reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;

theorem Th50:
  for n being Nat for a being Real
  for M being Matrix of n,REAL for x being FinSequence of REAL st
  n > 0 & len x = n holds M * (a * x) = (a * M) * x
  proof
    let n be Nat;
    let a be Real;
    let M be Matrix of n,REAL;
    let x be FinSequence of REAL;
    assume that
A1: n > 0 and
A2: len x = n;
    width M = n & len M = n by MATRIX_0:24; then
A3: width M = len ColVec2Mx x & len M > 0 & len ColVec2Mx x > 0 &
      width ColVec2Mx x > 0 by A1,A2,MATRIXR1:def 9;
    (a * M) * x = Col((a * M) * ColVec2Mx x, 1) by MATRIXR1:def 11
               .= Col( a * (M * ColVec2Mx x),1) by A3,MATRIXR1:41
               .= Col( M * (a * ColVec2Mx x),1) by A3,MATRIXR1:40
               .= Col( M * (ColVec2Mx (a * x)),1) by A1,A2,MATRIXR1:47
               .= M * (a * x) by MATRIXR1:def 11;
    hence thesis;
  end;
