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 Th101:
  |.z1 - z2.| = 0 iff z1 = z2
proof
  thus |.z1 - z2.| = 0 implies z1 = z2
  proof
    assume |.z1 - z2.| = 0;
    then z1 - z2 = 0c n by Th93;
    hence thesis by Th73;
  end;
  assume z1 = z2;
  then z1 - z2 = 0c n by Th72;
  hence thesis by Th92;
end;
