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 Th47:
  for n being Nat
  for a being Element of F_Real
  for A,B being Matrix of n,F_Real st n > 0 holds
  (a * A) * B = a * (A * B)
  proof
    let n be Nat;
    let a be Element of F_Real;
    let A,B be Matrix of n,F_Real;
    assume
A1: n > 0;
A2: width A = n & len A = n & width B = n & len B = n by MATRIX_0:24;
    reconsider ra = a as Real;
    reconsider rA = A, rB = B as Matrix of n,REAL;
A3: rA * rB = A * B by ANPROJ_8:17;
    a * A = ra * rA by Th46;
    then (a * A) * B = (ra * rA) * rB by ANPROJ_8:17
                    .= ra * (rA * rB) by A2,A1,MATRIXR1:41
                    .= a * (A * B) by A3,Th46;
    hence thesis;
  end;
