reserve a,b,s,t,u,lambda for Real,
  n for Nat;
reserve x,x1,x2,x3,y1,y2 for Element of REAL n;
reserve p1,p2,q1,q2 for Point of TOP-REAL n;

theorem
  for R being Subset of REAL,p1,p2,q1 being Point of TOP-REAL n st R={|.
  q1-p .| where p is Point of TOP-REAL n: p in Line(p1,p2)}
 ex q2 being
Point of TOP-REAL n st q2 in Line(p1,p2) & |. q1-q2 .| =lower_bound R & p1-p2,
  q1-q2 are_orthogonal
proof
  let R being Subset of REAL,p1,p2,q1 being Point of TOP-REAL n;
  reconsider y1 = q1 as Element of REAL n by EUCLID:22;
  consider x1,x2 being Element of REAL n such that
A1: p1=x1 & p2=x2 and
A2: Line(x1,x2)=Line(p1,p2) by Lm8;
A3: x1-x2 = p1-p2 by A1;
  consider y2 being Element of REAL n such that
A4: y2 in Line(x1,x2) and
A5: x1-x2,y1-y2 are_orthogonal and
A6: for x being Element of REAL n st x in Line(x1,x2) holds |.y1 - y2.|
  <= |.y1 - x.| by Lm7;
  reconsider q2 = y2 as Point of TOP-REAL n by EUCLID:22;
  assume
A7: R={|. q1-p .| where p is Point of TOP-REAL n: p in Line(p1,p2)};
A8: for s being Real st 0<s holds ex r being Real st r in R &
  r < |.q1-q2.|+s
  proof
    let s be Real;
    assume
A9: 0<s;
    take |.q1- q2.|;
    thus thesis by A7,A2,A4,A9,XREAL_1:29;
  end;
  p1 in Line(p1,p2) by RLTOPSP1:72;
  then
A10: |.q1-p1.| in R by A7;
A11: for r being Real st r in R holds |.q1-q2.| <=r
  proof
    let r be Real;
    assume r in R;
    then consider p0 being Point of TOP-REAL n such that
A12: r= |.q1-p0.| & p0 in Line(p1,p2) by A7;
    the carrier of Euclid n = the carrier of TOP-REAL n by EUCLID:67;
    then reconsider x = p0 as Element of REAL n;
    thus |.q1-q2.| <=r by A2,A6,A12;
  end;
   R is bounded_below
  proof
   take |.q1-q2.|;
    let r be ExtReal;
    assume r in R;
    then consider p0 being Point of TOP-REAL n such that
A13: r= |.q1-p0.| & p0 in Line(p1,p2) by A7;
    the carrier of Euclid n = the carrier of TOP-REAL n by EUCLID:67;
    then reconsider x = p0 as Element of REAL n;
    thus |.q1-q2.| <=r by A2,A6,A13;
  end;
  then y1-y2 = q1-q2 & |.q1-q2.| = lower_bound R by A10,A8,A11,SEQ_4:def 2;
  hence thesis by A2,A4,A5,A3;
end;
