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 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;
