 reserve n for Nat;
 reserve s1 for sequence of Euclid n,
         s2 for sequence of REAL-NS n;

theorem Th19:
  for a,b being Real, xa,xb being Point of Euclid 1 st
    xa = <*a*> & xb = <*b*> holds dist(xa,xb) = |.a - b.|
  proof
    let a,b be Real,xa,xb be Point of Euclid 1;
    assume that
A1: xa = <*a*> and
A2: xb = <*b*>;
    xa in Euclid 1 & xb in Euclid 1;
    then xa in TOP-REAL 1 & xb in TOP-REAL 1 by EUCLID:67;
    then reconsider ra = xa, rb = xb as Element of REAL 1 by EUCLID:22;
A3: Euclid 1 = MetrStruct(#REAL 1,Pitag_dist 1#) by EUCLID:def 7;
A4: ra = 1 |-> a by A1,FINSEQ_2:59;
    rb = 1 |-> b by A2,FINSEQ_2:59;
    then |.ra-rb.| = sqrt(1) * |.a-b.| by A4,TOPREALC:11
                  .= |.a - b.| by SQUARE_1:18;
    hence thesis by A3,EUCLID:def 6;
  end;
