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 Th6:
  for x,y being Point of RealSpace ex xr,yr being Real st x = xr & y = yr &
  dist(x,y) = real_dist.(x,y) & dist(x,y) = (Pitag_dist 1).(<*x*>,<*y*>) &
  dist(x,y) = |.xr - yr.|
  proof
    let x,y be Point of RealSpace;
    reconsider xr = x,yr = y as Real;
A1: real_dist.(x,y) = |.xr - yr.| by METRIC_1:def 12;
    reconsider x2 = <*xr*>,y2 = <*yr*> as Element of REAL 1;
    x2.1 = xr & y2.1 = yr by FINSEQ_1:def 8;
    then (Pitag_dist 1).(<*x*>,<*y*>) = |. xr - yr .| by SRINGS_5:99;
    hence thesis by A1;
  end;
