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 Th126:
  A <> {} & B <> {} implies dist(A,B) >= 0
proof
  defpred P[set,set] means $1 in A & $2 in B;
  deffunc f(Element of COMPLEX n,Element of COMPLEX n) = |.$1 - $2.|;
  deffunc g(Element of COMPLEX n,Element of COMPLEX n) = In(|.$1 - $2.|,REAL);
A1: for z1,z2 being Element of COMPLEX n holds f(z1,z2) = g(z1,z2);
A2:  {f(z1,z2) where z1,z2 is Element of COMPLEX n: P[z1,z2]}
   = {g(z1,z2)  where z1,z2 is Element of COMPLEX n: P[z1,z2]}
    from FRAENKEL:sch 7(A1);
  reconsider Z = {g(z1,z) where z1,z is Element of COMPLEX n:
  P[z1,z]} as Subset of REAL from DOMAIN_1:sch 9;
  assume that
A3: A <> {} and
A4: B <> {};
  consider z1 such that
A5: z1 in A by A3,SUBSET_1:4;
A6: now
    let r9;
    assume r9 in Z;
    then ex z1,z st r9 = |.z1 - z.| & z1 in A & z in B by A2;
    hence r9>= 0 by Th94;
  end;
  consider z2 such that
A7: z2 in B by A4,SUBSET_1:4;
A8: dist(A,B) = lower_bound Z by Def19,A2;
  |.z1 - z2.| in Z by A5,A7,A2;
  hence thesis by A8,A6,Th112;
end;
