theorem Th113:
  A <> {} implies dist(x,A) >= 0
proof
  defpred P[set] means $1 in A;
  deffunc f(Element of COMPLEX n) = In(|.x - $1.|,REAL);
    deffunc g(Element of COMPLEX n) = |.x - $1.|;
  reconsider X = {f(z) : P[z]} as Subset of REAL from DOMAIN_1:sch 8;
A1: f(z) = g(z);
A2:  {f(z1) where z1 is Element of COMPLEX n: P[z1]}

   = {g(z2)  where z2 is Element of COMPLEX n: P[z2]}
    from FRAENKEL:sch 5(A1);
  assume A <> {};
  then consider z1 such that
A3: z1 in A by SUBSET_1:4;
A4: |.x - z1.| in X by A3,A2;
A5: now
    let r9;
    assume r9 in X;
    then ex z st r9 = f(z) & z in A;
    hence r9>= 0 by Th94;
  end;
  dist(x,A) = lower_bound X by Def17,A2;
  hence thesis by A4,A5,Th112;
end;
