQ_1:1;
  then
A30: 1 in dom Q by FINSEQ_1:def 3;
A31: width Q=n by MATRIX_0:24;
  then
A32: len (Line(Q,1))=n by MATRIX_0:def 7;
  then
A33: 1 in dom (Line(Q,1)) by A28,A29,FINSEQ_1:def 3;
A34: for j st 1<j & j<=n holds P*(1,j)= -(a")*(Q*(1,j))
  proof
    let j;
    assume 1<j & j<=n;
    then -(a")*(Q*(1,j))= -(a")*( -a*(P*(1,j))) by A6;
    then -(a")*(Q*(1,j))= -(a")*( (-a)*(P*(1,j))) by VECTSP_1:9;
    then -(a")*(Q*(1,j))= (-(a"))*( (-a)*(P*(1,j))) by VECTSP_1:9;
    then -(a")*(Q*(1,j))= ((-(a"))*(-a))*(P*(1,j)) by GROUP_1:def 3;
    then -(a")*(Q*(1,j))= ((a)*(a"))*(P*(1,j)) by VECTSP_1:10;
    then -(a")*(Q*(1,j))= (1.K)*(P*(1,j)) by A2,VECTSP_1:def 10;
    hence thesis;
  end;
A35: for j st 1<j & j<=n holds (Q*P)*(1,j)= 0.K
  proof
    let j;
    assume that
A36: 1<j and
A37: j<=n;
A38: len (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))) =len ((-(a")*(Q*(1,j)))*(
    Base_FinSeq(K,n,1))) by Th6
      .=len (Base_FinSeq(K,n,1)) by MATRIXR1:16
      .=n by Th23;
A39: j in Seg n by A36,A37,FINSEQ_1:1;
A40: 1<=n by A36,A37,XXREAL_0:2;
    then
A41: 1 in Seg width Q by A31,FINSEQ_1:1;
    len Line(Q,1) = n by A31,MATRIX_0:def 7;
    then
A42: (Line(Q,1))/.j = (Line(Q,1)).j by A36,A37,FINSEQ_4:15;
A43: (Line(Q,1))/.1 = (Line(Q,1)).1 by A32,A40,FINSEQ_4:15
      .= a by A5,A41,MATRIX_0:def 7;
A44: len (Col(P,j))=len P by MATRIX_0:def 8
      .=n by MATRIX_0:24;
    reconsider p=Col(P,j),q=-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1)) +Base_FinSeq(
    K,n,j) as FinSequence of K;
A45: len (Base_FinSeq(K,n,j))=n by Th23;
A46: len ((-(a")*(Q*(1,j)))*(Base_FinSeq(K,n,1))) =len Base_FinSeq(K,n,1)
    by MATRIXR1:16
      .=n by Th23;
A47: for k be Nat st 1 <=k & k <= n holds p.k=q.k
    proof
A48:  len ((-(a")*(Q*(1,j)))*(Base_FinSeq(K,n,1))) = len (Base_FinSeq(K,
      n,1)) by MATRIXR1:16
        .=n by Th23;
      let k be Nat;
      assume that
A49:  1 <=k and
A50:  k <= n;
A51:  k in dom ((-(a")*(Q*(1,j)))*(Base_FinSeq(K,n,1))) by A46,A49,A50,
FINSEQ_3:25;
      len (Base_FinSeq(K,n,1))=n by Th23;
      then
A52:  ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A49,A50,FINSEQ_4:15
;
      k in dom P by A10,A49,A50,FINSEQ_3:25;
      then
A53:  p.k=P*(k,j) by MATRIX_0:def 8;
A54:  (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))).k =((-(a")*(Q*(1,j)))*(
      Base_FinSeq(K,n,1))).k by Th6
        .=(-(a")*(Q*(1,j)))*((Base_FinSeq(K,n,1))/.k) by A51,A52,FVSUM_1:50;
      per cases by A49,XXREAL_0:1;
      suppose
A55:    1=k;
        then 1 <= len (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))) by A50,A48,Th6;
        then
A56:    (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1)))/.1 =(-(a")*(Q*(1,j)))*((
        Base_FinSeq(K,n,1))/.1) by A54,A55,FINSEQ_4:15
          .=(-(a")*(Q*(1,j)))*(1.K) by A27,A52,A55,Th24
          .=-(a")*(Q*(1,j));
A57:    p.1=-(a")*(Q*(1,j)) by A34,A36,A37,A53,A55;
        (Base_FinSeq(K,n,j))/.1 = (Base_FinSeq(K,n,j)).1 by A45,A50,A55,
FINSEQ_4:15
          .= 0.K by A36,A50,A55,Th25;
        then q.1= -(a")*(Q*(1,j)) + 0.K by A38,A45,A50,A55,A56,Th5;
        hence thesis by A55,A57,RLVECT_1:4;
      end;
      suppose
A58:    1<k;
        [k,j] in Indices P by A36,A37,A49,A50,MATRIX_0:31;
        then
A59:    ex p2 being FinSequence of K st p2 = P.k & P*(k,j) = p2. j by
MATRIX_0:def 5;
        k in NAT by ORDINAL1:def 12;
        then
A60:    p.k= (Base_FinSeq(K,n,k)).j by A4,A50,A53,A58,A59;
        now
          per cases;
          suppose
A61:        k <> j;
            len (Base_FinSeq(K,n,1))=n by Th23;
            then ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A49,A50,
FINSEQ_4:15;
            then
A62:        (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))).k =(-(a")*(Q*(1,j)))*
            (0.K) by A50,A54,A58,Th25
              .= 0.K;
A63:        (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1)))/.k =(-(a")*(Q*(1,j))*
            (Base_FinSeq(K,n,1))).k by A38,A49,A50,FINSEQ_4:15;
A64:        p.k= 0.K by A36,A37,A60,A61,Th25;
            (Base_FinSeq(K,n,j))/.k = (Base_FinSeq(K,n,j)).k by A45,A49,A50,
FINSEQ_4:15
              .= 0.K by A49,A50,A61,Th25;
            then q.k= 0.K + 0.K by A38,A45,A49,A50,A63,A62,Th5;
            hence thesis by A64,RLVECT_1:4;
          end;
          suppose
A65:        k=j;
            then
A66:        p.k=1.K by A49,A50,A60,Th24;
A67:        (-(a")*(Q*(1,k))*(Base_FinSeq(K,n,1)))/.k =(-(a")*(Q*(1,k))*
            (Base_FinSeq(K,n,1))).k by A38,A49,A50,A65,FINSEQ_4:15;
            len (Base_FinSeq(K,n,1))=n by Th23;
            then ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A49,A50,
FINSEQ_4:15;
            then
A68:        (-(a")*(Q*(1,k))*(Base_FinSeq(K,n,1))).k =(-(a")*(Q*(1,k)))*
            (0.K) by A50,A54,A58,A65,Th25
              .= 0.K;
            (Base_FinSeq(K,n,k))/.k = (Base_FinSeq(K,n,k)).k by A45,A49,A50,A65
,FINSEQ_4:15
              .= 1.K by A36,A37,A65,Th24;
            then q.k= 0.K + 1.K by A38,A45,A49,A50,A65,A67,A68,Th5;
            hence thesis by A66,RLVECT_1:4;
          end;
        end;
        hence thesis;
      end;
    end;
    len (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))+Base_FinSeq(K,n,j)) =n by A38,A45
,Th2;
    then
A69: Col(P,j)=-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))+Base_FinSeq(K,n,j) by A44
,A47,FINSEQ_1:14;
    [1,j] in Indices (Q*P) by A27,A36,A37,MATRIX_0:31;
    then (Q*P)*(1,j)= |( Line(Q,1), Col(P,j) )| by A10,A31,MATRIX_3:def 4
      .= |( Line(Q,1), -(a")*(Q*(1,j))*(Base_FinSeq(K,n,1)) )| +|( Line(Q,1)
    , Base_FinSeq(K,n,j) )| by A32,A38,A45,A69,Th11
      .= |( Line(Q,1), (-(a")*(Q*(1,j)))*(Base_FinSeq(K,n,1)) )| +|( Line(Q,
    1), Base_FinSeq(K,n,j) )| by Th6
      .=(-(a")*(Q*(1,j)))* |( Line(Q,1), (Base_FinSeq(K,n,1)) )| +|( Line(Q,
    1), Base_FinSeq(K,n,j) )| by A32,A9,Th10
      .=(-(a")*(Q*(1,j)))* a +|( Line(Q,1), Base_FinSeq(K,n,j) )| by A32,A27
,A43,Th35
      .= -((a")*(Q*(1,j)))* a +|( Line(Q,1), Base_FinSeq(K,n,j) )| by
VECTSP_1:9
      .= -((Q*(1,j))*((a")* a)) +|( Line(Q,1), Base_FinSeq(K,n,j) )| by
GROUP_1:def 3
      .= -((Q*(1,j))*(1.K)) +|( Line(Q,1), Base_FinSeq(K,n,j) )| by A2,
VECTSP_1:def 10
      .= -((Q*(1,j))*(1.K)) + ( Line(Q,1))/.j by A32,A36,A37,Th35
      .= -(Q*(1,j)) + ( Line(Q,1))/.j
      .= -(Q*(1,j)) + Q*(1,j) by A31,A39,A42,MATRIX_0:def 7
      .= 0.K by RLVECT_1:5;
    hence thesis;
  end;
A70: 1 in Seg width Q by A31,A27,FINSEQ_1:1;
A71: for i,j st 1<i & i<=n & i=j holds (Q*P)*(i,j)=1.K
  proof
    let i,j;
    assume that
A72: 1<i and
A73: i<=n and
A74: i=j;
A75: len (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))) =len ((-(a")*(Q*(1,j)))*(
    Base_FinSeq(K,n,1))) by Th6
      .=len (Base_FinSeq(K,n,1)) by MATRIXR1:16
      .=n by Th23;
    [i,j] in Indices Q by A72,A73,A74,MATRIX_0:31;
    then consider p1 being FinSequence of K such that
A76: p1 = Q.i and
A77: Q*(i,j) = p1.j by MATRIX_0:def 5;
    p1=Base_FinSeq(K,n,i) by A7,A72,A73,A76;
    then
A78: p1.j=1.K by A72,A73,A74,Th24;
    [i,1] in Indices Q by A8,A72,A73,MATRIX_0:31;
    then consider p2 being FinSequence of K such that
A79: p2 = Q.i and
A80: Q*(i,1) = p2.1 by MATRIX_0:def 5;
A81: width Q=n by MATRIX_0:24;
A82: j in Seg n by A72,A73,A74,FINSEQ_1:1;
A83: len (Col(P,j))=len P by MATRIX_0:def 8
      .=n by MATRIX_0:24;
    reconsider p=Col(P,j),q=-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1)) +Base_FinSeq(
    K,n,j) as FinSequence of K;
A84: len (Base_FinSeq(K,n,j))=n by Th23;
A85: len ((-(a")*(Q*(1,j)))*(Base_FinSeq(K,n,1))) =len (Base_FinSeq(K,n,1
    )) by MATRIXR1:16
      .=n by Th23;
A86: for k be Nat st 1 <=k & k <= n holds p.k=q.k
    proof
A87:  len ((-(a")*(Q*(1,j)))*(Base_FinSeq(K,n,1))) = len (Base_FinSeq(K,
      n,1)) by MATRIXR1:16
        .=n by Th23;
      let k be Nat;
      assume that
A88:  1 <=k and
A89:  k <= n;
      k in Seg n by A88,A89,FINSEQ_1:1;
      then
A90:  k in dom ((-(a")*(Q*(1,j)))*(Base_FinSeq(K,n,1))) by A85,FINSEQ_1:def 3;
      len (Base_FinSeq(K,n,1))=n by Th23;
      then
A91:  ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A88,A89,FINSEQ_4:15
;
      k in dom P by A10,A88,A89,FINSEQ_3:25;
      then
A92:  p.k=P*(k,j) by MATRIX_0:def 8;
A93:  (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))).k =((-(a")*(Q*(1,j)))*(
      Base_FinSeq(K,n,1))).k by Th6
        .=(-(a")*(Q*(1,j)))*((Base_FinSeq(K,n,1))/.k) by A90,A91,FVSUM_1:50;
      per cases by A88,XXREAL_0:1;
      suppose
A94:    1=k;
        k <= len (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))) by A89,A87,Th6;
        then
A95:    (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1)))/.k =(-(a")*(Q*(1,j)))*((
        Base_FinSeq(K,n,1))/.k) by A88,A93,FINSEQ_4:15
          .=(-(a")*(Q*(1,j)))*(1.K) by A27,A91,A94,Th24
          .=-(a")*(Q*(1,j));
        (Base_FinSeq(K,n,j))/.k = (Base_FinSeq(K,n,j)).k by A84,A88,A89,
FINSEQ_4:15
          .= 0.K by A72,A74,A89,A94,Th25;
        then
A96:    q.k= -(a")*(Q*(1,j)) + 0.K by A75,A84,A88,A89,A95,Th5;
        p.k=-(a")*(Q*(1,j)) by A34,A72,A73,A74,A92,A94;
        hence thesis by A96,RLVECT_1:4;
      end;
      suppose
A97:    1<k;
        [k,j] in Indices P by A72,A73,A74,A88,A89,MATRIX_0:31;
        then
A98:    ex p2 being FinSequence of K st p2 = P.k & P*(k,j) = p2. j by
MATRIX_0:def 5;
        k in NAT by ORDINAL1:def 12;
        then
A99:    p.k=(Base_FinSeq(K,n,k)).j by A4,A89,A92,A97,A98;
        now
          per cases;
          suppose
A100:       k <> j;
            len (Base_FinSeq(K,n,1))=n by Th23;
            then ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A88,A89,
FINSEQ_4:15;
            then
A101:       (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))).k =(-(a")*(Q*(1,j)))*
            (0.K) by A89,A93,A97,Th25
              .= 0.K;
A102:       (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1)))/.k =(-(a")*(Q*(1,j))*
            (Base_FinSeq(K,n,1))).k by A75,A88,A89,FINSEQ_4:15;
A103:       p.k= 0.K by A72,A73,A74,A99,A100,Th25;
            (Base_FinSeq(K,n,j))/.k = (Base_FinSeq(K,n,j)).k by A84,A88,A89,
FINSEQ_4:15
              .= 0.K by A88,A89,A100,Th25;
            then q.k= 0.K + 0.K by A75,A84,A88,A89,A102,A101,Th5;
            hence thesis by A103,RLVECT_1:4;
          end;
          suppose
A104:       k=j;
            len (Base_FinSeq(K,n,1))=n by Th23;
            then ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A88,A89,
FINSEQ_4:15;
            then
A105:       (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))).k =(-(a")*(Q*(1,j)))*
            (0.K) by A89,A93,A97,Th25
              .= 0.K;
A106:       (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1)))/.k =(-(a")*(Q*(1,j))*
            (Base_FinSeq(K,n,1))).k by A75,A88,A89,FINSEQ_4:15;
A107:       p.k=1.K by A88,A89,A99,A104,Th24;
            (Base_FinSeq(K,n,j))/.k = (Base_FinSeq(K,n,j)).k by A84,A88,A89,
FINSEQ_4:15
              .= 1.K by A88,A89,A104,Th24;
            then q.k= 0.K + 1.K by A75,A84,A88,A89,A106,A105,Th5;
            hence thesis by A107,RLVECT_1:4;
          end;
        end;
        hence thesis;
      end;
    end;
    len (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))+Base_FinSeq(K,n,j)) =n by A75,A84
,Th2;
    then
A108: Col(P,j)=-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))+Base_FinSeq(K,n,j) by A83
,A86,FINSEQ_1:14;
A109: len Line(Q,i)=n by A31,MATRIX_0:def 7;
    then
A110: (Line(Q,i))/.1 = (Line(Q,i)).1 by A27,FINSEQ_4:15
      .= Q*(i,1) by A70,MATRIX_0:def 7;
A111: (Line(Q,i))/.j = (Line(Q,i)).j by A72,A73,A74,A109,FINSEQ_4:15
      .= Q*(i,j) by A82,A81,MATRIX_0:def 7;
    [i,j] in Indices (Q*P) by A72,A73,A74,MATRIX_0:31;
    then
A112: (Q*P)*(i,j)= |( Line(Q,i), Col(P,j) )| by A10,A81,MATRIX_3:def 4
      .= |( Line(Q,i), -(a")*(Q*(1,j))*(Base_FinSeq(K,n,1)) )| +|( Line(Q,i)
    , Base_FinSeq(K,n,j) )| by A75,A84,A108,A109,Th11
      .= |( Line(Q,i), (-(a")*(Q*(1,j)))*(Base_FinSeq(K,n,1)) )| +|( Line(Q,
    i), Base_FinSeq(K,n,j) )| by Th6
      .=(-(a")*(Q*(1,j)))* |( Line(Q,i), (Base_FinSeq(K,n,1)) )| +|( Line(Q,
    i), Base_FinSeq(K,n,j) )| by A9,A109,Th10
      .=(-(a")*(Q*(1,j)))* (Q*(i,1)) +|( Line(Q,i), Base_FinSeq(K,n,j) )| by
A27,A109,A110,Th35
      .= -((Q*(1,j))*(a")* (Q*(i,1))) +|( Line(Q,i), Base_FinSeq(K,n,j) )|
    by VECTSP_1:9
      .= -((Q*(1,j))*((a")* (Q*(i,1)))) +|( Line(Q,i), Base_FinSeq(K,n,j) )|
    by GROUP_1:def 3
      .= -(Q*(1,j))*((a")* (Q*(i,1))) + Q*(i,j) by A72,A73,A74,A109,A111,Th35;
A113: 1<=n by A72,A73,XXREAL_0:2;
    p2=Base_FinSeq(K,n,i) by A7,A72,A73,A79;
    hence (Q*P)*(i,j)= -(Q*(1,j))*((a")* ( 0.K)) + Q*(i,j) by A72,A113,A112,A80
,Th25
      .= -(Q*(1,j))*(( 0.K)) + Q*(i,j)
      .= -(0.K) + Q*(i,j)
      .= (0.K) + 1.K by A77,A78,RLVECT_1:12
      .= 1.K by RLVECT_1:4;
  end;
A114: Indices P=[: Seg n,Seg n :] by MATRIX_0:24;
A115: for i,j st 1<i & i<=n & 1<=j & j<=n & i<>j holds (Q*P)*(i,j)= 0.K
  proof
A116: len ((a")*(Base_FinSeq(K,n,1))) =len Base_FinSeq(K,n,1) by MATRIXR1:16
      .=n by Th23;
    let i,j;
    assume that
A117: 1<i and
A118: i<=n and
A119: 1<=j and
A120: j<=n and
A121: i<>j;
A122: [i,j] in Indices (Q*P) by A117,A118,A119,A120,MATRIX_0:31;
A123: j in Seg n by A119,A120,FINSEQ_1:1;
A124: len ((-(a")*(Q*(1,j)))*(Base_FinSeq(K,n,1))) =len (Base_FinSeq(K,n,1
    )) by MATRIXR1:16
      .=n by Th23;
A125: len (Col(P,j))=len P by MATRIX_0:def 8
      .=n by MATRIX_0:24;
A126: [i,1] in Indices Q by A27,A117,A118,MATRIX_0:31;
A127: len (Base_FinSeq(K,n,j))=n by Th23;
A128: len (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))) =len ((-(a")*(Q*(1,j)))*(
    Base_FinSeq(K,n,1))) by Th6
      .=len (Base_FinSeq(K,n,1)) by MATRIXR1:16
      .=n by Th23;
    then
A129: len (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))+Base_FinSeq(K,n,j)) =n by A127
,Th2;
A130: [i,j] in Indices Q by A117,A118,A119,A120,MATRIX_0:31;
    now
      per cases;
      suppose
A131:   j>1;
        reconsider p=Col(P,j),q=-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1)) +
        Base_FinSeq(K,n,j) as FinSequence of K;
        for k be Nat st 1 <=k & k <= n holds p.k=q.k
        proof
          let k be Nat;
          assume that
A132:     1 <=k and
A133:     k <= n;
          k in Seg n by A132,A133,FINSEQ_1:1;
          then
A134:     k
 in dom ((-(a")*(Q*(1,j)))*(Base_FinSeq(K,n,1))) by A124,FINSEQ_1:def 3;
          len (Base_FinSeq(K,n,1))=n by Th23;
          then
A135:     ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A132,A133,
FINSEQ_4:15;
          k in dom P by A10,A132,A133,FINSEQ_3:25;
          then
A136:     p.k=P*(k,j) by MATRIX_0:def 8;
A137:     (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))).k =((-(a")*(Q*(1,j)))*(
          Base_FinSeq(K,n,1))).k by Th6
            .=(-(a")*(Q*(1,j)))*((Base_FinSeq(K,n,1))/.k) by A134,A135,
FVSUM_1:50;
          len (Base_FinSeq(K,n,1))=n by Th23;
          then
A138:     ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A132,A133,
FINSEQ_4:15;
          per cases by A132,XXREAL_0:1;
          suppose
A139:       1=k;
            then
A140:       (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))).k =(-(a")*(Q*(1,j)))*(
            1.K) by A27,A135,A137,Th24;
A141:       (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1)))/.k =(-(a")*(Q*(1,j))*
            (Base_FinSeq(K,n,1))).k by A128,A132,A133,FINSEQ_4:15;
            (Base_FinSeq(K,n,j))/.k = (Base_FinSeq(K,n,j)).k by A127,A132,A133,
FINSEQ_4:15
              .= 0.K by A131,A133,A139,Th25;
            then q.k= (-(a")*(Q*(1,j)))*(1.K)+ 0.K by A128,A127,A132,A133,A141
,A140,Th5
              .= (-(a")*(Q*(1,j)))*(1.K) by RLVECT_1:4
              .=-(a")*(Q*(1,j));
            hence thesis by A34,A120,A131,A136,A139;
          end;
          suppose
A142:       1<k;
            [k,j] in Indices P by A119,A120,A132,A133,MATRIX_0:31;
            then
A143:       ex p2 being FinSequence of K st p2 = P.k & P*(k,j) = p2. j by
MATRIX_0:def 5;
            k in NAT by ORDINAL1:def 12;
            then
A144:       p.k=(Base_FinSeq(K,n,k)).j by A4,A133,A136,A142,A143;
            now
              per cases;
              suppose
A145:           k <> j;
                (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1)))/.k =(-(a")*(Q*(1,
j)))*((Base_FinSeq(K,n,1))/.k) by A128,A132,A133,A137,FINSEQ_4:15
                  .=(-(a")*(Q*(1,j)))*(0.K) by A133,A138,A142,Th25
                  .= 0.K;
                then q.k = 0.K +(Base_FinSeq(K,n,j))/.k by A128,A127,A132,A133
,Th5
                  .= (Base_FinSeq(K,n,j))/.k by RLVECT_1:4
                  .= (Base_FinSeq(K,n,j)).k by A127,A132,A133,FINSEQ_4:15
                  .= 0.K by A132,A133,A145,Th25;
                hence thesis by A119,A120,A144,A145,Th25;
              end;
              suppose
A146:           k=j;
                (-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1)))/.k =(-(a")*(Q*(1,
j)))*((Base_FinSeq(K,n,1))/.k) by A128,A132,A133,A137,FINSEQ_4:15
                  .=(-(a")*(Q*(1,j)))*(0.K) by A133,A138,A142,Th25
                  .= 0.K;
                then q.k = 0.K +(Base_FinSeq(K,n,j))/.k by A128,A127,A132,A133
,Th5
                  .= (Base_FinSeq(K,n,j))/.k by RLVECT_1:4
                  .= (Base_FinSeq(K,n,j)).k by A127,A132,A133,FINSEQ_4:15
                  .= 1.K by A119,A120,A146,Th24;
                hence thesis by A132,A133,A144,A146,Th24;
              end;
            end;
            hence thesis;
          end;
        end;
        then
A147:   Col(P,j)=-(a")*(Q*(1,j))*(Base_FinSeq(K,n,1))+Base_FinSeq(K,n,j)
        by A125,A129,FINSEQ_1:14;
A148:   1<=n by A117,A118,XXREAL_0:2;
A149:   width Q=n by MATRIX_0:24;
        then
A150:   len Line(Q,i) = n by MATRIX_0:def 7;
        then
A151:   (Line(Q,i))/.j = ( Line(Q,i)).j by A119,A120,FINSEQ_4:15
          .= Q*(i,j) by A123,A149,MATRIX_0:def 7;
A152:   (Line(Q,i))/.1 = (Line(Q,i)).1 by A27,A150,FINSEQ_4:15
          .= Q*(i,1) by A70,MATRIX_0:def 7;
A153:   (Q*P)*(i,j)= |( Line(Q,i), Col(P,j) )| by A10,A122,A149,MATRIX_3:def 4
          .= |( Line(Q,i), -(a")*(Q*(1,j))*(Base_FinSeq(K,n,1)) )| +|( Line(
        Q,i), Base_FinSeq(K,n,j) )| by A128,A127,A147,A150,Th11
          .= |( Line(Q,i), (-(a")*(Q*(1,j)))*(Base_FinSeq(K,n,1)) )| +|(
        Line(Q,i), Base_FinSeq(K,n,j) )| by Th6
          .=(-(a")*(Q*(1,j)))* |( Line(Q,i), (Base_FinSeq(K,n,1)) )| +|(
        Line(Q,i), Base_FinSeq(K,n,j) )| by A9,A150,Th10
          .=(-(a")*(Q*(1,j)))* (Q*(i,1)) +|( Line(Q,i), Base_FinSeq(K,n,j)
        )| by A27,A150,A152,Th35
          .= -((Q*(1,j))*(a")* (Q*(i,1))) +|( Line(Q,i), Base_FinSeq(K,n,j)
        )| by VECTSP_1:9
          .= -((Q*(1,j))*((a")* (Q*(i,1)))) +|( Line(Q,i), Base_FinSeq(K,n,j
        ) )| by GROUP_1:def 3
          .= -(Q*(1,j))*((a")* (Q*(i,1))) + Q*(i,j) by A119,A120,A150,A151,Th35
;
        consider p2 being FinSequence of K such that
A154:   p2 = Q.i and
A155:   Q*(i,1) = p2.1 by A126,MATRIX_0:def 5;
        consider p1 being FinSequence of K such that
A156:   p1 = Q.i and
A157:   Q*(i,j) = p1.j by A130,MATRIX_0:def 5;
        p1=Base_FinSeq(K,n,i) by A7,A117,A118,A156;
        then
A158:   p1.j= 0.K by A119,A120,A121,Th25;
        p2=Base_FinSeq(K,n,i) by A7,A117,A118,A154;
        hence (Q*P)*(i,j) = -(Q*(1,j))*((a")* (0.K)) + Q*(i,j) by A117,A148
,A153,A155,Th25
          .= -(Q*(1,j))*((0.K)) + Q*(i,j)
          .= -(0.K) + Q*(i,j)
          .= 0.K + 0.K by A157,A158,RLVECT_1:12
          .= 0.K by RLVECT_1:4;
      end;
      suppose
A159:   j<=1;
        reconsider p=Col(P,j),q=a"*(Base_FinSeq(K,n,1)) as FinSequence of K;
A160:   for k be Nat st 1 <=k & k <= n holds p.k=q.k
        proof
          let k be Nat;
          assume that
A161:     1 <=k and
A162:     k <= n;
A163:     len (Base_FinSeq(K,n,1))=n by Th23;
          then
A164:     ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A161,A162,
FINSEQ_4:15;
A165:     k in Seg n by A161,A162,FINSEQ_1:1;
          then k in dom ((a")*(Base_FinSeq(K,n,1))) by A116,FINSEQ_1:def 3;
          then
A166:     ((a")*(Base_FinSeq(K,n,1))).k = (a")*((Base_FinSeq(K,n,1))/.k)
          by A164,FVSUM_1:50;
          k in dom P by A10,A161,A162,FINSEQ_3:25;
          then
A167:     p.k=P*(k,j) by MATRIX_0:def 8;
          per cases by A161,XXREAL_0:1;
          suppose
A168:       1=k;
            then q.k= (a")*(1.K) by A162,A164,A166,Th24
              .=a";
            hence thesis by A3,A119,A159,A167,A168,XXREAL_0:1;
          end;
          suppose
A169:       1<k;
            [k,j] in Indices P by A114,A123,A165,ZFMISC_1:87;
            then
A170:       ex p2 being FinSequence of K st p2 = P.k & P*(k,j) = p2. j by
MATRIX_0:def 5;
            k in NAT by ORDINAL1:def 12;
            then
A171:       p.k=(Base_FinSeq(K,n,k)).j by A4,A162,A167,A169,A170;
            now
              per cases;
              suppose
A172:           k <> j;
                ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A161,A162
,A163,FINSEQ_4:15;
                then (a")*(Base_FinSeq(K,n,1))/.k = a"*(0.K) by A162,A169,Th25
                  .= 0.K;
                hence thesis by A119,A120,A166,A171,A172,Th25;
              end;
              suppose
                k=j;
                hence thesis by A159,A169;
              end;
            end;
            hence thesis;
          end;
        end;
A173:   1<=n by A117,A118,XXREAL_0:2;
A174:   width Q=n by MATRIX_0:24;
        then
A175:   len Line(Q,i) = n by MATRIX_0:def 7;
        then
A176:   (Line(Q,i))/.1 = (Line(Q,i)).1 by A27,FINSEQ_4:15
          .= Q*(i,1) by A70,MATRIX_0:def 7;
A177:   (Q*P)*(i,j)= |( Line(Q,i), Col(P,j) )| by A10,A122,A174,MATRIX_3:def 4
          .= |( Line(Q,i), a"*(Base_FinSeq(K,n,1)) )| by A125,A116,A160,
FINSEQ_1:14
          .= a"* |( Line(Q,i), (Base_FinSeq(K,n,1)) )| by A9,A175,Th10
          .= a"* (Q*(i,1)) by A27,A175,A176,Th35;
        consider p2 being FinSequence of K such that
A178:   p2 = Q.i and
A179:   Q*(i,1) = p2.1 by A126,MATRIX_0:def 5;
        p2=Base_FinSeq(K,n,i) by A7,A117,A118,A178;
        hence (Q*P)*(i,j)=a"* (0.K) by A117,A173,A177,A179,Th25
          .= 0.K;
      end;
    end;
    hence thesis;
  end;
  len (Col(P,1))=len P by MATRIX_0:def 8
    .=n by MATRIX_0:24;
  then
A180: Col(P,1)=(a")*(Base_FinSeq(K,n,1)) by A11,A12,FINSEQ_1:14;
A181: Indices (Q*P)=[: Seg n,Seg n :] by MATRIX_0:24;
A182: len ((a")*(Line(Q,1)))=len ((Line(Q,1))) by MATRIXR1:16
    .= n by A31,MATRIX_0:def 7;
  Indices Q=[: Seg n,Seg n :] & [1,1] in Indices Q by A27,MATRIX_0:24,31;
  then
A183: (Q*P)*(1,1)= |( Line(Q,1), Col(P,1) )| by A10,A31,A181,MATRIX_3:def 4
    .= (a")*( |( Line(Q,1), Base_FinSeq(K,n,1) )| ) by A32,A9,A180,Th10
    .=(a")*(Line(Q,1))/.1 by A32,A27,Th35
    .=((a")*(Line(Q,1)))/.1 by A33,POLYNOM1:def 1
    .=((a")*(Line(Q,1))).1 by A27,A182,FINSEQ_4:15
    .=(a")*(Q*(1,1)) by A70,A30,MATRIX12:3
    .=1.K by A2,A5,VECTSP_1:def 10;
  for i,j being Nat st [i,j] in Indices (Q*P) holds (Q*P)*(i,j)=(1.(K,n)
  )*(i,j)
  proof
    let i,j be Nat;
    reconsider i0=i,j0=j as Element of NAT by ORDINAL1:def 12;
    assume
A184: [i,j] in Indices (Q*P);
    then
A185: i in Seg n by A181,ZFMISC_1:87;
    then
A186: 1<=i by FINSEQ_1:1;
A187: i<=n by A185,FINSEQ_1:1;
A188: j in Seg n by A181,A184,ZFMISC_1:87;
    then
A189: 1<=j by FINSEQ_1:1;
A190: j<=n by A188,FINSEQ_1:1;
    per cases by A186,XXREAL_0:1;
    suppose
A191: 1<i;
      now
        per cases;
        suppose
A192:     i<>j;
A193:     (1.(K,n))*(i,j)=(Base_FinSeq(K,n,i0)).j0 by A186,A187,A189,A190,Th27
            .= 0.K by A189,A190,A192,Th25;
          thus (Q*P)*(i,j)=(Q*P)*(i0,j0)
            .=(1.(K,n))*(i,j) by A115,A187,A189,A190,A191,A192,A193;
        end;
        suppose
A194:     i=j;
A195:     (1.(K,n))*(i,j)=(Base_FinSeq(K,n,i0)).j0 by A186,A187,A189,A190,Th27
            .=1.K by A189,A190,A194,Th24;
          thus (Q*P)*(i,j)=(Q*P)*(i0,j0)
            .=(1.(K,n))*(i,j) by A71,A190,A191,A194,A195;
        end;
      end;
      hence thesis;
    end;
    suppose
A196: 1=i;
      now
        per cases;
        suppose
A197:     i<j;
A198:     (1.(K,n))*(i,j)=(Base_FinSeq(K,n,i0)).j0 by A186,A187,A189,A190,Th27
            .= 0.K by A189,A190,A197,Th25;
          thus (Q*P)*(i,j)=(Q*P)*(i0,j0)
            .=(1.(K,n))*(i,j) by A35,A190,A196,A197,A198;
        end;
        suppose
A199:     i>=j;
          then
A200:     i=j by A189,A196,XXREAL_0:1;
          (1.(K,n))*(i,j)=(Base_FinSeq(K,n,i0)).j0 by A186,A187,A189,A190,Th27
            .=1.K by A189,A190,A200,Th24;
          hence thesis by A183,A189,A196,A199,XXREAL_0:1;
        end;
      end;
      hence thesis;
    end;
  end;
  then
A201: Q*P=1.(K,n) by MATRIX_0:27;
A202: len Q=n by MATRIX_0:24;
A203: len Base_FinSeq(K,n,1)=n by Th23;
A204: len (a*(Base_FinSeq(K,n,1)))=len (Base_FinSeq(K,n,1)) by MATRIXR1:16
    .= n by Th23;
A205: for k be Nat st 1<=k & k<=n holds (Col(Q,1)).k=(a*(Base_FinSeq(K,n,1)))
  .k
  proof
    let k be Nat;
    assume that
A206: 1<=k and
A207: k<=n;
A208: ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A203,A206,A207,
FINSEQ_4:15;
A209: k in dom (a*(Base_FinSeq(K,n,1))) by A204,A206,A207,FINSEQ_3:25;
A210: now
      assume
A211: k<>1;
      thus (a*(Base_FinSeq(K,n,1))).k= a*((Base_FinSeq(K,n,1))/.k) by A208,A209
,FVSUM_1:50
        .= a*(0.K) by A206,A207,A208,A211,Th25
        .= 0.K;
    end;
A212: k in NAT by ORDINAL1:def 12;
    k in Seg n by A206,A207,FINSEQ_1:1;
    then
A213: k in dom Q by A202,FINSEQ_1:def 3;
A214: now
      [k,1] in Indices Q by A8,A206,A207,MATRIX_0:31;
      then
A215: ex p being FinSequence of K st p = Q.k & Q*(k,1) = p.1 by MATRIX_0:def 5;
      assume
A216: k<>1;
      then 1<k by A206,XXREAL_0:1;
      then Q*(k,1)=(Base_FinSeq(K,n,k)).1 by A7,A207,A212,A215
        .= 0.K by A8,A216,Th25;
      hence (Col(Q,1)).k= 0.K by A213,MATRIX_0:def 8;
    end;
A217: now
      assume
A218: k=1;
      thus (a*(Base_FinSeq(K,n,1))).k=a*((Base_FinSeq(K,n,1))/.k) by A208,A209,
FVSUM_1:50
        .= a*(1.K) by A207,A208,A218,Th24
        .= a;
    end;
    1<=n by A206,A207,XXREAL_0:2;
    then 1 in dom Q by A202,FINSEQ_3:25;
    hence thesis by A5,A210,A217,A214,MATRIX_0:def 8;
  end;
A219: 0+1<=n by A1,NAT_1:13;
A220: len P=n by MATRIX_0:24;
  then
A221: 1 in Seg len P by A219,FINSEQ_1:1;
  then
A222: 1 in dom P by FINSEQ_1:def 3;
A223: width P=n by MATRIX_0:24;
  then
A224: len Line(P,1)=n by MATRIX_0:def 7;
  then
A225: 1 in dom (Line(P,1)) by A220,A221,FINSEQ_1:def 3;
A226: 1 in Seg width P by A223,A219,FINSEQ_1:1;
A227: for j st 1<j & j<=n holds (P*Q)*(1,j)= 0.K
  proof
    let j;
    assume that
A228: 1<j and
A229: j<=n;
A230: len (-a*(P*(1,j))*(Base_FinSeq(K,n,1))) =len ((-a*(P*(1,j)))*(
    Base_FinSeq(K,n,1))) by Th6
      .=len (Base_FinSeq(K,n,1)) by MATRIXR1:16
      .=n by Th23;
    reconsider p=Col(Q,j),q=-a*(P*(1,j))*(Base_FinSeq(K,n,1)) +Base_FinSeq(K,n
    ,j) as FinSequence of K;
A231: len (Base_FinSeq(K,n,j))=n by Th23;
A232: len ((-a*(P*(1,j)))*(Base_FinSeq(K,n,1))) =len (Base_FinSeq(K,n,1))
    by MATRIXR1:16
      .= n by Th23;
A233: for k be Nat st 1 <=k & k <= n holds p.k=q.k
    proof
      let k be Nat;
      assume that
A234: 1 <=k and
A235: k <= n;
A236: k in dom Q by A202,A234,A235,FINSEQ_3:25;
      len (Base_FinSeq(K,n,1))=n by Th23;
      then
A237: ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A234,A235,
FINSEQ_4:15;
A238: k in dom ((-a*(P*(1,j)))*(Base_FinSeq(K,n,1))) & ((Base_FinSeq(K,n,
      1))/.k)=( (Base_FinSeq(K,n,1)).k) by A203,A232,A234,A235,FINSEQ_3:25
,FINSEQ_4:15;
A239: (-a*(P*(1,j))*(Base_FinSeq(K,n,1))).k =((-a*(P*(1,j)))*(Base_FinSeq
      (K,n,1))).k by Th6
        .=(-a*(P*(1,j)))*((Base_FinSeq(K,n,1))/.k) by A238,FVSUM_1:50;
      per cases by A234,XXREAL_0:1;
      suppose
A240:   1=k;
        (Base_FinSeq(K,n,j))/.k = (Base_FinSeq(K,n,j)).k by A231,A234,A235,
FINSEQ_4:15
          .= 0.K by A228,A235,A240,Th25;
        then q.k=(-a*(P*(1,j))*(Base_FinSeq(K,n,1)))/.k +0.K by A230,A231,A234
,A235,Th5;
        then
A241:   q.k=(-a*(P*(1,j))*(Base_FinSeq(K,n,1)))/.k by RLVECT_1:4;
A242:   (-a*(P*(1,j))*(Base_FinSeq(K,n,1))).k =(-a*(P*(1,j)))*(1_K) by A235
,A239,A237,A240,Th24
          .=-a*(P*(1,j));
        p.k=Q*(1,j) by A236,A240,MATRIX_0:def 8
          .=-a*(P*(1,j)) by A6,A228,A229;
        hence thesis by A230,A234,A235,A242,A241,FINSEQ_4:15;
      end;
      suppose
A243:   1<k;
        [k,j] in Indices Q by A228,A229,A234,A235,MATRIX_0:31;
        then consider p2 being FinSequence of K such that
A244:   p2 = Q.k and
A245:   Q*(k,j) = p2.j by MATRIX_0:def 5;
A246:   k in NAT by ORDINAL1:def 12;
A247:   p.k=p2.j by A236,A245,MATRIX_0:def 8
          .= (Base_FinSeq(K,n,k)).j by A7,A235,A243,A244,A246;
        now
          per cases;
          suppose
A248:       k <> j;
            (-a*(P*(1,j))*(Base_FinSeq(K,n,1)))/.k =(-a*(P*(1,j)))*((
            Base_FinSeq(K,n,1))/.k) by A230,A234,A235,A239,FINSEQ_4:15
              .=(-a*(P*(1,j)))*(0.K) by A235,A237,A243,Th25
              .= 0.K;
            then q.k = 0.K +(Base_FinSeq(K,n,j))/.k by A230,A231,A234,A235,Th5
              .= (Base_FinSeq(K,n,j))/.k by RLVECT_1:4
              .= (Base_FinSeq(K,n,j)).k by A231,A234,A235,FINSEQ_4:15
              .= 0.K by A234,A235,A248,Th25;
            hence thesis by A228,A229,A247,A248,Th25;
          end;
          suppose
A249:       k=j;
            (-a*(P*(1,j))*(Base_FinSeq(K,n,1)))/.k =(-a*(P*(1,j)))*((
            Base_FinSeq(K,n,1))/.k) by A230,A234,A235,A239,FINSEQ_4:15
              .=(-a*(P*(1,j)))*(0.K) by A235,A237,A243,Th25
              .= 0.K;
            then q.k = 0.K +(Base_FinSeq(K,n,j))/.k by A230,A231,A234,A235,Th5
              .= (Base_FinSeq(K,n,j))/.k by RLVECT_1:4
              .= (Base_FinSeq(K,n,j)).k by A231,A234,A235,FINSEQ_4:15
              .= 1.K by A234,A235,A249,Th24;
            hence thesis by A234,A235,A247,A249,Th24;
          end;
        end;
        hence thesis;
      end;
    end;
A250: width P=n by MATRIX_0:24;
    then
A251: len Line(P,1) = n by MATRIX_0:def 7;
    then
A252: (Line(P,1))/.1 = (Line(P,1)).1 by A219,FINSEQ_4:15
      .=a" by A3,A226,MATRIX_0:def 7;
A253: j in Seg n by A228,A229,FINSEQ_1:1;
A254: len (Col(Q,j))=len Q by MATRIX_0:def 8
      .=n by MATRIX_0:24;
    len (-a*(P*(1,j))*(Base_FinSeq(K,n,1))+Base_FinSeq(K,n,j)) =n by A230,A231
,Th2;
    then
A255: Col(Q,j)=-a*(P*(1,j))*(Base_FinSeq(K,n,1))+Base_FinSeq(K,n,j) by A254
,A233,FINSEQ_1:14;
A256: (Line(P,1))/.j = ( Line(P,1)).j by A228,A229,A251,FINSEQ_4:15
      .= P*(1,j) by A253,A250,MATRIX_0:def 7;
    [1,j] in Indices (P*Q) by A219,A228,A229,MATRIX_0:31;
    then (P*Q)*(1,j)= |( Line(P,1), Col(Q,j) )| by A202,A250,MATRIX_3:def 4
      .= |( Line(P,1), -a*(P*(1,j))*(Base_FinSeq(K,n,1)) )| +|( Line(P,1),
    Base_FinSeq(K,n,j) )| by A224,A230,A231,A255,Th11
      .= |( Line(P,1), (-a*(P*(1,j)))*(Base_FinSeq(K,n,1)) )| +|( Line(P,1),
    Base_FinSeq(K,n,j) )| by Th6
      .=(-a*(P*(1,j)))* |( Line(P,1), (Base_FinSeq(K,n,1)) )| +|( Line(P,1),
    Base_FinSeq(K,n,j) )| by A224,A203,Th10
      .=(-a*(P*(1,j)))* (a") +|( Line(P,1), Base_FinSeq(K,n,j) )| by A224,A219
,A252,Th35
      .= -(a*(P*(1,j)))* (a") +|( Line(P,1), Base_FinSeq(K,n,j) )| by
VECTSP_1:9
      .= -((P*(1,j))*(a* (a"))) +|( Line(P,1), Base_FinSeq(K,n,j) )| by
GROUP_1:def 3
      .= -((P*(1,j))*(1_K)) +|( Line(P,1), Base_FinSeq(K,n,j) )| by A2,
VECTSP_1:def 10
      .= -(P*(1,j)*(1.K)) + (Line(P,1))/.j by A224,A228,A229,Th35
      .= -(P*(1,j)) + P*(1,j) by A256
      .= 0.K by RLVECT_1:5;
    hence thesis;
  end;
A257: Indices Q=[: Seg n,Seg n :] by MATRIX_0:24;
A258: for i,j st 1<i & i<=n & i=j holds (P*Q)*(i,j)=1.K
  proof
    let i,j;
    assume that
A259: 1<i and
A260: i<=n and
A261: i=j;
A262: len (-a*(P*(1,j))*(Base_FinSeq(K,n,1))) =len ((-a*(P*(1,j)))*(
    Base_FinSeq(K,n,1))) by Th6
      .=len (Base_FinSeq(K,n,1)) by MATRIXR1:16
      .=n by Th23;
    reconsider p=Col(Q,j),q=-a*(P*(1,j))*(Base_FinSeq(K,n,1)) +Base_FinSeq(K,n
    ,j) as FinSequence of K;
A263: len (Base_FinSeq(K,n,j))=n by Th23;
A264: j in Seg n by A259,A260,A261,FINSEQ_1:1;
A265: for k be Nat st 1 <=k & k <= n holds p.k=q.k
    proof
A266: len ((-a*(P*(1,j)))*(Base_FinSeq(K,n,1))) = len (Base_FinSeq(K,n,1
      )) by MATRIXR1:16
        .=n by Th23;
      let k be Nat;
      assume that
A267: 1 <=k and
A268: k <= n;
A269: k in Seg n by A267,A268,FINSEQ_1:1;
      len ((-a*(P*(1,j)))*(Base_FinSeq(K,n,1))) =len (Base_FinSeq(K,n,1)
      ) by MATRIXR1:16
        .= n by Th23;
      then
A270: k in dom ((-a*(P*(1,j)))*(Base_FinSeq(K,n,1))) by A267,A268,FINSEQ_3:25;
A271: ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A203,A267,A268,
FINSEQ_4:15;
      len (Base_FinSeq(K,n,1))=n by Th23;
      then
A272: ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A267,A268,
FINSEQ_4:15;
      k in dom Q by A202,A267,A268,FINSEQ_3:25;
      then
A273: p.k=Q*(k,j) by MATRIX_0:def 8;
A274: (-a*(P*(1,j))*(Base_FinSeq(K,n,1))).k =((-a*(P*(1,j)))*(
      Base_FinSeq(K,n,1))).k by Th6
        .=(-a*(P*(1,j)))*((Base_FinSeq(K,n,1))/.k) by A270,A271,FVSUM_1:50;
      per cases by A267,XXREAL_0:1;
      suppose
A275:   1=k;
        k <= len (-a*(P*(1,j))*(Base_FinSeq(K,n,1))) by A268,A266,Th6;
        then
A276:   (-a*(P*(1,j))*(Base_FinSeq(K,n,1)))/.k = (-a*(P*(1,j)))*((
        Base_FinSeq(K,n,1))/.k) by A267,A274,FINSEQ_4:15
          .= (-a*(P*(1,j)))*(1.K) by A219,A272,A275,Th24;
        (Base_FinSeq(K,n,j))/.k = (Base_FinSeq(K,n,j)).k by A263,A267,A268,
FINSEQ_4:15
          .= 0.K by A259,A261,A268,A275,Th25;
        then q.k= (-a*(P*(1,j)))*(1.K)+ 0.K by A262,A263,A267,A268,A276,Th5
          .= (-a*(P*(1,j)))*(1.K) by RLVECT_1:4
          .= -a*(P*(1,j));
        hence thesis by A6,A259,A260,A261,A273,A275;
      end;
      suppose
A277:   1<k;
        [k,j] in Indices Q by A257,A264,A269,ZFMISC_1:87;
        then
A278:   ex p2 being FinSequence of K st p2 = Q.k & Q*(k,j) = p2. j by
MATRIX_0:def 5;
        k in NAT by ORDINAL1:def 12;
        then
A279:   p.k= (Base_FinSeq(K,n,k)).j by A7,A268,A273,A277,A278;
        now
          per cases;
          suppose
A280:       k <> j;
            then
A281:       p.k= 0.K by A259,A260,A261,A279,Th25;
A282:       (-a*(P*(1,j))*(Base_FinSeq(K,n,1)))/.k =(-a*(P*(1,j))*(
            Base_FinSeq(K,n,1))).k by A262,A267,A268,FINSEQ_4:15;
A283:       (-a*(P*(1,j))*(Base_FinSeq(K,n,1))).k =(-a*(P*(1,j)))*(0.K)
            by A268,A271,A274,A277,Th25
              .= 0.K;
            (Base_FinSeq(K,n,j))/.k = (Base_FinSeq(K,n,j)).k by A263,A267,A268,
FINSEQ_4:15
              .= 0.K by A267,A268,A280,Th25;
            then q.k=0.K + 0.K by A262,A263,A267,A268,A282,A283,Th5;
            hence thesis by A281,RLVECT_1:4;
          end;
          suppose
A284:       k=j;
            then
A285:       p.k=1.K by A267,A268,A279,Th24;
A286:       (-a*(P*(1,j))*(Base_FinSeq(K,n,1)))/.k =(-a*(P*(1,j))*(
            Base_FinSeq(K,n,1))).k by A262,A267,A268,FINSEQ_4:15;
A287:       (-a*(P*(1,j))*(Base_FinSeq(K,n,1))).k =(-a*(P*(1,j)))*(0.K)
            by A268,A271,A274,A277,Th25
              .= 0.K;
            (Base_FinSeq(K,n,j))/.k = (Base_FinSeq(K,n,j)).k by A263,A267,A268,
FINSEQ_4:15
              .= 1.K by A259,A260,A261,A284,Th24;
            then q.k= 0.K + 1.K by A262,A263,A267,A268,A286,A287,Th5;
            hence thesis by A285,RLVECT_1:4;
          end;
        end;
        hence thesis;
      end;
    end;
    [i,j] in Indices P by A259,A260,A261,MATRIX_0:31;
    then consider p1 being FinSequence of K such that
A288: p1 = P.i and
A289: P*(i,j) = p1.j by MATRIX_0:def 5;
    p1=Base_FinSeq(K,n,i) by A4,A259,A260,A288;
    then
A290: p1.j=1.K by A259,A260,A261,Th24;
A291: len (Col(Q,j))=len Q by MATRIX_0:def 8
      .=n by MATRIX_0:24;
    len (-a*(P*(1,j))*(Base_FinSeq(K,n,1))+Base_FinSeq(K,n,j)) =n by A262,A263
,Th2;
    then
A292: Col(Q,j)=-a*(P*(1,j))*(Base_FinSeq(K,n,1))+Base_FinSeq(K,n,j) by A291
,A265,FINSEQ_1:14;
A293: width P=n by MATRIX_0:24;
    then
A294: len Line(P,i) = n by MATRIX_0:def 7;
    then
A295: (Line(P,i))/.1 = (Line(P,i)).1 by A219,FINSEQ_4:15
      .= P*(i,1) by A226,MATRIX_0:def 7;
A296: (Line(P,i))/.j = (Line(P,i)).j by A259,A260,A261,A294,FINSEQ_4:15
      .= P*(i,j) by A264,A293,MATRIX_0:def 7;
    [i,j] in Indices (P*Q) by A259,A260,A261,MATRIX_0:31;
    then
A297: (P*Q)*(i,j)= |( Line(P,i), Col(Q,j) )| by A202,A293,MATRIX_3:def 4
      .= |( Line(P,i), -a*(P*(1,j))*(Base_FinSeq(K,n,1)) )| +|( Line(P,i),
    Base_FinSeq(K,n,j) )| by A262,A263,A292,A294,Th11
      .= |( Line(P,i), (-a*(P*(1,j)))*(Base_FinSeq(K,n,1)) )| +|( Line(P,i),
    Base_FinSeq(K,n,j) )| by Th6
      .=(-a*(P*(1,j)))* |( Line(P,i), (Base_FinSeq(K,n,1)) )| +|( Line(P,i),
    Base_FinSeq(K,n,j) )| by A203,A294,Th10
      .=(-a*(P*(1,j)))* (P*(i,1)) +|( Line(P,i), Base_FinSeq(K,n,j) )| by A219
,A294,A295,Th35
      .= -(a*(P*(1,j)))* (P*(i,1)) +|( Line(P,i), Base_FinSeq(K,n,j) )| by
VECTSP_1:9
      .= -((P*(1,j))*(a* (P*(i,1)))) +|( Line(P,i), Base_FinSeq(K,n,j) )| by
GROUP_1:def 3
      .= -(P*(1,j))*(a* (P*(i,1))) + P*(i,j) by A259,A260,A261,A294,A296,Th35;
A298: 1<=n by A259,A260,XXREAL_0:2;
    [i,1] in Indices P by A8,A259,A260,MATRIX_0:31;
    then consider p2 being FinSequence of K such that
A299: p2 = P.i and
A300: P*(i,1) = p2.1 by MATRIX_0:def 5;
    p2=Base_FinSeq(K,n,i) by A4,A259,A260,A299;
    hence (P*Q)*(i,j)= -(P*(1,j))*(a* (0.K)) + P*(i,j) by A259,A298,A297,A300
,Th25
      .= -(P*(1,j))*((0.K)) + P*(i,j)
      .= -((0.K)) + P*(i,j)
      .= 0.K + 1.K by A289,A290,RLVECT_1:12
      .= 1.K by RLVECT_1:4;
  end;
  len (Col(Q,1))=len Q by MATRIX_0:def 8
    .=n by MATRIX_0:24;
  then
A301: Col(Q,1)=a*(Base_FinSeq(K,n,1)) by A204,A205,FINSEQ_1:14;
A302: Indices (P*Q)=[: Seg n,Seg n :] by MATRIX_0:24;
A303: for i,j st 1<i & i<=n & 1<=j & j<=n & i<>j holds (P*Q)*(i,j)= 0.K
  proof
A304: len (a*(Base_FinSeq(K,n,1))) =len (Base_FinSeq(K,n,1)) by MATRIXR1:16
      .=n by Th23;
    let i,j;
    assume that
A305: 1<i and
A306: i<=n and
A307: 1<=j and
A308: j<=n and
A309: i<>j;
A310: [i,j] in Indices (P*Q) by A305,A306,A307,A308,MATRIX_0:31;
A311: j in Seg n by A307,A308,FINSEQ_1:1;
A312: len (Col(Q,j))=len Q by MATRIX_0:def 8
      .=n by MATRIX_0:24;
A313: [i,1] in Indices P by A8,A305,A306,MATRIX_0:31;
A314: len (Base_FinSeq(K,n,j))=n by Th23;
A315: len (-a*(P*(1,j))*(Base_FinSeq(K,n,1))) =len ((-a*(P*(1,j)))*(
    Base_FinSeq(K,n,1))) by Th6
      .=len (Base_FinSeq(K,n,1)) by MATRIXR1:16
      .=n by Th23;
    then
A316: len (-a*(P*(1,j))*(Base_FinSeq(K,n,1))+Base_FinSeq(K,n,j)) =n by A314,Th2
;
A317: [i,j] in Indices P by A305,A306,A307,A308,MATRIX_0:31;
    now
      per cases;
      suppose
A318:   j>1;
        reconsider p=Col(Q,j),q=-a*(P*(1,j))*(Base_FinSeq(K,n,1)) +Base_FinSeq
        (K,n,j) as FinSequence of K;
        for k be Nat st 1 <=k & k <= n holds p.k=q.k
        proof
          let k be Nat;
          assume that
A319:     1 <=k and
A320:     k <= n;
A321:     ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A203,A319,A320,
FINSEQ_4:15;
A322:     len ((-a*(P*(1,j)))*(Base_FinSeq(K,n,1))) = len (Base_FinSeq(K
          ,n,1)) by MATRIXR1:16
            .=n by Th23;
          then
A323:     k in dom ((-a*(P*(1,j)))*(Base_FinSeq(K,n,1))) by A319,A320,
FINSEQ_3:25;
A324:     (-a*(P*(1,j))*(Base_FinSeq(K,n,1))).k =((-a*(P*(1,j)))*(
          Base_FinSeq(K,n,1))).k by Th6
            .=(-a*(P*(1,j)))*((Base_FinSeq(K,n,1))/.k) by A323,A321,FVSUM_1:50;
          len (Base_FinSeq(K,n,1))=n by Th23;
          then
A325:     ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A319,A320,
FINSEQ_4:15;
          k in dom Q by A202,A319,A320,FINSEQ_3:25;
          then
A326:     p.k=Q*(k,j) by MATRIX_0:def 8;
          per cases by A319,XXREAL_0:1;
          suppose
A327:       1=k;
            k <= len (-a*(P*(1,j))*(Base_FinSeq(K,n,1))) by A320,A322,Th6;
            then
A328:       (-a*(P*(1,j))*(Base_FinSeq(K,n,1)))/.k = (-a*(P*(1,j)))*((
            Base_FinSeq(K,n,1))/.k) by A319,A324,FINSEQ_4:15
              .= (-a*(P*(1,j)))*(1.K) by A219,A325,A327,Th24;
            (Base_FinSeq(K,n,j))/.k = (Base_FinSeq(K,n,j)).k by A314,A319,A320,
FINSEQ_4:15
              .= 0.K by A318,A320,A327,Th25;
            then q.k= (-a*(P*(1,j)))*(1.K)+ 0.K by A315,A314,A319,A320,A328,Th5
              .= (-a*(P*(1,j)))*(1_K) by RLVECT_1:4
              .= (-a*(P*(1,j)));
            hence thesis by A6,A308,A318,A326,A327;
          end;
          suppose
A329:       1<k;
            [k,j] in Indices Q by A307,A308,A319,A320,MATRIX_0:31;
            then
A330:       ex p2 being FinSequence of K st p2 = Q.k & Q*(k,j) = p2. j by
MATRIX_0:def 5;
            k in NAT by ORDINAL1:def 12;
            then
A331:       p.k=(Base_FinSeq(K,n,k)).j by A7,A320,A326,A329,A330;
A332:       k <= len (-a*(P*(1,j))*(Base_FinSeq(K,n,1))) by A320,A322,Th6;
            now
              per cases;
              suppose
A333:           k <> j;
                (Base_FinSeq(K,n,j))/.k =(Base_FinSeq(K,n,j)).k by A314,A319
,A320,FINSEQ_4:15
                  .= 0.K by A319,A320,A333,Th25;
                then q.k = (-a*(P*(1,j))*(Base_FinSeq(K,n,1)))/.k +0.K by A315
,A314,A319,A320,Th5;
                then
A334:           q.k = (-a*(P*(1,j))*(Base_FinSeq(K,n,1)))/.k by RLVECT_1:4
                  .= (-a*(P*(1,j))*(Base_FinSeq(K,n,1))).k by A319,A332,
FINSEQ_4:15;
                (-a*(P*(1,j))*(Base_FinSeq(K,n,1))).k =(-a*(P*(1,j)))*(
                0.K) by A320,A324,A325,A329,Th25
                  .= 0.K;
                hence thesis by A307,A308,A331,A333,A334,Th25;
              end;
              suppose
A335:           k=j;
                then
A336:           p.k=1.K by A319,A320,A331,Th24;
A337:           (-a*(P*(1,j))*(Base_FinSeq(K,n,1)))/.k =(-a*(P*(1,j))*(
                Base_FinSeq(K,n,1))).k by A315,A319,A320,FINSEQ_4:15;
A338:           (-a*(P*(1,j))*(Base_FinSeq(K,n,1))).k =(-a*(P*(1,j)))*(
                0.K) by A320,A321,A324,A329,Th25
                  .= 0.K;
                (Base_FinSeq(K,n,j))/.k = (Base_FinSeq(K,n,j)).k by A314,A319
,A320,FINSEQ_4:15
                  .= 1.K by A319,A320,A335,Th24;
                then q.k= 0.K + 1.K by A315,A314,A319,A320,A337,A338,Th5;
                hence thesis by A336,RLVECT_1:4;
              end;
            end;
            hence thesis;
          end;
        end;
        then
A339:   Col(Q,j)=-a*(P*(1,j))*(Base_FinSeq(K,n,1))+Base_FinSeq(K,n,j) by A312
,A316,FINSEQ_1:14;
A340:   1<=n by A305,A306,XXREAL_0:2;
A341:   width P=n by MATRIX_0:24;
        then
A342:   len Line(P,i) = n by MATRIX_0:def 7;
        then
A343:   (Line(P,i))/.j = (Line(P,i)).j by A307,A308,FINSEQ_4:15
          .= P*(i,j) by A311,A341,MATRIX_0:def 7;
A344:   (Line(P,i))/.1 = (Line(P,i)).1 by A219,A342,FINSEQ_4:15
          .= P*(i,1) by A226,MATRIX_0:def 7;
A345:   (P*Q)*(i,j)= |( Line(P,i), Col(Q,j) )| by A202,A310,A341,MATRIX_3:def 4
          .= |( Line(P,i), -a*(P*(1,j))*(Base_FinSeq(K,n,1)) )| +|( Line(P,i
        ), Base_FinSeq(K,n,j) )| by A315,A314,A339,A342,Th11
          .= |( Line(P,i), (-a*(P*(1,j)))*(Base_FinSeq(K,n,1)) )| +|( Line(P
        ,i), Base_FinSeq(K,n,j) )| by Th6
          .=(-a*(P*(1,j)))* |( Line(P,i), (Base_FinSeq(K,n,1)) )| +|( Line(P
        ,i), Base_FinSeq(K,n,j) )| by A203,A342,Th10
          .=(-a*(P*(1,j)))* (P*(i,1)) +|( Line(P,i), Base_FinSeq(K,n,j) )|
        by A219,A342,A344,Th35
          .= -(a*(P*(1,j)))* (P*(i,1)) +|( Line(P,i), Base_FinSeq(K,n,j) )|
        by VECTSP_1:9
          .= -((P*(1,j))*(a* (P*(i,1)))) +|( Line(P,i), Base_FinSeq(K,n,j)
        )| by GROUP_1:def 3
          .= -(P*(1,j))*(a* (P*(i,1))) + P*(i,j) by A307,A308,A342,A343,Th35;
        consider p2 being FinSequence of K such that
A346:   p2 = P.i and
A347:   P*(i,1) = p2.1 by A313,MATRIX_0:def 5;
        consider p1 being FinSequence of K such that
A348:   p1 = P.i and
A349:   P*(i,j) = p1.j by A317,MATRIX_0:def 5;
        p1=Base_FinSeq(K,n,i) by A4,A305,A306,A348;
        then
A350:   p1.j= 0.K by A307,A308,A309,Th25;
        p2=Base_FinSeq(K,n,i) by A4,A305,A306,A346;
        then (P*Q)*(i,j)= -(P*(1,j))*(a* (0.K)) + P*(i,j) by A305,A340,A345
,A347,Th25
          .= -(P*(1,j))*(0.K) + P*(i,j)
          .= - (0.K) + P*(i,j)
          .= 0.K + 0.K by A349,A350,RLVECT_1:12
          .= 0.K by RLVECT_1:4;
        hence thesis;
      end;
      suppose
A351:   j<=1;
        reconsider p=Col(Q,j),q=a*(Base_FinSeq(K,n,1)) as FinSequence of K;
A352:   for k be Nat st 1 <=k & k <= n holds p.k=q.k
        proof
          let k be Nat;
          assume that
A353:     1 <=k and
A354:     k <= n;
A355:     k in dom Q by A202,A353,A354,FINSEQ_3:25;
A356:     len (Base_FinSeq(K,n,1))=n by Th23;
          then
A357:     ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A353,A354,
FINSEQ_4:15;
          k in dom (a*(Base_FinSeq(K,n,1))) by A204,A353,A354,FINSEQ_3:25;
          then
A358:     (a*(Base_FinSeq(K,n,1))).k =a*((Base_FinSeq(K,n,1))/.k) by A357,
FVSUM_1:50;
          per cases by A353,XXREAL_0:1;
          suppose
A359:       1=k;
            then
A360:       q.k= (a)*(1.K) by A354,A357,A358,Th24
              .=a;
            p.k=Q*(1,j) by A355,A359,MATRIX_0:def 8
              .=a by A5,A307,A351,XXREAL_0:1;
            hence thesis by A360;
          end;
          suppose
A361:       1<k;
            [k,j] in Indices Q by A307,A308,A353,A354,MATRIX_0:31;
            then
A362:       ex p2 being FinSequence of K st p2 = Q.k & Q*(k,j) = p2. j by
MATRIX_0:def 5;
A363:       k in NAT by ORDINAL1:def 12;
A364:       p.k=Q*(k,j) by A355,MATRIX_0:def 8
              .= (Base_FinSeq(K,n,k)).j by A7,A354,A361,A362,A363;
            now
              per cases;
              suppose
A365:           k <> j;
                ((Base_FinSeq(K,n,1))/.k)=((Base_FinSeq(K,n,1)).k) by A353,A354
,A356,FINSEQ_4:15;
                then a*(Base_FinSeq(K,n,1))/.k = a*(0.K) by A354,A361,Th25
                  .= 0.K;
                hence thesis by A307,A308,A358,A364,A365,Th25;
              end;
              suppose
                k=j;
                hence thesis by A351,A361;
              end;
            end;
            hence thesis;
          end;
        end;
A366:   1<=n by A305,A306,XXREAL_0:2;
A367:   len (Line(P,i))=n by A223,MATRIX_0:def 7;
        then
A368:   (Line(P,i))/.1 = (Line(P,i)).1 by A219,FINSEQ_4:15
          .= P*(i,1) by A226,MATRIX_0:def 7;
A369:   (P*Q)*(i,j)= |( Line(P,i), Col(Q,j) )| by A202,A223,A310,MATRIX_3:def 4
          .= |( Line(P,i), a*(Base_FinSeq(K,n,1)) )| by A312,A304,A352,
FINSEQ_1:14
          .= a* |( Line(P,i), (Base_FinSeq(K,n,1)) )| by A203,A367,Th10
          .= a* (P*(i,1)) by A219,A367,A368,Th35;
        consider p2 being FinSequence of K such that
A370:   p2 = P.i and
A371:   P*(i,1) = p2.1 by A313,MATRIX_0:def 5;
        p2=Base_FinSeq(K,n,i) by A4,A305,A306,A370;
        hence (P*Q)*(i,j)= a*(0.K) by A305,A366,A369,A371,Th25
          .=0.K;
      end;
    end;
    hence thesis;
  end;
A372: len (a*(Line(P,1)))=len ((Line(P,1))) by MATRIXR1:16
    .= n by A223,MATRIX_0:def 7;
  Indices P=[: Seg n,Seg n :] & [1,1] in Indices P by A219,MATRIX_0:24,31;
  then
A373: (P*Q)*(1,1)= |( Line(P,1), Col(Q,1) )| by A202,A223,A302,MATRIX_3:def 4
    .= a*( |( Line(P,1), Base_FinSeq(K,n,1) )| ) by A224,A203,A301,Th10
    .=a*(Line(P,1))/.1 by A224,A219,Th35
    .=(a*(Line(P,1)))/.1 by A225,POLYNOM1:def 1
    .=(a*(Line(P,1))).1 by A219,A372,FINSEQ_4:15
    .=a*(P*(1,1)) by A226,A222,MATRIX12:3
    .=1.K by A2,A3,VECTSP_1:def 10;
  for i,j being Nat st [i,j] in Indices (P*Q) holds (P*Q)*(i,j)=(1.(K,n)
  )*(i,j)
  proof
    let i,j be Nat;
    reconsider i0=i,j0=j as Element of NAT by ORDINAL1:def 12;
    assume
A374: [i,j] in Indices (P*Q);
    then
A375: i in Seg n by A302,ZFMISC_1:87;
    then
A376: 1<=i by FINSEQ_1:1;
A377: i<=n by A375,FINSEQ_1:1;
A378: j in Seg n by A302,A374,ZFMISC_1:87;
    then
A379: 1<=j by FINSEQ_1:1;
A380: j<=n by A378,FINSEQ_1:1;
    per cases by A376,XXREAL_0:1;
    suppose
A381: 1<i;
      now
        per cases;
        suppose
A382:     i<>j;
A383:     (1.(K,n))*(i,j)=(Base_FinSeq(K,n,i0)).j0 by A376,A377,A379,A380,Th27
            .= 0.K by A379,A380,A382,Th25;
          thus (P*Q)*(i,j)=(P*Q)*(i0,j0)
            .=(1.(K,n))*(i,j) by A303,A377,A379,A380,A381,A382,A383;
        end;
        suppose
A384:     i=j;
A385:     (1.(K,n))*(i,j)=(Base_FinSeq(K,n,i0)).j0 by A376,A377,A379,A380,Th27
            .=1.K by A379,A380,A384,Th24;
          thus (P*Q)*(i,j)=(P*Q)*(i0,j0)
            .=(1.(K,n))*(i,j) by A258,A380,A381,A384,A385;
        end;
      end;
      hence thesis;
    end;
    suppose
A386: 1=i;
      now
        per cases;
        suppose
A387:     i<j;
A388:     (1.(K,n))*(i,j)=(Base_FinSeq(K,n,i0)).j0 by A376,A377,A379,A380,Th27
            .= 0.K by A379,A380,A387,Th25;
          thus (P*Q)*(i,j)=(P*Q)*(i0,j0)
            .=(1.(K,n))*(i,j) by A227,A380,A386,A387,A388;
        end;
        suppose
A389:     i>=j;
          then
A390:     i=j by A379,A386,XXREAL_0:1;
          (1.(K,n))*(i,j)=(Base_FinSeq(K,n,i0)).j0 by A376,A377,A379,A380,Th27
            .=1.K by A379,A380,A390,Th24;
          hence thesis by A373,A379,A386,A389,XXREAL_0:1;
        end;
      end;
      hence thesis;
    end;
  end;
  then
A391: P*Q=1.(K,n) by MATRIX_0:27;
  hence P is invertible by A201,Th19;
  thus thesis by A391,A201,Th18;
end;
