
theorem
  for i,j being Nat,M1,M2 being Matrix of COMPLEX st len M2>0
holds ex s being FinSequence of COMPLEX st len s = len M2 & s.1=(M1*(i,1))*(M2*
(1,j)) & for k be Nat st 1<=k & k < len M2 holds s.(k+1)=s.k + (M1*(
  i,k+1))*(M2*(k+1,j))
proof
  let i,j be Nat,M1,M2 be Matrix of COMPLEX;
  defpred P[Nat] means ex q being FinSequence of COMPLEX st 1<=$1+1
  & $1+1<=len M2 implies ( len q=$1 +1 & q.1=(M1*(i,1))*(M2*(1,j)) &
   (for k be Nat st 1<=k & k<$1+1
     holds q.(k+1)=q.k+(M1*(i,k+1))*(M2*(k+1,j))));
  reconsider q0=<*(M1*(i,1))*(M2*(1,j))*> as FinSequence of COMPLEX
          by RVSUM_1:146;
A1: for k be Nat st 1<=k & k<1 holds q0.(k+1)=q0.k+(M1*(i,k+1))*(
  M2*(k+1,j));
A2: for i2 being Nat st P[i2] holds P[i2+1]
  proof
    let i2 be Nat;
    set k0=i2;
    assume P[i2];
    then consider q2 being FinSequence of COMPLEX such that
A3: 1<=i2+1 & i2+1<=len M2 implies len q2=i2+1 & q2.1=(M1*(i,1))*(M2*
(1,j)) & for k2 being Nat st 1<=k2 & k2<i2+1 holds q2.(k2+1)=q2.k2+(
    M1*(i,k2+1))*(M2*(k2+1,j));
    reconsider q3=q2^<* q2.(k0+1)+(M1*(i,k0+1+1))*(M2*(k0+1+1,j)) *>
       as FinSequence of COMPLEX by RVSUM_1:146;
    1<=i2+1+1 & i2+1+1<=len M2 implies len q3 = i2+1+1 & q3.1=(M1*(i,1))*
(M2*(1,j)) & for k be Nat st 1<=k & k< i2+1+1 holds q3.(k+1)=q3.k+(
    M1*(i,k+1))*(M2*(k+1,j))
    proof
      assume that
      1<=i2+1+1 and
A4:   i2+1+1<=len M2;
A5:     1 <= i2+1 by NAT_1:12;
       thus
A6:     len q3 = len q2+len (<* q2.(k0+1)+(M1*(i,k0+1+1))*(M2*(k0+1+1,j))
        *>) by FINSEQ_1:22
          .=i2+1+1 by A3,A4,A5,FINSEQ_1:39,NAT_1:13;
A7:     for k2 being Nat st 1<=k2 & k2<i2+1+1 holds q3.(k2+1)=
        q3.k2+(M1*(i,k2+1))*(M2*(k2+1,j))
        proof
          let k2 be Nat;
          assume that
A8:       1<=k2 and
A9:       k2<i2+1+1;
A10:      k2<=i2+1 by A9,NAT_1:13;
          per cases by A10,XXREAL_0:1;
          suppose
A11:        k2<i2+1;
            then k2 < len q3 by A6,NAT_1:13;
            then
            k2 < len q2 + len <* q2.(k0+1)+(M1*(i,k0+1+1))*(M2*(k0+1+1,j)
            )*> by FINSEQ_1:22;
            then k2 < len q2 + 1 by FINSEQ_1:39;
            then k2 <= len q2 by NAT_1:13;
            then k2 in dom q2 by A8,FINSEQ_3:25;
            then
A12:        q3.k2=q2.k2 by FINSEQ_1:def 7;
            k2+1 < i2 + 1+1 by A11,XREAL_1:6;
            then
            k2+1 < len q2 + len <* q2.(k0+1)+(M1*(i,k0+1+1))*(M2*(k0+1+1,
            j))*> by A6,FINSEQ_1:22;
            then k2+1 < len q2 + 1 by FINSEQ_1:39;
            then
A13:        k2+1 <= len q2 by NAT_1:13;
            1 <= k2+1 by A8,NAT_1:13;
            then k2+1 in dom q2 by A13,FINSEQ_3:25;
            then q3.(k2+1)=q2.(k2+1) by FINSEQ_1:def 7;
            hence thesis by A3,A4,A5,A8,A11,A12,NAT_1:13;
          end;
          suppose
A14:        k2=i2+1;
            then k2 < i2 + 1+1 by NAT_1:13;
            then
            k2 < len q2 + len <* q2.(k0+1)+(M1*(i,k0+1+1))*(M2*(k0+1+1,j)
            )*> by A6,FINSEQ_1:22;
            then k2 < len q2 + 1 by FINSEQ_1:39;
            then k2 <= len q2 by NAT_1:13;
            then k2 in dom q2 by A8,FINSEQ_3:25;
            then
A15:        q3.k2=q2.k2 by FINSEQ_1:def 7;
            1 in Seg 1 by FINSEQ_1:3;
            then
            (<* q2.(k0+1)+(M1*(i,k0+1+1))*(M2*(k0+1+1,j)) *>).1 = q2.(k0+
1)+(M1*(i,k0+1+1 ))*(M2*(k0+1+1,j)) & 1 in dom (<* q2.(k0+1)+(M1*(i,k0+1+1))*(
            M2*(k0+1+1,j)) *> ) by FINSEQ_1:def 8;
            hence thesis by A3,A4,A5,A14,A15,FINSEQ_1:def 7,NAT_1:13;
          end;
        end;
        1 < len (q2^<* q2.(k0+1)+(M1*(i,k0+1+1))*(M2*(k0+1+1,j)) *>) by A5,A6,
NAT_1:13;
        then 1 < len q2 + len <* q2.(k0+1)+(M1*(i,k0+1+1))*(M2*(k0+1+1,j))*>
        by FINSEQ_1:22;
        then 1 < len q2 + 1 by FINSEQ_1:39;
        then 1 <= len q2 by NAT_1:13;
        then 1 in dom q2 by FINSEQ_3:25;
        hence thesis by A3,A4,A5,A7,FINSEQ_1:def 7,NAT_1:13;
    end;
    hence thesis;
  end;
  len q0 = 1 & q0.1=(M1*(i,1))*(M2*(1,j)) by FINSEQ_1:39;
  then
A16: P[0] by A1;
A17: for j being Nat holds P[j] from NAT_1:sch 2(A16,A2);
  assume
A18: len M2>0;
  then 0+1<=len M2 by NAT_1:8;
  then 0+1-1<=len M2-1 by XREAL_1:9;
  then
A19: len M2-'1=len M2-1 by XREAL_0:def 2;
  then 0+1<=(len M2)-'1+1 by A18,NAT_1:8;
  hence thesis by A19,A17;
end;
