reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;
reserve k,n for Nat,
  r,r9,r1,r2 for Real,
  c,c9,c1,c2,c3 for Element of COMPLEX;
reserve z,z1,z2 for FinSequence of COMPLEX;
reserve x,z,z1,z2,z3 for Element of COMPLEX n,
  A,B for Subset of COMPLEX n;

theorem Th93:
  |.z.| = 0 implies z = 0c n
proof
  assume
A1: |.z.| = 0;
  now
    let j be Nat;
    assume
A2: j in Seg n;
    then reconsider c = z.j as Element of COMPLEX by Th57;
    0 <= Sum sqr abs z by RVSUM_1:86;
    then (abs z).j = (n|->0).j by A1,RVSUM_1:91,SQUARE_1:24;
    then |.c.| = (n|-> 0).j by A2,Th88;
    then c = 0c by COMPLEX1:45;
    hence z.j = (n|->0c).j;
  end;
  hence thesis by FINSEQ_2:119;
end;
