reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;
reserve n for non zero Nat;
reserve n for non zero Nat;
reserve n for Nat,
        X for set,
        S for Subset-Family of X;
reserve n for Nat,
        S for Subset-Family of REAL;
reserve n       for Nat,
        a,b,c,d for Element of REAL n;
reserve n for non zero Nat;
reserve n     for non zero Nat,
        x,y,z for Element of REAL n;

theorem Th66:
  for x,y being Element of REAL 1 holds (Pitag_dist 1).(x,y) = |.x.1-y.1.|
  proof
    let x,y be Element of REAL 1;
A1: (Pitag_dist 1).(x,y) = |.x-y.| by EUCLID:def 6;
    reconsider f = x - y as Element of TOP-REAL 1 by EUCLID:22;
    consider r being Real such that
A2: f = <*r*> by JORDAN2B:20;
    sqr (x-y) = <*r^2*> by A2,RVSUM_1:55;
    then Sum sqr (x-y) = r^2 by RVSUM_1:73; then
A3: sqrt Sum sqr (x-y) = |.r.| by COMPLEX1:72;
    f.1 = (x-y).1 = x.1 - y.1 by RVSUM_1:27;
    hence thesis by A1,A3,A2;
  end;
