reserve a,a1,a2,a3,b,b1,b2,b3,r,s,t,u for Real;
reserve n for Nat;
reserve x0,x,x1,x2,x3,y0,y,y1,y2,y3 for Element of REAL n;
reserve L,L0,L1,L2 for Element of line_of_REAL n;

theorem
  x in L iff dist(x,L) = 0
proof
  thus x in L implies dist(x,L) = 0
  proof
A1: for r being Real st r in dist_v(x,L) holds 0 <= r
    proof
      let r be Real;
      assume r in dist_v(x,L);
      then ex x0 being Element of REAL n st r= |.x-x0.| & x0 in L;
      hence thesis;
    end;
A2: dist_v(x,L) is bounded_below
    proof
     take 0;
      let r be ExtReal;
      assume r in dist_v(x,L);
      then ex x0 being Element of REAL n st r= |.x-x0.| & x0 in L;
      hence thesis;
    end;
    assume
A3: x in L;
    |.x - x.| = |.0*n.| by Th9
      .= sqrt |(0*n,0*n)| by EUCLID_4:15
      .= 0 by EUCLID_4:17,SQUARE_1:17;
    then
A4: 0 in dist_v(x,L) by A3;
    then for s being Real st 0<s holds ex r being Real st r in
    dist_v(x,L) & r < 0 + s;
    hence thesis by A4,A1,A2,SEQ_4:def 2;
  end;
  now
    consider x0 being Element of REAL n such that
A5: x0 in L and
A6: |.x-x0.| = dist(x,L) by Th57;
    assume dist(x,L) = 0;
    then |(x - x0, x - x0)| = 0 by A6,EUCLID_4:16;
    then x - x0 = 0*n by EUCLID_4:17;
    hence x in L by A5,Th9;
  end;
  hence thesis;
end;
