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;

theorem Th31:
  for N being Matrix of 3,REAL for uf being FinSequence of REAL st
  uf = 0.TOP-REAL 3 holds N * uf = 0.TOP-REAL 3
  proof
    let N be Matrix of 3,REAL;
    let uf be FinSequence of REAL;
    assume
A1: uf = 0.TOP-REAL 3;
    reconsider M = N as Matrix of 3,F_Real;
    consider n1,n2,n3,n4,n5,n6,n7,n8,n9 be Element of F_Real such that
A2: M = <* <* n1,n2,n3 *>,
    <* n4,n5,n6 *>,
    <* n7,n8,n9 *> *> by PASCAL:3;
    reconsider r1 =n1,r2 = n2, r3 = n3,
               r4 = n4, r5 = n5, r6 = n6,
               r7 = n7, r8 = n8, r9 = n9 as Element of REAL;
    reconsider z = 0.TOP-REAL 3 as Element of TOP-REAL 3;
    reconsider p = <* 0,0,0 *> as FinSequence of REAL by EUCLID:24,EUCLID_5:4;
    reconsider z1 = z`1, z2 = z`2, z3 = z`3 as Element of REAL
      by XREAL_0:def 1;
    z = <* 0,0,0 *> & z = <* z1,z2,z3 *> by EUCLID_5:3,4; then
A3: z1 = 0 & z2 = 0 & z3 = 0 by FINSEQ_1:78;
    |[z`1,z`2,z`3]| = p by EUCLID_5:3,4;
    then N * uf = <* r1 * z1 + r2 * z2 + r3 * z3 ,
                     r4 * z1 + r5 * z2 + r6 * z3 ,
                     r7 * z1 + r8 * z2 + r9 * z3 *>
                       by A2,A1,EUCLID_5:4,PASCAL:9
               .= 0.TOP-REAL 3 by A3,EUCLID_5:4;
    hence thesis;
  end;
