theorem Th46:
  for n being Nat for a being Element of F_Real
  for ra being Real for A being Matrix of n,F_Real
  for rA being Matrix of n,REAL st
  a = ra & A = rA holds
  a * A = ra * rA
  proof
    let n be Nat;
    let a be Element of F_Real;
    let ra be Real;
    let A be Matrix of n,F_Real;
    let rA be Matrix of n,REAL;
    assume that
A1: a = ra and
A2: A = rA;
    ra * rA = MXF2MXR( a * MXR2MXF rA) by A1,MATRIXR1:def 7
           .= MXF2MXR( a * A) by A2,MATRIXR1:def 1;
    hence thesis by MATRIXR1:def 2;
  end;
