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 Th56:
  for a being non zero Real
  for u being Element of TOP-REAL 3 st a * u = 0.TOP-REAL 3
  holds u is zero
  proof
    let a be non zero Real;
    let u be Element of TOP-REAL 3;
    assume a * u = 0.TOP-REAL 3;
    then |[ a * u`1, a * u`2,a * u`3 ]| = |[0,0,0]| by EUCLID_5:4,7;
    then u`1 = 0 & u`2 = 0 & u`3 = 0 by FINSEQ_1:78;
    hence thesis by EUCLID_5:3,4;
  end;
