reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;

theorem Th2:
  for A being Matrix of D for p being FinSequence of D st p=A.i & 1
  <=i & i<=len A & 1<=j & j<=width A & len p=width A holds A*(i,j)=p.j
proof
  let A be Matrix of D;
  let p be FinSequence of D;
  assume that
A1: p=A.i and
A2: 1<=i & i<=len A and
A3: 1<=j & j<=width A and
A4: len p=width A;
A5: j in Seg width A by A3;
  then j in dom p by A4,FINSEQ_1:def 3;
  then rng p c= D & p.j in rng p by FINSEQ_1:def 4,FUNCT_1:def 3;
  then reconsider xp=p.j as Element of D;
A6: xp=p.j;
  i in dom A by A2,FINSEQ_3:25;
  then [i,j] in Indices A by A5,ZFMISC_1:87;
  hence thesis by A1,A6,MATRIX_0:def 5;
end;
