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
  A meets B implies dist(A,B) = 0
proof
  assume A meets B;
  then consider z being object such that
A1: z in A and
A2: z in B by XBOOLE_0:3;
  reconsider z as Element of COMPLEX n by A1;
  dist(z,A) = 0 & dist(z,B) = 0 by A1,A2,Th115;
  then (0 qua Nat) + (0 qua Nat) >= dist(A,B) by A1,A2,Th128;
  hence thesis by A1,A2,Th126;
end;
