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

theorem
  for A being Matrix of n,REAL for x,y being FinSequence of REAL st len
x=n & len y=n & x*A=y holds for j being Nat st 1<=j & j<=n holds y.j
  =|(x,Col(A,j))|
proof
  let A be Matrix of n,REAL;
  let x,y be FinSequence of REAL;
  assume that
A1: len x=n and
A2: len y=n and
A3: x*A=y;
  let j be Nat;
  assume 1<=j & j<=n;
  then j in Seg len (x*A) by A2,A3;
  hence thesis by A1,A3,MATRIX_0:24,MATRPROB:40;
end;
