reserve            x for object,
               X,Y,Z for set,
         i,j,k,l,m,n for Nat,
                 r,s for Real,
                  no for Element of OrderedNAT,
                   A for Subset of [:NAT,NAT:];

theorem Th8:
  for x,y being Point of Euclid 1, r,s being Real st
  x = <*r*> & y = <*s*> holds dist(x,y) = |.r - s.|
  proof
    let x,y be Point of Euclid 1,r,s be Real;
    assume that
A1: x = <*r*> and
A2: y = <*s*>;
    consider x1,y1 being Point of RealSpace, xr,yr being Real such that
    x1 = xr and
    y1 = yr and
A3: x = <*xr*> and
A4: y = <*yr*> and
    dist(x1,y1) = real_dist.(xr,yr) and
A5: dist(x1,y1) = (Pitag_dist 1).(<*xr*>,<*yr*>) and
A6: dist(x1,y1) = |.xr - yr.| by Th7;
    xr = r & yr = s by A3,A1,A2,A4,FINSEQ_1:76;
    hence thesis  by A5,A6,A3,A4;
  end;
