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 Th49:
  M = symmetric_3(a,a,-a,0,0,0) implies
  M * M * M = (a * a * a) * symmetric_3(1,1,-1,0,0,0)
  proof
    assume
A1: M = symmetric_3(a,a,-a,0,0,0);
    reconsider ra2 = a * a as Element of F_Real;
    M * M * M = (ra2 * 1.(F_Real,3)) * M by A1,Th40
             .= ra2 * (1.(F_Real,3) * M) by Th47
             .= ra2 * M by MATRIX_3:18
             .= ra2 * (a * symmetric_3(1,1,-1,0,0,0)) by A1,Th48
             .= (a * a * a) * symmetric_3(1,1,-1,0,0,0) by MATRIX_5:11;
    hence thesis;
  end;
