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 Th7:
  for x,y being Point of TopSpaceMetr(Euclid 1)
  ex x1,y1 being Point of RealSpace, xr,yr being Real st
  x1 = xr & y1 = yr & x = <*xr*> & y = <*yr*> &
  dist(x1,y1) = real_dist.(xr,yr) &
  dist(x1,y1) = (Pitag_dist 1).(<*xr*>,<*yr*>) &
  dist(x1,y1) = |.xr - yr.|
  proof
    let x,y be Point of TopSpaceMetr(Euclid 1);
    reconsider xr1 = x as Point of Euclid 1;
    xr1 in 1-tuples_on REAL;
    then xr1 in the set of all <*r*> where r is Element of REAL by FINSEQ_2:96;
    then consider r1 be Element of REAL such that
A1: xr1 = <*r1*>;
    reconsider yr1 = y as Point of Euclid 1;
    yr1 in 1-tuples_on REAL;
    then yr1 in the set of all <*r*> where r is Element of REAL by FINSEQ_2:96;
    then consider r2 be Element of REAL such that
A2: yr1 = <*r2*>;
    reconsider xr2 = r1,yr2 = r2 as Element of RealSpace;
    reconsider x2 = <*r1*>,y2 = <*r2*> as Element of REAL 1;
A3: x2.1 = r1 by FINSEQ_1:def 8;
A4: (Pitag_dist 1).(<*r1*>,<*r2*>) = |.x2.1-y2.1.| by SRINGS_5:99
                                  .= |. r1 - r2 .| by A3,FINSEQ_1:def 8;
    take xr2,yr2;
    take r1,r2;
    thus thesis by A4,A1,A2,METRIC_1:def 12;
  end;
