reserve i,j for Nat;

theorem
  for a being Real,i being Nat,A being Matrix of REAL
    st len A>0 & i in
dom A holds (ex p being FinSequence of REAL st p=A.i) & for q being FinSequence
  of REAL st q=A.i holds (a*A).i=a*q
proof
  let a be Real,i be Nat,A be Matrix of REAL;
  assume that
A1: len A>0 and
A2: i in dom A;
  consider n3 being Nat such that
A3: for x being object st x in rng A ex s2 being FinSequence st s2=x & len
  s2 = n3 by MATRIX_0:def 1;
  A.i in rng A by A2,FUNCT_1:def 3;
  then
A4: ex qq0 being FinSequence st qq0=A.i & len qq0=n3 by A3;
  len (a*A)=len A by Th27;
  then consider s being FinSequence such that
A5: s in rng (a*A) and
A6: len s = width (a*A) by A1,MATRIX_0:def 3;
A7: width (a*A)=width A by Th27;
  consider s3 being FinSequence such that
A8: s3 in rng A and
A9: len s3 = width A by A1,MATRIX_0:def 3;
  consider n2 being Nat such that
A10: for x being object st x in rng (a*A) ex s2 being FinSequence st s2=x &
  len s2 = n2 by MATRIX_0:def 1;
  len (a*A)=len A by Th27;
  then
A11: dom (a*A) = dom A by FINSEQ_3:29;
  then (a*A).i in rng (a*A) by A2,FUNCT_1:def 3;
  then consider qq being FinSequence such that
A12: qq=(a*A).i and
A13: len qq=n2 by A10;
  consider n being Nat such that
A14: for x being object st x in rng A ex p being FinSequence of REAL st x =
  p & len p = n by MATRIX_0:9;
  A.i in rng A by A2,FUNCT_1:def 3;
  then ex p2 being FinSequence of REAL st A.i = p2 & len p2 = n by A14;
  hence ex p being FinSequence of REAL st p=A.i;
  let q be FinSequence of REAL;
  assume
A15: q=A.i;
A16: ex s4 being FinSequence st s4=s3 & len s4 = n3 by A8,A3;
  then
A17: len (a*q)=width A by A15,A9,A4,RVSUM_1:117;
A18: for j being Nat st 1<=j & j<=len (a*q) holds qq.j=(a*q).j
  proof
    let j be Nat;
    assume 1<=j & j<=len (a*q);
    then
A19: j in Seg width A by A17,FINSEQ_1:1;
    then
A20: [i,j] in Indices A by A2,ZFMISC_1:87;
    reconsider j as Element of NAT by ORDINAL1:def 12;
    [i,j] in Indices (a*A) by A2,A7,A11,A19,ZFMISC_1:87;
    then
A21: ex p being FinSequence of REAL st p= (a*A).i & (a*A)*(i,j ) =p.j by
MATRIX_0:def 5;
    ex p2 being FinSequence of REAL st p2= A.i & A*(i,j) =p2. j by A20,
MATRIX_0:def 5;
    then qq.j=a*(q.j) by A15,A12,A20,A21,Th29;
    hence thesis by RVSUM_1:44;
  end;
  ex s2 being FinSequence st s2=s & len s2 = n2 by A5,A10;
  then width A=len qq by A6,A13,Th27;
  hence thesis by A15,A9,A16,A12,A4,A18,FINSEQ_1:14,RVSUM_1:117;
end;
