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

theorem
  for a being Real,x being FinSequence of REAL,A being Matrix of REAL st
  width A=len x & len x>0 holds A*(a*x)=a*(A*x)
proof
  let a be Real,x be FinSequence of REAL,A be Matrix of REAL;
  assume that
A1: width A=len x and
A2: len x>0;
A3: len ColVec2Mx x=len x by A2,MATRIXR1:def 9;
  width ColVec2Mx x=1 by A2,MATRIXR1:def 9;
  then
A4: 1<=width(A*(ColVec2Mx x)) by A1,A3,MATRIX_3:def 4;
  thus A*(a*x)=Col(A*(a*ColVec2Mx x),1) by A2,MATRIXR1:47
    .=Col(a*(A*(ColVec2Mx x)),1) by A1,A3,MATRIXR1:40
    .=a*(A*x) by A4,MATRIXR1:56;
end;
