reserve a,b,s,t,u,lambda for Real,
  n for Nat;
reserve x,x1,x2,x3,y1,y2 for Element of REAL n;

theorem
  for R being Subset of REAL,x1,x2,y1 being Element of REAL n st
   R={|.y1-x.| where x is Element of REAL n: x in Line(x1,x2)}
 ex y2 being Element of REAL n
  st y2 in Line(x1,x2) & |.y1-y2.|=lower_bound R & x1-x2,y1-y2 are_orthogonal
proof
  let R being Subset of REAL,x1,x2,y1 being Element of REAL n;
  consider y2 being Element of REAL n such that
A1: y2 in Line(x1,x2) and
A2: x1-x2,y1-y2 are_orthogonal and
A3: for x being Element of REAL n st x in Line(x1,x2) holds |.y1 - y2.|
  <= |.y1 - x.| by Lm7;
  assume
A4: R={|.y1-x.| where x is Element of REAL n: x in Line(x1,x2)};
A5: for s being Real st 0<s holds ex r being Real st r in R &
  r < |.y1-y2.|+s
  proof
    let s be Real;
    assume
A6: 0<s;
    take |.y1- y2.|;
    thus thesis by A4,A1,A6,XREAL_1:29;
  end;
  x1 in Line(x1,x2) by Th9;
  then
A7: |.y1-x1.| in R by A4;
A8: for r being Real st r in R holds |.y1-y2.| <=r
  proof
    let r be Real;
    assume r in R;
    then ex x0 being Element of REAL n st r= |.y1-x0.| & x0 in Line(x1,x2) by
A4;
    hence thesis by A3;
  end;
  R is bounded_below proof take |.y1-y2.|;
    let r be ExtReal;
    assume r in R;
    then ex x0 being Element of REAL n st r= |.y1-x0.| & x0 in Line(x1,x2) by
A4;
    hence thesis by A3;
  end;
  then |.y1-y2.| = lower_bound R by A7,A5,A8,SEQ_4:def 2;
  hence thesis by A1,A2;
end;
