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 Th63:
  (Pitag_dist 2).(|[0,0]|,|[1,1]|) = sqrt 2
  proof
    reconsider A = |[0,0]|,B=|[1,1]| as Element of REAL 2 by EUCLID:22;
    |[0,0]|`1 = 0 & |[0,0]|`2 = 0 & |[1,1]|`1 = 1 & |[1,1]|`2 = 1; then
A1: (A-B).1 = 0 - 1 & (A-B).2 = 0 - 1 by RVSUM_1:27;
    reconsider f = A-B as FinSequence of REAL;
    len f = len(A-B) & len(A-B) = 2 by FINSEQ_2:132; then
A2: |.A-B.| = sqrt ((-1)^2+((-1)^2)) by A1,EUCLID_3:22;
    (-1)^2 = 1^2 by SQUARE_1:3 .= 1 * 1 by SQUARE_1:def 1 .= 1;
    hence thesis by A2,EUCLID:def 6;
  end;
