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 Th114:
  A <> {} implies dist(x + z,A) <= dist(x,A) + |.z.|
proof
  defpred P[set] means $1 in A;
  deffunc g(Element of COMPLEX n) = In(|.x + z - $1.|,REAL);
  deffunc h(Element of COMPLEX n) = |.x + z - $1.|;
  reconsider Y = {g(z1) : P[z1]} as Subset of REAL from DOMAIN_1:sch 8;
  deffunc f(Element of COMPLEX n) = In(|.x - $1.|,REAL);
  deffunc f1(Element of COMPLEX n) = |.x - $1.|;
A1: for z1 being Element of COMPLEX n holds h(z1) = g(z1);
A2:  {h(z1): P[z1]}

   = {g(z2)  where z2 is Element of COMPLEX n: P[z2]}
    from FRAENKEL:sch 5(A1);
A3: for z1 being Element of COMPLEX n holds f(z1) = f1(z1);
A4:  {f(z1): P[z1]}

   = {f1(z2)  where z2 is Element of COMPLEX n: P[z2]}
    from FRAENKEL:sch 5(A3);
A5: Y is bounded_below
  proof
    take 0;
    let r be ExtReal;
    assume r in Y;
    then ex z1 st r = |.x + z - z1 .| & z1 in A by A2;
    hence thesis by Th94;
  end;
  reconsider X = {f(z1) : P[z1]} as Subset of REAL from DOMAIN_1:sch 8;
  assume A <> {};
  then consider z2 such that
A6: z2 in A by SUBSET_1:4;
A7: dist(x+z,A) = lower_bound Y by Def17,A2;
A8: now let r9;
    assume r9 in X;
    then consider z3 such that
A9: r9 = |.x - z3.| and
A10: z3 in A by A4;
    |.x + z - z3.| = |.x - z3 + z.| by Th75; then
A11: |.x + z - z3.| <= r9 + |.z.| by A9,Th97;
    |.x + z - z3.| in Y by A10,A2;
    then |.x + z - z3.| >= dist(x + z,A) by A7,A5,Def2;
    then r9+ |.z.| >= dist(x + z,A) by A11,XXREAL_0:2;
    hence r9 >= dist(x + z,A) - |.z.| by XREAL_1:20;
  end;
A12: |.x - z2.| in X by A6,A4;
  dist(x,A) = lower_bound X by Def17,A4;
  then dist(x + z,A) - |.z.| <= dist(x,A) by A12,A8,Th112;
  hence thesis by XREAL_1:20;
end;
