reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);

theorem Th15:
  for A,B being Matrix of F_Real for RA,RB being Matrix of REAL st
  A = RA & B = RB holds A * B = RA * RB
  proof
    let A,B be Matrix of F_Real;
    let RA,RB be Matrix of REAL;
    assume that
A1: A = RA and
A2: B = RB;
    RA * RB = MXF2MXR (MXR2MXF RA * MXR2MXF RB) by MATRIXR1:def 6
           .= MXF2MXR (A * MXR2MXF RB) by A1,MATRIXR1:def 1
           .= MXF2MXR (A * B) by A2,MATRIXR1:def 1;
    then MXR2MXF (RA * RB) = A * B by Th14;
    hence thesis by MATRIXR1:def 1;
  end;
